Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction

Self-duality of the compactiﬁed Ruijsenaars-Sc hneider system from quasi-Hamilton i an redu c tion L. F eh ´ er a and C. Klim ˇ c ´ ık b a Departmen t of The oretical Ph ysics, WIGNER R CP , RMKI H-1525 Budap est, P .O.B. 49, Hungary , and Departmen t of The oretical Ph ysics, Univ ersit y of Szeged Tisza La jos krt 84-86, H-6720 Sze ged, Hungary e-mail: lfeher@rmki.kfki.h u b Institut de math ´ ematiques de Lumin y 163, Av en ue de Lumin y F-13288 Marseille, F rance e-mail: klimcik@univme d.fr Abstract The Delzan t theorem of symplectic top ology is used to deriv e the completely integ r able compactiﬁed Ruijsenaars-Schneider I I I b system fr om a quasi-Hamiltonian redu ction of the in tern ally f used double S U ( n ) × S U ( n ). In particular, the reduced sp ectral fu nctions dep end ing resp ectiv ely on the ﬁrst and s econd S U ( n ) factor of the double engender t wo toric momen t maps on the I I I b phase space C P ( n − 1) that pla y the roles of action-v ariables and particle-p ositions. A suitable cen tral extension of the S L (2 , Z ) mapp ing class group of the torus with one b oundary comp onen t is sh o wn to act on the quasi-Hamilt onian double by automorph ism s and, up on reduction, the standard generato r S of the mapp ing class group is pro ved to d escend to the Ruijsenaars self-dualit y symplectomorphism that exc hanges the toric momen t maps. W e giv e also tw o new presentati ons of this d u alit y map: one as the comp osition of tw o Delzan t symplectomorphisms and the other as the comp osition of th ree Dehn twist symplectomo rphisms realized b y Goldman t w ist ﬂo ws. Through the we ll-kno wn relation b et w een quasi-Hamiltonian manifolds and mo du li spaces, our results rigorously establish the v alidit y of the interpretation [going bac k to Gorsky and Nekraso v] of the I I I b system in terms of ﬂ at S U ( n ) connections on the one-holed torus. 1 Con te n ts 1 In tr o duction 3 2 Preliminaries 5 2.1 Quasi-Hamiltonian systems and their reductions . . . . . . . . . . . . . . . . . . 6 2.2 Ev olution ﬂow s on the in ternally fused double of S U ( n ) . . . . . . . . . . . . . . 7 3 Reduction of the in ternally fused double of S U ( n ) 11 3.1 The reduced phase space is smo oth and compact . . . . . . . . . . . . . . . . . . 12 3.2 The images of the Hamiltonians α j , β j restricted to µ − 1 ( µ 0 ) . . . . . . . . . . . 13 3.3 The reduced phase space is a Hamiltonian toric manifold . . . . . . . . . . . . . 19 3.4 The global structure o f the reduced systems ( P , ˆ ω , ˆ α ) and ( P , ˆ ω , ˆ β ) . . . . . . . . 22 4 Construction of the Delzant symplectomorphism s 25 4.1 Lo cal v ersion of the Delzan t map f β . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Global extension of t he Lax matrix . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Construction of the global Delzan t map f β . . . . . . . . . . . . . . . . . . . . . 33 4.4 The global D elzan t map f α and in v olution prop erties . . . . . . . . . . . . . . . 37 5 Self-duali t y of the compactiﬁed II I b system 41 6 Dualit y and the mapp ing class group 45 6.1 Flat connections and the quasi-Hamiltonian double . . . . . . . . . . . . . . . . 46 6.2 Mapping class g roup of the one-holed torus . . . . . . . . . . . . . . . . . . . . . 47 6.3 The dualit y map S as a mapping class symplectomorphism . . . . . . . . . . . . 49 6.4 The an ti-symplectomorphism R as a GL (2 , Z ) g enerato r . . . . . . . . . . . . . 51 7 Discussion 52 2 1 In tro ductio n A remark able feature of the no n-relativistic [3, 34, 2 2] and relativistic [33] in t egrable man y-b o dy systems of Calogero type is their dualit y relation discov ered b y Ruijsenaars [29]. The phase space of a classical in tegrable man y-b o dy system is alw ay s equipp ed with tw o Ab elian algebras of distinguished observ ables: the particle-p o sitions tied t o the ph ysical in terpretation and the action-v ariables tied to the Liouville in tegrabilit y . The ‘R uijsenaars dualit y’ b et we en systems (i) and (ii) requires the existence of a symplectomorphism b et w een the p ertinen t phase spaces that con v erts the particle-p ositions of system (i) in t o the actio n- v ariables of system (ii), and vice v ersa. One ta lks o f self-duality if the leading Hamiltonians of systems (i) and (ii), which underlie the man y-b o dy in terpretation, hav e the same form p ossibly with diﬀeren t c o upling constan ts. In addition to b eing fascinating in itself, the dualit y pro v ed useful for studying the dynamics, and it also app ears at the quan tum mec hanical lev el where its manifestation is the bisp ectral prop erty [6] of the man y-b o dy Hamiltonian op erators [30, 32]. The dualit y relation has b een established in [29, 31] with the help o f a direct metho d for all non-elliptic Calogero t yp e systems associat ed with t he A n ro ot system. The presen t pap er is part of the series [8, 9, 7] aimed at understanding all Ruijsenaars dualities b y means of the reduction a pproac h. The basic tenet o f this a pproac h, w hich o riginated from the pioneering pap ers [27, 17], is that the in tegrable many -b o dy sys tems descend from certain natural ‘free’ systems that can b e reduced using their large symmetries. Regarding t he dua lity , it is env isioned [10] that the starting phase spaces to b e reduced actually carry t w o ‘free’ systems that turn in to a dual pair in terms of tw o mo dels of a single reduced phase space. The existence of the symplectomorphism b et w een tw o mo dels (that arise in the simplest cases a s t w o gauge slices) of a single reduced phase space is entire ly automatic. In this w ay the r eduction approach may yield considerable tec hnical simpliﬁcation ov er the direct metho d, where the pro of of the symplectic c haracter of t he duality map is v ery lab orio us. Ho w ev er, nothing guaran tees that this a ppro ac h m ust alwa ys w ork; ﬁnding the correct reduction pro cedure relies on inspiration. T o this date, the reduction approac h has b een success f ully implemen ted for describing all but tw o cases of the kno wn Ruijsenaars dualities. The remaining t w o cases a re the self-dualities of the h yp erb olic and of the compactiﬁed trigonometric Ruijsenaars-Sc hneider systems . The ph ysical in t erpretat io n of the latter system (also called I I I b system in [31], with ‘b’ for ‘b ounded’) is based on its lo cal description v alid b efore compactiﬁcation. Since it is needed subseq uen tly , next w e brieﬂy summarize this lo cal description. The deﬁnition of the I II b system b egins with the lo cal Hamiltonian 1 H lo c y ( δ, Θ) ≡ n X j =1 cos p j n Y k 6 = j  1 − sin 2 y sin 2 ( x j − x k )  1 2 , (1.1) where y is a real non-v a nishing parameter, the δ j = e i2 x j ( j = 1 , ..., n ) are in terpreted as the p ositions of n ‘particles’ mo ving on the circle, and the canonically conjugate momenta p j enco de the compact v aria bles Θ j = e − i p j . Here, the cen ter of mass condition Q n j =1 δ j = Q n j =1 Θ j = 1 is also adopted. Denoting the standard maximal torus of S U ( n ) as S T n , the lo cal phase space is M lo c y ≡ { ( δ, Θ) | δ = ( δ 1 , ..., δ n ) ∈ D y , Θ = (Θ 1 , ..., Θ n ) ∈ S T n } , (1.2) 1 The index k in the next pro duct Q n k 6 = j runs ov er { 1 , 2 , ..., n } \ { j } , and similar notation is used thr o ughout. 3 where the domain D y ⊂ S T n (a so-called W eyl alco ve with thic k walls [35]) is c hosen in suc h a w a y to guarante e t ha t H lo c y tak es real v alues. The non-emptiness of D y is ensured b y the restriction | y | < π n . The symplec tic fo rm on M lo c y reads Ω lo c ≡ 1 2 tr  δ − 1 dδ ∧ Θ − 1 d Θ  = n X j =1 dx j ∧ dp j . (1.3) The Hamiltonian H lo c y can b e recast as the real part of the trace of t he unitary Lax matr ix L lo c y : L lo c y ( δ, Θ) j l ≡ e i y − e − i y e i y δ j δ − 1 l − e − i y W j ( δ, y ) W l ( δ, − y )Θ l (1.4) with the po sitiv e functions W j ( δ, y ) := n Y k 6 = j  e i y δ j − e − i y δ k δ j − δ k  1 2 . (1.5) The ﬂo ws generated b y the sp ectral inv ariants o f L lo c y comm ute, but are not complete on M lo c y . Ruijsenaars [3 1] has sho wn that o ne can realize ( M lo c y , Ω lo c ) as a dense op en submanifold of the complex pro jectiv e space C P ( n − 1) equipp ed with a multiple of the F ubini-Study sy mplectic form, and thereb y the comm uting lo cal ﬂo ws generated by L lo c y extend to complete Hamiltonia n ﬂo ws on the compact phase space C P ( n − 1). The self-duality of the resulting compactiﬁed II I b system w as also prov ed in [31 ]. Besides its app earance in solito n theory , the imp ortance of the I II b system reside s mainly in its in terpretation in terms o f a n appropriate sym plectic reduction of the space of S U ( n ) connections on the torus with one b oundary comp onen t (i.e. the one-holed to rus). In fact, the lo cal v ersion of the I I I b system had b een deriv ed by means of suc h a symplectic reduction by Gorsky and his collab orat o rs [14, 10] who moreo ver conj ectured that the Ruijsenaars dualit y originates f rom t he geometrically natural action of t he S L (2 , Z ) mapping class gro up of the torus on the reduced phase space. Ho w ev er, imp ortant global issues suc h as the compactiﬁcation of the lo cal phase space and the problem of the completenes s of the Liouville ﬂo ws w ere no t a ddressed in their approach, and they hav e not pro v ed that the Ruijsenaars self-dualit y symplectomorphism of [31] indeed origina tes from the action of the standard mapping class generator S ∈ S L ( 2 , Z ). The principal ac hiev emen t of the presen t pap er is a complete, glo bal reduction treatmen t of the compactiﬁed I II b system including a simple pro of of it s self-dualit y . The self-duality map will automatically arise as the comp osition of t wo Delzan t symplectomorphisms, whic h will pa v e the w a y to also prov e its conjectured relation [10] to the mapping class group. T o obta in t hese results, we do not pro ceed b y dev eloping further the inﬁnite-dimensional a pproac h of [14], but shall rat her w ork in a suitable ﬁnite-dim ensional framew ork based on a non-trivial generalization of the Marsden-W einstein symplec tic reduction, called quasi-Hamiltonia n reduction [1]. The quasi-Hamiltonian reduction w a s in ven ted [1] as a ﬁnite-dimensional alternativ e for de- scribing the symplec t ic structures on v a r io us mo duli spaces o f ﬂat connections on Riemann surfaces whose in ve stiga tion w as initiated b y A tiy ah a nd Bott in the inﬁnite-dimensional reduc- tion con text (see e.g. the b o ok [18] and references therein). F rom this angle, it is not surprising 4 that quasi-Hamiltonian metho ds can b e a pplied for ﬁnite-dimensional reduction treatmen t of in tegrable systems 2 . Nev ertheless, w e ﬁnd it remark able how naturally the quasi-Hamiltonian geometry together with the D elzant theorem of symplectic top ology [5] lead to an understanding of the g lobal structure of the compactiﬁed Ruijsenaars-Schne ider system. In fact, the Delzan t theorem will b e applied t o establish t he existen ce of tw o suitably equiv ariant symplectomor- phisms, f α and f β , that b oth map C P ( n − 1 ) on to the quasi-Hamiltonian reduced phase space. By utilizing their main features, w e also will b e able to construct thes e Delzant symple cto- morphisms ex plicitly , and then shall recov er the Ruijsenaars self-duality symplectomorphism as the composition S = f − 1 α ◦ f β : C P ( n − 1) → C P ( n − 1). A re- phrasing of this form ula will allow us to interpret the self-duality of t he compactiﬁed Ruijsenaars-Sc hneider system as a direct consequence of the ‘mapping class demo cracy’ betw een the S U ( n ) factors of the quasi- Hamiltonian double. In fact, w e shall pro v e the presen tation S = f − 1 β ◦ S P ◦ f β where S P stands for the natural action of S ∈ S L (2 , Z ) on the quasi-Hamiltonian reduced phase space. Inspired b y resu lts of Goldman [12], S P itself will b e decomposed in to a product of three Dehn twis t sympletomorphisms realized as special cases of certain Hamiltonian ﬂows. The pap er is essen tia lly self-con tained and its o rganization is as f o llo ws. In Section 2, w e ﬁrst recall the concept of quasi-Hamiltonian dynamic s and the method of quasi-Hamiltonian reduction. Then w e desc rib e the in ternally fused quasi-Hamiltonian double of t he group S U ( n ) and deﬁne t w o torus actions on it that will descend to the reduced phase space of our in terest. In Section 3, w e p erform the reduction, we prov e that the reduced phase space is a Hamiltonian toric manifold (in tw o alternativ e but equiv a len t w ay s) and we ﬁnd it s top ology and symplectic structure b y iden tifying the Delzan t polytop e corresp onding to the momen t map of the torus action. In Section 4, w e construct the Delzan t symplectomorphisms f α and f β explicitly . The lo cal La x matrix (1.4) a nd its global extension will arise naturally as building blo c ks of these maps. In Section 5, we reco v er the compactiﬁe d Ruijsenaars-Sc hneider sy stem and its self- dualit y from our reduction. In Section 6, w e demons trate tha t the action of the Ruijsenaars self-dualit y symplectomorphism on the I II b phase space is the standard action of the mapping class generator S ∈ S L (2 , Z ). Theorems 7 , 8 of Section 5 and Theorem 9 of Section 6 are our main results represen ting the ﬁnal outcome of our analysis. Their implications ar e further discusse d in Section 7, together with an outlo ok on op en problems. 2 Preliminaries Quasi-Hamiltonian system s can be useful sinc e they can b e reduced to honest Hamiltonian systems b y a generalization of the standard Marsden-W einstein r eduction pro cedure, and this can giv e a n eﬀectiv e to ol f or studying the resulting reduce d systems . Belo w w e ﬁrst recall fro m [1] the rele v an t notions, and then describ e those quasi-Hamiltonian dynamical system s that later will b e sho wn to yield the compactiﬁed Ruijsenaars-Sc hneider system up on reduction. 2 It w as remarked b y O blomko v [26] that the F o ck-Rosly trea tmen t [11] of the complexiﬁe d trigo nometric Ruijsenaars-Schneider sys tem co uld be replaced b y quasi- Hamiltonian reduction based on GL ( n, C ). This is very close in spirit to o ur framework, but the compact cas e that w e conside r is very diﬀerent tec hnically . 5 2.1 Quasi-Hamiltonian syste ms and their r edu ctions Let G b e a compact Lie group with Lie algebra G . Fix an in v arian t scalar pro duct h· , ·i on G and denote b y ϑ and ¯ ϑ , respectiv ely , t he left- and right-in v arian t Maurer-Cartan forms on G . F or a G -manifold M with action Ψ : G × M → M , w e use Ψ g ( m ) := Ψ( g , m ) and let ζ M denote the ve ctor ﬁeld on M t ha t corr espo nds to ζ ∈ G ; w e hav e [ ζ M , η M ] = − [ ζ , η ] M for all ζ , η ∈ G . The adjoint action of G on itself is giv en b y Ad g ( x ) := g xg − 1 , a nd Ad g denotes also the induced action on G . By deﬁnition [1], a quasi-Hamiltonian G -space ( M , G, ω , µ ) is a G -manifold M equipped with an in v arian t 2-form ω ∈ Λ( M ) G and with an equiv ariant map µ : M → G , µ ◦ Ψ g = Ad g ◦ µ , in suc h w a y that the following conditio ns hold. (a1) The diﬀerential of ω is given b y dω = − 1 12 µ ∗ h ϑ, [ ϑ, ϑ ] i . (2.1) (a2) The inﬁnitesimal action is related to µ and ω by ω ( ζ M , · ) = 1 2 µ ∗ h ϑ + ¯ ϑ, ζ i , ∀ ζ ∈ G . (2.2) (a3) A t eac h x ∈ M , the k ernel of ω x is pro vided by Ker( ω x ) = { ζ M ( x ) | ζ ∈ Ker(Ad µ ( x ) + Id G ) } . (2.3) The map µ is called the momen t map. A quasi-Hamilto nian dynamical system ( M , G, ω , µ, h ) is a quasi-Hamiltonian G -space with a distinguished G -in v arian t function h ∈ C ∞ ( M ) G , t he Hamiltonian. It follows from the axioms that there exists a unique v ector ﬁeld v h on M determined b y the follo wing tw o requiremen ts: ω ( v h , · ) = dh, L v h µ = 0 . (2.4) The ‘quasi-Hamiltonian vec tor ﬁeld’ v h is G -in v arian t a nd it pres erves ω , L v h ω = 0. Th us, G -in v ariant Hamiltonians on a quasi-Hamiltonia n G -space deﬁne ev olution ﬂows in m uch the same w ay as arbitrar y Hamiltonians do o n symplectic manifo lds. One can also in tro duce a n honest P oisson brac ke t on C ∞ ( M ) G . Naturally , if f and h are G -inv ariant functions and v f and v h the correspo nding quasi-Hamiltonian v ector ﬁelds, then this Poiss on brack et is giv en b y { f , h } := ω ( v f , v h ) . (2.5) Indeed, it is not diﬃcult to chec k tha t the result { f , h } is a gain an inv ariant function and all the usual prop erties (including the Jacobi identit y) are veriﬁed b y this Poisson brack et. It is w orth emphasizing that the general quasi-Hamiltonian manifold M is not symplectic and the quasi-Hamiltonian form ω do es not induce a prop er P oisson algebra on the smo oth functions on M but just on the G -inv ariant smo ot h functions. The quasi-Hamilto nian r eduction of a quasi-Hamiltonian dynamical system ( M , G, ω , µ, h ) that in terests us is determined b y c ho osing an elemen t µ 0 ∈ G . W e say that µ 0 is str ong ly r e gular if it satisﬁes the follow ing tw o conditions: 6 1. The subset µ − 1 ( µ 0 ) := { x ∈ M | µ ( x ) = µ 0 } is an embedded submanifold of M . 2. If G 0 ⊂ G is the isotropy group of µ 0 with resp ect to the adjoin t action, then the quotien t µ − 1 ( µ 0 ) /G 0 is a manifo ld f or whic h the canonical pro j ection p : µ − 1 ( µ 0 ) → µ − 1 ( µ 0 ) /G 0 is a smoot h submersion. The result of the reduction based on a strongly r egula r elem en t µ 0 is a standard Hamiltonian system, ( P , ˆ ω, ˆ h ). The reduced phase space P is the manifold P ≡ µ − 1 ( µ 0 ) /G 0 , (2.6) whic h carries the reduced symplectic form ˆ ω and reduced Hamiltonian ˆ h uniquely deﬁned b y p ∗ ˆ ω = ι ∗ ω , p ∗ ˆ h = ι ∗ h, (2.7) where ι : µ − 1 ( µ 0 ) → M is the tautological em b edding. W e stress that ˆ ω is a symplectic fo r m in the usual sense, whilst ω is neither closed nor g lobally non-degenerate in general. It follo ws from the a b ov e deﬁnitions that the Hamiltonia n v ector ﬁeld and the ﬂo w deﬁned by ˆ h o n P can b e obtained b y ﬁrst restricting the quasi-Hamiltonian v ector ﬁeld v h and its ﬂo w to the ‘constrain t surface’ µ − 1 ( µ 0 ) and t hen applying the canonical pro jection p . The P oisson brac k ets on ( P , ˆ ω ) are inherited from the P oisson brac k ets (2.5) of the G -in v ariant functions as in standard symplectic reduction. 2.2 Ev olution ﬂo ws on the in ternally fused double of S U ( n ) Consider a quasi-Hamiltonian space M and a set of k distinguished G -in v ariant functions on it. In the sens e o f the preceding subs ection, the se data deﬁne a fa mily of quasi-Hamiltonian dynamical systems. W e shall sp eak ab out a ‘comm uting k -family’ if the correspo nding quasi- Hamiltonian v ector ﬁelds all comm ute among eac h ot her. In this pap er, w e shall deal with two commuting ( n − 1)-f amilies of quasi-Hamiltonian dy- namical systems, whic h b oth liv e on a single quasi-Hamiltonian G -space. The quasi-Hamiltonian G -space in question is the so-called internal ly f use d double of the group G := S U ( n ) [1], whic h as a ma nif o ld is pro vided b y the direct pro duct D := G × G = { ( A, B ) | A, B ∈ G } . (2.8) The in v arian t scalar pro duct o n G := su ( n ) is g iv en by h η , ζ i := − 1 2 tr ( η ζ ) , ∀ η , ζ ∈ G . (2.9) The group G acts on D b y comp onen t wise conjugation 3 Ψ g : ( A, B ) 7→ ( g Ag − 1 , g B g − 1 ) . (2.10) 3 Later in some equations w e a pply g ∈ U ( n ) in the formu la of the action, which is harmles s since o nly the factor gr oup S U ( n ) / Z n ≃ U ( n ) /U (1) acts eﬀectively . 7 The 2-form ω of M := D reads 2 ω = h A − 1 dA ∧ , dB B − 1 i + h dAA − 1 ∧ , B − 1 dB i − h ( AB ) − 1 d ( AB ) ∧ , ( B A ) − 1 d ( B A ) i , (2.11) and the G -v alued momen t map µ is deﬁned by µ ( A, B ) = AB A − 1 B − 1 . (2.12) Consider a real class function h ∈ C ∞ ( G ) G . D eﬁne the deriv ative ∇ h ∈ C ∞ ( G, G ) G b y the equation d dt     t =0 h ( e tζ g ) = h ζ , ∇ h ( g ) i , ∀ g ∈ G, ∀ ζ ∈ G . (2.13) Asso ciate to h the following G -in v arian t functions on D , h 1 ( A, B ) := h ( A ) , h 2 ( A, B ) := h ( B ) . (2.14) The ev olution ﬂo w of t he quasi-Hamiltonian syste m ( D , G, ω , µ, h 1 ) throug h the initial v alue ( A 0 , B 0 ) is furnished by ( A ( t ) , B ( t )) = ( A 0 , B 0 e − t ∇ h ( A 0 ) ) , (2.15) while the system ( D , G, ω , µ, h 2 ) has the ﬂo w ( A ( t ) , B ( t )) = ( A 0 e t ∇ h ( B 0 ) , B 0 ) . (2.16) Indeed, the ev olutio n v ector ﬁeld give n b y the t -deriv ativ e of the ﬂo w (2.15) at the p oin t ( A ( t ) , B ( t )) of the do uble equals (0 ⊕ − B ( t ) ∇ h ( A 0 )) and one can easily v erify that it satis- ﬁes t he deﬁning relations (2 .4) of the quasi-Hamiltonian v ector ﬁeld b elonging to the function h 1 . In order to sp ecify the Hamiltonians of tw o comm uting ( n − 1)-families of quasi-Hamiltonian dynamical systems on the double, w e ha v e to in tro duce the so-called sp ectral functions on the group G . As a preparation, w e deﬁne the alc ove A b y A := n ( ξ 1 , ..., ξ n ) ∈ R n    ξ j ≥ 0 , j = 1 , ..., n, n X j =1 ξ j = π o , (2.17) and the op en alco v e A 0 b y A 0 := n ( ξ 1 , ..., ξ n ) ∈ R n    ξ j > 0 , j = 1 , ..., n, n X j =1 ξ j = π o . (2.18) W e then consider the inj ective map δ from A in to the subgroup S T n of the diagonal elemen ts of S U ( n ) giv en by δ 11 ( ξ ) := e 2i n P n j =1 j ξ j , δ k k ( ξ ) := e 2i P k − 1 j =1 ξ j δ 11 ( ξ ) , k = 2 , ..., n. (2.19) 8 The image of δ is a fundamen tal domain for the action of the W ey l group of S U ( n ) (i.e. the p erm utat io n group) on S T n , whic h is often called a W eyl alcov e. F or this reason, w e ma y also refer to A as a W eyl alco ve . With the aid of the fundamen tal w eights λ k of su ( n ) represen ted b y the diagonal matrices λ k ≡ P k j =1 E j,j − k n 1 n , the matrix δ ( ξ ) can b e recast in the f o rm δ ( ξ ) = exp − 2i n − 1 X k =1 ξ k λ k ! . (2.20) Here w e denoted by E j,j the n × n -matrix featuring 1 in the inte rsection of the j th -row with the j th -column and 0 ev erywhere else. Ev ery elemen t A ∈ S U ( n ) can b e written as A = g ( A ) − 1 δ ( ξ ) g ( A ) , (2.21 ) for some g ( A ) ∈ S U ( n ) and unique ξ ∈ A . Moreo v er, whene ver ξ ∈ A 0 , t he elemen t A is r e gular and g ( A ) is then determined up to left-m ultiplicatio n by an elemen t of S T n . By deﬁnition, t he j th comp onen t of the alcov e elemen t ξ en tering the decomp osition (2.21 ) is the v alue of the sp e ctr al function Ξ j on A ∈ S U ( n ). In other w ords, the conjugatio n inv ariant function Ξ j on G = S U ( n ) is c haracterized by the equation Ξ j ( δ ( ξ ) ) = ξ j , ∀ ξ ∈ A , j = 1 , ..., n. (2.22) It is easily seen that the sp ectral function Ξ j is smo oth on G reg ⊂ G , but it dev elops singularities at the non-regular p oin ts of G . Note also that Ξ n = π − P n − 1 j =1 Ξ j according to (2.17). We ar e now in the p osition to deﬁn e the 2 ( n − 1) distinguishe d G -invariant Hamiltonians α j , β j on the double D as fol lows: α j ( A, B ) := Ξ j ( A ) , β j ( A, B ) := Ξ j ( B ) , j = 1 , ..., n − 1 . (2.23) W e call α j and β j ‘sp ectral Hamiltonians’ and our next task is to show that they resp ectiv ely deﬁne c ommuting ( n − 1)- families of quasi-Hamiltonian dynamical systems. T o b e more precise, it m ust b e noted that the domain of the α j -Hamiltonians (resp. β j -Hamiltonians) is the dense op en subset D a ⊂ D (resp. D b ⊂ D ) consisting of the couples ( A, B ) ∈ D with A ∈ G reg (resp. B ∈ G reg ), whic h is stable under the corresp onding ﬂo ws. In order to describ e the ﬂo ws, w e now prov e the follo wing lemma. Lemma 1. The derivative of the sp e ctr al function Ξ j ∈ C ∞ ( G reg ) ( j = 1 , ..., n − 1 ) r e ads ∇ Ξ j ( A ) = g ( A ) − 1 (i( E j +1 ,j +1 − E j,j )) g ( A ) , ∀ A ∈ G reg . (2.24) Pr o of. The G -inv ariance of Ξ j implies t he G -equiv aria nce of ∇ Ξ j , and therefore it is enough to calculate ∇ Ξ j at the po in ts of the op en W eyl alcov e. But at the p oints of the W eyl a lco v e ∇ Ξ j m ust b e a diagonal matrix, b ecause of in v ariance under S T n inherited from the G - equiv ariance on G reg . The n ( ∇ Ξ j )( δ ( ξ )) is readily calculated t o b e i( E j +1 ,j +1 − E j,j ), whic h implies (2 .2 4) on accoun t of (2.21). 9 Note that ∇ Ξ j ( A ) is w ell-deﬁned by form ula (2.24) since g ( A ) is determined up to left- m ultiplication b y the elemen ts of the maximal to r us, and its smo o t hness on G reg follo ws directly from the smoothness of Ξ j . By com bining the formulae (2.1 5 ), (2.16) and (2.24), w e ﬁnd that the follo wing 2 π -p erio dic curv e in D a passing through ( A, B ) is the in tegral curv e of t he quasi-Hamiltonian v ector ﬁeld v α j :  A, B g ( A ) − 1 diag(1 , 1 , ..., 1 , e i t , e − i t , 1 , ..., 1) g ( A )  , t ∈ R , (2.25) and the follo wing 2 π - p erio dic curv e in D b is an in tegral curv e of t he ve ctor ﬁeld v β j :  Ag ( B ) − 1 diag(1 , 1 , ..., 1 , e − i t , e i t , 1 , ..., 1) g ( B ) , B  , t ∈ R . (2.26) In particular, the form ulae (2.25 ) and (2.26) trivially imply t ha t the α -ﬂows c ommute am ong themselves and so do the β -ﬂows. In other w ords, the inﬁnitesimal actions of the comm uting quasi-Hamiltonian v ector ﬁelds v α j in tegrate to a (smo ot h fr ee) action of the torus T n − 1 := U (1) ( n − 1) (2.27) on D a ⊂ D . The formula (2.25) giv es the action of the j th U ( 1) factor of T n − 1 , the phase e i t sits in the j th place of the diagonal and g ( A ) is giv en b y the decomp osition (2.21). In spite of the ambiguit y in the deﬁnition of g ( A ), the curv e (2.25) is deﬁned unambiguously . T o display the action map Ψ a : T n − 1 × D a → D a more explicitly , w e in tro duce ρ ( τ ) := exp  i n − 1 X j =1 t j ( E j,j − E j +1 ,j +1 )  for all τ = ( e i t 1 , ..., e i t n − 1 ) ∈ T n − 1 . (2.28) Then w e ha ve Ψ a τ : ( A, B ) 7→ ( A, B g ( A ) − 1 ρ ( τ ) g ( A )) . (2.29) Similarly , t he comm uting quasi-Hamiltonian v ector ﬁelds v β j generate a T n − 1 -action on the dense op en subset D b ⊂ D , and the corresp o nding action map Ψ b : T n − 1 × D b → D b reads Ψ b τ : ( A, B ) 7→ ( Ag ( B ) − 1 ρ ( τ ) − 1 g ( B ) , B ) . (2.30) W e observ e also that { α j , α l } = 0 = { β j , β l } , (2.31) where the P oisson brac k et of G -inv ariant functions w as deﬁned in Eq. (2.5). Indeed, w e hav e { α j , α l } ≡ ω ( v α j , v α l ) ≡ L v α l α j = 0 , (2.32) where the last equalit y holds since the α l -generated ﬂow acts only on the B -comp onen t o f the double (see Eq. (2.15) ) lea ving therefore the α j -functions inv arian t. The Poiss o n-comm utativity (2.32) o f the s p ectral Hamiltonians α j (and that of the β j ) surviv es a ny quasi-Hamiltonian reduction, and t his fact will provide o ne of the underpinnings of our approac h to the compactiﬁed Ruijsenaars-Sc hneider syste m. 10 Remark 1. The sp ectral Hamiltonians α j , β j can b e view ed a s the resp ectiv e generator s of the P oisson-comm ut a tiv e rings C a and C b consisting of smo o th in v arian t functions deﬁned with the help of Eq. ( 2 .14): C a := { h 1 ∈ C ∞ ( D a ) G | h ∈ C ∞ ( G reg ) G } , C b := { h 2 ∈ C ∞ ( D b ) G | h ∈ C ∞ ( G reg ) G } . (2.33) The rings C a and C b can b e of course generated also b y other generators, e.g. b y t he in v arian ts H m ( A, B ) ≡ ℜ tr ( A m ), H − m ( A, B ) ≡ ℑ tr ( A m ) and, resp ectiv ely , by F m ( A, B ) ≡ ℜ tr ( B m ), F − m ( A, B ) ≡ ℑ tr ( B m ) for m ∈ N . Although the generators H ± m , F ± m ha ve the apparent adv an tage of b eing globally smo oth on G , it is more suited for our purp ose t o use t he generators α j and β j since their ﬂows are 2 π - p erio dic (this circumstance will b e crucial for o ur argumen ts in Subsec tions 3.3 and 3.4). It will b e sho wn tha t after our quasi-Hamiltonian reduction the matrix A yields t he Lax mat r ix of the R uijsenaars-Sc hneider system, the generato r s H ± m b ecome the Ruijsenaars-Schne ider Hamiltonians, the α j b ecome the a ctio n- v ariables and the β j will parametrize the particle-p ositions. W e shall a lso establish a dual in t erpretatio n of the reduction, where B yie lds the L a x matrix, the g enerators F ± m b ecome the Hamiltonians, the β j b ecome the action-v ariables and the α j the parameters of the particle-p ositions. Remark 2. W e note that from the viewpoint of the corresp onding mo duli spaces of ﬂat connections the ﬂows (2.15 ) and (2.16) are sp ecial cases of the Goldman ﬂo ws [12]. The fact that the sp ectral functions are not smo o th at the non-r egula r p o in ts of G will cause no pro blem, since w e shall consider a quasi-Hamiltonian r eduction for whic h the constrain t surface µ − 1 ( µ 0 ) turns out to be a submanifold of G reg × G reg ⊂ D . 3 Reduction of the in ternally fused double of S U ( n ) As w e already know , the starting p oint of the reduction is the c hoice of an elemen t µ 0 ∈ G , and the corresp onding constrain t surface µ − 1 ( µ 0 ) is the space of those ( A, B ) ∈ D tha t solv e the momen t map constrain t 4 AB A − 1 B − 1 = µ 0 . (3.1) The simplest non- trivial p o ssibilit y is to take µ 0 from a conjug a cy class of minimal but non- zero dimension. As seen from simple coun ting, in this case w e ma y hop e to obtain a non-trivial reduced system of dimension 2( n − 1). Obviously , diﬀeren t c hoices from the same conjug a cy class yield equiv alen t reduced systems. W e here c ho ose µ 0 diagonal of the form µ 0 = diag( e 2i y , ..., e 2i y , e 2(1 − n )i y ) , y ∈ R . (3.2) An ticipating its ev entual identiﬁcation with the parameter of the Hamiltonian (1.1), in the next subsection w e restrict y to the r a nge 0 < | y | < π n and then pro ve that µ 0 (3.2) leads to a smo oth, compact reduced phase space P = µ − 1 ( µ 0 ) /G 0 . In the end, w e shall iden tify the reduced phase space w it h the c omplex pro jectiv e space C P ( n − 1) and shall also obtain a full c haracterization of the reduced spectral Hamiltonians ˆ α j and ˆ β j in t erms o f the standar d par a metrization of C P ( n − 1). 4 A similar constra int equation w as studied previously in a diﬀerent lo ca l con text [1 4, 10] and in complex holomorphic settings [11, 2 6]. 11 3.1 The re duced ph ase space is smo oth a nd compact Theorem 1. Co nsider the di a gonal matrix µ 0 = diag( e 2i y , ..., e 2i y , e 2(1 − n )i y ) ∈ S U ( n ) with a r e al p ar ameter y verifying 0 < | y | < π n . (3.3) A n y such µ 0 is a str ongly r e gular value of the moment map µ (2.12), and the c orr esp onding r e duc e d phase sp ac e P = µ − 1 ( µ 0 ) /G 0 is a smo oth, c o mp act manifold o f dimens i on 2( n − 1) . Pr o of. W e ﬁrst remark that µ − 1 ( µ 0 ) is non-empty since ev ery elemen t of any connected, compact semi-simple Lie group can be written as a comm utator [15 ]. T o contin ue, note that the action (2 .1 0) of G on the do uble naturally descends to an action of the factor group ¯ G := G/ Z n , whe re Z n is the cen ter of G = S U ( n ). Similarly , the action of the adjoint isotropy group G 0 ⊂ G of µ 0 on the constraint surface µ − 1 ( µ 0 ) descends to an action of the factor g roup ¯ G 0 := G 0 / Z n . (3.4) It is suﬃcien t to pro v e that this latter action is free. Indeed, the free action of ¯ G 0 implies the em b edded nature of µ − 1 ( µ 0 ) b y statemen t 3 of Prop o sition 4.1 of [1] (whic h sho ws that the lo cally free nature of the action of the isotropy group on the constrain t surface is equiv alen t to the regularity of the moment map v alue). The f a ct that the compact Lie group ¯ G 0 acts freely on the smo oth compact manifold µ − 1 ( µ 0 ) then ensures t ha t µ − 1 ( µ 0 ) /G 0 = µ − 1 ( µ 0 ) / ¯ G 0 (3.5) also b ecomes a smo oth compact manifold. As f o r it s dimension, w e hav e dim( ¯ G 0 ) = ( n − 1) 2 , since G 0 = S ( U ( n − 1 ) × U (1)) , (3.6) and therefore dim  µ − 1 ( µ 0 ) / ¯ G 0  = ( n − 1)( n + 1) − ( n − 1) 2 = 2( n − 1) . (3.7) It remains to pro ve that if ( A, B ) ∈ µ − 1 ( µ 0 ) is ﬁxed b y some g ∈ S U ( n ), then g b elongs to the central subgroup Z n . F or this, suppo se that ( g Ag − 1 , g B g − 1 ) = ( A, B ) ho lds for some ( A, B ) ∈ µ − 1 ( µ 0 ) and g ∈ G . This implies that b oth A and B b elong to the centralize r subgroup G ( g ) := { η ∈ S U ( n ) | η g η − 1 = g } ⊆ S U ( n ) , (3.8) and µ 0 = AB A − 1 B − 1 b elongs to the corresp onding deriv ativ e subgroup G ( g ) ′ that con ta ins the group-comm uta tors in G ( g ). Now observ e that if g is not cen tral, then it is conjugate to an elemen t g 0 of the maximal torus of S U ( n ) whose cen tralizer G ( g 0 ) is a blo c k-diagonal subgroup G ( g 0 ) = S ( U ( n 1 ) × U ( n 2 ) × · · · × U ( n k )) , (3.9) for some k ≥ 2 and p ositiv e integers for whic h n 1 + n 2 + · · · + n k = n ( k = 1 o ccurs for g ∈ Z n ). Accordingly , if g is not cen tral, then µ 0 m ust b e conjugate to an eleme n t of the commutator subgroup G ( g 0 ) ′ of G ( g 0 ) (3.9). It is r eadily seen that G ( g 0 ) ′ is pro vided by G ( g 0 ) ′ = S U ( n 1 ) × S U ( n 2 ) × · · · × S U ( n k ) , (3.10) 12 whic h leads to a con tradiction. Indeed, it follows from (3.3 ) that in whatev er w ay w e partition the n eigen v alues of µ 0 in to k > 1 parts, the pro duct of the eigen v alues in a t least one par t (actually in eac h part) will not b e equal to 1. Th us µ 0 cannot b e conjugate to an elemen t of G ( g 0 ) ′ (3.10) for k > 1. W e remark in passing that the ab ov e argumen ts sho w also the strong regularit y of any suc h momen t map v alue from S U ( n ) whic h is not conjugate to a blo c k-dia g onal S U ( n ) matrix whose blo c ks t hemselv es ha v e determinant 1. 3.2 The i mages of the Hamiltonians α j , β j restricted to µ − 1 ( µ 0 ) Ha ving established t ha t the reduced phase space µ − 1 ( µ 0 ) /G 0 is a compact smo oth manifold, the next step is to determin e the reduced symplectic form ˆ ω on it. Remark ably , the shortest wa y to this goa l leads through the study of the images of the sp ectral Hamiltonians α j , β j (2.23) restricted to the constraint surface µ − 1 ( µ 0 ). Theorem 2. F or µ 0 = diag ( e 2i y , ..., e 2i y , e 2(1 − n )i y ) w ith 0 < | y | < π n , the c onvex p olytop e P y := n ( ξ 1 , ..., ξ n − 1 ) ∈ R n − 1    ξ j ≥ | y | , j = 1 , ..., n − 1 , n − 1 X j =1 ξ j ≤ π − | y | o (3.11) is the c ommon image of the ve ctor-value d Hamiltonian functions ( α 1 , ..., α n − 1 ) and ( β 1 , ..., β n − 1 ) r estricte d to the c onstr aint surfac e µ − 1 ( µ 0 ) . Pr o of. The form ula t ion of Theorem 2 in terms of the con v ex p olytop e P y will pla y an imp ortant role in Section 3.4. Ho w ev er, from the tec hnical p oint of view, it is more con v enien t to include in to the analysis also the f unctions α n ( A, B ) := Ξ n ( A ), β n ( A, B ) := Ξ n ( B ) (cf. (2.22)–(2.23)) and to pro v e the follo wing equiv alen t statemen t: The c ommon image of the ve ctor-value d functions ( α 1 , ..., α n − 1 , α n ) and ( β 1 , ..., β n − 1 , β n ) r e- stricte d to the c onstr aint surfac e µ − 1 ( µ 0 ) i s the se t A y := n ( ξ 1 , ..., ξ n ) ∈ R n    ξ j ≥ | y | , j = 1 , ..., n , n X j =1 ξ j = π o . (3.12) The constraint AB A − 1 B − 1 = diag ( e 2i y , ..., e 2i y , e 2(1 − n )i y ) is inv ariant under the in terchange of A and B a ccompanied with a simultaneous c hange of the sign of the parameter y . Since A y do es not dep end on the sign of y , it is enough to show that the ima g e of ( β 1 , ..., β n − 1 , β n ) restricted to µ − 1 ( µ 0 ) is A y . Part 1 : First we sho w that if ξ ∈ A y then there exist g ( ξ ) ∈ S U ( n ) and A ( ξ ) ∈ S U ( n ) suc h that A ( ξ ) and B ( ξ ) := g ( ξ ) − 1 δ ( ξ ) g ( ξ ) solve the momen t map constraint (3.1). (Recall that t he map δ : A → S T n w as deﬁned in Eq. (2.19) in connection with the decomp o sition ( 2 .21); b elow w e use δ j := δ j j .) Consider an arbitrary ξ = ( ξ 1 , ..., ξ n ) ∈ A y and deﬁne ξ k n + j := ξ j , k ∈ Z , j = 1 , ..., n. (3.13) 13 As an immediate consequence o f Eq. (2.19), note the v alidit y o f the f o llo wing relation: δ j ( ξ ) δ l ( ξ ) − 1 = exp  2i j − 1 X k = l ξ k  , 1 ≤ l < j ≤ n. (3.14) Moreo ve r , using the con v en tion (3.13), w e hav e cot | y | ≥ | cot( j − 1 X k = l ξ k ) | , l = 1 , ..., n ; j = l + 1 , ..., l + n − 1 , (3.15) b ecause P j − 1 k = l ξ k in (3 .15) alw ay s lies in the closed in terv al [ | y | , π − | y | ]. Th us fo r ξ ∈ A y and l = 1 , ..., n we obtain the reality and non-n e gativity of the quan tities z l ( δ ( ξ ) , y ) deﬁned by z l ( δ ( ξ ) , y ) := e 2i y − 1 e 2 n i y − 1 n Y j 6 = l δ j ( ξ ) − e 2i y δ l ( ξ ) δ j ( ξ ) − δ l ( ξ ) = (sin | y | ) n sin ( n | y | ) l + n − 1 Y j = l +1  cot | y | − y | y | cot( j − 1 X k = l ξ k )  . (3.16) Note that the second equality in (3.1 6) follows fr om (3.14) and from the following trig onometric iden tity : cot y − cot β ≡ sin ( β − y ) sin y sin β ≡ 2i e 2i β − e 2i y ( e 2i y − 1)( e 2i β − 1) . (3.17) No w consider an arbitrary map v : A y → C n suc h that | v l ( ξ ) | 2 := z l ( δ ( ξ ) , y ) . (3.18) Let us sho w then that || v ( ξ ) || 2 := n X l =1 | v l ( ξ ) | 2 = 1 . (3.19) F or this, w e ﬁrst chec k the equalit y o f the following t w o p olynomials in an auxiliary complex v ariable λ : n Y j =1 ( δ j ( ξ ) − λ ) = n Y j =1 ( δ j ( ξ ) ǫ 2i y − λ ) + ( e 2i(1 − n ) y − e 2i y ) n X k =1  | v k ( ξ ) | 2 δ k ( ξ ) n Y j 6 = k ( δ j ( ξ ) e 2i y − λ )  . (3.20) Indeed, it is easy to v erify (3.20) for the n (all distinc t) v alues λ j = δ j ( ξ ) e 2i y , j = 1 , ..., n . Consequen tly , (3 .2 0) holds t r ue for an y λ , and w e obtain (3.19) b y ev aluating (3.20) for λ = 0. W e note that the p olynomial iden tit y (3.20) can b e understoo d as the equalit y of the c har- acteristic p olynomials of the diagonal matrix δ ( ξ ) and of the mat r ix µ v ( ξ ) δ ( ξ ), det( δ ( ξ ) − λ 1 n ) = det( µ v ( ξ ) δ ( ξ ) − λ 1 n ) , (3.21) where the matrix µ v ( ξ ) reads µ v ( ξ ) := e 2i y 1 n + ( e 2i(1 − n ) y − e 2i y ) v ( ξ ) v ( ξ ) † . (3.22) 14 Because of the normalization prop ert y (3.19), there certainly exists an S U ( n ) matrix g ( ξ ) ha ving the v ector v ( ξ ) as its last column, i.e., v j ( ξ ) = g ( ξ ) j n . It is then easily seen that the diag o nal momen t map v alue µ 0 (3.2) can b e written as µ 0 = g ( ξ ) − 1 µ v ( ξ ) g ( ξ ) , (3.23) and the determinan t iden tit y (3.21) can be therefore rewritten as det( g ( ξ ) − 1 δ ( ξ ) g ( ξ ) − λ 1 n ) = det( µ 0 g ( ξ ) − 1 δ ( ξ ) g ( ξ ) − λ 1 n ) . (3.24) This means tha t t he matrix B ( ξ ) := g ( ξ ) − 1 δ ( ξ ) g ( ξ ) has the same sp ectrum as the matrix µ 0 B ( ξ ), whic h implies the existence of a matrix A ( ξ ) ∈ S U ( n ) suc h that A ( ξ ) B ( ξ ) A ( ξ ) − 1 = µ 0 B ( ξ ) . (3.25) P art 1 of the pro of of Theorem 2 is thus complete. Part 2 : It remains to show that if ( A, B ) ∈ D satisﬁes the momen t map constrain t (3.1), then B can b e written as B = g − 1 δ ( ξ ) g (3.26) with some g ∈ S U ( n ) a nd some ξ ∈ A y (3.12). Using that an y B ∈ S U ( n ) has the form (3.26) with uniquely determined ξ ∈ A (2.17), it will b e con v enien t to distinguish tw o cases: i) ξ is in the op en W eyl alco v e A 0 (2.18); ii) ξ / ∈ A 0 . W e consider ﬁrst i) a nd then ii). More precisely , w e shall ﬁrst pro v e the statemen t: i) If ( A, B ) ∈ µ − 1 ( µ 0 ) , B = g − 1 δ ( ξ ) g for so me g ∈ S U ( n ) and ξ ∈ A 0 , then ξ ∈ A y . Then w e pro v e the statemen t: ii) If ξ / ∈ A 0 then whatever is g ∈ S U ( n ) , t h e matrix B = g − 1 δ ( ξ ) g c annot b e the se c ond c omp onent of s ome solution ( A, B ) of the c onstr aint (3.1). Pr o of of statement i): Deﬁne A g and µ g as A g := g Ag − 1 , µ g := g µ 0 g − 1 . (3.27) The v alidity of (3.1 ) implies A g δ ( ξ ) A − 1 g = µ g δ ( ξ ) . (3.28) Note that the matrix µ g dep ends only on the last column o f the matrix g . T o see t his w e rewrite µ 0 as µ 0 = e 2i y 1 n + ( e 2i(1 − n ) y − e 2i y ) v 0 v † 0 , (3.29) where the v ector v 0 ∈ C n is deﬁned by its components ( v 0 ) n := 1 , ( v 0 ) j := 0, j = 1 , ..., n − 1. This means that µ g can b e written as µ g = e 2i y 1 n + ( e 2i(1 − n ) y − e 2i y )( g v 0 )( g v 0 ) † = e 2i y 1 n + ( e 2i(1 − n ) y − e 2i y ) v v † , (3.30) where v := g v 0 is the last column of the matrix g , i.e., v j = g j n . W e observ e fro m (3.28) that the sp ectrum of the matrix µ g δ ( ξ ) m ust b e equal to t he sp ectrum of δ ( ξ ), whic h entails the equality o f the characteristic po lynomials det( δ ( ξ ) − λ 1 n ) = det( µ g δ ( ξ ) − λ 1 n ) . (3.31) 15 Both determinan ts in (3.31) can b e easily ev a luated so that (3.31) becomes n Y j =1 ( δ j ( ξ ) − λ ) = n Y j =1 ( δ j ( ξ ) ǫ 2i y − λ ) + ( e 2i(1 − n ) y − e 2i y ) n X k =1  | v k | 2 δ k ( ξ ) n Y j 6 = k ( δ j ( ξ ) e 2i y − λ )  . (3.32) Due to the assumption ξ ∈ A 0 , we know that the elemen ts of the diagonal matrix δ ( ξ ) hav e n distinct v alues. By ev aluating the relation (3.32) f or the n distinct v alues λ l = δ l ( ξ ) e 2i y , w e immediately ﬁnd | v l | 2 = e 2i y − 1 e 2 n i y − 1 n Y j 6 = l δ j ( ξ ) − e 2i y δ l ( ξ ) δ j ( ξ ) − δ l ( ξ ) = (sin | y | ) n sin ( n | y | ) l + n − 1 Y j = l +1  cot | y | − y | y | cot( j − 1 X k = l ξ k )  . (3.33) No w we ha v e to distinguish whether y > 0 or y < 0. W e start with y > 0. Then the ﬁrst term in the last pro duct in (3.33) is cot y − cot ξ l . If ξ l w as strictly inferior t o | y | for a certain l , w e w ould hav e ob viously cot | y | − cot ξ l < 0 . (3.34) If w e then had cot | y | − cot( ξ l + ξ l +1 ) > 0 , this would imply cot | y | − cot( ξ l + ξ l +1 + ξ l +2 ) > 0 etc, whic h w ould give | v l | 2 < 0 b y (3.33). In order t o a void suc h a contradiction, we see that the assumption ξ l < | y | leads to cot | y | − cot( ξ l + ξ l +1 ) ≤ 0 , (3.35) and hence to ξ l + ξ l +1 ≤ | y | . (3.36) Because ξ ∈ A 0 , w e ha ve ξ l > 0 and ξ l +1 > 0. This fact together with (3.36) giv es ξ l +1 < | y | . (3.37) Th us, w e ha ve sho wn that if ξ l < | y | for some l then ξ l +1 < | y | , and hence ξ j < | y | for all j = 1 , ..., n . This is a contradiction sinc e w e know, resp ectiv ely , from (2.17) and (3.2) that P n j =1 ξ j = π and n | y | < π hold. W e conclude that ξ l ≥ | y | for all l = 1 , ..., n , whereb y statemen t i) is pro v ed for y > 0. If y < 0, note that the last term in the last pro duct in (3.33) is equal to cot | y | − cot ξ l − 1 . If ξ l − 1 w as strictly inferior to | y | , w e w ould hav e ob viously cot | y | − cot ξ l − 1 < 0 . (3.38) If moreov er the nex t to last term, cot | y | − cot( ξ l − 1 + ξ l − 2 ), w a s strictly p ositiv e, this w o uld giv e | v l | 2 < 0 because all prece ding terms would ha v e to b e strictly p ositiv e, too. Th us, the assumption ξ l − 1 < | y | leads to cot | y | − cot( ξ l − 1 + ξ l − 2 ) ≤ 0. This implies ξ l − 2 < | y | , and consequen tly ξ j < | y | for all j . But this creates the same contradiction as in the case y > 0, whereb y the pro of of statemen t i) is complete. Pr o of of statement ii) : T o start, w e note that the condition that ξ ∈ A but ξ / ∈ A 0 (2.18) means that there exists at least one index l ∈ { 1 , ..., n } for which ξ l = 0. W e call suc h a conﬁguration 16 ξ de gener ate , since it is c haracterized by the fact that the phases δ j ( ξ ) tak e only r < n distinct v alues. W e ﬁnd it more conv enien t to describ e the degenerate conﬁgurations directly by their phases δ j ( ξ ). Since the map δ : A → S T n is injectiv e, suc h a description is equiv alent to the previous description in terms ξ and from no w on w e simply write δ j instead of δ j ( ξ ). Fixing an arbitrary degenerate conﬁguration, we partition n as a sum of 1 ≤ r < n p o sitiv e in tegers, n = k 1 + k 2 + ... + k r , (3.39) in suc h a w ay that δ 1 = δ 2 = ... = δ k 1 , δ k 1 +1 = δ k 1 +2 = ... = δ k 1 + k 2 , . . . , δ P r − 1 i =1 k i +1 = δ P r − 1 i =1 k i +2 = ... = δ n . (3.40) Plainly , at least o ne in teger k s (1 ≤ s ≤ r ) m ust b e sup erior or equal to 2. Deﬁne the matrices A g , µ g and the vec tor v in the same wa y as in the pro of of statemen t i). Then the assumed v alidit y of the relation A g δ A − 1 g = µ g δ en tails the equalit y of the c haracteristic p olynomials of the matrices δ and µ g δ , whic h no w yields r Y j =1 (∆ j − λ ) k j = r Y j =1 (∆ j e 2i y − λ ) k j + ( e 2i(1 − n ) y − e 2i y ) r X m =1 Z m ∆ m (∆ m e 2i y − λ ) k m − 1 r Y j 6 = m (∆ j e 2i y − λ ) k j . (3.41) Here w e in t r o duced r distinct v ariables ∆ s ( s = 1 , ..., r ), ∆ 1 := δ k 1 , ∆ 2 := δ k 1 + k 2 , . . . , ∆ r := δ k 1 + k 2 + ... + k r ≡ δ n , (3.42) and r n o n-ne gative real v ariables Z s , Z 1 := | v 1 | 2 + | v 2 | 2 + ... + | v k 1 | 2 , . . . , Z j +1 := | v k 1 + ... + k j +1 | 2 + | v k 1 + ... + k j +2 | 2 + ... + | v k 1 + k 2 + ... + k j +1 | 2 (3.43) for all j = 1 , ..., r − 1. Due to the degeneracy of δ , the implications of (3.41) are qualitative ly diﬀeren t from the implications of its relativ e (3.32) obtained in case i). T o see this, w e no w rewrite equation (3.41) as a relat io n b et w een t wo ratio nal functions of λ : Q (∆ , y , λ ) := r Y j =1 (∆ j − λ ) k j (∆ j e 2i y − λ ) k j − 1 = r Y j =1 (∆ j e 2i y − λ ) + ( e 2i(1 − n ) y − e 2i y ) r X m =1 Z m ∆ m r Y j 6 = m (∆ j e 2i y − λ ) . (3.44) Eq. (3.44) sa ys that the v aria bles ∆ s (3.42) m ust b e suc h that the rational function Q (∆ , y , λ ) is a po lynomial in λ . This me ans that all putative p oles of Q (∆ , y , λ ) m ust b e cancelled b y appropriate mo no mials presen t in the n umerator. The necessary and suﬃcien t condition for this to o ccur is easily seen to b e the follow ing: F or every ind e x m ∈ { 1 , ..., r } such that k m > 1 , ther e must exist an index s ∈ { 1 , ..., r } such that ∆ s = ∆ m e 2i y and k s ≥ k m − 1 . (*) F rom now on we consider only the admissible degenerate δ -conﬁgurations that, b y deﬁnition, satisfy the condition (*). (W e saw that other degenerate conﬁgurations cannot o ccur in the sp ectrum of the matrix B solving t he constraint (3.1).) T aking any such conﬁguration, with ∆ in (3.42), w e can ﬁnd the uniquely determined quan tities Z m = Z m (∆ , y ) ( m = 1 , ..., r ) for 17 whic h the relation (3.44) is satisﬁed. F or this, it is suﬃcien t to use r diﬀeren t v alues of the parameter λ giv en b y λ m = ∆ m e 2i y , whe reb y w e obtain Z m (∆ , y ) from (3.44). How ev er, in distinction to the non-degenerate cases, here three p ossibilities ma y o ccur. First, if k m = 1 and there exists no such index s for whic h ∆ s = ∆ m e 2i y , then w e ﬁnd Z m (∆ , y ) = e 2i y − 1 e 2 n i y − 1 r Y j 6 = m  ∆ j − e 2i y ∆ m ∆ j − ∆ m  k j 6 = 0 . (3.45) Second, if k m > 1 and k s = k m − 1, then Z m (∆ , y ) = ( − 1) k m +1 e 2( k m − 1)i y e 2i y − 1 e 2 n i y − 1 r Y j 6 = m,s  ∆ j − e 2i y ∆ m ∆ j − ∆ m  k j 6 = 0 . (3.46) Here and b elo w, it should not cause a n y confusion that w e suppressed the dep endence of s on m as give n by the condition (* ). Third, in the rest of the case s, for whic h either k m = 1 and there exists an index s with ∆ s = ∆ m e 2i y or k m > 1 and k s > k m − 1, we obtain Z m (∆ , y ) = 0 . (3.47) Let S ( δ ) denote the set of the in tegers m that o ccur in Eq s. (3.45) and (3.46). This set cannot b e empt y , since o therwise all comp o nen ts of the v ector v = g v 0 of unit norm w ere zero (cf. (3.43)). W e are going to ﬁnish the pro of o f statemen t ii) by showin g that Eqs. (3.45) and (3.46) imply that at least one of the a priori non-negativ e quan tities Z m (∆ , y ) is necessarily strictly negativ e, whatev er is the admissible degenerate δ -conﬁguration that we consider. T o this end, w e in tr o duce a real p ositiv e parameter ǫ and asso ciate to ev ery admissible degenerate δ - conﬁgur a tion a contin uous ǫ -family o f conﬁgurations δ ǫ in the op en W eyl a lco ve δ ( A 0 ) ⊂ S T n : δ ǫ,p := ∆ 1 e i pǫ , p = 1 , ..., k 1 ; δ ǫ,k 1 + ... + k j − 1 + p := ∆ j e i pǫ , p = 1 , ..., k j , j = 2 , ..., r , (3.48) where we use the partition (3.39). It is evide n t that for suﬃcien tly small v alues of ǫ > 0 the conﬁgurations δ ǫ are non-de gene r ate , i.e., they sit in the δ -image o f A 0 (deﬁned in (2.18) and (2.19)). Consider now for l = 1 , ..., n the quantities z l ( δ ǫ , y ), z l ( δ ǫ , y ) := e 2i y − 1 e 2 n i y − 1 n Y j 6 = l δ ǫ,j − e 2i y δ ǫ,l δ ǫ,j − δ ǫ,l , (3.49) whic h appeared also in the fo rm ula (3.16). F rom the fact that the conﬁgura tion is admissible (*) it fo llo ws easily that for some l ’s the quan tities z l ( δ ǫ , y ) v anish. More precisely , w e ﬁr st observ e that z l ( δ ǫ , y ) = 0 if l / ∈ { k 1 , k 1 + k 2 , ..., k 1 + ... + k r } (3.50) and also z k 1 + k 2 + ... + k m ( δ ǫ , y ) = 0 ∀ m for whic h ∃ s suc h that ∆ s = e 2i y ∆ m and k s ≥ k m . (3.51) Moreo ve r , it turns out that the li m its lim ǫ → 0 z l ( δ ǫ , y ) exist and do not v anish for all other l . That is they do not v anish fo r all l = k 1 + k 2 + ... + k m for whic h one of the following alternativ es 18 o ccurs: a) k m = 1 and there is no s suc h that ∆ s = e 2i y ∆ m ; b) k m > 1, ∆ s = e 2i y ∆ m and k s = k m − 1. Those non-v anishing limits read lim ǫ → 0 z k 1 + ... + k m ( δ ǫ , y ) = Z m (∆ , y ) , (3.52) where Z m (∆ , y ) is g iven b y Eqs. (3.45) or (3.46) for the cases a) and b), resp ectiv ely . In other w ords, the m -v alues o ccurring in (3.52) form the set S ( δ ) deﬁned after (3.47). No w turning to the crux o f the argumen t, w e note that for small ǫ the conﬁgur a tion δ ǫ do es not b elong to the δ - image of A y (3.12). Indeed, whenev er k j > 1 (recall that such 1 ≤ j ≤ r exists), w e observ e that in the ξ -para metrization ( 2.19) o f the conﬁguration δ ǫ w e ha ve ξ k j − 1 = ǫ/ 2 < | y | . F o llo wing the pro of of statemen t i), the quan tity z l ( δ ǫ , y ) m ust b e therefore strictly negativ e at least for one (in principle ǫ -dependen t) v alue of l . As an ob vious consequence, there m ust also exist a ﬁxed index l and a decreasing series ǫ p → 0 suc h that z l ( δ ǫ p , y ) is strictly negativ e for all p ositiv e in t egers p . F rom (3.50) and (3.5 1), w e conclude the existence of an in teger m ∗ ∈ S ( δ ) suc h that the ab ov e men tioned ǫ -indep endent l is given by l = k 1 + k 2 + ... + k m ∗ . W e kno w from (3.52) that the limit lim p →∞ z k 1 + ... + k m ∗ ( δ ǫ p , y ) = Z m ∗ (∆ , y ) (3.53) do es not v anish, whic h implies that Z m ∗ (∆ , y ) mus t b e strictly nega t ive. This is a con t radiction with the non-negativit y of the v ariables Z m (3.43). 3.3 The re duced ph ase space is a Hamiltonian toric manifold By deﬁnition, a Hamiltonian toric manifold 5 is a compact, connected sy mplectic manifold of dimension 2 d equipp ed with a n eﬀectiv e, Hamiltonian action of a torus of dimen sion d . W e already know that t he reduced phase space P = µ − 1 ( µ 0 ) /G 0 is a compact symplectic manifold. The follo wing Lemmas 2 a nd 3 show t ha t P is a Hamiltonian toric manifold. Lemma 2 refers to the β -generated torus action (2.30), but of c ourse an analogous result holds also f o r the α -generated action (2.29); and ev en tua lly this will explain the Ruijsenaars self-duality . Lemma 2. The β -gener ate d T n − 1 -action o n the op en submanifold D b of the internal ly fuse d double D , given by (2 . 3 0), desc ends to the r e duc e d phas e sp ac e P = µ − 1 ( µ 0 ) /G 0 , w her e it b e c o m es Hamiltonian and eﬀe ctive. Pr o of. It follows from Theorem 2 that the constraint surface µ − 1 ( µ 0 ) ⊂ D lies completely in t he op en submanifold D b ⊂ D (recall that D b is the set o f pairs ( A, B ) ∈ D for whic h B is regular). Th us, t he statemen t that the torus action Ψ b (2.30) descends to a Hamiltonian torus action on t he reduced phase space follow s immediately from the general theory of quasi-Hamiltonian reduction [1], whic h w e brieﬂy summarized ar ound equation ( 2 .7). In fact, the reduced torus action ˆ Ψ b : T n − 1 × P → P c an b e deﬁned b y means of the equalit y ˆ Ψ b τ ◦ p = p ◦  Ψ b τ | µ − 1 ( µ 0 )  , ∀ τ ∈ T n − 1 , (3.54) 5 The gener al theo ry of these c o mpact completely in tegr able systems is reviewed, for example, in [2 ]. 19 where p : µ − 1 ( µ 0 ) → P is the canonical pro jection. The corresp onding inﬁnitesimal torus action on P is generated by the v ector ﬁelds ˆ v ˆ β j ( j = 1 , ..., n − 1) that a r e the pro jections of the v ector ﬁelds v β j (2.26) restricted to µ − 1 ( µ 0 ). These pro jected v ector ﬁelds are Hamiltonian, ˆ ω ( ˆ v ˆ β j , · ) = d ˆ β j , (3.55) where ˆ ω is the reduce d symple ctic form on P and the reduced Hamiltonians ˆ β j ∈ C ∞ ( P ) are c haracterized by ˆ β j ◦ p = β j ◦ ι using the em b edding ι : µ − 1 ( µ 0 ) → D . In o t her words, ˆ β ≡ ( ˆ β 1 , ..., ˆ β n − 1 ) : P → R n − 1 (3.56) is the momen t map for the T n − 1 -action ˆ Ψ b on P . Supp ose now that the T n − 1 -action ˆ Ψ b on P is not eﬀectiv e. W e observ e from (2.30 ) that this is equiv alen t to the exis tence of a non-unit elemen t ρ ∈ S T n suc h that for all ( A, B ) ∈ µ − 1 ( µ 0 ) there exists an elemen t h ( A, B ) ∈ G 0 satisfying ( Ag ( B ) − 1 ρg ( B ) , B ) = ( h ( A, B ) Ah ( A, B ) − 1 , h ( A, B ) B h ( A, B ) − 1 ) . (3.57) This means that ˆ Ψ b τ ( p ( A, B )) = p ( A, B ) for the elemen t τ ∈ T n − 1 for whic h ρ = ρ ( τ ) according to (2.28). Note from (3.57) that h ( A, B ) m ust commu te with B . Because ( A, B ) ∈ µ − 1 ( µ 0 ), B = g ( B ) − 1 δ ( ξ ) g ( B ) is regular b y Theorem 2 and therefore there exists some ( A, B )-dep enden t ζ = dia g ( ζ 1 , ..., ζ n ) ∈ S T n suc h that h ( A, B ) = g ( B ) − 1 ζ g ( B ) . (3.58) The fact that h ( A, B ) ∈ G 0 then sa ys that g ( B ) − 1 ζ g ( B ) µ 0 = µ 0 g ( B ) − 1 ζ g ( B ) , (3.59) or, equiv alen t ly , ζ g ( B ) µ 0 g ( B ) − 1 = g ( B ) µ 0 g ( B ) − 1 ζ . (3.60) W e kno w from the pro of of Theorem 2 that the last column of the matrix g ( B ) is giv en b y a v ector v ( ξ ) verifyin g | v l ( ξ ) | 2 = (sin | y | ) n sin ( n | y | ) l + n − 1 Y j = l +1  cot | y | − y | y | cot( j − 1 X k = l ξ k )  , ∀ l = 1 , ..., n, (3.61) where ξ ∈ A y (3.12). If ξ belongs to the in terior A 0 y of A y , A 0 y := n ( ξ 1 , ..., ξ n ) ∈ R n    ξ j > | y | , j = 1 , ..., n, n X j =1 ξ j = π o , (3.62) then all compo nen ts of the v ector v ( ξ ) are non- v anishing. In this case w e c ompare the last columns o f t he matrices on the t wo sides of Eq. (3.60). By using the form ula of µ 0 , this leads to the relation ζ v ( ξ ) = v ( ξ ) ζ n , ( 3 .63) from whic h w e conclude that ζ = ζ n 1 n . It then follo ws f r o m (3.58) that h ( A, B ) = ζ n 1 n , a nd thereb y (3 .57) implies that ρ = 1 n . This contradicts our assumption that ρ is a non-unit elemen t of S T n . Therefore the T n − 1 -action ˆ Ψ b on P is eﬀectiv e. 20 The statement of the follow ing lemma can b e obtained as an immediate consequence of Theorem 7 .2 of [1] (the pro of o f whic h itself is based on results o f [20]). W e give here a direct pro of since w e shall need some details of it subse quently . Lemma 3. The r e duc e d ph a se sp ac e P = µ − 1 ( µ 0 ) /G 0 is c o n ne cte d. Pr o of. It is enough to prov e that an y t w o po in ts o f P can b e connected by a con tinuous path. W e ﬁx a p oint x ∈ P a nd deﬁne P x := { z ∈ P | ˆ β ( z ) = ˆ β ( x ) } , where ˆ β = ( ˆ β 1 , ..., ˆ β n ) is the R n -v alued function on P that desc ends from the R n -v alued 6 in v arian t function β on µ − 1 ( µ 0 ). W e pic k an arbitrary z ∈ P x , and next show that x can b e connected to z . T o b egin, denote some represen tativ es of x and z in µ − 1 ( µ 0 ) b y ( A, B ) and ( A ′ , B ′ ), resp ectiv ely . Referring to Section 2.2, w e ha v e g ( B ) B g ( B ) − 1 = δ ( ˆ β ( x )) = g ( B ′ ) B ′ g ( B ′ ) − 1 for some S U ( n ) matr ices g ( B ) and g ( B ′ ). W e see f rom the pro of of Theorem 2 that g ( B ) a nd g ( B ′ ) can be c hosen to ha ve the same last column (in fa ct, one may tak e g ( B ) j n = g ( B ′ ) j n = z j ( δ ( ˆ β ( x )) , y ) 1 2 deﬁned in (3.16)). T hen it follows that g ( B ) µ 0 g ( B ) − 1 = g ( B ′ ) µ 0 g ( B ′ ) − 1 , whic h in turn implies that h := g ( B ) − 1 g ( B ′ ) is in G 0 . This tells us that the represen tativ e o f z in µ − 1 ( µ 0 ) can be replaced b y ( A ′′ , B ) = ( hA ′ h − 1 , hB ′ h − 1 ). Then, it must b e true that A ′′ = AM where M = g ( B ) − 1 ζ g ( B ) for some ζ ∈ S T n . This holds b ecause the momen t map constrain t for ( A, B ) and ( A ′′ , B ) implies that AB A − 1 = A ′′ B A ′′− 1 , and B is regular b y Theorem 2. Next, by using the T n − 1 - action (2.30), w e can rewrite the equalit y ( A ′′ , B ) = ( AM , B ) as ( A ′′ , B ) = Ψ b η ( A, B ) , where η ∈ T n − 1 is deﬁned b y the relation ζ − 1 = ρ ( η ) with (2.28). Finally , we c ho ose a contin uous curv e [0 , 1] ∋ s 7→ τ ( s ) ∈ T n − 1 for whic h τ (0) is the identit y and τ ( 1) = η , whereb y w e obtain the con t in uous path ˆ Ψ b τ ( s ) ( x ) in P that connects x to z . Notice tha t ˆ Ψ b τ ( s ) ( x ) ∈ P x for all s , and th us w e ha v e also sho wn that the ˆ Ψ b action (3.54) of T n − 1 is transitiv e on P x . W e no w take t wo arbitrary po in ts x 0 , x 1 ∈ P fo r whic h ˆ β ( x 0 ) 6 = ˆ β ( x 1 ), and prov e the existence of a contin uous path [0 , 1] ∋ s 7→ x ( s ) ∈ P f o r whic h x (0) = x 0 and x (1) = x 1 . The follo wing arg umen t relies on the ﬁrst part of the pro of of Theorem 2 . W e b egin by c ho osing a con tinuous pat h ξ ( s ) ∈ A y in suc h a wa y that ξ (0) = ˆ β ( x 0 ) a nd ξ (1 ) = ˆ β ( x 1 ). Next w e deﬁne the v ector function v ( s ) by putting v l ( s ) := z l ( δ ( ξ ( s )) , y ) 1 2 using (3 .1 6). Since v ( s ) is con tinuous in s , w e can ﬁnd (actually could giv e explic itly) a n S U ( n )- v alued contin uous function g ( s ) that solv es µ 0 = g ( s ) − 1 µ v ( s ) g ( s ), w here µ v ( s ) is obtained by replacing v ( ξ ) in (3.22) by v ( s ) (see also ( 3 .23)). W e con tin ue by deﬁning B ( s ) := g ( s ) − 1 δ ( ξ ( s ) ) g ( s ), and then note the existence of a con tin uous function A ( s ) ∈ S U ( n ) for whic h A ( s ) B ( s ) A ( s ) − 1 = µ 0 B ( s ). Suc h function exists since B ( s ) is similar to µ 0 B ( s ), as can be seen from the discussion around equations (3.21)-(3.25), and the eigen ve ctors of B ( s ) and that of µ 0 B ( s ) can b e ch osen as contin uous functions of s . No w the pro jection o f the curv e ( A ( s ) , B ( s )) ∈ µ − 1 ( µ 0 ) yields a con tinuous curv e ˜ x ( s ) := p ( A ( s ) , B ( s )) ∈ P for whic h ˆ β ( ˜ x (0)) = ˆ β ( x 0 ) and ˆ β ( ˜ x (1)) = ˆ β ( x 1 ). By the previous part of the pro of, it is ob viously p ossible to ﬁnd a con tinuous curv e τ ( s ) ∈ T n − 1 suc h that x ( s ) := ˆ Ψ b τ ( s ) ( ˜ x ( s )) giv es the pa t h connecting x 0 with x 1 . Remark 3. The main message of the presen t subsection is that the reduced phase space ( P , ˆ ω ) is naturally equipped with two eﬀectiv e Hamiltonian a ctions of the tor us T n − 1 . The ﬁrst is 6 The fact that we here consider ˆ β and β as R n -v alued functions, but elsewhere view them as R n − 1 -v alued functions should no t lead to any confusion; we hav e ˆ β n ≡ π − P n − 1 k =1 ˆ β k and similarly for β . 21 the action ˆ Ψ b (3.54), whic h w e call the ˆ β -generated action since its momen t map is giv en b y ˆ β (3.56). The second is the ˆ α -generated action, whic h can b e deﬁned by the f o rm ula ˆ Ψ a τ ◦ p = p ◦  Ψ a τ | µ − 1 ( µ 0 )  , ∀ τ ∈ T n − 1 , (3.64) where Ψ a is giv en by (2.29). The corresp onding moment map is ˆ α ≡ ( ˆ α 1 , ..., ˆ α n − 1 ) : P → R n − 1 , (3.65) where the f unctions ˆ α j descend from t he sp ectral Hamiltonians α j in tro duced in (2 .23). The data ( P , ˆ ω , ˆ α ) a nd ( P , ˆ ω , ˆ β ) b oth enco de Hamiltonian toric manifolds, or in o ther w o r ds we ha ve t wo completely integrable sys t ems on the reduced phase space ( P , ˆ ω ). Remark 4. Let us in tro duce the follo wing op en submanifolds o f P : P a 0 := ˆ α − 1 ( P 0 y ) and P b 0 := ˆ β − 1 ( P 0 y ) . (3.66) Observ e from the pro ofs of Lemmas 2 and 3 that the ˆ Ψ b action is fr e e and tr ansitive on P x for all x ∈ P b 0 (the transitivit y holds for all x ∈ P ). Th us P b 0 is a principal T n − 1 -bundle ov er the base P 0 y . This bundle is top olog ically t r ivial since its base is con tractible. C o nsider no w x 0 , x 1 ∈ P suc h that x 0 / ∈ P b 0 and x 1 ∈ P b 0 . Then x 0 can b e connected to x 1 b y a curve x ( s ) as in the pro of of L emma 3 in suc h a wa y that ˆ β ( x ( s )) ∈ P 0 y for all 0 < s ≤ 1 (since P y is a conv ex p olytop e). This in turn implies that P b 0 is a dense op en submanifold of P . By the reasoning used b elow Eq. ( 3 .27), P a 0 is a also a dense op en submanifold of P and, equipped with the ˆ Ψ a action, it is a principal T n − 1 -bundle o ve r P 0 y . 3.4 The global structure of the reduced systems ( P , ˆ ω , ˆ α ) and ( P , ˆ ω , ˆ β ) W e b elow iden tify the reduced systems b y utilizing a celebrated result of Delzan t [5] that c har- acterizes Hamiltonian toric manifolds in terms o f the imag e of the moment map. Delzan t’s ﬁrst theorem (Th. 2.1 of [5]). L et ( M 1 , ω 1 ) and ( M 2 , ω 2 ) b e 2 d -dimensional Hamiltonian toric ma n ifolds with moment maps Φ 1 : M 1 → T ∗ and Φ 2 : M 2 → T ∗ , w her e T is the Lie alge b r a of the d -dimensional torus T acting on M 1 and on M 2 . If the image s Φ 1 ( M 1 ) and Φ 2 ( M 2 ) c oincide, then ther e exists a T -e quivariant symple ctomorphism ϕ : M 1 → M 2 such that Φ 2 ◦ ϕ = Φ 1 . According to an earlier result of A tiy ah, Guillemin and Sternberg, the images in question are con v ex p o lytop es. Delzan t a lso obtained full classiﬁcation of the momen t p olytop es associated with Hamiltonian toric manifolds, whic h are now routinely called Delzan t p olytop es [2]. By no w, w e ha v e exhibited t wo eﬀectiv e Hamiltonian actions of the torus T n − 1 = U (1) n − 1 on the compact connected r educed phase space ( P , ˆ ω ). Referring to a ﬁxed basis 7 of the Lie algebra T n − 1 of T n − 1 , the resp ectiv e momen t maps ar e ˆ α : P → R n − 1 (3.65) and ˆ β : P → R n − 1 (3.56). The Delzan t p o lytop es are pro vided in b oth c ases by P y (3.11). 7 Our base elements, X 1 , ..., X n − 1 , cor resp ond to a ﬁxed pro duct structure (2.27), a nd realize T n − 1 as T n − 1 factored by the lattice span Z { 2 π X 1 , ..., 2 π X n − 1 } , i.e., the corresp onding Hamiltonian ﬂows are 2 π -p erio dic. 22 Sp ecialists o f symplectic geometry can immediately recognize the Delzant p olytop e P y (3.11) as the one asso ciated with a v ery standard Hamiltonian toric manifold: C P ( n − 1) equipp ed with a multiple of the F ubini-Study form a nd the familiar ‘rotational action’ of T n − 1 . F or the sak e of k eeping our pap er self-con ta ined, and also since we need to ﬁx notations, we next explain ho w this Hamiltonian toric manifold comes ab out. Let us start with t he symplectic v ector space C n ≃ R 2 n endo w ed with the Darb o ux fo r m Ω C n = i n X k =1 d ¯ u k ∧ du k , (3.67) where u k ( k = 1 , ..., n ) are the comp onen ts of the v ector u that runs ov er C n . Then conside r the Hamiltonian action ψ of the gr oup U (1) on C n op erating as ψ e i γ ( u ) := e i γ u. (3.68) This action is generated b y the momen t map χ : C n → R , χ ( u ) ≡ n X k =1 | u k | 2 , (3 .69) since dχ = Ω C n ( V , · ) holds for the v ector ﬁeld V = i P n k =1 ( u k ∂ ∂ u k − ¯ u k ∂ ∂ ¯ u k ) a sso ciated with the inﬁnitesimal a ction. F or any ﬁxed v alue χ 0 > 0, usual symplectic reduction yields the reduced phase space χ − 1 ( χ 0 ) /U (1) ≡ C P ( n − 1) . (3.70) F or χ 0 = 1, the reduced symplectic form is the standard F ubini- Study form ω FS of C P ( n − 1). On the C n − 1 c hart corresponding to those u ∈ χ − 1 (1) fo r whic h u n 6 = 0, the reduced symplectic form b ecomes ω FS ( C n − 1 ) = i P n − 1 k =1 d ¯ z k ∧ d z k 1 + | z | 2 − i P n − 1 j,k =1 z j ¯ z k d ¯ z j ∧ d z k (1 + | z | 2 ) 2 = i ¯ ∂ ∂ log ( | z | 2 + 1) , (3.71) where w e use the ‘inhomogeneous co or dina t es’ z j := u j u n and | z | 2 ≡ P n − 1 k =1 | z k | 2 . It is well- kno wn that ω FS tak es the form (3.71) in terms of all the n p o ssible systems of inhomog eneous co ordinates that together co v er C P ( n − 1). F or arbitrary χ 0 > 0, one has the follo wing result. Lemma 4. The r e duc e d symple ctic man i f old χ − 1 ( χ 0 ) /U (1) obtaine d fr om ( C n , Ω C n ) as describ e d ab ove is the c omplex pr oje ctive sp ac e C P ( n − 1) e quipp e d with the symple ctic fo rm χ 0 ω FS . No w fo cus on the action R : T n − 1 × C n → C n of the torus T n − 1 on C n furnished b y R τ ( u 1 , ..., u n − 1 , u n ) := ( τ 1 u 1 , ..., τ n − 1 u n − 1 , u n ) , ∀ τ ∈ T n − 1 , ∀ u ∈ C n . (3.72) Deﬁning J k := | u k | 2 , ∀ k = 1 , ..., n − 1 , (3.73) the corresp onding momen t map can b e tak en to b e J = ( J 1 , ..., J n − 1 ) : C n → R n − 1 . Of course, the momen t map o f the torus a ction is unique only up to a shift by an arbitrary constant, whic h w e shall ﬁx b y conv enienc e. 23 The a b o v e T n − 1 -action and momen t map surv iv e the sy mplectic reduction by the U (1)- action (3.68) and descend to the rotational T n − 1 -action on ( C P ( n − 1) , χ 0 ω FS ), whic h th us b ecomes a Hamilto nia n toric manifold. This means that the ro tational T n − 1 -action, denoted as R : T n − 1 × C P ( n − 1) → C P ( n − 1 ) , op erates according to the rule R τ ◦ π χ 0 = π χ 0 ◦ R τ (3.74) where π χ 0 : χ − 1 ( χ 0 ) → C P ( n − 1) is the canonical pro jection. W e deﬁne its momen t map J = ( J 1 , ..., J n − 1 ) : C P ( n − 1) → R n − 1 b y the formula J k ◦ π χ 0 = J k + J 0 k , k = 1 , ..., n − 1 , (3.75) where the J 0 k are constan ts. It is obv io us that 0 ≤ J k and n − 1 X k =1 J k ≤ χ 0 . (3.76) The p o int to note is that if we cho ose χ 0 := ( π − | y | n ) and J 0 k := | y | , then the Delza n t p olytop e J ( C P ( n − 1)) of the r otational T n − 1 -action c oincides with the p olytop e P y (3.11) . Therefore w e o btain the following main result of this section b y com bining Delzant’s theorem with the statemen ts prov ed previously . Theorem 3. Cho ose y ∈ R for whi c h 0 < | y | < π n . Conside r the Hamiltonian toric mani fold ( C P ( n − 1) , ( π − | y | n ) ω FS , J ) , wher e J deﬁne d by J k ◦ π χ 0 = J k + | y | is the moment map of the r otational T n − 1 -action, an d c onsider also the Hamiltonian toric manifolds ( P , ˆ ω , ˆ α ) and ( P , ˆ ω , ˆ β ) that r esult fr om the quasi-Hamiltonian r e duction ac c or ding to The or em 1 and R emark 3 . Th en any two of these thr e e Hamiltonian toric manifolds ar e T n − 1 -e quivariantly symple ctomorphi c . Mor e pr e cisely, ther e exists a diﬀe omorphism φ α : P → C P ( n − 1 ) s uch that φ ∗ α (( π − | y | n ) ω FS ) = ˆ ω , ˆ α = J ◦ φ α , (3 .77) and also a d iﬀe omorphism φ β : P → C P ( n − 1) such that φ ∗ β (( π − | y | n ) ω FS ) = ˆ ω , ˆ β = J ◦ φ β . (3.78) The c omp ose d d iﬀe omorphism φ := φ − 1 β ◦ φ α : P → P c onverts ( P , ˆ ω , ˆ α ) into ( P , ˆ ω , ˆ β ) . Remark 5. The o rem 3 sa ys that b oth completely inte grable Hamiltonian systems ( P , ˆ ω , ˆ α ) and ( P , ˆ ω, ˆ β ) obta ined from the quasi-Hamiltonian reduction can b e iden tiﬁed with the syste m on ( C P ( n − 1) , ( π − | y | n ) ω FS ) pro vided b y the simple Hamiltonians J k that generate the rotational action of T n − 1 . As w e shall see later, the functions J k ∈ C ∞ ( C P ( n − 1)) pla y the role of particle- p ositions in t he compactiﬁed I I I b system [31]. It will turn out that if one con ve rts ˆ β in to the particle-p ositions J of the I I I b system b y the symplectomorphism φ β , then ˆ α is conv erted by the same symplectomorphism in to the action-v ariables of the system. Ro ughly sp eaking, the exc hange of the ro les of ˆ α and ˆ β will then explain the Ruijsenaars self-dualit y since the other symplectomorphism φ α con v erts ˆ α in to the particle-p o sitions and ˆ β into the a ction-v ariables. F rom no w on, the symplectomorphisms φ α , φ β that app ear in Theorem 3 as w ell as their inv erses 24 and comp ositions will b e referred to as De l z a nt symple ctomorphism s , or simply as Delzan t maps. In the next section, we explicitly construct the Delzant maps f α := φ − 1 α and f β := φ − 1 β , (3.79) whic h will b e utilized in Section 5 where the stat ements of this remark will b e elab ora ted. 4 Constr u ction of th e Del zan t symplectomorphis ms The aim of this section is t o construct explicitly the Delzan t maps f α , f β : C P ( n − 1) → P whose existence ha s b een established b y Theorem 3. W e shall see that the Ruijsenaars-Sc hneider Lax matrix L lo c y app ears a s a principal building blo ck o f these symplectomorphisms. This remark able fact will b e further exploited in Section 5 where the emergence of the compactiﬁed I I I b system as the fr uit of the quasi-Hamiltonian reduction will b e established and the Ruijsenaars self-dua lity map of the I I I b system will b e expre ssed in terms of the Delzan t maps. F rom the tec hnical p oin t of view, b elo w w e ﬁrst describe a lo cal ve rsion of the map f β deﬁned in some dense op en subset of C P ( n − 1), and then w e construct its global extension tha t will in v olv e the g lobal extension of the local Lax matrix (appearing already in [31]). Finally , w e shall construct f α out of f β and certain in v o lutions. 4.1 Lo cal v ersion of the Delzan t map f β Let us denote b y C P ( n − 1) 0 the dense op en submanifold of C P ( n − 1) = χ − 1 ( π − n | y | ) /U (1) where none of the n homogeneous co or dina t es u k can v anish (cf. Eq. (3.70)). In what follows w e construct a symplectomorphism f 0 : C P ( n − 1) 0 → P b 0 , (4.1) where P b 0 = ˆ β − 1 ( P 0 y ) is the dense op en submanifold o f P intro duced in (3.66). On the subset of t he constraint surface χ − 1 ( π − n | y | ) co v ering C P ( n − 1 ) 0 w e ma y imp ose the gauge ﬁxing condition u n > 0, and then u j ( j = 1 , ..., n − 1) parametrize C P ( n − 1) 0 . Adopting this condition, w e no w in tro duce Darb oux co o rdinates ξ j , τ j ( j = 1 , ..., n − 1) on C P ( n − 1 ) 0 b y setting u j := τ j p ξ j − | y | for j = 1 , ..., n − 1, where ξ ∈ P 0 y (3.11), τ ∈ T n − 1 . That is w e parametrize C P ( n − 1) 0 using the diﬀeomorphism E : P 0 y × T n − 1 → C P ( n − 1) 0 (4.2) giv en by E ( ξ , τ ) := π χ 0 ( τ 1 p ξ 1 − | y | , ..., τ n − 1 p ξ n − 1 − | y | , p ξ n − | y | ) with ξ n ≡ π − n − 1 X k =1 ξ k (4.3) and π χ 0 deﬁned in Subsection 3.4 . An easy calculation sho ws that the F ubini-Study symplectic structure on C P ( n − 1) 0 tak es the Da rb oux form in the v ariables ξ j , τ j . Speaking more precisely , 25 with the parametrization τ k := e i θ k ( k = 1 , ..., n − 1), there holds t he relation ( π − n | y | ) E ∗ ( ω FS ) = i n − 1 X k =1 dξ k ∧ dτ k τ − 1 k = n − 1 X k =1 dθ k ∧ dξ k . (4.4) As further pieces of preparation, recall the isomorphism ρ : T n − 1 → S T n (2.28), ρ ( τ ) ≡ exp  i n − 1 X j =1 θ j ( E j,j − E j +1 ,j +1 )  for τ = ( e i θ 1 , ..., e i θ n − 1 ) , (4.5) and consider the vec t or v j ( ξ , y ) :=  sin y sin ny  1 2 W j ( δ ( ξ ) , y ) , ∀ ξ ∈ P y , j = 1 , ..., n, (4.6) with W j in (1.5) where non-negativ e squ are r o ots are ta k en. Observ e that if ξ ∈ P 0 y , then all v j ( ξ , y ) are strictly po sitiv e since their squares are the same as the rig ht-hand side of (3.16). It is also imp ortant to no t ice that these are C ∞ functions on the op en alco v e P 0 y , but their ﬁrst deriv a tiv es dev elop some singularities at the b oundary of P 0 y . It is readily chec k ed that the follow ing form ulae yield a unitary ma t r ix g ( v ) ∈ U ( n ) fo r any v ector v ∈ R n that has unit norm a nd comp onen t v n 6 = − 1: g ( v ) j n := − g ( v ) nj := v j , ∀ j = 1 , ..., n − 1 , g ( v ) nn := v n , g ( v ) j l := δ j l − v j v l 1 + v n , ∀ j, l = 1 , ..., n − 1 . ( 4 .7) Equations (1.5), (3.1 6)–(3 .19) imply that the v ector v ( ξ , y ) in (4.6) has unit norm, and using this w e now in tro duce the unitary (actually real-orthog o nal) matrix g y ( ξ ) ∈ U ( n ) by setting g y ( ξ ) := g ( v ( ξ , y )) , ∀ ξ ∈ P y . (4.8) Theorem 4. We c an de ﬁ ne a map f 0 : C P ( n − 1) 0 → P b 0 by the formula ( f 0 ◦ E )( ξ , τ ) := p  g y ( ξ ) − 1 L lo c y ( δ ( ξ ) , ρ ( τ ) − 1 ) g y ( ξ ) , g y ( ξ ) − 1 δ ( ξ ) g y ( ξ )  , (4.9) wher e L lo c y is the L a x matrix given by (1.4) and p : µ − 1 ( µ 0 ) → P is the c anonic al pr oje ction. This map is a symple ctic diﬀe omorphism with r esp e ct to the r estricte d symple ctic fo rm s, f ∗ 0 ( ˆ ω ) = ( π − n | y | ) ω FS , (4.10) and it intertwines the r estrictions o f the c orr esp onding toric moment maps , f ∗ 0 ( ˆ β ) = J . (4.11) The hardest part of the pro of will b e the v eriﬁcation of Eq. (4.10), and b efore dealing with this w e presen t t w o lemmas. 26 Lemma 5. The S U ( n ) m atrix L lo c y ( δ ( ξ ) , Θ) give n by Eq. (1.4) veriﬁes the r elation L lo c y ( δ ( ξ ) , Θ) δ ( ξ ) L lo c y ( δ ( ξ ) , Θ) − 1 =  e 2i y 1 n + ( e 2i(1 − n ) y − e 2i y ) v ( ξ , y ) v ( ξ , y ) †  δ ( ξ ) (4.12) for al l ξ ∈ P 0 y and Θ ∈ S T n , w ith the ve ctor (4.6). Pr o of. W e kno w from the pro of of Theorem 2 ( cf. t he discussion a round Eqs. (3.28) and (3.3 0)) that the unitary matrices δ ( ξ ) and  e 2i y 1 n + ( e 2i(1 − n ) y − e 2i y ) v ( ξ , y ) v ( ξ , y ) †  δ ( ξ ) ha ve the same sp ectra, and hence there exis ts a unitary matrix N ( ξ , y ) suc h that N ( ξ , y ) δ ( ξ ) N ( ξ , y ) − 1 =  e 2i y 1 n + ( e 2i(1 − n ) y − e 2i y ) v ( ξ , y ) v ( ξ , y ) †  δ ( ξ ) . (4.13) By conjugating the la st relation b y N ( ξ , y ) − 1 w e obtain δ ( ξ ) = h e 2i y 1 n + ( e 2i(1 − n ) y − e 2i y ) N ( ξ , y ) − 1 v ( ξ , y )  N ( ξ , y ) − 1 v ( ξ , y )  † i N ( ξ , y ) − 1 δ ( ξ ) N ( ξ , y ) (4.14) and b y in verting the term in square bra ck ets w e arrive at N ( ξ , y ) − 1 δ ( ξ ) N ( ξ , y ) = h e − 2i y 1 n + ( e 2i( n − 1) y − e − 2i y ) N ( ξ , y ) − 1 v ( ξ , y )  N ( ξ , y ) − 1 v ( ξ , y )  † i δ ( ξ ) . (4.15) Let us rewrite Eq. (4.1 3) for − y a s N ( ξ , − y ) δ ( ξ ) N ( ξ , − y ) − 1 =  e − 2i y 1 n + ( e 2i( n − 1) y − e − 2i y ) v ( ξ , − y ) v ( ξ , − y ) †  δ ( ξ ) . (4.16) W e ha ve learned in pro ving Theorem 2 ( cf. Eq. (3.33)) that the equalit y of the spectra of the matrices δ ( ξ ) and  e 2i y 1 n + ( e 2i(1 − n ) y − e 2i y ) w w †  δ ( ξ ) ﬁxes the absolute v alues o f the comp onen ts of the v ector w to b e giv en by t he right-hand side of (4.6). Comparing (4.16) with (4.15), we therefore see that the absolute v alues of the comp onents of the v ector N ( ξ , y ) − 1 v ( ξ , y ) are the same as the (strictly p ositiv e) comp onen ts of the v ector v ( ξ , − y ). Note that the matrix N ( ξ , y ) v erifying Eq. (4.13) is not unique b ecause it can b e m ultiplied from t he right b y an y diago nal elemen t of U ( n ) while k eeping (4.1 3) v alid. Ho w ev er, this am biguity can b e completely ﬁxed b y requiring that the v ector N ( ξ , y ) − 1 v ( ξ , y ) has all comp onents real and strictly p ositive . W e denote the unique ma t rix N ( ξ , y ) satisfying t his requiremen t b y ˜ N ( ξ , y ). Th us we hav e v ( ξ , y ) = ˜ N ( ξ , y ) v ( ξ , − y ) and hence ˜ N ( ξ , − y ) = ˜ N ( ξ , y ) − 1 . (4.17) By considering it for N ( ξ , y ) := ˜ N ( ξ , y ), let us rewrite ( 4.13) as ˜ N ( ξ , y ) δ ( ξ ) − e 2i y δ ( ξ ) ˜ N ( ξ , y ) = ( e 2i(1 − n ) y − e 2i y ) v ( ξ , y )  ˜ N ( ξ , y ) − 1 v ( ξ , y )  † ˜ N ( ξ , y ) − 1 δ ( ξ ) ˜ N ( ξ , y ) . (4.18) With the help o f (4.17 ) , w e can further rewrite the last relation as ˜ N ( ξ , y ) δ ( ξ ) − e 2i y δ ( ξ ) ˜ N ( ξ , y ) = ( e 2i(1 − n ) y − e 2i y ) v ( ξ , y ) v ( ξ , − y ) † ˜ N ( ξ , − y ) δ ( ξ ) ˜ N ( ξ , − y ) − 1 . (4.19) Expressing ˜ N ( ξ , − y ) δ ( ξ ) ˜ N ( ξ , − y ) − 1 from Eq. (4.1 6) and using subsequen tly Eq. (3.19), w e deriv e ˜ N ( ξ , y ) δ ( ξ ) − e 2i y δ ( ξ ) ˜ N ( ξ , y ) = (1 − e 2 n i y ) v ( ξ , y ) v ( ξ , − y ) † δ ( ξ ) . (4.20) 27 By solving this for the comp o nen ts of ˜ N ( ξ , y ) w e obtain the equalit y ˜ N ( ξ , y ) = e i( n − 1) y L lo c y ( δ ( ξ ) , 1 n ) . ( 4 .21) This implies the desired relation (4 .12). The ab ov e argumen t also sho ws that L lo c y ( δ ( ξ ) , Θ) is unitary , and the fa ct that its determinan t equals 1 is easily che c k ed b y the determinan t form ula of Cauc h y matrices. Lemma 6. Every element ( A, B ) ∈ µ − 1 ( µ 0 ) such that p ( A, B ) b elongs to P b 0 (3.66) has the form ( A, B ) = Ψ ( g y ( ξ ) η ) − 1  L lo c y ( δ ( ξ ) , ρ ( τ ) − 1 ) , δ ( ξ )  (4.22) with ξ = β ( A, B ) , τ ∈ T n − 1 and η ∈ U ( n ) for wh ich η − 1 µ 0 η = µ 0 , using the notation (2.10). By this formula, the p air ( ξ , τ ) ∈ P 0 y × T n − 1 uniquely p ar a metrizes the pr oje ction p ( A, B ) ∈ P b 0 . Pr o of. Conjugating the relation (4.1 2) b y g y ( ξ ) − 1 , a nd putting Θ := ρ ( τ ) − 1 (4.5), we conclude b y using Eq. (3.23) that the pair (4.2 2 ) b elongs to the constraint surface µ − 1 ( µ 0 ). It follows b y tracing the deﬁnitions that, after the pro j ection p , τ parametrizes a T n − 1 orbit in P b 0 under the ˆ Ψ b -action. As w as no t ed in Remark 4, this action is transitiv e and free on P b 0 . Hence the ab ov e solution of the constraint (3.1) pro jects to the most g eneral elemen t of the reduced phase space P for whic h the v alue of the function ˆ β equals ξ , and specifying the pair ( ξ , τ ) is equiv alen t to sp ecifying the pro jection p ( A, B ) ∈ P b 0 . Pr o of of The or e m 4: It follows directly from Lemma 6 that the form ula ( F 0 ◦ E )( ξ , τ ) :=  g y ( ξ ) − 1 L lo c y ( δ ( ξ ) , ρ ( τ ) − 1 ) g y ( ξ ) , g y ( ξ ) − 1 δ ( ξ ) g y ( ξ )  , (4.23) deﬁnes a smo oth map F 0 : C P ( n − 1) 0 → p − 1 ( P b 0 ) , (4.24 ) whic h is injectiv e and its imag e interse cts ev ery ga uge or bit in p − 1 ( P b 0 ) precisely in o ne p oin t. Th us f 0 = p ◦ F 0 : C P ( n − 1 ) 0 → P b 0 is an injectiv e and surjectiv e smo oth map. On accoun t of (4.10) (pro v ed in what follows), the corresp onding Jacobian determinant cannot v anish a nd hence f 0 is a diﬀeomorphism. Since the v alidity of (4.11) is ob vious, it remains to sho w that f 0 satisﬁes (4.1 0). Bec ause of Eqs. (2.7) and (4.4), this amoun ts to pro ving tha t the restriction of the quasi-Hamiltonian 2-form ω on p − 1 ( P b 0 ) pulled bac k b y the map F 0 ◦ E on P 0 y × T n − 1 is the Darb oux 2-form: ( F 0 ◦ E ) ∗ ω | p − 1 ( P b 0 ) = i n − 1 X k =1 dξ k ∧ dτ k τ − 1 k . (4 .2 5) In fact, w e here v erify this b y direct c omputation, b y inserting the form ula (4.23) into the form ula (2 .11): ( F 0 ◦ E ) ∗ ω | p − 1 ( P b 0 ) = 1 2 h A − 1 ξ ,τ dA ξ ,τ ∧ , d B ξ B − 1 ξ i + 1 2 h dA ξ ,τ A − 1 ξ ,τ ∧ , B − 1 ξ dB ξ i (4.26) 28 where A ξ ,τ := g y ( ξ ) − 1 L lo c y ( δ ( ξ ) , ρ ( τ ) − 1 ) g y ( ξ ) a nd B ξ := g y ( ξ ) − 1 δ ( ξ ) g y ( ξ ). Not e that w e ha ve omitted in (4.26) the third term of the form ω displa y ed in (2.11). This term do es not con tribute since, due to the momen t map constrain t (3.1), we hav e µ 0 B ξ A ξ ,τ = A ξ ,τ B ξ and hence h ( A ξ ,τ B ξ ) − 1 d ( A ξ ,τ B ξ ) ∧ , ( B ξ A ξ ,τ ) − 1 d ( B ξ A ξ ,τ ) i = h ( B ξ A ξ ,τ ) − 1 d ( B ξ A ξ ,τ ) ∧ , ( B ξ A ξ ,τ ) − 1 d ( B ξ A ξ ,τ ) , (4.27) whic h v anishes as the scalar pro duct is symm etric and the w edge pro duct is anti-symme tric. In the fo llowing calculation, w e set for simplicit y g ≡ g y ( ξ ), L ≡ L lo c y ( δ ( ξ ) , ρ ( τ ) − 1 ), ρ ≡ ρ ( τ ) and δ ≡ δ ( ξ ). Th us we obta in f rom (4.26) ( F 0 ◦ E ) ∗ ω | p − 1 ( P b 0 ) = 1 2 h L − 1 dL + dg g − 1 − L − 1 dg g − 1 L ∧ , dδ δ − 1 + δ dg g − 1 δ − 1 − dg g − 1 i − ( L ↔ δ ) , (4.28) where ( L ↔ δ ) means the ﬁrst term on the r.h.s. of (4.28) with the role of L a nd δ in terc hanged. By using the in v ariance of t he scalar pro duct h ., . i and the fact that h φ ∧ , ψ i = −h ψ ∧ , φ i for an y su ( n )-v alued diﬀerential forms φ a nd ψ , w e can rewrite Eq. (4.28) as ( F 0 ◦ E ) ∗ ω | p − 1 ( P b 0 ) = 1 2 h δ − 1 2 ( L − 1 dL − L − 1 dg g − 1 L ) δ 1 2 + δ 1 2 ( dLL − 1 + Ldg g − 1 L − 1 ) δ − 1 2 ∧ , κ + κ t i . (4.29) Here w e ha v e intro duced the su ( n )-v alued diﬀeren tial form κ b y κ := 1 2 dδ δ − 1 + δ 1 2 dg g − 1 δ − 1 2 , (4.30) κ t denotes the transp osed matrix, and b y using (2.20) w e hav e δ 1 2 ≡ exp  − i P n − 1 k =1 ξ k λ k  . W e can write the matrix L as L ≡ L 1 ( ξ ) ρ ( τ ) − 1 . (4.31) Th us in Eq. (4.29) the dep endence of the form ( F 0 ◦ E ) ∗ ω | p − 1 ( P b 0 ) on the v a r iable τ is hidden in the (diagonal) matrix ρ ( τ ). It will b e con ve nien t to emplo y also the decomp osition ( F 0 ◦ E ) ∗ ω | p − 1 ( P b 0 ) ≡ V + ˆ V , (4.32) where V dep ends on ρ diﬀeren tially , i.e., V collects the terms that con tain d ρ . The part V is easily singled out from (4 .29) as V = − 1 2 h dρρ − 1 ∧ , d δ δ − 1 + δ dg g − 1 δ − 1 − dg g − 1 + L − 1 1 ( dδ δ − 1 + dg g − 1 − δ − 1 dg g − 1 δ ) L 1 i . (4.33) F or g is real orthogonal, d g g − 1 is an t i- symmetric. Because the trace of the pro duct of a sym- metric matrix with an a nti-symmetric one v anishes, we obtain h dρρ − 1 ∧ , dg g − 1 i = 0 , (4.34) and then V can b e rewritten as V = − 1 2 h dρρ − 1 ∧ , dδ δ − 1 + L − 1 1 dg g − 1 L 1 − L − 1 1 dL 1 + ( δ L 1 ) − 1 d ( δ L 1 ) − ( δ L 1 ) − 1 dg g − 1 ( δ L 1 ) i . (4.35) 29 No w note that the constrain t ( 3.1) implies δ L 1 = ζ − 1 L 1 δ , ζ := g µ 0 g − 1 . (4.36) Inserting this in the last tw o terms of (4.35) giv es directly the Darb oux form: V = −h dρρ − 1 ∧ , dδ δ − 1 i − 1 2 h dρρ − 1 − δ d ρρ − 1 δ − 1 ∧ , L − 1 1 dζ ζ − 1 L 1 − L − 1 1 dL 1 i = −h dρρ − 1 ∧ , d δ δ − 1 i . (4.37) The restricted form ω | p − 1 ( P b 0 ) is closed b ecause p − 1 ( P b 0 ) is a subset of a lev el set of the momen t map, a nd thus its pull-bac k (4.32) is closed a s w ell. Moreo v er, the Darb oux diﬀerential f o rm V is also closed. Then we observ e that the part ˆ V in (4.32) cannot dep end on ρ b ecause otherwise it w ould not b e closed. By taking this in to account, (4.2 9) giv es ˆ V = 1 2 h δ − 1 2 ( L − 1 1 dL 1 − L − 1 1 dg g − 1 L 1 ) δ 1 2 + δ 1 2 ( dL 1 L − 1 1 + L 1 dg g − 1 L − 1 1 ) δ − 1 2 ∧ , κ + κ t i . (4.38) No w w e again use the constrain t (4.36) to deriv e L − 1 1 dL 1 − L − 1 1 dg g − 1 L 1 = δ  L − 1 1 dL 1 + L − 1 1 ( δ − 1 dδ − δ − 1 dg g − 1 δ ) L 1  δ − 1 − d δ δ − 1 . (4.39) Inserting the express ion (4.39 ) in to (4.38) yields immediately ˆ V = 1 2 h L − 1 1 dL 1 + L − 1 1 ( δ − 1 dδ − δ − 1 dg g − 1 δ ) L 1 + dL 1 L − 1 1 + L 1 dg g − 1 L − 1 1 ∧ , δ − 1 2 ( κ + κ t ) δ 1 2 i . (4.40) Coming bac k to the form ula (1.4), w e notice that the unitary matrix L 1 can b e cast as L 1 = δ − 1 2 M δ 1 2 (4.41) where M is a real orthogona l matrix. With this represen tation of L 1 , the expression ˆ V can b e rewritten as ˆ V = 1 2 h d MM − 1 + M − 1 d M + M κ M − 1 + M − 1 κ t M ∧ , κ + κ t i . (4.42) By using aga in that the trace of the pro duct of a symmetric matrix with an anti-symmetric o ne v anishes , and using also the in v ariance of the scalar pro duct and that h φ ∧ , ψ i = h φ t ∧ , ψ t i for an y s u ( n )-v alued forms, the last equation implies that ˆ V = 0. Ha ving calculated V and ˆ V in (4.32), w e ﬁnally obtain the desired equalit y: ( F 0 ◦ E ) ∗ ω | p − 1 ( P b 0 ) = −h dρρ − 1 ∧ , dδ δ − 1 i = i n − 1 X k =1 dξ k ∧ d τ k τ − 1 k . (4.43)  Remark 6. W e know from the theory of the quasi-Hamilto nian reduction that the reduced sp ectral Hamiltonians ˆ α i P oisson commute, and The o rem 4 p ermits t o iden tify the ˆ α i on P b 0 with t he spectral functions of the Ruijsenaars-Sc hneider Lax matrix. The pro of o f Theorem 4 sho ws tha t this comm utativit y prop ert y of the ˆ α i can b e view ed as a consequence of the 30 Darb oux form o f the reduced symple ctic structure. The fact that the sp ectral in v ariants of the Ruijsenaars-Sc hneider Lax matrix P oisson comm ut e with resp ect to the Darb oux structure w a s also pro ved previously by means of diﬀeren t metho ds (see [33, 3 2, 25] and references therein). It follo ws easily from The orem 4 that the lo cal Delzan t map f 0 con v erts ˆ β in to particle- p osition v ariables and con v erts ˆ α in to action-v ar ia bles of the lo c a l I I I b system. Consequen tly , the ful l reduced phase space P mus t carry a c ompletion of the lo cal I I I b system. Ev en tually this completion will b e identiﬁe d with the one in tro duced by Ruijsenaars, but b efore explaining this further eﬀort is needed in order to w ork out certain details of our picture that will enable us t o giv e precise comparison with the results o f [31] regarding also the self-dualit y of the completed I I I b system. In particular, w e need to prov e tha t f 0 extends to a g lobal Delzan t map. 4.2 Global e xtension of the Lax matrix The lo cal Lax matrix L lo c y ( δ ( ξ ) , ρ ( τ ) − 1 ), view ed as a function on C P ( n − 1) 0 , is the cruc ia l ingredien t of the lo cal Delzan t map of The orem 4. The globa l Delzan t map that w e shall construct later will in volv e an extension of (a conjugate of ) this Lax mat r ix to a smo oth function on C P ( n − 1). W e here presen t this extension, whic h app ears also in [31]. In order to sa v e space, from no w on w e a ssume that y > 0 . (4.44) W e con tin ue to realize C P ( n − 1) as the fa cto r space C P ( n − 1) = S 2 n − 1 χ 0 /U (1) with S 2 n − 1 χ 0 = { ( u 1 , ..., u n ) ∈ C n | n X k =1 | u k | 2 = χ 0 } , χ 0 = π − n | y | . (4.45) In the subse quen t argumen ts we identify the U (1) - in v ariant functions deﬁned on S 2 n − 1 χ 0 b y r i := | u i | and ξ i := | u i | 2 + | y | , i = 1 , ..., n, (4.46) as functions on C P ( n − 1). Regarded in this w ay , ξ i b elongs to C ∞ ( C P ( n − 1)) , while r i is not ev en diﬀeren tiable at its zero lo cus. (F or i = 1 , ..., n − 1, the function ξ i is just another name for the momen t map comp onen t J i .) No w we giv e a simple tec hnical lemma, whose pro of con tains t he esse n tial observ ation that will lead to the global La x matrix. Its statemen t will b e utilized also in Subse ction 4.3. Lemma 7. By c ombining e quations (1. 5), (2.19) and (4.46), with 0 < y < π n , c onsid er the expr essions W k ( δ ( ξ ) , ± y ) as functions on C P ( n − 1 ) . Then W k ( δ ( ξ ) , y ) c an b e written as W k ( δ ( ξ ) , y ) = r k w y k ( ξ ) , (4.47) wher e w y k ( ξ ) r epr esents a p ositive C ∞ function on C P ( n − 1) for e ach k = 1 , ..., n . Similarly, W k ( δ ( ξ ) , − y ) = r k − 1 w − y k ( ξ ) , r 0 := r n , (4.48) wher e the function w − y k has the same p r op erties as those mentione d f o r w y k . 31 Pr o of. First restricting to C P ( n − 1) 0 where r k 6 = 0, w e directly sp ell out W k ( δ ( ξ ) , y ) in the form (4.47) with w y k ( ξ ) =  sin( r 2 k ) r 2 k sin( ξ k )  1 2 R y k ( ξ ) , (4.49) where, in tro ducing the shorthand ξ i,l := P l m = i ξ m for all 1 ≤ i ≤ l ≤ n , w e hav e R y k ( ξ ) = Y 1 ≤ j ≤ k − 1  sin( ξ j,k − 1 + y ) sin( ξ j,k − 1 )  1 2 ! Y k +2 ≤ j ≤ n  sin( ξ k ,j − 1 − y ) sin( ξ k ,j − 1 )  1 2 ! . ( 4 .50) By using (4.46), it is easily c heck ed that all argumen ts of the sin us-f unctions in v olved in (4 .50) lie strictly in the in terv al (0 , π ) ev en when running o v er the full C P ( n − 1), whic h immediately implies that R y k ( ξ ) represen ts a p o sitiv e C ∞ -function on C P ( n − 1 ). Since the function sin x/x remains smo ot h and positive at x = 0, and y ≤ ξ k ≤ π − ( n − 1) y , w e then see from (4.49) that w y k ( ξ ) also represen ts a p ositiv e C ∞ function on C P ( n − 1). The claim ab out W k ( δ ( ξ ) , − y ) can b e v eriﬁed in an analogous manner. Next, using L lo c y giv en in (1.4), we in tro duce the functions Λ y k ,l ( ξ ) (1 ≤ k , l ≤ n ) b y the equations r k r l − 1 Λ y k ,l ( ξ ) ≡ L lo c y ( δ ( ξ ) , 1 n ) k ,l for l 6 = k + 1 , ( k , l ) 6 = ( n, 1) , (4.51) Λ y k ,k +1 ( ξ ) ≡ L lo c y ( δ ( ξ ) , 1 n ) k ,k +1 , Λ y n, 1 ( ξ ) ≡ L lo c y ( δ ( ξ ) , 1 n ) n, 1 . (4.52) These equations directly deﬁne Λ y k ,l as functions on C P ( n − 1) 0 , where all r i are non-zero. Then, b y using Lemma 7 and a similar analysis for t he denominators in the formula (1.4), we ﬁnd that Λ y k ,l ( ξ ) extends to a C ∞ function o n C P ( n − 1) for eac h 1 ≤ k , l ≤ n . The extended function, whic h w e denote b y the same letter, v anishes nowhere on C P ( n − 1). The last stat ement follow s b y easy insp ection, and will b e utilized later. Lemma 8. By using the ab ove functions Λ y k ,l and (4.4 6 ), a nd setting u 0 := u n , w e c an de ﬁne C ∞ functions L y k ,l on S 2 n − 1 χ 0 by the formulae L y k ,l ( u ) := ¯ u k u l − 1 Λ y k ,l ( ξ ) for l 6 = k + 1 , ( k , l ) 6 = ( n, 1) , (4.53) L y k ,k +1 ( u ) := Λ y k ,k +1 ( ξ ) , L y ( u ) n, 1 := Λ y n, 1 ( ξ ) . (4.54) The functions L y k ,l ar e U (1) -i n variant and thus yield C ∞ functions on C P ( n − 1) that to gether form an S U ( n ) -value d C ∞ function on C P ( n − 1) , also denote d as L y . The r estriction of this matrix function to C P ( n − 1) 0 satisﬁes the fol lowing identity: ( L y ◦ E )( ξ , τ ) = ∆( τ ) − 1 L lo c y ( δ ( ξ ) , ρ ( τ ) − 1 )∆( τ ) , (4.55) wher e E denotes the p ar a metrization intr o duc e d in (4 .3) and ∆( τ ) := diag( τ 1 , ..., τ n − 1 , 1) . (4.56) 32 Pr o of. It f ollo ws from what w e established b efo r e that the form ulas (4.53) and (4.54) yield C ∞ functions on S 2 n − 1 χ 0 , whic h are ob viously in v ariant under the U (1)-action u 7→ e i γ u . By using (4.5) and (4.56), one readily c hec ks tha t the right-hand-side of (4 .5 5) is equal to the matrix diag( τ − 1 1 , ..., τ − 1 n − 1 , 1) L lo c y ( δ ( ξ ) , 1 n ) diag(1 , τ 1 , ..., τ n − 1 ) . (4.57) On accoun t of (4.51) and (4.52), this is further equal to the matrix L y ( u ) at u given b y u i = τ i p ξ i − | y | ( i = 1 , ..., n − 1) , u n = p ξ n − | y | . (4.58) By the deﬁnition of the map E (4.3), this prov es the equalit y (4.55). F ina lly , note that L y is S U ( n ) v alued since its restriction t o C P ( n − 1) 0 is S U ( n ) v alued. Remark 7. The C ∞ function L y : C P ( n − 1 ) → S U ( n ) sp eciﬁed b y Lemma 8 will b e r eferred to a s the glob al L ax matrix . Since by (4.55) it reduces to a conjuga t e o f the lo cal La x matrix L lo c y on C P ( n − 1 ) 0 , L y can serv e as the Lax matrix of a n in tegrable system deﬁned on C P ( n − 1 ) . Apart from sligh t diﬀerences of conv en tions, our L y actually coincides with the Lax matrix of the compactiﬁed I I I b system constructed in [31] (pages 3 1 1-312 lo c. cit.) relying on argumen ts similar to the ab ov e. It might b e w ort h noting L y is not only C ∞ but real-a nalytic on C P ( n − 1), as follo ws b y inspection of the ab o v e pro of and establishe d also in [31]. 4.3 Construction of the global Delzan t map f β Let us remem b er that µ − 1 ( µ 0 ) is the total space of a principal bundle with pro jection φ β ◦ p : µ − 1 ( µ 0 ) → P → C P ( n − 1) . (4.59) W e established the c haracteristic prop erties (3 .78) of φ β , but no t y et its explicit form. No w w e wish to give a construction of the in v erse map f β = φ − 1 β . Our pla n t o achie ve this is as follo ws. W e ﬁrst co v er C P ( n − 1) b y n ‘co or dinat e charts’ C P ( n − 1) j , where C P ( n − 1) j ⊂ C P ( n − 1 ) ( j = 1 , ..., n ) is by deﬁnition the set of those U (1) orbits in S 2 n − 1 χ 0 (4.45) f or whic h u j 6 = 0. B y an explic i t formula , w e then in tro duce a map F j : C P ( n − 1) j → µ − 1 ( µ 0 ) , (4.60 ) whic h turns out to deﬁne a lo cal section of the principal bundle ov er C P ( n − 1) j . These maps ha ve the prop ert y that the pro jected maps p ◦ F j : C P ( n − 1) j → P (4 .6 1) coincide on the o ve rlaps of their domains, and engender the desired global symplectomorphism f β : C P ( n − 1) → P . (4.62) W e shall also see that f β extends the map f 0 : C P ( n − 1) 0 → P de scrib ed in Theorem 4. T his will b e shown b y using that C P ( n − 1) 0 = ∩ n j =1 C P ( n − 1) j and f 0 has the fo r m f 0 = p ◦ F 0 with the lo cal section F 0 : C P ( n − 1) 0 → µ − 1 ( µ 0 ) giv en in equation (4.23). 33 T o b egin, w e in tro duce co ordinates on C P ( n − 1) j for eac h j b y considering the n -tuples u j := ( u j 1 , ..., u j n ) (4.63) sub j ect to the conditions n X k 6 = j | u j k | 2 < χ 0 , u j j := v u u t χ 0 − n X k 6 = j | u j k | 2 , χ 0 = π − n | y | . (4.64) Since u j j is a function of the other comp o nents of u j , w e ma y think of u j as a v ariable running o ver the op en ball B χ 0 ⊂ C n − 1 deﬁned b y B χ 0 := n ( z 1 , ..., z n − 1 ) ∈ C n − 1 | n − 1 X k =1 | z k | 2 < χ 0 o . (4.65) Accordingly , w e let ( C P ( n − 1) j , B j χ 0 ) (4.66) denote the dense o p en subs et C P ( n − 1) j of C P ( n − 1) eq uipp ed with the coordinates u j k . F or not a tional con v enience, w e also k eep the comp onen t u j j , alt ho ugh it is a function of the true co ordinates u j k ( k 6 = j ) on C P ( n − 1) j . In this notation the formu la for the change of co ordinates b et w een the c harts in esp ecially simple. F or example, the n alternativ e co ordinates u j ( j = 1 , ..., n ) of the same p oin t E ( ξ , τ ) ∈ C P ( n − 1) 0 (4.3) can b e written brieﬂy as u j k = r k ¯ τ j τ k with r k = p ξ k − | y | , k = 1 , ..., n, τ n := 1 . (4.67) Since it is a reduction of Ω C n (3.67), the symplec tic form χ 0 ω FS is represen ted by the Darb oux form i P k 6 = j d ¯ u j k ∧ d u j k on the chart ( 4.66) Consider now the comp o nen t v j of the v ector (4.6). Notice f rom Lemma 7 that (since y > 0) v j ( ξ , y ) yields a C ∞ function on C P ( n − 1) j . Then, for eac h j = 1 , ..., n , deﬁne the U ( n ) v alued function g j y ( ξ ) as follows . First of all, set g n y := g y in (4.8). F or 1 ≤ j < n , let T j denote the n b y n transp osition matrix giv en explicitly b y T j := 1 n − E j,j − E n,n + E j,n + E n,j . (4.68) Then using the form ula (4.7) and the v ector v ( ξ , y ) in (4.6) deﬁne g j y ( ξ ) := T j g ( T j v ( ξ , y )) . (4.69) It is clear that g j y ( ξ ) is actually a real-orthogonal matrix for all j . Moreo v er, we p oint out that g j y ( ξ ) = g n y ( ξ ) η j y ( ξ ) with η j y ( ξ ) ∈ U ( n ) for whic h η j y ( ξ ) µ 0 η j y ( ξ ) − 1 = µ 0 , (4.70) whic h holds simply b ecause the matrices g j y ( ξ ) ha v e the same last column fo r all j . As examples that illustrate w ell the general case, for n = 3 w e displa y the matrices g 1 y =    − v 3 − v 2 v 1 − v 2 v 3 d 1 (1 − v 2 2 d 1 ) v 2 (1 − v 2 3 d 1 ) − v 3 v 2 d 1 v 3    and g 2 y =    (1 − v 2 1 d 2 ) − v 1 v 3 d 2 v 1 − v 1 − v 3 v 2 − v 3 v 1 d 2 (1 − v 2 3 d 2 ) v 3    , (4.71) 34 where d 1 := 1 + v 1 , d 2 := 1 + v 2 and v := v ( ξ , y ) in (4.6 ). The p oin t is that, in g eneral, g j y con tains t he denominator d j = 1 + v j , whic h yields a C ∞ function on C P ( n − 1 ) j . The rationale b ehind the deﬁnition of g j y is that using ∆ n := ∆ in (4.56) and intro ducing ∆ j ( τ ) := T j ∆ n ( τ ) T j = diag ( τ 1 , ..., 1 , ..., τ n − 1 , τ j ) , ∀ j = 1 , ..., n − 1 , (4.72) where the en try 1 app ears in the j j p osition, one can v erify the follo wing lemma. Lemma 9. The U ( n ) -va l ue d C ∞ function G j y deﬁne d on C P ( n − 1) 0 by the formula ( G j y ◦ E )( ξ , τ ) := ∆( τ ) − 1 g j y ( ξ )∆ j ( τ ) , j = 1 , ..., n, (4.73) extends to a C ∞ function on C P ( n − 1 ) j . T h e extende d function is denote d by the same letter, G j y : C P ( n − 1) j → U ( n ) . Pr o of. This is a simple insp ection of the matrix elemen ts of G j y based on the prop erties of w y j ( ξ ) in (4.47) and the formula (4.67). Indeed, if u j is represen tativ e of E ( ξ , τ ) according to (4.6 7), then for j 6 = n one ﬁnds for example tha t G j y ( u j ) k n ∼ ¯ u j k , G j y ( u j ) nj ∼ 1 , G j y ( u j ) j j ∼ u j n , G j y ( u j ) k l ∼ δ k l + (1 − δ k l ) ¯ u j k u j l for k , l / ∈ { j, n } , (4.74) where the sym b ol ∼ means prop ortionalit y b y a function of ξ that extends to a C ∞ , now here zero function on C P ( n − 1) j . The relat io ns (4.74), and similar relations that hold for all matrix elemen ts of G j y ( u j ), including j = n , imply that G j y extends from C P ( n − 1) 0 to a C ∞ function on C P ( n − 1) j . F or an y ﬁxed j = 1 , ..., n and y > 0, deﬁne the S U ( n ) × S U ( n ) v alued C ∞ map F j on the c hart ( C P ( n − 1 ) j , B j χ 0 ) b y the follow ing form ula: F j ( u j ) :=  G j y ( u j ) − 1 L y ( u j ) G j y ( u j ) , G j y ( u j ) − 1 δ ( ξ ) G j y ( u j )  , (4.75) where ξ i = | u j i | 2 + y for ev ery i = 1 , ..., n and L y is the global Lax matrix giv en in Lemma 8. Theorem 5. The maps F j (4.75) en joy the fol lowing pr op erties: 1. F j ( u j ) b elon gs to µ − 1 ( µ 0 ) a nd p ◦ F j : C P ( n − 1) j → P is a smo oth map. 2. p ◦ F j c oincides w i th p ◦ F k on C P ( n − 1 ) j ∩ C P ( n − 1) k and it c oincide s with f 0 of The or em 4 on C P ( n − 1) 0 = ∩ n j =1 C P ( n − 1 ) j . 3. O ne c an deﬁne a smo o th map f β : C P ( n − 1) → P by r e q uiring that f β c oincides with p ◦ F j on C P ( n − 1) j . T h e so-obtaine d map satisﬁes f ∗ β ( ˆ ω ) = χ 0 ω FS and f ∗ β ( ˆ β ) = J . (4.76) 4. T he map f β is surje ctive and inje ctive. 35 Conse quently, f β is a symple ctomorphism that extends the lo c a l Delzant map f 0 of The o r em 4 and its inverse φ β := f − 1 β is a Delz a nt symple ctomorphism sa tisfying e quation (3. 7 8). Pr o of. It follo ws directly from the deﬁnitions that ( F j ◦ E )( ξ , τ ) = Ψ η j y ( ξ ,τ ) − 1 (( F 0 ◦ E )( ξ , τ ) ) , ∀ ( ξ , τ ) ∈ P 0 y × T n − 1 , (4.77) where we use (4.23) and the deﬁnition η j y ( ξ , τ ) := η j y ( ξ )∆ j ( τ ) with (4 .7 0). Since η j y ( ξ , τ ) b elong s to the lit t le group of µ 0 in U ( n ), this en tails that ( F j ◦ E )( ξ , τ ) ∈ µ − 1 ( µ 0 ). Then w e obtain prop ert y 1 since µ − 1 ( µ 0 ) ⊂ D is closed, the v alues E ( ξ , τ ) co ver C P ( n − 1) 0 whic h is dense in C P ( n − 1 ) j , and F j : C P ( n − 1) j → D is a C ∞ map as sho wn by its form ula ( 4.75). Prop ert y 2 holds since C P ( n − 1) 0 ⊂ C P ( n − 1) j ∩ C P ( n − 1) k is dense, and p ◦ F j coincides with f 0 = p ◦ F 0 on C P ( n − 1) 0 b ecause of (4.77). Prop ert y 3 is immediate from the preceding pro p erties and the fa ct that f 0 is a lo cal D elzan t map satisfying (4.10) and (4.11). T o establish the surjectivit y of f β , notice that equiv ariance with resp ect to the torus actions (3.74) and (3.54), f β ◦ R τ = ˆ Ψ b τ ◦ f β , ∀ τ ∈ T n − 1 , (4.78) follo ws from (4.76). Then f ∗ β ( ˆ β ) = J and (4.78) imply t hat the image of f β con tains each T n − 1 -orbit in P . This en tails the surjectivit y . Our ﬁnal task it to demonstrate the injectivit y of f β . T o do this, we remark that if f β tak es the same v alues on tw o elemen ts of C P ( n − 1), then those elemen ts m ust b elong to the same c hart C P ( n − 1) j at least for one j . Indeed, this is a consequence of the second equ alit y in (4.76) and t he deﬁnition of C P ( n − 1 ) j . Then, assume that f β ( u j ) = f β ( z j ) for tw o elemen ts u j , z j ∈ B j χ 0 . By the deﬁnition of f β , this is equiv alent to the exis tence o f an elemen t η from the little group o f µ 0 in U ( n ) suc h that  η − 1 G j y ( u j ) − 1 L y ( u j ) G j y ( u j ) η , η − 1 G j y ( u j ) − 1 δ ( ξ ) G j y ( u j ) η  = =  G j y ( z j ) − 1 L y ( z j ) G j y ( z j ) , G j y ( z j ) − 1 δ ( ξ ) G j y ( z j )  , (4.79) where ξ is given b y ξ i = | u j i | 2 + | y | = | z j i | 2 + | y | . W e see from the second compo nent of (4.79) that T := G j y ( u j ) η G j y ( z j ) − 1 (4.80) m ust b elong to the t o rus T n ⊂ U ( n ). Then the ﬁrst compo nen t of (4.7 9) and the fact that L y k ,k +1 (4.54) depends only on ξ and nev er v anishes for an y k imply t hat T = λ 1 n for some λ ∈ U (1 ) . Up on re-substitution in to the ﬁrst comp onen t of (4.79), this g iv es the equalit y L y ( u j ) = L y ( z j ) . ( 4 .81) By using t ha t the comp onen ts of Λ y ( ξ ) in (4.53) are non-zero, w e infer from the inspection of L y k ,l ( u j ) = L y k ,l ( z j ) for the ﬁxed index l = j + 1 ( l := 1 if j = n ) t ha t ¯ u j k u j j = ¯ z j k z j j holds fo r eac h k . Since u j j = z j j as this comp onen t dep ends o nly on ξ , and u j j 6 = 0, w e conclude that z j = u j , whereb y the injectivit y of f β follo ws. 36 As an alternativ e to the ab o ve self-con tained reasoning, we can also giv e a shorter pro of of the injectivit y of f β b y in v o king that for any Hamiltonian tor ic manifold a nd the pre-image of an y momen t map v alue there exists a certain subtorus that acts fr e ely on that pre-image [2 ]. The subtorus in question (whic h is the whole to rus for t he in terior o f the Delzant p olytop e) dep ends only on the momen t map v alue, and b y using this the injectivit y of f β follo ws easily from f ∗ β ( ˆ β ) = J and (4.7 8). Remark 8. It is readily seen from the ab o v e that the maps F j : C P ( n − 1) j → µ − 1 ( µ 0 ) deﬁned by (4.75) are indeed lo cal sections of the principal bundle in (4.59) whose base is C P ( n − 1 ) and to tal space is the constrain t surface µ − 1 ( µ 0 ). W e constructed the glob al Delzan t symplectomorphism f β : C P ( n − 1) → P b y patc hing together the pro jected maps p ◦ F j . 4.4 The g lobal De lzant map f α and inv olution prop erties W e here presen t the construction of f α in terms of f β , and establish the in v olution prop erties (4.92) and (4.118) for t he maps (4.117) whose signiﬁcance will b ecome clear in Section 5. W e need some preparatio ns. First, let us deﬁne the map ν on the unreduced double b y ν ( A, B ) := ( ¯ B , ¯ A ) , (4.82) where ‘bar’ means complex conj ug ation. This is an in v o lution of D that enjoys the prop erties ν ∗ ( ω ) = − ω , ν ∗ ( µ ) = ¯ µ − 1 , ν ◦ Ψ g = Ψ ¯ g ◦ ν ( ∀ g ∈ G ) . (4.83) By using these prop erties and the fact that ( ¯ µ 0 ) − 1 = µ 0 , w e see that ν maps µ − 1 ( µ 0 ) to itself and it induces an an ti-symp l e ctic involution , ˆ ν , of the reduced phase space P . Second, let us deﬁne the a nti-symplec tic inv olution Γ of the sym plectic v ector space C n b y Γ( u 1 , ..., u n − 1 , u n ) := ( ¯ u n − 1 , ..., ¯ u 1 , ¯ u n ) , (4.84) whic h acts a s ‘reﬂection comp osed with complex conjugation’ on the ﬁrst ( n − 1) co ordinat es. Straigh t forw ardly , Γ induces an anti-symple ctic involution , ˆ Γ, o f ( C P ( n − 1) , χ 0 ω FS ). W e also need the a n ti-symplectic inv olution ˆ C : C P ( n − 1) → C P ( n − 1) that descend s fro m comp onen- t wise complex conjugation on C n , i.e. for whic h ˆ C ◦ π χ 0 ( u 1 , ..., u n − 1 , u n ) = π χ 0 ( ¯ u 1 , ..., ¯ u n − 1 , ¯ u n ) with u ∈ S 2 n − 1 χ 0 , (4.85) as w ell the symplectic in volution ˆ σ furnished b y 8 ˆ σ := ˆ C ◦ ˆ Γ = ˆ Γ ◦ ˆ C . (4.86) Third, note that t he sp ectral functions Ξ k deﬁned in Section 2 (Eq. (2.22)) v erify the iden tity Ξ k ( ¯ A ) = Ξ n − k ( A ) ∀ k = 1 , ..., n − 1 , Ξ n ( ¯ A ) = Ξ n ( A ) , ∀ A ∈ G. (4.8 7) 8 The v alue n = 2 is sp ecial since in this c a se ˆ C = ˆ Γ and ˆ σ (as well as σ in (4.8 8)) b ecomes the identit y map. 37 This can b e c hec k ed by direct calculation star t ing from (2.19), and also follows fro m w ell-kno wn group theoretic facts via the formula (2 .20). F ourth, let σ denote t he in v olutive map on C n − 1 giv en by σ ( x ) k := x n − k , ∀ x ∈ C n − 1 , ∀ k = 1 , ..., n − 1 . (4.88) As a result of (4.87), the R n − 1 -v alued sp ectral Hamiltonians, a nd their resp ectiv e reductions, are sub ject to the relatio ns α = σ ◦ β ◦ ν , ˆ α = σ ◦ ˆ β ◦ ˆ ν . (4.89) Fifth, the momen t map J : C P ( n − 1 ) → R n − 1 of the rotational T n − 1 -action used in Theorem 3 ob eys J ◦ ˆ Γ = J ◦ ˆ σ = σ ◦ J . (4.90) No w w e can construct f α in t erms o f f β . Theorem 6. If f β is a Delzant m ap in the sense of The or e m 3 (cf. also Eq. (3.79)), then f α := ˆ ν ◦ f β ◦ ˆ Γ (4.91) is also a Delzant ma p in the sense of T he or em 3. Equation (4.91 ) implies the invol ution p r op erty ( ˆ Γ ◦ f − 1 α ◦ f β ) 2 = id C P ( n − 1) . (4.92) Pr o of. It is clear that f α as deﬁned b y (4.91) is a symp lectomorphism prov ided t ha t f β is a symplectomorphism. By assuming that ˆ β ◦ f β = J , (4.91) and the previous relations en tail ˆ α ◦ f α = ˆ α ◦ ˆ ν ◦ f β ◦ ˆ Γ = σ ◦ ˆ β ◦ f β ◦ ˆ Γ = σ ◦ J ◦ ˆ Γ = J , (4.93) whic h is the required prop erty of the Delzant map f α . Eq. (4.92) follows directly from (4.91 ) . It is w orth noting that w e did not use the explicit form ula of f β to establish Theorem 6. Lemma 10. It fol low s fr om the lo c al formula (4.9) o f f β that the r estriction of the map f α (4.91) to C P ( n − 1) 0 op er ates ac c or ding to ( f α ◦ E )( ξ , τ ) =  p ◦ Ψ g − y ( ξ ) − 1   δ ( ξ ) , L lo c − y ( δ ( ξ ) , ρ ( τ ))  , (4.94) using the same no tations as in (4.9) and Ψ in (2.10). Mor e over, the fol lowing ide ntity holds: f − 1 β ◦ f α = ˆ Γ ◦ ( f − 1 β ◦ f α ) ◦ ˆ C . (4.95) 38 Pr o of. F or t yp ographic reasons, in this pro o f w e use b oth alternat ive notations M ∗ ≡ ¯ M (4.96) to denote the complex conjugate of an y matr ix M ; M t denotes transp ose a nd M † adjoin t . W e b egin the pro of of form ula (4.94) b y remarking that the map E (4.3) satisﬁes ( ˆ Γ ◦ E )( ξ , τ ) = E ( σ ( ξ ) , σ ( ¯ τ )) , ∀ ( ξ , τ ) ∈ P 0 y × T n − 1 , (4.97) with ˆ Γ and σ deﬁned earlier. Then t he combination of equations (4.9), ( 4 .91) and (4.97) g ives ( f α ◦ E )( ξ , τ ) = p ◦ Ψ g y ( σ ( ξ )) − 1  δ ( σ ( ξ )) ∗ , L lo c y  δ ( σ ( ξ )) , ρ ( σ ( ¯ τ )) − 1  ∗  . (4.98) Here, w e ha ve tak en into accoun t that g y ( σ ( ξ )) ∗ = g y ( σ ( ξ )) holds since g y (4.7) is real. Let η 0 b e the n × n matrix whose non-zero en tries are ( η 0 ) j,n +1 − j = 1 for all j = 1 , ..., n ; η 0 = η − 1 0 = η t 0 . It is not diﬃcult, a lthough somewhat long, to c hec k that δ ( σ ( ξ )) ∗ = η 0 δ ( ξ ) η 0 (4.99) and L lo c y  δ ( σ ( ξ )) , ρ ( σ ( ¯ τ )) − 1  ∗ = η 0 L lo c − y ( δ ( ξ ) , ρ ( τ )) η 0 . (4.100) In the course of deriving these relations w e utilized that W j ( δ ( σ ( ξ )) , y ) = W n +1 − j ( ξ , − y ) , ∀ j = 1 , ..., n, (4.101) and ρ ( σ ( ¯ τ )) − 1 = ρ ( σ ( τ )) = η 0 ρ ( τ ) − 1 η 0 . (4.102) By using (4.99) and (4.100), we can rewrite (4.98) as ( f α ◦ E )( ξ , τ ) = p ◦ Ψ ( η 0 g y ( σ ( ξ ))) − 1  δ ( ξ ) , L lo c − y ( δ ( ξ ) , ρ ( τ ))  . (4.10 3) It follo ws from (4.101) that η 0 g y ( σ ( ξ )) is a unitary matrix whose last column is given b y t he v ector v ( ξ , − y ) deﬁned in (4.6). This p ermits to conclude that η 0 g y ( σ ( ξ )) = g − y ( ξ ) g with some ( ξ and y -dep enden t) g ∈ U ( n ) for whic h g µ 0 g − 1 = µ 0 . T aking in to accoun t that Ψ g is a ga ug e transformation, equation (4.103) implies the desired form ula (4.94). No w are are going to prov e the identit y (4.95). It is enough to v erify this iden tit y on the dense op en submanifold of C P ( n − 1) 0 whose imag e under the map f − 1 β ◦ f α is also contained in C P ( n − 1) 0 . F or an y E ( ξ , τ ) from this submanifold, w e deﬁne ( ξ ′ , τ ′ ) ∈ P 0 y × T n − 1 b y the equation ( f − 1 β ◦ f α )( E ( ξ , τ )) = E ( ξ ′ , τ ′ ) , (4.104) whic h can b e rewritten equiv a lently as f α ( E ( ξ , τ ) ) = f β ( E ( ξ ′ , τ ′ )) . (4.105) Then the claim ( 4 .95) can b e reform ulated as the statemen t that the relation (4 .105) is equiv alent to the relation ( ˆ Γ ◦ ( f − 1 β ◦ f α ) ◦ ˆ C )( E ( ξ , τ ) ) = E ( ξ ′ , τ ′ ) , (4.106) 39 whic h (by taking into account (4.9 7 )) is in turn equiv alen t t o f α ( E ( ξ , ¯ τ )) = f β ( E ( σ ( ξ ′ ) , σ ( τ ′ ) − 1 )) . (4.107) Consequen tly , we hav e to sho w that (4.105) is equiv alen t to (4 .1 07). W e no w in tro duce the notatio n ( A 1 , B 1 ) ∼ ( A 2 , B 2 ) for elemen ts of the double D for whic h there exists g ∈ U ( n ) suc h that g A 1 g − 1 = A 2 and g B 1 g − 1 = B 2 . W e no t ice that t wo pairs ( A 1 , B 1 ) and ( A 2 , B 2 ) in µ − 1 ( µ 0 ) represen t the same elemen t of P if and only if ( A 1 , B 1 ) ∼ ( A 2 , B 2 ). There fore, b y using the lo cal f o rm ulae (4.9) of f β and (4.94 ) of f α , w e can reform ulate the equiv alence o f (4.105) and (4.1 07) as the equiv alence b et w een t he relation  δ ( ξ ) , L lo c − y ( δ ( ξ ) , ρ ( τ ))  ∼  L lo c y ( δ ( ξ ′ ) , ρ ( τ ′ ) − 1 ) , δ ( ξ ′ )  (4.108) and the relation  δ ( ξ ) , L lo c − y ( δ ( ξ ) , ρ ( ¯ τ ))  ∼  L lo c y ( δ ( σ ( ξ ′ )) , ρ ( σ ( τ ′ ))) , δ ( σ ( ξ ′ ))  . (4.109) By applying (4.99) and (4.100), we o bserv e that (4.109) is equiv alen t to  δ ( ξ ) , L lo c − y ( δ ( ξ ) , ρ ( ¯ τ ))  ∼  L lo c − y ( δ ( ξ ′ ) , ρ ( τ ′ )) ∗ , δ ( ξ ′ ) ∗  . (4.110 ) T o ﬁnish the pro of, we need t he iden tities L lo c − y ( δ ( ξ ′ ) , ρ ( τ ′ )) ∗ = ρ ( τ ′ )  L lo c y ( δ ( ξ ′ ) , ρ ( τ ′ ) − 1 )  t ρ ( τ ′ ) − 1 , (4.111) L lo c − y ( δ ( ξ ) , ρ ( τ )) ∗ = δ ( ξ ) L lo c − y ( δ ( ξ ) , ρ ( τ ) − 1 ) δ ( ξ ) − 1 , (4.112) whic h can b e readily v eriﬁed. No w supp ose that (4.108) holds (for some arbitrarily ﬁxed ( ξ , τ )). This assumption is equiv- alen t to the existence of a unitary matrix g for which g δ ( ξ ) g − 1 = L lo c y ( δ ( ξ ′ ) , ρ ( τ ′ ) − 1 ) and g − 1 δ ( ξ ′ ) g = L lo c − y ( δ ( ξ ) , ρ ( τ )) . (4.113) Then, b y means of (4.111), the v alidit y of the ﬁrst equality in ( 4 .113) implies that L lo c − y ( δ ( ξ ′ ) , ρ ( τ ′ )) ∗ = ρ ( τ ′ )( g δ ( ξ ) g − 1 ) t ρ ( τ ′ ) − 1 = [ ρ ( τ ′ ) ¯ g δ ( ξ )] δ ( ξ )[ ρ ( τ ′ ) ¯ g δ ( ξ )] − 1 , (4.114) whic h can b e recognized as the ‘ﬁrst comp onen t’ of the relation (4.110). By using (4.112), the second equalit y in (4.113) b ecomes δ ( ξ ′ ) ∗ = ¯ g L lo c − y ( δ ( ξ ) , ρ ( τ )) ∗ ¯ g − 1 = ¯ g δ ( ξ ) L lo c − y ( δ ( ξ ) , ρ ( τ ) − 1 )( ¯ g δ ( ξ )) − 1 , (4.115) whic h (since δ and ρ tak e v alues in T n ) en tails that δ ( ξ ′ ) ∗ = [ ρ ( τ ′ ) ¯ g δ ( ξ )] L lo c − y ( δ ( ξ ) , ρ ( τ ) − 1 )[ ρ ( τ ′ ) ¯ g δ ( ξ ) ] − 1 . (4.116) Th us w e hav e deriv ed (4.114) and (4.1 1 6) from (4.11 3), which tells us that (4.108) im plies (4.109). The con v erse imp lication can b e demonstrated b y follow ing the ab o ve equations in rev erse or der, whereb y the pro of is complete. 40 In Section 5, w e shall iden tify t he maps S := f − 1 α ◦ f β and R := ˆ C ◦ S (4.117) as the symplectic and resp ectiv ely the anti-sy mplectic v ersion of Ruijsenaars’ self-duality map of the compactiﬁed II I b system. Then the prop erties (4.9 2) and (4.95) will repro duce certain relations established in [3 1]. The same is true regarding the follo wing iden tities that can be deriv ed easily from the ab ov e: S 2 = ˆ σ and R 2 = id C P ( n − 1) . (4.118) As for t heir deriv ation, the ﬁrst iden tity in (4.1 18) is obtained b y recasting (4.92) as id C P ( n − 1) = ˆ Γ ◦ S ◦ ˆ Γ ◦ S = ˆ Γ ◦ ˆ C ◦ S 2 = ˆ σ ◦ S 2 , (4.119) where w e applied (4.95) to establish the second equalit y . Similarly , w e can write R = ˆ C ◦ S = S ◦ ˆ Γ = ˆ Γ ◦ ( ˆ Γ ◦ S ) ◦ ˆ Γ , (4.120) whereb y the second iden t ity in (4.118) f o llo ws fro m ( 4 .119). Finally , let us record also the following useful iden tities: δ ◦ J ◦ ˆ C = δ ◦ J , δ ◦ J ◦ ˆ Γ = δ ◦ J ◦ ˆ σ = η 0 ( δ ◦ J ) † η 0 , (4.121) L y ◦ ˆ C = ¯ L y , L y ◦ ˆ Γ = η 0 ( L y ) t η 0 , L y ◦ ˆ σ = η 0 ( L y ) † η 0 , (4.122) with η 0 deﬁned ab ov e (4.99). The iden tities in (4.121) are equiv a lent to (4 .90) and (4.99), and those in (4.122) can b e ve riﬁed directly by using the deﬁnition of the global Lax matrix L y . 5 Self-duality of the compactiﬁed I I I b system The construction o f the compactiﬁed I I I b system and the discov ery of its self-dualit y prop er- ties are due to Ruijsenaars [31 ]. Belo w w e ﬁrst recall the deﬁnition of this system from [31 ], in terlaced with some explanatory commen ts in terms of our presen t w ork. Then we establish that our reduction yields the compactiﬁed I I I b system, whic h is the message of Theorem 7 . The subseque n t Theorem 8 repro duces the symplectic version of Ruijsenaars’ self-dualit y map a s an automatic conseque nce of our construction. Finally , w e brieﬂy explain how the an t i- symplectic self-dualit y inv olution of Ruijsenaars ﬁts in to our fra mew ork. Theorems 7 and 8 represen t the ﬁrst main result of this pap er (the second will b e encapsu- lated in Theorem 9 of Section 6). W e can b e relativ ely brief here, since these theorems are j ust easy coro llaries of o ur preceding technic al results. In particular, they fo llow fr o m our detailed description of the Delzan t symple ctomorphisms f β and f α that incorp orate the Lax matrix (1.4) and its glo bal extension (4.55). Altho ug h w e gained inspiration fr om the seminal pap er [3 1] with whic h w e m ust compare o ur r esults, w e wish to emphasize that our approac h is self-con tained. The space of particle-p ositions D y ⊂ S T n of the lo cal I I I b system can b e iden tiﬁed with the op en p olytop e P 0 y b y means o f the map δ (cf. (2.20)). Although the true particle-p ositions 41 are the components of δ ( ξ ), w e may (and often do) regard the equiv alen t ξ ∈ P 0 y as the lo cal particle-p osition v ariable of the system. The canonical conjuga t es of the co ordinates ξ k are the θ k that parametrize Θ = ρ ( e i θ ) − 1 b y (4.5). By inserting these parametrizations into (1.3) one indeed obtains Ω lo c = − 1 2 tr  δ ( ξ ) − 1 dδ ( ξ ) ∧ ρ ( τ ) − 1 dρ ( τ )  = i n − 1 X k =1 dξ k ∧ τ − 1 k dτ k = n − 1 X k =1 dθ k ∧ d ξ k . ( 5 .1) T aking ξ and τ = e i θ as the basic v ariables, from now on w e iden t if y the lo cal phase space as M lo c y ≡ P 0 y × T n − 1 = { ( ξ , τ ) } . (5.2) The same v ariables w ere use d in [31 ] to describe the relativ e motion of the particles gov erned b y the Hamiltonian (1.1). The lo cal I I I b system of [31] is thus encapsu lated b y the triple ( M lo c y , Ω lo c , L lo c y ) , (5.3 ) where the v alue of the Lax matrix L lo c y at ( ξ , τ ) ∈ M lo c y is deﬁned to b e L lo c y ( ξ , τ ) := L lo c y ( δ ( ξ ) , ρ ( τ ) − 1 ) (5.4) with the expression (1 .4). The comm uting Hamiltonians of the system are prov ided by the functions h ◦ L lo c y for all h ∈ C ∞ ( G reg ) G . The comp osed function h ◦ L lo c y b elongs to C ∞ ( M lo c y ) since L lo c y : M lo c y → G reg , as was shown in [31] and follow s also from our Theorem 2 com bined with Lemma 6. Expresse d in our no t a tions, the compactiﬁed I I I b system w as deﬁned in [31] as the triple ( C P ( n − 1) , χ 0 ω FS , L y ) , χ 0 = π − n | y | , (5.5) where L y is the global Lax matrix describ ed in Subsection 4.2 . The comm uting Hamiltonians of this system w ere iden tiﬁed in [31] as the C ∞ functions 9 of t he form h ◦ L y for all h ∈ C ∞ ( G reg ) G . In particular, it w as established in [31] (and is o b vious in our setting) that the functions Ξ k ◦ L y , k = 1 , ..., n − 1 , (5.6) represen t action- v ariables for the system (5.5), where Ξ k ∈ C ∞ ( G reg ) G w as deﬁned in (2 .22). The crucial f act [3 1] is that b y the identiﬁcation (5.2) the map E (4.2) em b eds the lo cal I I I b system into C P ( n − 1), con v erting M lo c y in to the dens e op en s ubmanifold C P ( n − 1) 0 . As seen up on comparison of (4.4) and (5.1), this em b edding is symple ctic. Because of Lemma 8, the comm uting Hamiltonians of the lo cal system extend to those of the compactiﬁed o ne. The compactiﬁed system has complete ﬂows , simply since C P ( n − 1) is compact and the Ha milto nians of in terest are smo oth. Theorem 7. F or any h ∈ C ∞ ( G reg ) G , let ˆ h 1 and ˆ h 2 denote the r e duc e d Hamiltonians deﬁn e d by me ans of e quation (2.7) with (2.14). Consider the Delzant symple ctomorphisms f β and f α 9 Actually in [31] (E q. (5.50) lo c. cit.) r eal-analy tic Hamiltonians were s tudied, which is a negligible diﬀerence. 42 that map ( C P ( n − 1) , χ 0 ω FS ) to the r e duc e d phas e s p ac e ( P , ˆ ω ) as given by The or ems 5 and 6. Then, using the pr e c e ding notations , the fo l lo w ing r elations hold for f β : ˆ h 1 ◦ f β = h ◦ L y , ˆ h 2 ◦ f β = h ◦ δ ◦ J . (5.7) In p articular, with h = Ξ k for any k = 1 , ..., n − 1 , ˆ α k ◦ f β = Ξ k ◦ L y , ˆ β k ◦ f β = J k . (5.8) R e gar di ng f α , ther e hold the analo gous r ela tion s, ˆ h 1 ◦ f α = h ◦ δ ◦ J , ˆ h 2 ◦ f α = h ◦ ( L y ) † , (5.9) and in p articular ˆ α k ◦ f α = J k , ˆ β k ◦ f α = Ξ n − k ◦ L y . ( 5 .10) Pr o of. After recalling the underlying deﬁnitions, all these relatio ns follo w in a direct and straigh tfo rw ard w ay from the form ulae for the maps f α and f β . Actually it is enoug h to use the lo cal f o rm ulae (4.9) for f β and ( 4 .94) for f α , since a n y smo oth function is determined b y its restrictions t o a dense op en submanifold. What was non-trivial to establish is that the lo cal form ulae just men t io ned represen t the restrictions of glo b al Delzan t ma ps. Let us discuss the meaning of Theorem 7. First note that the function δ ◦ J (o r equiv alently just J ) on C P ( n − 1) represe n ts t he global analogue of t he particle-p ositions, since under t he em b edding E : M lo c y → C P ( n − 1) w e hav e ( J k ◦ E )( ξ , τ ) = ξ k . (5.11) According to (5 .7), f β con v erts the reductions o f the inv ariant functions o f the fo r m h 1 , where h 1 ( A, B ) = h ( A ) b efore reduction, in to the resp ectiv e functions of the global Lax matrix L y , and con v erts the r eductions o f the in v arian t functions h 2 , where h 2 ( A, B ) = h ( B ) b efo r e reduction, in to the resp ectiv e functions of the particle-p osition matrix δ ◦ J . In particular, f β con v erts the reduced sp ectral Hamiltonian ˆ α k in to the action-v ariable Ξ k ◦ L y of the compactiﬁed I I I b system, and at the same time it con v erts the reduced sp ectral Hamilto nia n ˆ β k in to the global particle-p osition v ariable J k , resp ectiv ely for eac h k = 1 , ..., n − 1. The form ula (5.9) for the map f α w orks similarly , but in addition to the exc hang e of the subscripts 1 a nd 2 on the function ˆ h , whic h go es bac k to the exc hange of the t w o f actors o f the double, it applies the adjoin t of the unitary Lax matr ix L y instead of L y . This implies that, for each k = 1 , ..., n − 1, f α con v erts ˆ α k in to the particle-p o sition v ariable J k , and at the same time con v erts ˆ β k in to the ‘ﬂipped action-v ariable’ Ξ n − k ◦ L y . Such a ﬂip, which arises from the in v olutions inevitably inv olv ed in the relation (4 .91) b et we en f α and f β , is neces sary in order to ensure the symplectic prop ert y of the map. The phase space C P ( n − 1) of the compactiﬁed I I I b system is equipped with the t wo Ab elian P oisson algebras for med by the resp ectiv e functions of the f orm h ◦ δ ◦ J and h ◦ L y with h ∈ C ∞ ( G reg ) G . (5.12) 43 These Ab elian algebras are generated resp ectiv ely by the global particle-p osition v ariables J k and action- v ariables Ξ k ◦ L y . W e no w describ e the b eha viour o f these algebras under the Delzant symplectomorphism S , S ≡ f − 1 α ◦ f β : C P ( n − 1) → C P ( n − 1 ) (5.13) as intro duced in (4.117). In the follow ing t heorem, we shall use the ‘ﬂip-in volution’ h 7→ h ♯ of C ∞ ( G reg ) G deﬁned b y h ♯ ( g ) := h ( g † ) , ∀ g ∈ G reg . (5.14) With this notation, the pro p ert y (4.87) of Ξ k ∈ C ∞ ( G reg ) G implies Ξ ♯ k = Ξ n − k , k = 1 , ..., n − 1 . (5.15 ) Theorem 8. The Delzant symple ctomorphism S g i v en by (5.13) sa tisﬁ e s the id entities ( h ◦ ( δ ◦ J )) ◦ S = h ◦ L y and ( h ◦ L y ) ◦ S = h ♯ ◦ ( δ ◦ J ) (5.16) for every h ∈ C ∞ ( G reg ) G . I n p articular, with h = Ξ k ( k = 1 , ..., n − 1 ) this yields J k ◦ S = Ξ k ◦ L y and (Ξ k ◦ L y ) ◦ S = J n − k . (5.17) In this way, S c onverts the p article-p osition variables into the action-varia bles and c onverts the action-variables into the ﬂipp e d p article-p osition variable s . Pr o of. Both equalities in (5.16) follow by trivial one line calculations fro m our pr eceding results. In fact, by using the deﬁnition of S and Theorem 7 w e can write ( h ◦ δ ◦ J ) ◦ S = ( ˆ h 1 ◦ f α ) ◦ ( f − 1 α ◦ f β ) = ˆ h 1 ◦ f β = h ◦ L y , (5.18) and similarly ( h ◦ L y ) ◦ S = ( h ♯ ◦ ( L y ) † ) ◦ ( f − 1 α ◦ f β ) = ( h ♯ ◦ ( L y ) † ◦ f − 1 α ) ◦ f β = h ♯ 2 ◦ f β = h ♯ ◦ ( δ ◦ J ) . (5.19) The equalities in (5.17) ar e sp ecial cases o f those in (5.1 6), using a lso (5.15). The message of Theorem 8 is that S is the symple c tic version of Ruijsenaars’ self-duali ty map 10 of the c omp actiﬁe d I I I b system . The sym plectic prop erty of this map is an automatic consequenc e of our construction, while it w as pro v ed in [31] in a very diﬀerent manner. It is w ort h not ing that, as follows immediately f r om Theorem 8, the symplectomorphism S 2 satisﬁes the relations ( h ◦ ( δ ◦ J )) ◦ S 2 = h ♯ ◦ ( δ ◦ J ) , ( h ◦ L y ) ◦ S 2 = h ♯ ◦ L y , (5.20) 10 T o g ive a pr ecise co mparison, our map S corr esp onds to f = k ◦ φ in Eq. (4.1 2 5) of [31]. The anti-symplectic inv olution R = ˆ C ◦ S deﬁned here in Eq. (4.117) co rresp onds to φ in [31]. The inv olutions ˆ C , ˆ Γ , ˆ σ that we int ro duced in Subsec tio n 4.4 cor resp ond resp ectively to k , ˆ k, p in Subsec tio n 4.4 o f [31], which also contains the equiv alents o f the relations (4.92), (4.9 5) and (4.118) enjoy ed by S a nd R . The iden tiﬁcation of k and p with time reversal a nd parity is further discussed in [32 ] (page 69). 44 J k ◦ S 2 = J n − k , ( Ξ k ◦ L y ) ◦ S 2 = Ξ n − k ◦ L y . (5.21) These are consisten t with the fact that S 2 is e qual to the symple ctic involution ˆ σ of C P ( n − 1) according to (4.118). Ruijsenaars fo cused mainly o n the anti-symple ctic v ersion of his self-duality map that has somewhat simpler (but equiv a len t) prop erties as the symplectic map S . As we no w explain, this map is pro vided in our setting b y R ≡ ˆ C ◦ S , deﬁned already in (4.117 ). O ur map R is ob viously an ti-symplectic, R ∗ ω FS = − ω FS , and it is in v o lutiv e according t o (4 .1 18). F or any h ∈ C ∞ ( G reg ) G , w e readily deriv e t hat ( h ◦ ( δ ◦ J )) ◦ R = h ◦ L y , ( h ◦ L y ) ◦ R = h ◦ ( δ ◦ J ) , (5.22) whic h in particular implies that J k ◦ R = Ξ k ◦ L y and (Ξ k ◦ L y ) ◦ R = J k for all k = 1 , ..., n − 1 . (5.23) These are precisely the characteristic prop erties of the an ti-symplectic self-dualit y inv olution o f Ruijsenaars [31]. The deriv a t ion of the ﬁrst relatio n in (5.22) go es simply a s h ◦ δ ◦ J ◦ ˆ C ◦ S = h ◦ δ ◦ J ◦ S = h ◦ L y , (5.24) b y com bining (5.16) with J ◦ ˆ C = J . The second relation in (5.22) can b e deriv ed similarly using (5.16) and the equalit y L y ◦ ˆ C = ¯ L y (4.122). The relativ e adv an tage of R o ve r S is t hat it simply in terc hanges the resp ectiv e action and particle-p osition v ariables, with the price that R is an ti-symplectic while S is symplectic. T he relationship b etw een S and R is similar to that b et w een the symplectic map ( q , p ) 7→ ( p, − q ) and the an ti-symplectic map ( q , p ) 7→ ( p, q ) on t he canonical phase space R 2 n . W e note that the seminal pap er [31] contains further in teresting properties of the maps S , R and ‘discrete symmetries ’ ( including but not exhausted by the time rev ersal ˆ C and the parity ˆ σ ), whic h w e could also repro duce. This, p erhaps together with the in v estigation of op en questions men tioned in [31], will b e rep or t ed in a surv ey that we plan to write ab out the reduction approac h to Ruijsenaars dualities in a f ew y ears. 6 Dualit y and the mapping class group As was already men tioned, quasi-Hamiltonian geometry pro vides univ ersal ﬁnite-dimensional ‘master’ ob jects b y reductions of whic h o ne ma y obtain the symplectic mo duli spaces of non- Ab elian ﬂat connections on Riemann surfaces with prescrib ed b oundary conditions [1]. Origi- nally those mo duli spaces w ere deriv ed by inﬁnite-dimensional reductions of inﬁnite-dimensional symplectic manifo lds A (Σ) consisting of all (suitably smo oth) non-Ab elian connections on the Riemann surface Σ. If Σ is a torus with a n op en disc remov ed (called ‘one-holed torus’), then the relev an t ﬁnite- dimensional master ob ject is precisely the inte rnally fused quasi-Hamiltonian double of the non- Ab elian group. The treatmen t of the compactiﬁed I I I b system b y quasi-Hamiltonian reduction no w p ermits us to establish the v alidit y of a remark able in terpretation of this system that go es bac k to 45 Gorsky and Nekraso v [14]. Namely , it is immediate from [1] and our results tha t t he compactiﬁed Ruijsenaars-Sc hneider system is the natura l in tegra ble system on t he mo duli space of ﬂat S U ( n ) connections o n the one-holed torus, where the holonom y around the hole is constrained to the conjugacy class of the moment map v alue µ 0 that w e used to deﬁne our reduction. In this picture, the pair ( A, B ) in the double represen ts the holo nomies of the ﬂat connection along the standard cycles on the torus. T his in terpretation o f the I I I b system w as antic ipated in [14] based on some f ormal, non-rigoro us argumen ts. Utilizing the quasi-Hamiltonian framework, w e here further dev elop this in terpretatio n a nd clarif y the connection (conjectured in [10]) b et wee n the mapping class group o f the o ne-holed torus and the Ruijsenaars duality . In the next tw o subsections 6.1 and 6.2 w e brieﬂy summarize necessary bac kground material [1, 13], and presen t new results in the last tw o subsec tions 6.3 and 6.4. In the bac kground material G can b e tak en t o b e any connected and simply connected compact Lie g roup, with in v arian t scalar pro duct h , i on its Lie algebra G , but when dealing with our example it will b e understo o d without further notice that G ≡ S U ( n ). 6.1 Flat connections and the quasi-Hamiltonia n double Let A (Σ) b e the space of smo oth G -v alued connection forms on the one-holed torus Σ. The space A (Σ) carries a natural symplectic form Ω with respect to whic h the gauge gro up G ( Σ) of smo oth maps from Σ to G acts in the Hamiltonian w ay . Th e symplectic form reads Ω( a, b ) = Z Σ h a ∧ , b i , (6.1) where a and b are G -v alued 1-forms on Σ in terpreted as t a ngen t v ectors to A (Σ). Consider then the space A ﬂat (Σ) of connections φ v erifying the ﬂatness condition dφ + [ φ, φ ] = 0 and the subgroup G res (Σ) of G (Σ) mapping a c hosen p oin t p 0 on the b oundary of the hole t o the unit eleme n t of the group G . The quotient A ﬂat (Σ) /G res (Σ) can be iden tiﬁed with the ﬁnite-dimensional manifold G × G because ev ery ﬂat connection is completely determined b y its ( G res (Σ)-indep enden t) ho lonomies alo ng t wo generators of the fundamental g roup π 1 (Σ , p 0 ). F urthermore, as sho wn in [1], the symplectic form Ω induces the quasi-Hamiltonian for m (2.11) on the double G × G suc h that ev ery quasi-Hamiltonian reduction of the double ( G × G, ω ) g iv es the same outcome as a corresp onding sy mplectic reduction of ( A (Σ) , Ω). In particular, the quasi-Hamiltonian reduction based o n t he momen t map v alue µ 0 (3.2) that w e ha v e p erformed in Section 3 giv es the symplectic manifold P that can b e also in terpreted as the mo duli space A ﬂat (Σ , C ) /G (Σ) consisting of ﬂat connections with holono mies around the b oundary of the hole b elonging to the conjugacy class C of µ 0 mo dulo G (Σ) gauge transformations. Since the concepts o f quasi-Hamiltonian geometry and quasi-Hamiltonian reduction mak e no reference to ﬂat connections on Riemann surfaces, the following question app ears. Despite tec hnical adv an tag es of the quasi-Hamiltonian v an ta g e p oint, ar e there some relev an t structural insigh ts concerning the symplectic geometry o f the mo duli spaces that are still more transparen t in the inﬁnite-dimensional reduction approach? The answ er to this question is aﬃrmative and this section is dev oted to an imp ortant illustration of this fact. Indeed, w e shall sho w that the symplectic v ersion S of the Ruijsenaars dualit y map in tro duced in Section 4 is nothing but the 46 natural symplec tomorphism induced by a particular elemen t of the mapping class group of the one-holed torus. Recall that the group D iﬀ + (Σ) of orien tatio n- preserving diﬀeomorphisms acts symplectically on the space of all connections ( A ( Σ) , Ω) b y simply associating to a connection 1 - form φ its pull-bac k b y the diﬀeomorphism. This inﬁnite-dimensional group of symp lectomorphisms has an immense ‘reduction k ernel’ consisting of all diﬀeomorphisms whic h can b e connected to the iden tity . This means that eﬀectiv ely only the corresponding discrete quotien t group (kno wn as the mapping class group) acts o n the mo duli space A ﬂat (Σ , C ) /G (Σ) = P . W e shall explain how this symplectic action on the reduced phase space P can be understoo d en tirely in the quasi- Hamiltonian con text as well. In fact, it turns o ut tha t an appropriate cen tral extension of the mapping class group acts b y auto morphisms on the quasi-Hamiltonian double, and up o n quasi- Hamiltonian reduction the cen ter factors out yielding the symplectic a ctio n of the orien tation- preserving mapping classes on t he reduced phase space. 6.2 Mapping cla ss group of the one-holed torus The diﬀeomorphisms considered b elo w include also those whic h rev erse the orientation of the Riemann surface. W e shall need this generalit y later on in order to ex plain the geometrical origin of b oth the symplec t ic v ersion S and the anti-sy mplectic v ersion R o f the duality map. W e recall ( see e.g. [13]) that the fundamen tal group π 1 := π 1 (Σ , p 0 ) of the one-ho led to r us is a free group with t wo generators X and Y corresp onding to the pathes passing thro ugh the p oin t p 0 and f o rming the standard homolo gy basis o f the torus. It admits a redundan t prese ntation in terms of three g enerators X, Y , K : π 1 =  X , Y , K | X Y X − 1 Y − 1 = K  , ( 6 .2) where K correspo nds to t he generator of the b oundary fundamen tal group π 1 ( ∂ Σ , p 0 ). Ev ery elemen t from the g roup of diﬀeomorphisms Diﬀ p 0 (Σ) preserving the p oint p 0 induces an auto - morphism of the fundamen tal gro up π 1 and tw o diﬀeomorphisms f rom D iﬀ p 0 (Σ) whic h can b e connected b y a path in Diﬀ (Σ) induce automorphisms of π 1 diﬀering b y an inner automor phism. Let Diﬀ p 0 , 0 (Σ) b e the normal subgroup of Diﬀ p 0 (Σ) consisting of the elemen ts connected to the iden tity by a path in Diﬀ (Σ). The mapping class group can be deﬁned as MCG p 0 (Σ) ≡ D iﬀ p 0 (Σ) / Diﬀ p 0 , 0 (Σ) , (6.3) and there exists a na tural homomorphism N : MCG p 0 (Σ) → Out( π 1 ) ≡ Aut( π 1 ) / Inn( π 1 ) . (6.4) Nielsen [24 ] has pro ve d that N is actually an isomorphism iden tifying MCG p 0 (Σ) with Out( π 1 ). F urthermore one has t he standard Hurewicz homomorphism from π 1 in to the homology group H 1 (Σ; Z ) = π 1 / [ π 1 , π 1 ] ∼ = Z 2 . Since H 1 (Σ; Z ) is Ab elian, its outer automor phisms o b viously form the discre te linear group GL (2 , Z ). Hence one obtains a homomorphism H : Out( π 1 ) → GL (2 , Z ) , (6.5) 47 whic h aga in turns out to b e an isomorphism [23]. In addition to the isomorphisms N and H , there exists another isomorphism (see e.g. [13]) I : GL (2 , Z ) → Aut( π 1 , K ) / h ι K i , (6.6) whic h will b e esp ecially useful for us. Here Aut( π 1 , K ) denotes the subgroup of auto mo r phisms of π 1 whic h take K either to K or to K − 1 and h ι K i is the normal subgroup of Aut( π 1 , K ) formed b y conjuga tions b y p ow ers of K . T he existence of the isomorphism I is based on the fact t hat the cen tralizer of K in π 1 is t he inﬁnite cyclic group h K i and on the result of Nielsen [23] stating that an y automorphism of π 1 tak es K to a conjugate of either K itself or to a conjugate of its in v erse K − 1 . Notice that a s the one-holed torus has just one b oundary comp onen t, whic h is therefore preserv ed b y ev ery elemen t o f Diﬀ p 0 (Σ), in the image of N (6.4) there ma y o ccur only suc h elemen ts of Out( π 1 ) that preserv e the union of the conjugacy classes of K and of K − 1 in π 1 . The orien ta tion-preserving mapping classe s are mapp ed by N in to outer automorphisms resp ecting the conjugacy class of K alone. (F or t he results recalled so far, the reader may also consult the review s [16, 19].) By comp osing the isomorphisms N , H and I w e iden tify the mapping class group MCG p 0 (Σ) with t he group Aut( π 1 , K ) / h ι K i . F o ur particular elemen ts F, S , T and ˜ T of Aut( π 1 , K ) will feature subseque ntly . W e deﬁne them as follo ws: F : X → Y , Y → X ; S : X → Y − 1 , Y → Y X Y − 1 ; T : X → X Y , Y → Y ; ˜ T : X → X , Y → Y X − 1 . (6.7) These elemen ts (redundan tly) generate Aut( π 1 , K ) [13]. Since N (6.4) is an isomorphism, there exist diﬀeomorphisms inducing the actions of their equiv a lence classes on π 1 . The diﬀeomor- phisms (and their mapping classes) that yield [ T ] and [ ˜ T ] are commonly referred to as Dehn t wists around the cycles Y and X , respectiv ely . By an easy calculation w e ﬁnd F 2 = ( F S ) 2 = ( F S T ) 2 = Id , S 2 = ( S T ) 3 , S 4 : X → K − 1 X K , Y → K − 1 Y K , (6.8) and therefore the equiv alence classes [ F ] , [ S ] and [ T ] in Aut( π 1 , K ) / h ι K i v erify [ F ] 2 = ([ F ][ S ]) 2 = ([ F ][ S ][ T ]) 2 = Id , [ S ] 2 = ([ S ][ T ]) 3 , [ S ] 4 = Id . (6.9) It is w ell-know n ([4 ], Eq. (7.21) ) t hat the relations in (6.9) are deﬁning relations of the g r oup GL (2 , Z ) and the last t wo of them deﬁne the subgroup S L (2 , Z ). Correspondingly , w e ha ve the iden tiﬁcations S L (2 , Z ) ≃ Aut + ( π 1 , K ) / h ι K i ≃ MCG + p 0 (Σ) , (6.10) where Aut + ( π 1 , K ) is generated b y S and T and MCG + p 0 (Σ) con tains the or ientation-preserving mapping classes. It is helpful to displa y the inte ger matrices asso ciated to the classes [ F ] , [ S ] and [ T ] b y the isomorphism I (6.6). F ollowing [13], w e obtain [ F ] →  0 1 1 0  , [ S ] →  0 1 − 1 0  , [ T ] →  1 0 1 1  . (6.11) 48 W e did not need all four automorphisms F , S, T and ˜ T to c haracterize the mapping class group, since the relation ˜ T = ( T S T ) − 1 is v alid. Ins tead of F , S and T w e can use as generators F , T and ˜ T , whereb y [ S ] can b e expressed as the composition of three successiv e Dehn t wists: [ S ] = ( [ T ][ ˜ T ][ T ]) − 1 . (6.12) 6.3 The duali t y map S as a mapping class symplectomorphism By deﬁnition, an automorphism of a quasi-Hamiltonian space ( M , G, ω , µ ) is a diﬀeomor phism of M that preserv es ω and µ and commutes with the G -a ction. W e next exhibit a na t ura l homo- morphism from the group Aut + ( π 1 , K ) g enerated by S and T (6.7) in to the group Aut( D ( G )) of automorphisms of the do uble. F o r this purp ose, w e set S ( A, B ) := ( B − 1 , B AB − 1 ) , T ( A, B ) := ( AB , B ) , ˜ T ( A, B ) := ( A, B A − 1 ) . (6.13) One readily v eriﬁes that the maps S , T and ˜ T deﬁned b y (6.13) preserv e the basic structures (2.10), ( 2 .11) and (2.12) of t he double, g iving automorphisms o f D ( G ). Note t ha t w e sligh tly abuse the not a tion b y using the same sym b o ls S , T , ˜ T for the elemen t s of Aut + ( π 1 , K ) and of Aut( D ( G )), and later w e shall use them ev en for correspo nding elemen ts of factor groups of Aut + ( π 1 , K ) and of Aut( D ( G )). W e b eliev e ho w ev er that the reader prefers to ﬁgure o ut from the contex t whic h group w e hav e in mind rather than to remem b er whic h of se v eral diﬀerent putativ e not ations corresponds to whic h group. The similarity of the f o rm ulae (6.7) and (6.13) implies that in (6.13) w e ha v e cons tructed a homomorphism of Aut + ( π 1 , K ) in to Aut( D ( G )). This similarit y is not acciden tal, of course. Indeed, the assignmen t of a pair of G -elemen ts A and B to a ﬂat connection is not unique but it is ﬁxed b y the c hoice of the homology cycle s along whic h o ne computes the holonomies of the ﬂat connection. It is the standard c hoice of homology basis whic h w e hav e adopted previously . The t r ansition fr om the standard homology basis t o another one follo wing the matrices (6.1 1) induces the automorphisms (6.13) of t he double. As w e kno w, the f o urth p o w er S 4 in Aut + ( π 1 , K ) is the conj ug ation b y the elemen t K − 1 , whic h b elong s to the cen ter of Aut + ( π 1 , K ). On the other hand, calculating S 4 (6.13) in Aut( D ( G )) giv es the so-called tw ist automorphism Q in tro duced in [1]: Q ( A, B ) = Ψ µ ( A,B ) − 1 ( A, B ) . (6.14) It is easy to c hec k directly tha t the t wist automorphism is in t he cen ter of the group Aut( D ( G )) (this is b y the w a y true for an y quasi-Hamiltonian manifold). The Ab elian group h Q i generated b y Q is th us a distinguishe d normal subgroup of Aut( D ( G )), and Aut( D ( G )) can b e view ed as a cen tral extension of the gro up Aut( D ( G )) / h Q i . On a ccoun t of (6.10), w e may conclude that the orientation-preserv ing subgroup of the mapping class group acts on the double D ( G ) ‘pro jectiv ely’. T urning to quasi-Hamiltonian reduction, note that the automorphisms S , T and ˜ T deﬁned in (6.13) resp ect the constrain t surface µ − 1 ( µ 0 ). Moreov er, µ 0 b elongs to the isotrop y gr o up G 0 and therefore the automorphism Q descends to the trivial iden tity ma p on the reduced phase space P = µ − 1 ( µ 0 ) /G 0 . Th us S , T and ˜ T descended on P g enerate a true action of the 49 orien tatio n-preserving subgroup of the mapping class group, MCG + p 0 (Σ) giv en b y (6.10). By construction, this action op erates via symplectomorphisms of ( P , ˆ ω ). W e a re now going to demonstrate that in the C P ( n − 1) parametrization o f t he reduced phase space P of our in terest the mapping class generator S yie lds just the Ruijsenaars self-dualit y symplectomorphism S of C P ( n − 1) . Theorem 9. With the choic e µ 0 in (3.2), let us de note by S P : P → P the mapping class symple ctomorphism that desc ends fr om the automorphism S (6.13). Then S = f − 1 β ◦ S P ◦ f β , (6.15) wher e f β : C P ( n − 1 ) → P is the Delzant symple ctomorphis m c onstructe d in Se ction 4 and S : C P ( n − 1 ) → C P ( n − 1 ) is the Ruijsenaars symple ctomorphism d eﬁne d in (4.117). Pr o of. W e ha v e o b viously ( S P ◦ p )( A, B ) = p ( B − 1 , B AB − 1 ) , ∀ ( A, B ) ∈ µ − 1 ( µ 0 ) , (6.16) where p is the pro jection from µ − 1 ( µ 0 ) to P = µ − 1 ( µ 0 ) /G 0 . Recall the inv olution ˆ ν that veriﬁes ( ˆ ν ◦ p )( A, B ) = p ( ¯ B , ¯ A ) and descends from the map ν ( A, B ) = ( ¯ B , ¯ A ) deﬁned on the double in (4.82). W e can thus write ( ˆ ν ◦ S P ◦ p )( A, B ) = p ( ¯ B ¯ A ¯ B − 1 , ¯ B − 1 ) , ∀ ( A, B ) ∈ µ − 1 ( µ 0 ) . (6.17) Let us sho w that ˆ ν ◦ S P ◦ f β = f β ◦ ˆ C , (6.18) where ˆ C is the complex conjugation on C P ( n − 1) in tro duced in (4.85). By con tinuit y , it is suﬃcien t to ve rify (6.18) on C P ( n − 1) 0 , where (thanks to the f o rm ula (4.9)) it can b e rewritten as the equalit y p ◦ Ψ g y ( ξ ) − 1  δ ( ξ ) − 1 L lo c y ( δ ( ξ ) , ρ ( τ ) − 1 ) ∗ δ ( ξ ) , δ ( ξ )  = p ◦ Ψ g y ( ξ ) − 1  L lo c y ( δ ( ξ ) , ρ ( τ )) , δ ( ξ )  , (6.19) for all ( ξ , τ ) ∈ P 0 y × T n − 1 . But this holds simply on accoun t of equation (4.1 1 2) applied to − y instead of y . No w w e use the in volutivit y of ˆ ν , then Theorem 6, whic h states that f α = ˆ ν ◦ f β ◦ ˆ Γ, then also equation (4.119), rewritten as S 2 = ˆ Γ ◦ ˆ C , and the deﬁnition of the map S := f − 1 α ◦ f β (4.117) to conclude from ( 6 .18) f − 1 β ◦ S P ◦ f β = f − 1 β ◦ ˆ ν ◦ f β ◦ ˆ C = f − 1 β ◦ f α ◦ ˆ Γ ◦ ˆ C = f − 1 β ◦ f α ◦ S 2 = S − 1 ◦ S 2 = S . (6.20) As is w ell-know n, and is eviden t from (6.12), the Dehn twists T and ˜ T can b e used as alternativ e generators o f the (o r ientation-preserving) mapping class gro up instead o f T and S . This directly leads to the decomp osition S P = ( T P ◦ ˜ T P ◦ T P ) − 1 . (6.21) 50 Applying ideas of Goldman [12] (see also [21]), w e no w sho w that the D ehn twis t symplecto- morphisms T P and ˜ T P themselv es a r e sp ecializations o f simple Ha milto nian ﬂo ws on P . More precisely , we can realize already the automorphisms T and ˜ T of D given in (6.13) b y means of quasi-Hamiltonian ﬂo ws as stated b y the follo wing lemma. Lemma 11. Employing the notations of Subse ction 2.2, deﬁne the functions h ∈ C ∞ ( D b ) G and ˜ h ∈ C ∞ ( D a ) G by h = tr ( n − 1 X k =1 β k λ k ) 2 , ˜ h = tr ( n − 1 X k =1 α k λ k ) 2 , (6.22) and le t φ h,s and φ ˜ h,s b e the c orr esp o nding quasi-Hamiltonian ﬂow s . Then, r esp e ctively on D b and on D a , ther e hold the e qualities T ( A, B ) = φ h, 1 ( A, B ) and ˜ T ( A, B ) = φ ˜ h, 1 ( A, B ) . Pr o of. Using the deﬁnitions of Subsection 2.2, in t r o duce arbitrary real p ow ers of an y C ∈ G reg b y setting C s := g ( C ) − 1 exp − 2i s n − 1 X k =1 Ξ k ( C ) λ k ! g ( C ) , ∀ s ∈ R . (6.2 3) Then it follo ws from equations (2.15), (2.16) and (2.24) that φ h,s ( A, B ) = ( AB s , B ) , φ ˜ h,s ( A, B ) = ( A, B A − s ) , (6.24) and comparison with (6.13) en tails the claim. The f unctions h and ˜ h descend to P , and whe n transferred to the mo del C P ( n − 1) they b ecome functions of the global particle-p ositions, ˆ β k ◦ f β , and action-v ariables, ˆ α k ◦ f β . The decomp osition of the Ruijsenaars duality map S implied b y (6.15) and (6.2 1 ) represen ts a new result. This is a simple b y-pro duct of the reduction approac h, whic h w ould ha v e b een diﬃcult to notice in the direct appro ac h [31] to the compactiﬁed I I I b system. 6.4 The a n ti-symplectomorphism R as a GL (2 , Z ) generator In Section 6.3 , w e hav e implemen ted the generato r s S and T of the orien tat io n-preserving S L (2 , Z ) part of the full mapping class gro up GL (2 , Z ) as automorphisms of the do uble that descend up on reduc t io n to the symplectomorphisms S P and T P of our red uced phase space P . W e no w observ e that, on the one hand, the third generator [ F ] of GL (2 , Z ) implemen ted, according to (6.7), as the map F D ( A, B ) := ( B , A ) is not an automorphism of the double and it do es not surviv e the quasi-Hamilto nia n reduction. On the other hand, if w e consider instead of F D the related map ν : D ( G ) → D ( G ) deﬁne d b y (4.8 2) as ν ( A, B ) = ( ¯ B , ¯ A ), then ν maps µ − 1 ( µ 0 ) to itself and induces the a nti-symplec tic in v o lution ˆ ν of the reduced phase space. Moreo ve r , it is readily c hec ke d tha t under the assignmen t [ F ] 7→ ˆ ν , [ S ] 7→ S P , [ T ] 7→ T P (6.25) ˆ ν , S P and T P fulﬁl the generating relations (6.9 ) of GL (2 , Z ), and th us they induce an action of GL (2 , Z ) on P . Note also that GL (2 , Z ) can b e written as the semi-direct pro duct GL (2 , Z ) = Z 2 ⋉ S L (2 , Z ) , (6.26) 51 where the Z 2 subgroup is generated b y [ F ]. Correspondingly , there are tw o kinds of elemen ts of GL (2 , Z ): (+ , ρ ) and ( − , ρ ) where ρ ∈ S L (2 , Z ) and ± is the sign of the determinan t o f the GL (2 , Z ) matrix. The elemen ts (+ , ρ ) acting on P are symplectomorpisms and ( − , ρ ) are an ti- symplectomorphisms . This follo ws from the fact that ˆ ν ≡ ( − , e ) rev erses the sign of the symplectic form on P ( since ν rev erses the sign of ω o n D ( G )) while S P and T P preserv e it. P arametrizing our reduced phase space P as C P ( n − 1) b y means of the Delzan t symp lecto- morphism f β , as b efore, the generator ˆ ν of GL (2 , Z ) b ecomes f − 1 β ◦ ˆ ν ◦ f β and it is directly related to the in volutiv e Ruijsenaars an ti-symplectomorphism R : C P ( n − 1) → C P ( n − 1). Indeed, w e ﬁnd the following identitie s b y com bining Theorem 6, whic h states that f α = ˆ ν ◦ f β ◦ ˆ Γ, the deﬁnition of the map S := f − 1 α ◦ f β (4.117) and Eq. (4.120) saying that S ◦ ˆ Γ = R : R = S ◦ ˆ Γ = f − 1 α ◦ f β ◦ ˆ Γ = f − 1 α ◦ ˆ ν ◦ f α = S ◦ f − 1 β ◦ ˆ ν ◦ f β ◦ S − 1 . (6.27) Hence, the Ruijsenaars map R can b e view ed as a n alternativ e an ti-symplectic generator o f the GL (2 , Z ) action on C P ( n − 1). W e hav e exhibited an (an ti) - symplectic action of the full group GL (2 , Z ) on our reduced phase space. This action do es no t descend from the (pro jectiv e) action of the full mapping class group o n the double since we ha v e replaced the generator F D b y the new generator ν . It is th us natural to ask the following question: Do es the GL (2 , Z ) generator ˆ ν has its origin in some natural (anti)-symplec tomorphism of the space of connections ( A (Σ) , Ω)? It turns out that the answ er to this question is p ositiv e. Indeed, tak e an y orien ta tion-rev ersing diﬀeomorphism of Σ whic h is in the class F of the mapping class group and compose it with the complex conjugation acting on the su ( n )-v alued connection 1 - forms (without touc hing their argument). Since the complex conjugatio n is an automorphism of the group S U ( n ) and of its Lie alg ebra, this comp osed map is a n a n ti-symplectomorphism o f ( A (Σ) , Ω) whic h descends to the in v olutiv e an ti- symplectomorphism ˆ ν . 7 Discuss ion In this pap er w e ha v e demonstrated that an appropriate quasi-Hamiltonian reduc tion [1] of the in ternally fused double D = S U ( n ) × S U ( n ) yields a reduced phase space P that turns in to a Hamilto nia n toric manifold (i.e. a compact completely in tegrable system) in tw o diﬀer- en t but equiv ar ian tly symplectomorphic wa ys. The underlying t w o toric moment maps on P , with respectiv e componen ts ˆ α k and ˆ β k , ar ise from the reduc tions of the tw o sets of sp ectral Hamiltonians on D generated b y the tw o comp onen ts of the pair ( A, B ) ∈ D . O n the other hand, the phase space C P ( n − 1) also carries tw o distinguished toric structures, with momen t maps J k and Ξ k ◦ L y that enco de, resp ectiv ely , the particle-p ositions a nd the action-v ariables of the compactiﬁed I II b system as disco ve red in [31]. We have explicitly c onstructe d two ‘Delzant symple ctomorphisms’ f α and f β fr om C P ( n − 1 ) to P that r elate these toric moment maps ac- c or ding to Eqs. (5.8) and (5.10), and have identiﬁe d the c omp ose d map S = f − 1 α ◦ f β as the symple ctic se lf-duality map [31] of the c omp actiﬁe d I I I b system, which satisﬁes Eq. (5.17) . In our setting the symplectic prop erty of the p ertinen t self-dualit y map is ob vious, while in the original approach of [3 1] it required a sp ecial pro of. W e ha v e also reco vere d the an ti-symplectic v ersion R = ˆ C ◦ S of Ruijsenaars’ self-dualit y ma p, which satisﬁes Eq. (5 .23). 52 In addition, w e hav e rigorously established the in terpretatio n o f the compactiﬁed I I I b system in terms of ﬂat S U ( n ) connections o n the one-holed torus suggested b y Gorsky and Nekraso v [14] and, b y pro ving the form ula S = f − 1 β ◦ S P ◦ f β (6.15), we have demonstr ate d that t h e Ru i j senaars self - duality map S r epr esents the natur al action of the mapping cla ss g e ner ator S ∈ S L (2 , Z ) on C P ( n − 1 ) ≃ P . As for the map R , w e hav e sho wn that it ar ises from a GL (2 , Z ) extension o f the S L (2 , Z ) mapping class group action o n our reduced phase space. The in terpretation of the Ruijsenaars self-dualit y as the reduction remnant of the S L (2 , Z ) mapping class generator S is a long-exp ected result that w e ﬁnally succeeded to pro ve thanks to the quasi-Hamiltonian techniq ue. F o r the sak e of ob j ectivit y , w e should men tion that Gorsky and his collab orators w ere v ery close to establish this interpretation; form ula (4.31) of their pap er [10] coincides essen tially with our form ula (6.13). Ho wev er, they remark ed tha t their deﬁnition (4.31) of S violates the S L (2 , Z ) relations and they could reco v er a true S L (2 , Z ) action only fo r the rat io nal Calogero limiting case of the reduced system [10]. Since they hav e not furnished more quan tita t ive details w e cannot extract from their pap er the precise cause of the trouble, but we b eliev e that it may b e related to the fact that our fo rm ula (6.1 3) a lso deﬁnes only the action o f a suitable cen tral extension of S L (2 , Z ) on the double and not a true S L (2 , Z ) action. Our p oint is, how ev er, that up on the quasi-Hamiltonian reduction this pro jectiv e action descends to a true S L (2 , Z ) action on the reduced phase space. Besides the coupling constan t, y , a second parameter, Λ, can b e in tro duced in to the I I I b system b y replacing t he symplectic form (1.3) b y Ω lo c Λ := ΛΩ lo c . The lo cal Darb oux v ariables p j and x j then parametrize δ and Θ in (1.2) a s δ j = e 2i x j / Λ and Θ j = e − i p j , whereb y x j b ecomes x j / Λ also in the Hamiltonian (1.1). The parameter Λ can b e enco ded in the reduction approac h b y choo sing the inv ariant scalar pro duct on su ( n ) to b e − Λ 2 tr , whic h scales t he 2- form ω (2.11) as w ell as t he reduced symplectic form a nd the corresp onding toric momen t p olytop e. Being a mere scale parameter of the symplectic structure, Λ essen tially play s no role at the classical lev el, and w e o mitt ed it to simplify the notations. How ev er, this pa rameter is imp ortant at the quan tum leve l (see [35]). After the presen t pap er, just one from the list of the kno wn Ruijsenaars dualities remains to b e deriv ed in the reduction approac h. It is the self-duality of the h yp erb olic R uijsenaars- Sc hneider system fo r whic h a suitable ‘double’ to b e reduced is still to b e disco ve red. It ma y app ear tempting to searc h for distinct r eal forms of the complex holomorphic constructions of [11, 26], but this scenario do es not seem to w ork and the problem is wide op en. An intriguing reform ulatio n of the problem is to enquire whether the know n self-duality map of the hyperb olic system [29] can b e factorized similarly to the represen tation S = f − 1 α ◦ f β that we obtained here in the case of the compactiﬁed I I I b system. Another interesting problem for the future is to study the Ruijsenaars dualit y in relation to ro ot systems diﬀeren t from A n . F or progress in this direction, we refer to the pap er [28]. 53 Ac knowledgem en ts. W e wish to thank A. Alekseev for relev an t remarks and for bringing reference [20] to our atten tion, and are also grateful to J. 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Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction

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