The symplectic Verlinde algebras and string K-theory

THE SYMPLECTIC VERLINDE ALGEBRAS AND STRING K -THEOR Y IGOR KRIZ AND CRAIG WESTERLAND WITH CONTRIBUTIONS BY JOSHUA T. LEVIN Abstract. W e const ruct string topology operations i n twisted K - theory . W e study the examples giv en b y symplectic Grassmannians, computing K τ ∗ ( L H P ℓ ) in detail. Via the work of F reed-Hopkins-T eleman, these computations are re- lated to completions of t he V erlinde algebras of S p ( n ). W e compute these com- pletions, and other relev ant i nformation ab out the V erli nde algebras. W e also iden tify the completions with the t wisted K -the ory of the Gruher-Salv atore pro-sp ectra. F urther commen ts on the ﬁeld theoretic nature of these construc- tions are made. Much of the recent history of alg ebraic top o logy h as bee n concer ned with mani- festations o f ideas from mathematical ph ysics within to po logy . A stunning exa mple is Chas-Sulliv an’s theory of string top o logy [4 ], which provides a family of algebraic structures ana lo gous to conformal ﬁeld theor y on the ho mology H ∗ ( LM ) of the fr e e lo op space LM o f a closed orien table manifold M ([16]). The w o rk of Chas a nd Sulliv a n started an entirely new ﬁeld of alg ebraic to po logy , and led to paper s too nu merous to quote. Equally interesting a s this analo gy howev er is the fact that it is not quite precise: while the notion of conformal ﬁeld theor y is s uppo sed to be completely se lf-dual, the string top olog y copr o duct in H ∗ LM has no co-unit. The inspir ation for this pap er came from tw o sour ces: one is the pap er of Cohen and Jo nes [6], gener alizing string to po logy to an arbitra r y M -or iented generaliz e d cohomolog y theo ry . The other is the work of F reed, Hopkins, and T eleman [14] which identiﬁed the famous V erlinde alg ebra of a co mpa ct Lie group G with its equiv ariant twisted K -theo ry . It follo ws that a co mpletio n of the V erlinde algebra is isomorphic to the (non-equiv ar iant) t wisted K -theory of LB G . In more detail, in [14], F reed-Hopkins -T eleman establish a ring isomor phism (1) G K ∗ τ ( G ) ∼ = V ( τ − h ( G ) , G ) betw een the twisted equiv ar ia nt K -theory of a simple, simply co nnected, compact Lie group G acting on itself b y conjuga tion a nd the V erlinde algebra of pos itiv e energy representations of LG at level τ − h ( G ). It is w ell known that the Borel construction for the conjugation actio n G × G E G is homotopy equiv alent to LB G . So, using a twisted version of the A tiy ah-Segal completion theorem due to C. Dwyer [11] and Lahtinen [21], we may co nclude (2) K ∗ τ ( LB G ) ∼ = V ( τ − h ( G ) , G ) ∧ I I.K. w as partially supported by the NSA. C.W. w as partial l y supp orted b y the NSF under agreemen t DMS-0705428. J.T.L. was partially supp orted by the NSF under the REU program. 1 2 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN where the V erlinde a lg ebra is completed at its augmentation ideal. Now a str ik ing prop erty o f the V erlinde alg ebra is that it is a Poincare alg ebra, which is the same thing a s a 2-dimensional topolo gical ﬁeld theory (with no asymmetry). As far as we know, there is no analogo us result inv olving ordinary homolo gy: in so me sense, while the K -theory info r mation s hould be o n the level of a mo dula r functor of a conformal ﬁeld theory , the ﬁeld theory in homolog y should b e on the level of the conformal ﬁeld theo ry itself. F rom CFT, one ca n pro duce ﬁnite mo dels in c haracteris tic 0 in the case of N = 2 supers ymmetry , the A -mo del and the B - mo del. Ho wever, N = 2- supe rsymmetry corr esp onds to extra data beyond topo lo gy (Calabi-Y au s tructure). (Indeed, F an, Jar v is and Ruan [12] give a construction of A -mo dule TQFT, co upled with compactiﬁed g ravit y , in the case of a Landau- Ginzburg mo del o rbifolds, which is related to the Calabi- Y a u case, using Gromov- Witten theory . W e return to this in the Concluding remar k s.) Is it then p o s sible that by consider ing an analogue of string top olog y on twisted K -theor y , we could mimic, using string top olog y constr uctions, so me o f the prope r ties of the V er linde algebra, and in pa rticular, p erhaps, construct a self-dual top olog ical ﬁeld theory in some s e nse? The purp ose of this pap er was to inv estiga te this question. What we found was partially sa tisfactory , partially not. First of a ll, it turns out that twisted K -theory of lo op spaces is quite diﬃcult to c o mpute, even in very simple cases. W e orig inally thoug h t that the q uestion may b e easier to tackle for quotients of symplectic gro ups (s uch as quaternionic Grassmania ns), b ecause of their relatively sparse homolo g y . This led us to sp ecializ e to the symplectic ca se (it is, of course, only an example, analogo us discussions should exist for other compact Lie groups). Ho wev er, it turns out tha t the q uestion is quite hard, and relatively little co uld b e said b eyond the case of pro jectiv e spaces using our metho ds. E ven for H P ℓ , wher e one has a collapsing sp ectra l s equence, it app ear s that one can only describ e the exa ct extensio ns by considering , indeed, the str ing top olo gy pr o duct. Exploring the co nnection with the V erlinde alg ebra turned out to b e mo re of a success: indeed, we show that the pro d uct in the V erlinde a lg ebra is connected with the string top olog y pr o duct in the twisted K -theory of free lo op s pace. A key step is the inv estiga tion of Gr uher and Salv a tore [17, 18], who in tr o duced spaces int erp olating b etw een the lo op spaces of ﬁnite and inﬁnite Grassmannia ns. On the other hand, it tur ne d out tha t the copro duct is n ot related in the same wa y to the V er linde alg ebra copr o duct, and in fa ct ex hibits the same asymmetry as elsewher e in string top olo g y . In particular , the “g enus stabiliza tion” string top olog y op eratio n is 0, while it is alwa y s injective for the V erlinde a lg ebra. The genus stabilization T is interesting for the following rea son: Godin has r ecently shown [1 6] that H ∗ ( LM ) admits the structure of a “non- counital homological conformal ﬁeld theory”: this means that H ∗ ( LM ) is an a lgebra ov er the homolo gy H ∗ ( S ) of the Segal- Tillmann surface op erad, built out of the clas s ifying spaces of ma pping class gr oups. In [2 3], it is shown tha t for a to po logical algebra A over S , the g roup completion of A is an inﬁnite lo op space. This g roup completion involv es inv er ting T . In the string top ology setting, this trivializes the algebra , since T = 0 . O n the other hand, one can show for ex ample that a K -module whos e 0-homotopy is the V er linde algebr a with T inverted admits the str ucture of an E ∞ -ring sp ectrum (this will b e discussed in a subseq uent pap er). W e sp ecula te in the Co ncluding remarks of this pap er that THE SYMPLE CTIC VERLINDE ALGEBRAS 3 per haps a Gr omov-Witten type co nstruction can lea d to top o lo gical ﬁeld theories in this context, bridg ing the gap betw een b o th c o nt exts. Computationally , we deter mine the twisted K - homology string pro duct and co - pro duct for qua ternionic pro jective spaces , exhibiting tha t these struc tur es contain, in a cute way , more information than the corres po nding str uctures in ordinar y ho - mology . Also , one gets the sens e that the connection with the V er linde algebra makes the t wis ted K -theory constructio n so mehow “smaller” tha n the un twisted case (by virtue of the d 3 diﬀerential in the t wisted Atiy ah-Hirzebruch sp ectr al se- quence), althoug h the answer is not ﬁnite. The present pap er is organized as fo llows: In Section 1 , we will start, as a “warm- up ca se”, the computation of twisted K -theory of the free lo op spac es of symplectic pro jectiv e spaces and the rela ted Gruher-Salv ato re spaces. W e will s ee in particular that even here, the free lo op space is tricky , a nd to reso lve it further , string top ology structure is ne e ded. This, in some sense, ser ves as a motiv a tion for the remainder of the pap er. In Section 2, we will attempt to extend this c a lculation to the c ase of symplectic Gr assmanians of S p ( n ) with n > 1. W e will see that the situation ther e is still mu ch more complicated, a nd r aises pur ely algebra ic questions a b o ut the symplectic V erlinde algebra s, for example the structure of their as so ciated g raded rings under the Atiy ah- Hir zebruch ﬁltration. These and related algebr a ic questions will be treated in Se c tio ns 3, 4. In Section 3, w e will sp eciﬁca lly consider the case of V er linde a lgebras o f represe ntation level 1, where more pr ecise infor mation can be obtained. In Section 4, we will co ns ider the g e neral case. O ur results include a complete explicit computation o f the Douglas-Br aun num b er, the completion o f the symplectic V er linde algebr as, a complete computatio n of T for the case n = 1 and an estimate (and prec ise conjecture) for the n > 1 ca se. In Sec tio n 5 , we ﬁnally return to string top ology , constructing the string pro duct in t wisted K -theory , a nd computing the string top olog y ring structure on the t wisted K -theory of L H P ℓ , and as a result, we determine the extensions in its additive s tructure. W e will see that in s o me sense, the ring structure is more non-trivial a nd interesting than in homology . In Section 6, we consider the string copr o duct, sho w the failure of stabilization, and determine the s tring copro duct fo r quaternionic pro jectiv e s paces (with a no n-trivial result). Finally , in Section 7, we discuss the compariso n of the pro duct with the V er linde a lgebra pro duct via Gruher-Salv a tore spa ces, and the failure of a na logous b ehaviour in the case of the c o pro duct. Section 8 co nt ains concluding remarks, including the sp eculations on Gromov-Witten theor y . In the Appendix 9, we r eview very brieﬂy some foundational mater ia l needed in this pa pe r . W e w o uld like to thank Chr is Do uglas, Nora Ganter, Nitu Kitc hlo o, and Arun Ram for helpful conversations on this material. 1. The case n = 1 . In this pape r , we fo cus on symplectic groups. Denote b y G S p ( ℓ, n ) the symplectic Grassmania n of n -dimensional H -submo dules of H ℓ + n where H is the algebr a of quaternions. By S p ( n ), we shall mea n its co mpact form, whic h is the group o f automorphisms of the H -mo dule H n which preserve the norm. Of course, there is a canonical map G S p ( ℓ, n ) → B S p ( n ). Conseque ntly , we can induce K -theor y t wistings on LG S p ( ℓ, n ) fro m K - theory twistings on L B S p ( n ). Here L denotes the free lo op space. Twistings ar e classiﬁed by H 3 (? , Z ). W e ha ve H 3 ( LB S p ( n ) , Z ) ∼ = 4 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN H 3 S p ( n ) ( S p ( n ) , Z ) where the r ight hand side denotes B o rel co homology , and S p ( n ) acts on itself by conjugation. As reviewed in [1 4], this g r oup is Z , and there is a canonical gene r ator ι s uch that the twisting τ = mι corres po nds to level m − n − 1 lo west weigh t t wisted representations o f LS p ( n ), when this num b er is p os itive (since n + 1 is the dual Co xeter num ber of S p ( n )). This is the t wisting w e will be considering her e. W e b egin with a concr e te co mputation of K τ ∗ ( L H P ℓ ). W e pro ceed in stages, beg inning with a co mputation of the completion of the V erlinde algebr a fo r S p (1). Using this, w e compute the twisted K -theory of an intermediate spac e Y ( ℓ , 1) w hich, in combination with the lo op pro duct develop e d in section 5, allows us to c o mpute K τ ∗ ( L H P ℓ ). The V er linde algebra of t wisting level m ≥ 3 (lo op gro up r e pr esentation level m − 2 ) is isomo rphic to (3) V m = V ( m, 1) = Z [ x ] /S y m m − 1 ( x ) where x is the tautologica l repres ent ation of S p (1) = S U (2) and the p olynomials S ym k ( x ) ar e deﬁned inductively by (4) S ym 0 ( x ) = 1 , S y m 1 ( x ) = x, xS ym k ( x ) = S y m k +1 ( x ) + S y m k − 1 ( x ) . The augmentation ideal I o f V m is g enerated by x − 2. If we ch ange v a r iables to y = x − 2, and deﬁne σ m − 1 ( y ) = S y m m − 1 ( y + 2 ), then the completion o f V m at I is the quotient of the power se r ies r ing Z [[ y ]]: (5) ( V m ) ∧ I = Z [[ y ]] /σ m − 1 ( y ) A str aightforw a rd induction shows that σ m − 1 (0) = S y m m − 1 (2) = m , so m lies in the idea l ( y ). Consequently , equation (5) implies (6) ( V m ) ∧ I = Y p | m Z p [[ y ]] /σ m − 1 ( y ) W e would like to know how ma n y copies of Z p app ears in this co mpletion. W e will prov e a gener alization of the fo llowing result in Section 4 b elow. How ever, it is instructive to prove this sp ecial case immediately . Prop ositio n 1. L et p b e a prime and let p i || m (by which we me an that p i is the p -primary c omp onent of m ). L et δ ( p, m ) =  ( p i − 1) / 2 if p 6 = 2 2 i − 1 if p = 2 . Then the p -primary c omp onent of ( V m ) ∧ I is ( Z p ) δ ( p,m ) . Notice that δ ( p, m ) is the num b er of 2 m -th r o ots of unity that have po s itive imaginary pa rt and also are p -p ow er ro ots of unity . Pro of: The p olynomials (7) S ym m − 1 ( x ) are Chebyshev polyno mials of the sec ond kind applied to 2 x . This means that if we let ζ m to b e the primitive m -th ro ot of unit y , then the ro ots of (7) are (8) ζ k 2 m + ζ − k 2 m , k = 1 , ..., m − 1 . THE SYMPLE CTIC VERLINDE ALGEBRAS 5 This means that if we denote b y W a ring of Witt vectors at the prime p in which (7) splits, we obtain an injective homomorphis m (9) V m ⊗ W → m − 1 Q k =1 W [ x ] / ( x − ζ k 2 m − ζ − k 2 m ) with ﬁnite cokernel. Completing this at the ideal ( x − 2) will therefor e additively give r ise to a pro duct of as many co pie s of W as there a re num b ers k = 1 , ..., m − 1 such that (10) ζ k 2 m + ζ − k 2 m − 2 has p os itive v alua tion. But now (10) is e q ual to ζ − k 2 m ( ζ k 2 m − 1) 2 , so (10) has p o sitive v aluatio n if a nd only if ζ k 2 m − 1 do es. It is sta nda rd tha t ζ ℓ − 1 has p ositive v a luation if and only if ℓ = p i for some i . Indeed, suﬃciency follows from the fact that (( x + 1) p i − 1) / (( x + 1) p i − 1 − 1) is an Eisenstein p oly nomial with ro ot ζ p i − 1. T o see necess ity , if ζ − 1 has positive v aluatio n, so do es ζ p − 1, so it suﬃces to co nsider the case when ℓ is not divisible by p . But then we hav e a ﬁeld F p [ ζ ℓ ], in which ζ ℓ − 1 is inv er tible, so its lift to W cannot have p ositive v aluation.  It follows from the completion theorem that the t wis ted K -theo r y of a simply connected simple compact Lie gr oup is isomorphic to the completio n of the V erlinde algebra (at a t wisting w her e the V erlinde a lgebra exists) at the augmentation ideal with a shift equa l to dimension and is equa l to 0 in dimensions o f opp osite par it y , so in particular (11) K 0 τ ( L H P ∞ ) = 0 , K 1 τ ( L H P ∞ ) ∼ = ( V m ) ∧ I . Next, we will compute the t wisted K - theory of L H P ℓ . W e will see that the answer is actually quite co mplicated; this motiv ates the structure which we shall introduce later, a nd to which we will pa rtially ne e d to refer to complete the calculation. Demonstrating a nother theme o f this paper, w e shall also see that before con- sidering L H P ℓ , it helps to consider the “intermediate” space (12) Y ( ℓ, 1) = L H P ∞ × H P ∞ H P ℓ . Note that (12) has the homotopy type of a ﬁnite-dimens io nal manifold: a n S p (1)- bundle over H P ℓ . W e will see in Se c tion 2 that Y ( ℓ, 1) is or ient able with resp ect to K -theory . Thus, its twisted K - homology a nd co homology with the sa me twisting are isomo r phic, with a shift in dimensions, which in this case is o dd. F or now, how ever, the mo s t imp or ta n t prop erty for us is that the twisted K - theo ry of Y ( ℓ , 1) is easier to determine. Let us be gin with a deﬁnition. Let i b e such that p i || m . Let ℓ b e a p o s itive int eger. Let r b e a po sitive integer and let p be a prime. Suppo se further that r ≤ ℓ + 1 and 2 j − 1 − 1 < r ≤ 2 j − 1 if p = 2 a nd p j − 1 − 1 2 < r ≤ p j − 1 2 6 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN if p > 2 for so me j = 1 , ..., i if p = 2. Then put ǫ ( p, ℓ, r ) = ⌈ 2 + ℓ − r 2 j − 1 ⌉ if p = 2 a nd ǫ ( p, ℓ, r ) = ⌈ 2(2 + ℓ − r ) ( p − 1 ) p j − 1 ⌉ if p > 2. Let ǫ ( p, ℓ, r ) = 0 in all other case s . Theorem 2. We have (13) K τ 1 Y ( ℓ, 1) = K 0 τ Y ( ℓ, 1) = 0 and (14) K τ 0 Y ( ℓ, 1 ) = K 1 τ Y ( ℓ, 1) = L p,r Z /p ǫ ( p,ℓ,r ) . A too l that we will need is the Serre spectr al seq uenc e in t wis ted K -theo ry for a ﬁbra tion F → E → B with twisting τ ∈ H 3 ( F ) (and pulled back to E ). It takes the for m H ∗ ( B ; K ∗ τ ( F )) = ⇒ K ∗ τ ( E ) . Pro of: First, the equality betw e en twisted K -homolog y and cohomo logy follows from the fact that Y ( ℓ, 1) is an o dd-dimensio na l, K -o rientable manifold. The canon- ical inclusio n Y ( ℓ, 1) → L H P ∞ induces a map (15) V ( m, 1) ∧ I → K 1 τ Y ( ℓ, 1 ) . F ur thermore, (15) is a map o f K 0 ( H P ∞ ) = Z [[ y ]]-modules. Clear ly y ℓ +1 annihilates the target (since the right ha nd side is a ctually a K 0 ( H P ℓ )-mo dule), so (15) induces a map of Z [[ y ]]-mo dules (16) Z [ y ] / ( σ m − 1 ( y ) , y ℓ +1 ) → K 1 τ Y ( ℓ, 1) . W e cla im that (16) is actua lly an isomorphism. T o this end, conside r the ma p of ﬁbration sequences (17) S p (1) = / /   S p (1)   Y ( ℓ, 1)   / / L H P ∞   H P ℓ / / H P ∞ . Since K 1 τ ( S p (1)) = Z /m , the induced map on E 2 -terms of twisted K -cohomology Serre sp ectra l sequences is a 3-fold susp ension of (18) ( Z /m )[ u, u − 1 ][ y ] → ( Z /m )[ u, u − 1 ][ y ] / ( y ℓ +1 ) . THE SYMPLE CTIC VERLINDE ALGEBRAS 7 Here u is the Bott class. Clear ly , the spectra l sequences collapse, so (18) induces an isomorphism betw e e n a s so ciated g raded ob jects to a (ﬁnite) ﬁltration on (16), and hence (16) is an isomorphis m. Thu s, we are reduced to computing the p - c o mpletion R ( m, ℓ ) of the left hand side of (16). Le t us assume that p i || m , i ≥ 1. The key p oint is to c o nsider the Eisenstein po lynomial Φ = Φ p,m = Q ζ ( y + 2 − ζ − ζ − 1 ) where the product is ov er all p i ’th ro ots of unit y ζ with I m ( ζ ) ≥ 0 such that ζ is not a p i − 1 ’st ro ot of unit y . Then m ultiplication by Φ deﬁnes an embedding of ﬁltered Z p [ y ]-mo dules (19) R ( m/p, ℓ ) → R ( m, ℓ ) , which, on the asso c iated graded ob jects, is given by multiplication by p (since Φ = p mo d y ). By induction on i , this then determines ho w multiples by p of the generator s (20) 1 , y , . . . , y δ ( p,m /p ) − 1 are r epresented in (18): the p -multiples the genera tors (20) are in the same ﬁltration degree, and are represented as p -multiples. Multiples of (20) by higher p ow ers p j are given b y taking the image under the ass o ciated graded map o f (19) o f the p j − 1 - m ultiples o f the c o rresp onding g enerator (2 0) in the asso cia ted gra ded ob ject of the left hand side of (19). Since the asso ciated g raded ob ject o f the co kernel of (19) is therefore annihila ted by p , on the remaining generato rs y δ ( p,m /p ) , . . . , y δ ( p,m ) − 1 m ultiplication by p is simply repres ent ed by the r eduction mo dulo p of Φ, which is y δ ( p,m ) − δ ( p, m /p ) = y p i − 1 / 2 . Recording these ex tens ions in closed form gives the sta temen t o f the theorem.  It turns o ut that actually fully determining the twisted K -theory of L H P ℓ is subtle, and s eems to requir e full use of the string top olo gy pro duct, and even then, the ex tens io ns do not seem to follow as simple a pattern as in the case o f Y ( ℓ, 1). A complete answer will be p ostp oned to Section 5. F r om a mere ex istence of the pro duct, howev er, we can now state the following Theorem 3. (21) K τ 1 ( L H P ℓ ) = K 0 τ ( L H P ℓ ) = 0 , Ther e exist s a de cr e asing ﬁltr ation on K τ 0 ( L H P ℓ ) such that the asso ciate d gr ade d obje ct is (22) E 0 K τ 0 ( L H P ℓ ) = K τ 0 ( Y ( ℓ, 1)) ⊗ Z [ t ] . Similarly, ther e is an incr e asing ﬁltr ation on K 1 τ ( L H P ℓ ) such that the asso ciate d gr ade d obje ct is (23) E 0 K 1 τ ( L H P ℓ ) = H om Z ( Z [ t ] , K τ 0 ( Y ( ℓ, 1))) . 8 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN Pro of: W e hav e a canonical ﬁbr ation (24) Ω S 4 ℓ +3 → L H P ℓ → Y ( ℓ, 1) where the second map is the pro jection. Consider the a sso ciated sp ectral seque nce in t wisted K -homology . The E 2 -term then is concentrated in e ven deg rees, a nd hence the sp ectral sequence collaps es. On the other hand, the second ma p in (2 4), as we will see in Section 5, is a ma p of r ings, and in fact the entire sp ectral sequence is a sp ectral sequence of r ings. It then follows that K τ 0 L H P ℓ is generated, as a ring , by tw o gener ators y and t where y is a s in (16), and t is the gener a tor of K 0 Ω S 4 ℓ +3 = Z [ t ] . F ur ther, w e see that K τ 0 L H P ℓ satisﬁes the r elation y ℓ +1 , a nd a relation which is congruent to σ m − 1 ( y ) mo dulo ( t ). The ﬁrst sta temen t follows. The second statement now follows from the universal co eﬃcient theorem.  2. On the case n > 1 . There are a num b er o f sp ectral sequences we may attempt to use for calculating the twisted K -theory of LG S p ( ℓ, n ) for n > 1. L et W ( ℓ, n ) denote the spa ce of symplectic n -frames in H ℓ + n . The mo st promising sp ectral sequence seems to b e the twisted K -theor y Serre sp ectral sequence asso ciated with the ﬁbratio n (25) Ω W ( ℓ, n ) → LG S p ( ℓ, n ) → Y ( ℓ , n ) . Here (26) Y ( ℓ, n ) = LB S p ( n ) × B S p ( n ) G S p ( ℓ, n ) . The reason this is adv a nt ageous is that Y ( ℓ, n ) is a homotopy equiv alent to a ﬁnite-dimensional manifold, namely a ﬁb er bundle ov er G S p ( ℓ, n ) with ﬁb er S p ( n ) (although not a principal bundle). Additionally , the manifolds Y ( ℓ, n ) are orientable with r esp ect to K -theor y b eca use of the following res ult: Lemma 4 . The K - the ory (or or dinary) homolo gy or c ohomolo gy sp e ctra l se qu enc es asso ciate d to the ﬁ br ations (27) S p ( n ) → Y ( ℓ, n ) → G S p ( ℓ, n ) , (28) S p ( n ) → LB S p ( n ) → B S p ( n ) c ol lapse to t heir r esp e ctive E 2 (r esp. E 2 ) t erms. Pro of: The ﬁbration (27) ma ps in a n o b vious wa y in to (28) which induces a n injection on E 2 (resp. surjection on E 2 ) terms o f the sp ectra l sequences in ques tion. Thu s, it suﬃces to consider (28). F or the s ame rea son, it s uﬃces to co nsider n = ∞ in (28). But for n = ∞ , (28) is a ﬁbra tion of inﬁnite lo op spa ces with the maps inﬁnite lo op maps. Since (28) splits, it is therefore a product in this ca s e, which implies the desired collaps e .  W e may therefore exp ect to use Poincar´ e duality together with the canonical compariso n map (29) Y ( ℓ, n ) → L B S p ( n ) to help calcula te the t wis ted K - theo ry Atiy ah-Hir zebruch spectr al sequenc e for Y ( ℓ, n ) (see the next section for a n example). THE SYMPLE CTIC VERLINDE ALGEBRAS 9 Unfortunately , the twisted AHSS for L B S p ( n ) is extremely tricky , even thoug h we know its targ et b y [14]. If this calculation can b e do ne, thoug h, we can so lve the spectral sequence of (25) b y the following discussion. First no te that by the Eilenberg-Mo ore sp ectral sequence, H ∗ Ω W ( ℓ , n ) is the divided p ow er algebra o n bo ttom gener ators in dimensions 4 ℓ + 2 , 4 ℓ + 6 , . . . , 4 ℓ + 4 n − 2 . Since the dimensions are all even, the AHSS for Ω W ( ℓ, n ) m ust co llapse (and b e- sides, there is no twisting when ℓ > 1 by connectivity). The spectral sequenc e is completely deter mined as the tenso r pro duct of the twisted AHSS for Y ( ℓ, n ) and H ∗ Ω W ( ℓ , n ) by the fo llowing result. Theorem 5. F or ℓ ≥ n , the t wiste d Serr e sp e ctr al se quenc e of the ﬁbr ation (25) is a sp e ctr al se qu en c e of H ∗ Ω W ( ℓ , n ) -mo dules. Pro of: W e will use the diagona l map from (2 5) to its pr o duct with itself for the mo dule structure. On the pro duct, w e can take the t wis ting tr ivial o n one factor , and eq ua l to the given twisting on the other. Th us, it suﬃces to show that the un- twiste d K -theory coho mology Ser re sp ectral s equence asso cia ted with the ﬁbratio n (25) collapses to the E 2 -term. Supp osed this is not the cas e. Since the E 2 -term is torsion free, the ﬁrst non-trivial diﬀere n tial must als o app ear in the cor resp onding K Q -cohomo logy Ser re sp ectra l sequence, and hence in the ordinar y cohomolo gy sp ectral s equence. So, we must prove tha t the or dinary ra tional co homology Serr e sp ectral s equence a sso ciated with (25) collapses. T o this end, ﬁrs t note that for ℓ >> n , the ordina ry cohomolog y Serre se q uence asso ciated with the ﬁbratio n (30) Ω W ( ℓ , n ) → Ω G S p ( ℓ, n ) → S p ( n ) collapses. Indeed, (30) is a principal ﬁbra tio n (the next term to the r ight is W ( ℓ, n )) so the corres po nding homology sp ectra l sequence is a sp ectral sequence of H ∗ Ω W ( ℓ , n )-mo dules, but the elemen ts H ∗ S p ( n ) ar e p ermanent cycles, since they hav e no po s sible target). Now co nsider the Serre spe c tr al sequence in o rdinary cohomolog y a sso ciated with the ﬁbration (31) Ω G S p ( ℓ, n ) → L G S p ( ℓ, n ) → G S p ( ℓ, n ) . W e cla im that this a lso co llapses to E 2 . Indeed, ma p into the c o rresp onding Serre sp ectral s equence (28). By Lemma 4, this co llapses, so it suﬃces to show that the po lynomial g enerator s of H ∗ (Ω W ( ℓ, n ) , Q ) are pe r manent cycles in the cohomolo g y Serre spec tral sequence as so ciated with (3 1). F o r example, we may map (25) to the case of n = ∞ , in which ca se we get (32) Ω S p/S p ( ℓ ) → L B S p ( ℓ ) → L B S p × B S p B S p ( ℓ ) . By mapping into (32) the case ℓ = 0, which is the ﬁbr ation (33) Ω S p → ∗ → S p, and tak ing o rdinary ho mology Serre sp ectra l seq uences, we know that for (33 ), the exterior generator s of H ∗ S p transgre ss to the co r resp onding p o ly nomial generator s of the homo logy o f the ﬁb er. This implies that in (32), the exterior generator s of H ∗ S p of dimens ions 4 ℓ + 3 , 4 ℓ + 7 , ... transg ress and the ones in dimensions 3 , ..., 4 ℓ − 1 are p er manent cycles. This shows that the E ∞ term of the homolo gy Ser r e sp ectra l sequence o f (32) is H ∗ S p ( ℓ ) ⊗ H ∗ B S p ( ℓ ), all o n the ho rizontal line. 10 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN Now dualizing, we see that in the H Q ∗ -cohomolo gy Serre sp ectral sequence of (32), the polyno mial gener ators of H ∗ (Ω S p/S p ( ℓ ) , Q ) tra nsgress. Hence, the same is true of the po lynomial g enerators o f H ∗ (Ω W ( ℓ, n ) , Q ) in the rationa l cohomo lo gy Serre sp ectral sequence of (25). What we need to show is that the tr ansgres sions o f the g enerator s of H ∗ (Ω W ( ℓ, n ) , Q ) to H ∗ ( Y ( ℓ, n ) , Q ) a re 0 . T o this end, we claim: (34) F o r n ≤ ℓ , the map induced on homo logy b y the nat- ural inclusion Y ( ℓ, n ) = LB S p ( n ) × B S p ( n ) G S p ( ℓ, n ) → LB S p × B S p B S p ( ℓ ) factors throug h the map induced in ho- mology b y the natura l inclus ion LB S p ( ℓ ) → L B S p × B S p B S p ( ℓ ). Note tha t if (34) is proved, then the same statement is true on r ational cohomolog y by the universal co eﬃcient theor e m. Then, the images of the transgre ssions of the generator s of H ∗ (Ω S p/S p ( ℓ ) , Q ) map to 0 in H ∗ ( Y ( ℓ, n ) , Q ), since they certainly map to 0 in H ∗ ( LB S p ( ℓ ) , Q ) by the edge map theor em. T his will conclude the pro of o f our theorem. T o prov e (34), note that we can further comp ose with the map LB S p × B S p B S p ( ℓ ) → L B S p , since this map is injective on homology . But the natura l inclusion Y ( ℓ, n ) = LB S p ( n ) × B S p ( n ) G S p ( ℓ, n ) → L B S p certainly factor s thro ugh the inclusion (35) LB S p ( n ) → LB S p. While note that this is induced in our setup by a map S p ( n ) → S p ( ℓ ) induced b y a qua ternion-linear map H n → H ∞ with the set of co ordina tes disjoint from thos e inv olved in the inclusion H ℓ → H ∞ inv olved in the inclusion S p ( ℓ ) → S p which induces the map L B S p ( ℓ ) → L B S p inv olved in the statement o f (34), nevertheless we have n ≤ ℓ , so the map (35) fa ctors throug h a map induced by some inclu- sion S p ( ℓ ) → S p induced by inclusio n o f co o r dinates, and a ny t wo s uch ma ps a re homotopic as maps of group. The sta temen t of (34) follows.  3. The represent a tion level 1 symplectic Verlinde algebra Let us consider the ca se of twisting level n + 2 for S p ( n ) (repr esentation level 1). The adv a nt age is that in this ca s e, we know by ra nk-level dua lit y that the V er linde algebra is is o morphic to the V erlinde alg ebra fo r S p (1) at the same t wisting level (i.e. repres ent ation level n ). More explicitly , the g enerator s o f the V er linde a lg ebra are the fundamen ta l (=level 1) r epresentations of S p ( n ). T o this end, we ha ve a “deﬁning” repres ent ation o f dimension 2 n , which we will for the moment deno te by x . Then the other fundamental representations are (36) v i = Λ i ( x ) − Λ i − 2 ( x ) , i = 2 , ..., n. Note tha t there is a canonica l co nt raction map Λ i → Λ i − 2 . It turns out that xv i ( x ) for i ≥ 1 (we put v 0 = 1, v 1 = x ) contains v i − 1 and v i +1 as subrepresentations, and their co mplement is a n irr educible r epresentation of level 2. This g ives the rela tion (37) v i x = v i +1 + v i − 1 , i = 1 , ..., n − 1 , v n x = v n − 1 . There a re mo r e level 2 representations, but the cor resp onding rela tions a re r edun- dant. THE SYMPLE CTIC VERLINDE ALGEBRAS 11 W e s ee immediately that (37) implies (38) v i = S y m i ( x ) where S y m i are the po lynomials from Section 1. The B raun-Dougla s num b er d ( n ) = d ( n + 2 , n ) is the greatest common divisor o f the diﬀerences o f dimen- sion of the left and r ight hand side of each relatio n (3 7). This num b er is contained in the aug mentation ideal of the V erlinde algebra . Perhaps sur prisingly , it turns out to b e very s mall, making the co mpletion trivial in mos t cases. W e will, again, prov e a ge ner alization of the following Theo rem in Section 4 b elow. Theorem 6. When n ≥ 2 , we have d ( n ) = 2 when n = 2 ℓ − 2 , and d ( n ) = 1 else. Pro of: Let s = P i ≥ 0 S y m i ( x ) t i be the genera ting ser ies of the p olynomials S y m i . Let us reca ll that from the recursive re lation (37) it follows that sxt = st 2 + s − 1 , or (39) s = 1 / ( t 2 − tx + 1) . W e may think that by (38), this is identiﬁed with the gener a ting series fo r the v i ’s as deﬁned by (36), which is (40) (1 + t ) 2 n (1 − t 2 ) , but that is not quite right. The p oint is , (4 0) has non-trivia l co eﬃcients a lso a t t i with i = n + 2 , ..., 2 n + 2 which (39) misses . The correct s eries equa l to (40) is then (41) s ( t ) − t 2 n +2 s ( t − 1 ) = (1 − t 2 n +4 ) s ( t ) . Thu s, if a prime p divides the Braun-Dougla s n umber for S p ( n ), repr esentation level 1 , then ov er F p , (42) t 2 n +4 − 1 = ( t 2 − 2 nt + 1)(1 + t ) 2 n +1 ( t − 1) . First let us no te that fo r p = 2 , the rig h t ha nd side is (1+ t ) 2 n +4 , so (42) oc c ur s if and only if 2 n + 4 is a p ower o f 2. T o see that in this case, the Braun-Do uglas num b er cannot be divisible by 4 , co nsider the diﬀerences of the diﬀerence of dimension betw een the tw o sides of (37) for i = 1 . Then the left hand side is divisible by 4, while the right hand side is n (2 n − 1), which is not divisible by 4. Now let us conside r a prime p 6 = 2. Let p j || 2 n + 4, let h = (2 n + 4) /p j . Then h is relatively prime to p , so F p actually has h diﬀerent r o ots of t h − 1 . By lo oking at the right hand side of (4 2), which has a t most 4 diﬀerent ro ots, we have h ≤ 4 . But h is divisible b y 2, so h = 2 , 4 . If j = 0, it is v er iﬁed b y direct computation that the Br a un-Douglas num b er is 1. When j > 0, we see that the left hand side contains a t least p factors of t − 1, while the righ t hand side contains at most 3. So we would hav e to have p = 3, j = 1. Thus, 2 n + 4 is equal to 6 or 12. 6 gives n = 1, which is excluded by ass umption. The o ther case actually g ives 6 copies of t − 1 on the left hand s ide o f (42), which cannot o ccur o n the r ight ha nd side.  W e se e therefore that the completion of the V erlinde a lg ebra is a profound op- eration which ca n lo se infor mation. In the c ase n = 2 ℓ − 2, it is also interesting to know the A tiyah-Hirzebruch ﬁltration on the V erlinde alg ebra completion. First, 12 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN we alrea dy k now that the repres e ntation lev el 1 V erlinde algebr a for S p ( n ) with n = 2 ℓ − 1 is (43) Z / 2[ x ] / ( x 2 n − 1 ) . Next, we will construct p olynomial ge ne r ators γ 1 , ..., γ n for the r epresentation ring R ( S p ( n )) of S p ( n ) such that γ i is in Atiy a h- Hirzebruch ﬁltration i . O bviously , for i = 1, we can just put (44) γ 1 = x − 2 n. Next, we put (45) γ i +1 = Λ i +1 ( x − 2 n + 2 i ) − Λ i − 1 ( x − 2 n + 2 i ) , i = 1 , ..., n − 1 . The element (45) is of ﬁltration degre e ≥ i + 1 b ecause it v anishes when restricted to S y m ( i ) (where it is equal to S y m i +1 ( x ′ ), where x ′ is the b ottom level 1 repr e - sentation of S p ( i )). Theorem 7. The asso ciate d gr ade d ring of the r epr esentation 1 level V erlinde algebr a of S p (2 r − 2) with r esp e ct of the Atiyah-Hirzebruch ﬁltra tion is isomorp hic to (46) Z [ γ 1 , ..., γ 2 r − 2 ] / (2 , γ 2 + γ 2 1 , ..., γ 2 r − 1 − 1 + γ 2 r − 1 − 1 1 , γ 2 r − 1 +1 + γ 2 r − 1 γ 1 , ..., γ 2 r − 2 + γ 2 r − 1 γ 2 r − 1 − 2 1 , γ 2 r − 1 γ 2 r − 1 − 1 1 ) The element 2 is r epr esente d by (47) γ 2 r − 1 + γ 2 r − 1 1 . Pro of: Let the genera ting function o f the Λ i ( x )’s b e λ . Then (45) is equal to (48) the co eﬃcient at t i +1 of λ ( x )(1 − t 2 ) / (1 + t ) 2 n − 2 i . But we know λ ( x )(1 − t 2 ) = v = 1 / ( t 2 − tx + 1) , so (4 8) is equal to (49) the co eﬃcient at t i +1 of 1 / ((1 + t ) 2 r +1 − 4 − 2 i ( t 2 − tx + 1)). Let us ﬁrst exa mine these p o ly nomials mo d 2. Fir st, note that (4 9) is the co eﬃ- cient at t i +1 of (1 + t ) 4+2 i / (1 − tx + t 2 ) . Upo n expanding the denominator in the v ariable t ( x − 2), we further get that this is the sum of co eﬃcients at t i +1 − j of (50) (1 + t ) 2+2 i − 2 j ( x − 2) j , which is  2 + 2 i − 2 j 1 + i − j  , which is o dd if and o nly if j = i + 1 . Let us also o bserve that (51) The co eﬃcient a t t i +1 − j of (50) is 2 mo d 4 if and only if i + 1 − j is a p ower of 2. Thu s, (51) will be the exac t case s when the co eﬃcient of (49) at x j is 2 mo d 4 , with the exception of the cas e when j = i , in which the co eﬃcient at x i is “a no malously” divisible by 4 when i is even. THE SYMPLE CTIC VERLINDE ALGEBRAS 13 Now let us exa mine the p olynomials (52) γ i + γ i 1 . W e just prov ed that the p oly nomials (52) are relations in the V erlinde algebra mo d 2. This is what we got fro m the po lynomial q i obtained from γ i by subtr a cting (49), substituting γ 1 + 2 r +1 − 4 for x , and reducing mo d 2 . T o pro ceed further, let us next lo ok at the p oly no mial (53) q 2 r − 1 . Since the co eﬃcient at γ 0 1 of (49 ) is 2 mod 4 and all the c o eﬃcients at γ j 1 , 0 < j < 2 r − 1 are even, we see that a dding recursively multiples of q 2 r − 1 γ j 1 , 0 < j < 2 r − 1 , we o bta in a relation in the V er linde algebra of the form (54) 2 + γ 2 r − 1 1 + higher ﬁltr ation ter ms. This implies that 2 is a relatio n in the asso ciated g raded ring, and 2 is repr esented by (47). Next, pro cessing in the sa me way q i with 1 < i < 2 r − 1 (i.e. adding recursively m ultiples of q 2 r − 1 γ j 1 , 0 < j < i ), w e g et (5 2) plus ter ms of higher ﬁltration deg ree, which shows that (52) is a relation in the asso ciated g raded ring. F o r q i with 2 r − 1 < i ≤ 2 r − 2 , we use (51): the low est co eﬃcient of q i at a p ow er of γ 1 which is not divisible by 4 is at γ i − 2 r − 1 1 . Then add γ i − 2 r − 1 1 q 2 r − 1 to mak e all co eﬃcients at γ j 1 , j < i , divisible by 4. But 4 is represented in ﬁltra tion a t least 2 r , so we obtain the relation (55) γ i + γ i − 2 r − 1 1 γ 2 r − 1 1 in the asso cia ted gr a ded ring as r equired. Finally , the relation S y m 2 r − 1 ( x ) in the V erlinde algebr a can b e treated a s γ k +1 = γ 2 r − 1 , giving the desired rela tion in this ca se also. W e now see by a counting argument that the ring (46) is indeed additively the asso ciated gr aded abelia n group of the Z 2 -mo dule Z 2 r − 1 2 with g e ne r ators in degrees 0 , ..., 2 r − 2 , and 2 in degree 2 r − 1 , s o there fo re o ur list of relations is complete.  Example: Let us lo ok at the low est no n-trivial case of Theorem 7 , r = 2, so n = 2. Let us ﬁrst co mpute the diﬀerentials of the t wisted K - theory cohomolog y Serre sp ectr al sequence of the ﬁbration (28). W e see from Theorem 7 that the only relation in the E ∞ term is (56) γ 1 γ 2 . The vertical part of the sp ectr al sequence is the twisted K - theory of S p (2), which is the susp ension b y 3 (an o dd nu mber) of the K ∗ / (2)-exterio r algebra on one generator ι in dimension 7. This is tensored with the horizo nt al part, whic h is R = Z [ γ 1 , γ 2 ]. Thus, the E 2 =term is (an o dd susp ension of ) (57) Z / 2[ γ 1 , γ 2 ] { 1 , ι } . But now the R -submo dule of (57) g enerated by 1 m ust disapp ea r (to confor m with the res ult of [1 4]), while the R -mo dule gener ated by ι needs the sing le relation (56) (times ι ). This means that we must hav e (58) d 12 (1) = ιγ 1 γ 2 . 14 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN (The right hand side of (58) is actually s till multiplied by an appro priate power of the Bo tt element.) Th e s pec tr al sequence is a spectra l s e quence of modules ov er the ho rizontal pa rt by Lemma 4 . Now let us co nsider the twisted K -theory Serr e sp ectral s e q uence (representation level 1) of the ﬁbra tion (2 7). W e claim that the diﬀerential (58) is s till the only one present. One sees in this case that the Bor el w ords ar e S y m ℓ + i ( γ 1 ), i = 1 , 2 homogenized b y multiplying by the appropriate p ow ers o f γ 2 . Th us, by co mparison with the ca s e of (28), no diﬀerentials d r , r < 12 ar e p o s sible, a nd the the ι -multiple of the E 13 term is (59) Z / 2[ γ 1 , γ 2 ] / ( γ 1 γ 2 , γ i 1 , γ j 2 ) where for ℓ even, j = ( ℓ/ 2 ) + 1, i = ℓ + 1, and for ℓ o dd, j = ( ℓ + 1) / 2, i = ℓ + 2. Additionally , the multiple of 1 is the Poincar´ e dual o f (59) times the top element of H ∗ ( G S p ( e ≪ , 2)). But by compa rison with (28), the elemen ts (59) are p ermanent cycles, a nd by Poincar´ e duality (see Section 2), so a re the corr esp onding multiples of 1. Thus, the sp ectr al seq ue nc e colla ps es to E 13 in this ca se. 4. More obser v a tions about the symplectic V erlinde algebras and their completions In this section, let V ( m, n ) denote the S p ( n ) V erlinde a lgebra of level m . If τ denotes the coho mologica l twisting asso cia ted to this level, then (60) K i τ ( LB S p ( n )) ∼ = V ( m, n ) ∧ I when i ≡ n mo d 2 w he r e I is the augmentation ideal of R S p ( n ). F or i ≡ n + 1 mo d 2, the left ha nd s ide of (60 ) is 0. V ( m, n ) is a quotient of the repre s ent ation ring RS p ( n ) b y the ideal gener ated by the ir reducible represe ntations of level m − n . Explicitly , let x 1 be the deﬁnining representation of S p ( n ) of dimension 2 n . Then for n ≥ k ≥ 2, there is a natural contraction (61) Λ k ( x 1 ) → Λ k − 2 ( x 2 ) using the symplectic form. The map (61) is o nto, and its kernel x k is an irreducible representation of S p ( n ). x 1 , ..., x n are pr ecisely the irreducible repr esentations o f level 1 . A descr iption of irreducible representations o f S p ( n ) o f level q is g iven as p oly - nomials in the v ariable s x 1 , ..., x n in [13], Pro p. 24.2 4. Let A = ( a 0 , a 1 , ... ) be the seq uence 1 , x 1 , x 2 , ..., x n , 0 , − x n , ..., − x 1 , 1 . The unsp eciﬁed v a lues o f a i are deﬁned to b e 0. Then deﬁne the se quenc e A b ent at i as the sequence a i , a i +1 + a i − 1 , a i +2 + a i − 2 , ... . Then irreducible re pr esentations of S p ( n ) of level q cor resp ond to Y oung dia grams with exactly q columns and at most n rows. Le t the lengths of the columns b e µ 1 ≥ µ 2 ≥ ...µ n (to reca ll, a Y oung dia g ram is precisely such seq uence of num b ers where µ 1 ≤ n ). Then the co rresp onding irreducible repres en tation is, in R S p ( n ), the determinant of the matrix whos e i ’th row is given by the ﬁrst q ter ms of the sequence A b ent at µ i − i + 1. THE SYMPLE CTIC VERLINDE ALGEBRAS 15 In pr inciple, the ab ov e descriptio n turns all algebra ic questions a bo ut the S p ( n )- V er linde alge bra in to problems of commutativ e algebr a. How ever, from this de- scription, it is not alwa ys easy to see what is happ ening, a nd for ge ne r al m, n , the algebra V ( m, n ) and its completion are not completely understo o d. F or example, there is a conjecture of Gepner [15] that the V erlinde a lgebra is a global complete int ersection ring . Cummins [9] exhibited a ‘level-rank’ duality isomorphism (62) V ( m, n ) ∼ = V ( m, m − n − 1) . The map (62) in terchanges rows and columns in Y oung diagrams, so it sends x i to the level i irreducible repre sentation with µ i = ... = µ 1 = 1. Completion of the S p ( n )- V er linde a lgebra at the augment ation idea l of RS p ( n ) do es not preserve the lev e l-rank. In fact, we saw an exa mple in Prop osition 1 and Theorem 6 ab ov e. Mor e generally , let d ( m, n ) b e the gr eatest common diviso r of the dimensio ns of the irreducible representations of S p ( n ) of level m − n . Prop ositio n 8 . We have (63) d ( m, n ) = ± g cd { − 1 P j = − m  2 j + 2( i − 1) 2( i − 1)  | 1 ≤ i ≤ n } . (The right hand side is the numb er c alculate d by C.Dougla s [10] .) Pro of: Denote, for the moment, the num b er on the right hand side of (63) b y e . Then the twisted K -theor y Serre sp ectral sequence a sso ciated with the ﬁbration S p ( n ) → L B S p ( n ) → B S p ( n ) , along with the calculation [10], and the fact that the ﬁltra tion on K 0 B S p ( n ) asso- ciated with the At iyah-Hirzebruch sp ectra l sequence is the ﬁltration by p ow ers of the a ugmentation ideal, shows that (64) E 0 I ( R ( S p ( n )) V ( m, n ) is a Z /e -mo dule , so in other words d | e, since the 0-slice of the ﬁltration of the V er linde algebra by I ( RS ( n )) is obtained by equating e ach x i to its dimension, which g ives Z /d . On the other hand, K 0 τ ( G ) is a mo dule o ver K G, 0 τ ( G ) by restriction, which is a map of rings, while the a ugmentation ideal of R ( G ) maps to 0 , by considering the restriction K G, 0 ( ∗ ) → K 0 ( ∗ ), which is the augmentation. this implies e | d.  As remarked ab ove, the completion of V ( m, n ) is a dditively a direct sum of a certain num ber of copies of Z p ov er pr imes p which divide the nu mber d ( m, n ). Observe that by (63), (65) d ( m, n + 1) | d ( m, n ) . By Nak ay a ma’s lemma, the n umber of c o pies of Z p is the same as the num b er o f copies of Z /p in the completion of V ( m, n ) /p . If w e deno te b y u i the element x i min us its dimension, then this is the same as taking the quotient of the p ow er series 16 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN ring Z /p [[ u 1 , ..., u n ]] by the ideal J m − n generated by r epresentations o f level m − n . Since this ideal has dimension 0, some pow er s ( u i ) N m ust be in the ideal J m − n . Since V ( m, n ) /p itself how e ver is a ﬁnite-dimensiona l Z /p -vector spa ce, N can b e taken as its dimension, which is  m − 1 n  . Using Maple, one can compute the Gr¨ obner basis of the ideal ge nerated b y the level m − n r epresentations and ( u i ) N . This was done by J.T.Levin [20] in a num b er of examples. Even tually , we detected a pattern, which allow ed us to compute the c ompletion of the symplectic V er linde algebra in general, as well as the num b er d ( m, n ). The results a re contained in the following tw o theor ems. Theorem 9. F or n < m − 1 , the c ompletion of V ( m, n ) at the au gmen t ation ide al of RS p ( n ) is additively isomorp hic to a sum of (66)  δ ( p, m ) n  c opies of Z p over al l primes p . Pro of: When all s ymmetric p olynomials of a ﬁnite collection of algebra ic int egers hav e p ositive p -v alua tion, so do es each of them. Consider the maximal tor us in S p ( n ) which is given b y em b edding the pro duct of n copies of a c ho s en maximal torus of S p (1 ) via the standard embedding (67) S p (1) × ... × S p (1) ⊂ S p ( n ) . If we cho ose a genera ting weigh t t o f S p (1), (67) g ives g enerating weigh ts t 1 , ..., t n of S p ( n ). Now the Grothendieck group o f level 1 r epresentations of S p ( n ) is eas ily seen to hav e basis consisting of the e lemen tary symmetric p o ly nomials σ 1 , ..., σ n in (68) t i + t − 1 i − 2 , i = 1 , ..., n. By the r e sults of F reed-Hopkins- T eleman, the V erlinde alg ebra, when extended to a large enough r ing of Witt vectors W , injects, with a ﬁnite cokernel, int o a pro duct of r ings where in eac h individual factor, w e quo tient out b y a rela tion setting σ i equal to the i ’th symmetric p olynomia l in the num ber s N 1 , ..., N n obtained from (68) by setting (69) t i = ζ j i 2 m , 1 ≤ j 1 < ... < j n ≤ m − 1 . Since the σ i ’s generate the aug men tation ideal, the p -primary comp onent of the completion o f each of the factors is W if (70) All the num b ers N 1 , ..., N n hav e p ositive p -v aluation and 0 otherwise . Now by the ab ove remarks, (70) o ccurs if and only if a ll the num b ers obtained by plugging in (69 ) into (68 ) have p os itive p -v aluation. Now by recalling the arg ument in the pro of of P rop osition 1 , this o ccur s if and only if each ζ j i 2 m is a p j ’th r o ot of unit y for some j . The num ber of such combinations is (66).  THE SYMPLE CTIC VERLINDE ALGEBRAS 17 Note that in particular it fo llows that the p -co mpo nent of the completio n o f V ( m, n ) is iso morphic to the p -c omp o nent of V ( m, n ) if and only if p = 2 and m = 2 r for so me r . W e also have a more explicit ev aluatio n of d ( m, n ). Theorem 10. We have (71) d ( m, n ) = n/ g cd ( n, K ) wher e for e ach prime p , K is divisible by the lar gest p ower of p such that δ ( p, K ) < n . Pro of: Despite the fact that Theor em 9 implies a part of Theorem 10 (namely , it detects when p | d ( m, n )), we do not have a pr o of a long the same lines at this p oint. Instead, we need to app eal to P rop osition 8 . Let (72) S ( m, i ) =  2 m − 1 1  +  2 m − 3 i  + ... +  1 i  Then by (63), (73) d ( m, n ) = g cd { S ( m, 0) , S ( m, 2) , ..., S ( m, 2 ( n − 1 )) } . Compute (74) P i ≥ 0 S ( m, i ) x i = (1 + x ) 2 m − 1 + (1 + x ) 2 m − 3 + ... + (1 + x ) = (1 + x ) (1+ x ) 2 m − 1 (1+ x ) 2 − 1 = (1+ x x ( x +2) ((1 + x ) 2 m − 1) . W e s ee that the ro ots are (75) − 1 , ζ k 2 m − 1 , k = 1 , ..., 2 m − 1 , k 6 = m. Now w e will distinguish t wo cas e s . The ﬁrst case is p = 2. Then consider the po lynomial (76) P ℓ = 2 ℓ − 1 − 1 Q j =0 ( x − ( ζ 2 j +1 2 ℓ − 1)) . The co eﬃcients o f (76) (with the exception of the leading co eﬃcient) a re divis ible by 2. W e have deg ( P ℓ ) = 2 ℓ − 1 . The p olyno mials P ℓ hav e no common ro o ts. F ur thermore, the p olynomia l P ℓ divides (74) precisely when 1 ≤ ℓ ≤ N where 2 N || m . Now S ( m, 2( i − 1)) is a symmetric p olyno mial of degr ee 2 m + 1 − 2( i − 1) in the ro ots (75). Decomp os e S ( m, 2( i − 1)) as a p olynomial in the co e ﬃcient s of P 1 , ..., P N . O bserve that if (77) i < 2 ℓ , then (78) 2 m + 1 − 2( i − 1) > (2 m + 1 ) − 2 1 − ... − 2 s , so each mo nomial of S ( m, 2( i − 1)) conside r ed as a p oly no mial in the ro o ts of (74) contains ro ots of a ll the poly nomials P ℓ except, at most, s − 1 of them. It then follows that S ( m, 2( i − 1)) is divisible by 2 n − ℓ +1 . On the other hand, if i = 2 s , we see that (78) turns into a n equality , so there e x ists precis ely one mono mial of S ( m, 2( i − 1)) considered as a po lynomial in the ro ots of (74) which do es not 18 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN contain ro ots of s of the polyno mials P ℓ (and contains all the other ro ots). Since S ( m, 0) = m , w e conclude then that 2 N − ℓ || S ( m, 2( i − 1)). This is what w e were aiming to prove. Let us now cons ide r the case p > 2 . In this case, w e m ust consider the p olyno- mials (79) Q ℓ = Y { ( x − ( ζ j p ℓ − 1))( x − ( ζ − 1 p ℓ − 1)) | j = 1 , ..., p ℓ − 1 , j not divisible by p ℓ − 1 } . Then deg ( Q ℓ ) = ( p − 1 ) p ℓ − 1 , the p olynomials Q ℓ hav e no common ro o ts and Q 1 , ..., Q N divide (74) wher e p N || m . O bserve that when (80) p s − 1 2 < i, then (81) 2 m + 1 − 2( i − 1) > (2 m + 1) − ( p − 1) − p ( p − 1 ) − ...p s − 1 ( p − 1) , so aga in each mono mial of S ( m, 2( i − 1 )), considere d a s a p olynomial in the r o ots of (74), will co nt ain r o ots of a ll the Q ℓ ’s, except at most s − 1 of them. Therefore, S ( m, 2( i − 1)) is divisible by p N − s +1 . On the other hand, when (80) tur ns into an equality , (81) turns into a n equality , and therefore there is pr ecisely one monomial in S ( m, 2( i − 1)) cons idered as a po lynomial in the ro ots of (74) whic h doe s not contain the ro ots of precisely s of the polynomia ls Q ℓ (and contains all the other roots of (74)). Conse q uent ly , we can co nclude that p N − s || S ( m, 2( i − 1)), as needed.  Let us collect one mo re result, which will come to use in the con text of the next section. Recall that the V erlinde alg e bra is a Poincar´ e (=clo sed commutativ e F r ob enius) ring where the aug ment ation ǫ is deﬁned by ǫ (1) = 1, and ǫ ( a ) = 0 if a is a label diﬀerent from 1. The o nly axio m of a Poincar´ e ring V (other than commutativit y) is that the map (82) M : V ⊗ V → Z deﬁned by ǫ ( ab ) for v ariables a, b ∈ V deﬁne an isomor phism (83) V ∼ = H om ( V , Z ) . One then has an inv erse of (83), which ca n b e interpreted as a map (84) N : Z → V ⊗ V . The comp ositio n (8 4) with the triple pro duct is the “1- lo op transla tio n op era tor”, which we denote by T . Theorem 11. F or every m ≥ n + 2 , (85) det ( T ) 6 = 0 . In V ( m, 1) , we have (86) det ( T ) = ( − 2 ) m − 1 m m − 3 . Pro of: By the V erlinde conjecture (which is known to b e true fo r S p ( n )), the C - Poincar´ e a lgebra V ( m, n ) ⊗ C is isomo rphic to a pro duct of 1-dimensiona l algebr as, which are then automatica lly Poincar´ e algebr as, which implies that T ⊗ C is always an inv ertible matrix, which implies (85). T o prov e (86), w e recall the formula (3). THE SYMPLE CTIC VERLINDE ALGEBRAS 19 Let ζ be the primitive 2 m ’th ro ot of unity . In the m − 1 direct facto r s o f V ( m, 1), we will then hav e (87) x = ζ j + ζ − j , j = 1 , ..., m − 1 . The lab els ar e (88) 1 , S y m 1 ( x ) , ....S y m m − 2 ( x ) , and they are self-co n tragre dient, so the restriction T i of T to the i ’th summand (87), we hav e (89) T i = m − 2 P j =0 S y m j ( ζ i + ζ − i ) 2 . Using the well known formula (90) S y m j (2 x ) = ( x + √ x 2 − 1) j +1 − ( x − √ x 2 − 1) j +1 2 √ x 2 − 1 , we g e t (91) S y m j ( ζ i + ζ − i ) = ζ i ( j +1) − ζ − i ( j +1) ζ i − ζ − i , which gives (92) T i = − 2 m ( ζ i − ζ − i ) 2 . The determinant of T is then the pro duct of the num b ers (88) ov er i = 1 , ..., m − 1, which is (86).  It is worth noting that computer calculations using Maple sugges t the conjecture (93) de t ( T ) = 2 ( m − 1) 0 @ m − 3 n − 2 1 A m ( m − 3) 0 @ m − 3 n − 2 1 A . W e s hall prove here a weak er statement. Theorem 12. When for a prime p , P | d ( m, n ) , m > 3 , then p | det ( T ) in V ( m, n ) . Pro of: Let ζ be a pr imitive 2 m ’th r o ot of unity . Then by [14], V ( m, n ) ⊗ Q splits as a product of Poincare algebra s V I where I = (1 ≤ i 1 < ..., i n < m ) a nd V I is the quotient o f V ( m, n ) ⊗ Q by the idea l g e nerated b y x i − α i , i = 1 , ..., n where x i are the lev el 1 irr educible repr esentations, and α i are the n um ber s obtained b y expressing i as a p olynomia l in the standard weigh ts t i , a nd ev alua ting t j = ζ i j . (Here t j corres p o nd to choos ing a maximal tor us T in S p (1) and then T n ⊂ S p (1) n ⊂ S p ( n ) where the la tter is the standard embedding.) Now V I as an Q -algebr a, is isomor phic to Q . T o sp ecify its structure as a Poincare algebra, one must ev aluate (94) e I = ǫ (1) where ǫ is the augmentation. On the other ha n, in a 1 -dimensional Poincar e algebr a, it is easy to chec k that (95) T = 1 /ǫ (1) , 20 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN So (96) T I = 1 /e I where T I is the T -oper ator on V I . Now let us change from Q to Q p . Then one als o has the for m ula T = P k ∈ K x k x ∗ k where x k , k ∈ K , are the irreducible represe n tations of representation level ≤ m − n . This shows that, if we denote by v the p -v alua tion, then v ( T I ) ≥ 0 , so , by (96), we have (97) v ( e I ) ≤ 0 . W e nee d to show that when (98) p | d ( m, n ) , the ineq uality (97) is sharp for a t least one I . Assume ther e fo re (98), a nd that we hav e (99) v ( e I ) = 0 for a ll I . Recall now that the augmentation ǫ satisﬁes (100) ǫ (1) = 1 , ǫ ( x k ) = 0 , x k 6 = 1 , k ∈ K . Denote by x k,I the num ber obtained from k k by plugging in ζ i j for t j j = 1 , ..., n (using the W eyl c har a cter formula for x k ). Then B = ( x k , I ) is a s quare matrix, and (10 0), (99) signify that the eq uation (101) B x = (1 , 0 , ..., 0 ) T (the right ha nd side has 1 in k ’th p osition wher e x k = 1, and 0 elsewhere ) has a solution in Z p (the s olution b eing ( e I ) T ). But now note that the “twisted augmen- tation” ǫ k given b y (102) ǫ k ( x k ) = 1 , ǫ k ( x ℓ ) = 0 , ℓ 6 = k is given simply by (103) ǫ k ( x ) = ǫ ( x · x ∗ k ) . Therefore, al l equa tions (104) B x = (0 , ...., 0 , 1 , 0 , ..., 0) T where 1 is in the k ’th p o s ition for any k ∈ K , hav e a solution in Z p . Therefor e, B is an inv er tible matrix, and (105) v detB = 0 . But now m > 3 , so 1 + 2 m p < m (f or p > 2 ). Now consider I = ( i 1 < i 2 ... < i n ) where i 1 = 1 , i j 6 = 1 + 2 m p for a n y j = 1 , ..., n, THE SYMPLE CTIC VERLINDE ALGEBRAS 21 and let J b e o bta ined from I b y repla cing i 1 with 1 + 2 m p . Since v ( ζ − ζ 1+2 m/p ) > 0 , the I ’th and J ’t column of B are congr uent mo dulo v > 0, a nd hence v ( det ( B )) > 0 , which is a contradiction. F or p = 2 , repla ce 2 m p by m 2 = 2 m 4 .  5. String topology opera tions in twisted K -theor y: the product In [16], Go din deﬁnes a family o f string top olog y o p e rations on H ∗ ( LM ), pa- rameterized by the homology H ∗ (Γ g,n ) o f mapping class groups. This e xtended Chas-Sulliv an’s pro of in [4] that H ∗ ( LM ) is a Batalin-Vilko visky algebra . It seems likely that analogues of these ope r ations are present in the t wisted K - homology K τ ∗ ( LM ), for suitable c hoices of τ . In this se ction we fo cus on the most basic op eration – the lo op pro duct – a nd co mpute it for K τ ∗ ( L H P n ). 5.1. Basic recoll ections. Let X b e a top olo g ical space and τ ∈ H 3 ( X ) a twisting. Recall that τ deﬁnes a bundle E τ = ( E i ) i ∈ Z of sp ectra over X with ﬁbre the K - theory sp ectrum. The twisted K -homolog y is K τ n ( X ) = π n ( E τ /X ) = lim − → i π i + n ( E i /X ) where X is regar ded as a subspac e of E i via a section. F uncto riality is not as straightforward as for un twisted theorie s . F or a n y map f : Y → X , there is a pullback bundle E f ∗ ( τ ) ov er Y , equipped with a map to E τ cov er ing f . Consequently ther e is an induced map f ∗ : K f ∗ ( τ ) n ( Y ) → K τ n ( X ) Cross pro ducts also require so me care . If σ ∈ H 3 ( Z ), consider the element ( τ , σ ) := p ∗ 1 ( τ ) + p ∗ 2 ( σ ) ∈ H 3 ( X × Z ) where p i are pro jections o nt o X and Z . This deﬁnes a bundle of K -theory sp ectra E ( τ ,σ ) ov er X × Z . The smash pr o duct E τ ∧ E σ also deﬁnes a bundle of s p ectr a ov er X × Z ; in this case the ﬁbr e is K ∧ K . Multiplication in this r ing sp ectrum gives a ma p E τ ∧ E σ → E ( τ ,σ ) , which in turn deﬁnes an exter ior cro ss pro duct × : K τ n ( X ) ⊗ K σ m ( Z ) → K ( τ ,σ ) m + n ( X × Z ) Now assume X is a homo to p y as s o ciative, ho motopy unital H -spa ce with mul- tiplication µ : X × X → X . Comp osing the external cross pro duct with µ ∗ gives the following: Lemma 13. If τ is primitive; i.e., µ ∗ ( τ ) = ( τ , τ ) , then µ makes K τ ∗ ( X ) into a ring. 5.2. The lo o p pro duct. W e will need the following, adapted fro m, e.g., Section 3.6 of [14]: Prop ositio n 14. L et f : Y → X b e an emb e dding of ﬁn ite c o dimension, with K -orientable normal bund le N of dimension d . Ther e is an umkehr map f ! : K τ n ( X ) → K f ∗ ( τ ) n − d ( Y ) for any τ ∈ H 3 ( X ) . 22 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN As usual, this is obtained from a P ontrjagin-Thom c o llapse and Tho m isomor- phism (which is just a shift des usp e ns ion since we are taking N to b e K - orientable). Putting this together with the work of Chas-Sulliv a n and Cohen-J o nes [4, 6], gives us a loop pro duct in twisted K -theor y . Namely , let M d be a clo sed smoo th manifold of dimension d , and write LM = M ap ( S 1 , M ) for the spa ce of piecewise smo oth maps S 1 → M . F o r simplicity , assume that M is 3-co nnected; a s a conseq uence, it is spin and hence K -or ientable. Let τ ′ ∈ H 4 ( M ); then connectivity ensures that there is a well-deﬁned tr ansgres s ed class τ ∈ H 3 (Ω M ). Through the Serre sp ectral sequence τ gives rise to a well-deﬁned cla ss (repres e n ted by 1 ⊗ τ ) which we will also denote τ ∈ H 3 ( LM ). Connectivity implies, further, that τ ∈ H 3 (Ω M ) is primitive. Theorem 15. Assume that M is 3 -c onne cte d. Then K τ ∗ + d ( LM ) is a u n ital, asso- ciative ring. Pr o of. As usual, we co nsider the following commutativ e diagram in which the low er left sq uare is car tesian: Ω M × Ω M   Ω M × Ω M   = o o µ / / Ω M   LM × L M ev × ev   LM × M LM ev ∞   ˜ ∆ o o concat / / LM ev   M × M M = / / ∆ o o M Primitivity of τ mea ns tha t µ ∗ ( τ ) = ( τ , τ ) on Ω M . Therefore concat ∗ ( τ ) = ˜ ∆ ∗ ( τ , τ ) on LM . The inclusion ˜ ∆ is of ﬁnite codimens ion with normal bundle N ∼ = ev ∗ ∞ ( T M ), which is K -orientable, by a s sumption. Therefore, w e may for m the compos ite m := concat ∗ ◦ ˜ ∆ ! ◦ × : K τ n ( LM ) ⊗ K τ m ( LM ) m / / ×   K τ n + m − d ( LM ) K ( τ ,τ ) n + m ( LM × LM ) ˜ ∆ ! / / K ˜ ∆ ∗ ( τ ,τ ) n + m − d ( LM × M LM ) concat ∗ O O Then the ar guments given in [6] in homology show that m is an as s o ciative and unital pro duct.  This mult iplication intertwines with the cup pr o duct in the unt wisted K -theo ry of M in the following wa y: ev : LM → M has a r ig ht in verse c : M → LM ; c ( p ) is the constant lo op a t p . Then c induces a ma p c ∗ : K c ∗ ( τ ) ∗ ( M ) → K τ ∗ ( LM ). How ever, since M is 3- connected, c ∗ ( τ ) = 0, so this is simply c ∗ : K ∗ ( M ) → K τ ∗ ( LM ) If we give the K ∗ ( M ) a ring str ucture via intersection theory (Poincar´ e dual to the cup pro duct), this is clear ly a r ing homo morphism: THE SYMPLE CTIC VERLINDE ALGEBRAS 23 Prop ositio n 1 6 . K τ ∗ ( LM ) is a mo dule over K ∗ ( M ) via the map c ∗ . Notice that unless τ = 0, ther e is no class τ ′ ∈ H 3 ( M ) with the pr op erty tha t ev ∗ ( τ ′ ) = τ . C o nsequently , ev do e s not induce a map ev ∗ : K τ ∗ ( LM ) → K ∗ ( M ) which splits the target oﬀ of the source (as is the case in the unt wisted setting). Indeed, we will see in examples that the source is often torsio n, while the targ et is often not. 5.3. A Cohen-Jones - Y an t yp e s p ectral sequence. In [7], C o hen-Jones- Y a n constructed a s p ectr al s equence c o nv erg ing to H ∗ + d ( LM ) as an algebra . Combining their arguments with the Atiy ah-Hir zebruch sp ectral s equence for twisted K - theory gives the following: Theorem 17 . L et M b e a 3 -c onne cte d, close d d -manifold, and cho ose τ ′ ∈ H 4 ( M ) with asso ciate d t ra nsgr esse d twisting τ ∈ H 3 ( LM ) . Ther e is a left half-p age sp e ct ra l se quenc e { E r p,q : − d ≤ p ≤ 0 } satisfying: (1) The diﬀer ent ials d r : E r p,q → E r p − r,q + r − 1 ar e derivations. (2) The s p e ctr al se quenc e c onver ges to K τ ∗ + d ( LM ) as an algebr a (wher e we us e the lo op pr o duct on Ω M ). (3) Its E 2 term is given by E 2 p,q := H − p ( M , K τ q (Ω M )) 5.4. An example: H P ℓ . Since H P ℓ is ev e n dimensio nal, there is no degree shift in the ring K τ 0 ( L H P ℓ ). W e let τ ′ ∈ H 4 ( H P ℓ ) ∼ = Z co rresp ond to m ∈ Z . Theorem 18. F or any m 6 = 0 , K τ ∗ ( L H P ℓ ) is c onc entr ate d in even de gr e es, and K τ 0 ( L H P ℓ ) , e qu ipp e d with the lo op pr o duct, is isomorphic to Y p | m Z p [ t, y ] / ( y ℓ +1 , σ m − 1 ( y ) − ( ℓ + 1) y ℓ t ) W e will prov e this using the spectr al sequence from the pr evious s e ction. First we need: Lemma 19. Ther e is a ring isomorphism K τ ∗ (Ω H P ℓ ) ∼ = ( K ∗ /m )[ t ] wher e | t | = 4 ℓ + 2 . Pro of: W e w ill use the twisted K -theory Ser re sp ectral sequence asso c ia ted with the ﬁbr ation (106) Ω S 4 ℓ +3 → Ω H P ℓ → S p (1) . The E 2 -term is (107) K ∗ Ω S 4 ℓ +3 ⊗ Λ[ x 3 ] . The ﬁrst factor is vertical, the seco nd is horizontal. The torsion on the vertical factor disapp ear s b ecause of co nnectivity . F urther, us ing the A tiyah-Hirzebruc h sp ectral s equence, (108) K ∗ Ω S 4 ℓ +3 = K ∗ ⊗ H ∗ (Ω S 4 ℓ +3 ) = K ∗ ⊗ Z { t q (4 ℓ +2) | q = 0 , 1 , 2 ... } ∧ . (The ? ∧ on the r ight hand side indicates that in K -co ho mology , w e hav e the pro duct-completion, i.e. a pro duct o f copies of Z r a ther than a direct sum. Actu- ally , (1 08) is the pr o duct-completion of a divided p ow er algebra . No extensions are 24 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN po ssible, since there is no torsion. Re g arding the diﬀerentials in (107), we k now that d 3 is multiplication by the twisting cla ss, w hich (b y our choice) is mx 3 . Thus, the E 4 = E ∞ term is a susp ension by x 3 of (109) K ∗ Ω S 4 ℓ +3 / ( m ) . Regarding extensions, ﬁrst note that the ele men t repres ent ed b y x 3 really is m - torsion, since our sp ectr a l s equence is a s pec tr al sequence of modules ov e r the A tiyah-Hirzebruc h sp ectral sequence for the twisted K -theor y of S p (1). Next, (10 7) is alwa ys a sp ectra l sequenc e of mo dules ov e r its unt wisted analog. Howev er, there d 3 = 0 by the s ame argument, a nd further the t q (4 ℓ +2) ’s are per manent cycles by ﬁltration considerations (there ar e no elemen ts in ﬁltra tion degrees > 3). Th us, the elemen t r e presented by t q (4 ℓ +2) x 3 is a product of the elemen ts represented by t q (4 ℓ +2) and x 3 in the unt wisted and twisted sp ectra l sequences resp ectively , and hence is m -to rsion. Additiv ely , the res ult in t wisted K -homolo gy follows by the Universal Co eﬃcient Theorem. Multiplica tiv ely , it follows from the fact that H ∗ (Ω S 4 ℓ +3 ) = Z [ t ].  Pr o of of The or em 18 The pr evious le mma implies that the s pe c tr al seq uenc e for K τ ∗ ( L H P ℓ ) has E 2 - term given by (110) E 2 ∗ , ∗ = K ∗ [ t, y ] / ( m, y ℓ +1 ) where y has ﬁltratio n deg ree − 4. The spectra l sequence collaps es at E 2 since it is concentrated in even degrees. W e ther efore know that the ring structure is given by (111) K τ 0 ( L H P ℓ ) = Z [ y , t ] / ( y ℓ +1 , σ m − 1 ( y ) − tp ( y , t )) for so me p olynomial p ( y , t ), since, by P rop osition 2 7, the map ˜ h : L H P ℓ → Y ( ℓ, 1) induces a ring map in K τ ∗ . Now recall (say , [24]) the ordinar y homology Serre sp ectral s equence o f the ﬁbration Ω H P ℓ → L H P ℓ → H P ℓ . The only diﬀerentials ar e d ( t j y ℓ u ) = ( ℓ + 1 ) t j +1 where y i denotes the gener ator of H 4 i ( H P ℓ ). Therefore, w e can conclude tha t Σ − 4 ℓ H ∗ ( L H P ℓ , Z ), with its string top olog y multip lication, is given b y (112) Z [ y , t, v ] / ( y ℓ +1 , v 2 , v y ℓ , ( ℓ + 1) ty ℓ ) where dim ( y ) = − 4, dim ( v ) = − 1, dim ( t ) = 4 ℓ + 2 . Now applying the twisted K -homology A tiyah-Hirzebruch sp ectral sequence to (112), we get the twisting diﬀerentials (113) d 3 ( v y i ) = my i +1 . W e s ee that the o dd-dimensiona l subgroup of E 4 is generated by (114) q i = ℓ + 1 g cd ( ℓ + 1 , m ) t i y ℓ − 1 , i > 0 . THE SYMPLE CTIC VERLINDE ALGEBRAS 25 Let us consider the case of i = 1 in (114). The lesser ﬁltration degree par t of the sp ectral s equence is (115) Z /m { y , y 2 , ..., y ℓ } ⊕ Z { 1 } . By mapping in to the t wisted K - homology AHSS for Y ( ℓ , 1), we know that no element of (116) Z /m { 1 , y , y 2 , ..., y ℓ } can b e the targ et of a diﬀerential. W e conc lude therefore that (117) d 5 ( q 1 ) = N . 1 , N 6 = 0 . for some num b er N . Additionally , r e calling the elements (116) are not targets of diﬀerentials, we see that we m ust hav e (118) m | N . In fact, by the m ultiplicative str ucture, the diﬀerential (117) remains v alid when we multiply b y a p ow er of the p ermanent cycle t , so we see that the AHSS co llapses to E 6 . W e conc lude that we must hav e (119) ( ℓ + 1 ) ty ℓ = q ( y ) for some p olynomia l q ( y ). This mea ns that the poly nomial ( ℓ + 1) ty ℓ − q ( y ) must belo ng to the ideal generated b y y ℓ +1 and σ m − 1 ( y ) + tp ( y , t ). By reducing p ( y , t ) to degr ee ≤ ℓ in the y v ar iable, this clearly implies (120) p ( y , t ) | ( ℓ + 1 ) y ℓ . On the o ther ha nd, by our computation of the AHSS, the only divisors of ( ℓ + 1) y ℓ which ca n hav e low er ﬁltration degree a re multiples o f (121) g cd ( m, ℓ + 1) y ℓ . But now note that a ny such p olyno mial is congruent to Z /m -unit times ( ℓ + 1) y ℓ mo dulo the re la tion my ℓ , whic h will be v alid once we know p ( y , t ) is divisible b y (121). Finally , we ma y change basis b y m ultplying t by a Z /m - unit, making the unit equa l to 1.  Comment: One now sees that N = m 2 /g cd ( m, ℓ + 1). W e can now determine the precise additive structure of K τ 0 ( L H P ℓ ) ( p ) . This is essentially a standard exerc ise in abelia n extensions o f p -groups. Despite so me simpliﬁcations co ming fr o m the ring structure, there are many even tualities, and we ﬁnd it easies t to use a g e ometric pattern to represe nt the answer. Let us , ﬁr st, represent in this way the additiv e structure of K τ 0 ( Y ( ℓ, 1) ( p ) , as calculated in Section 1. Let d be ( p − 1 ) / 2 if p > 2 and 1 if p = 2 . Imagine a table T with k rows indexed by 0 ≤ i < k and ℓ + 1 columns indexed by 0 ≤ j ≤ ℓ . The ﬁeld ( i, j ) represents the element p i y j in the AHSS. The extensions ar e determined by paths (a step in the path represents m ultiplication by p ). The paths lo ok as follows: w e sta rt from a ﬁeld (0 , j ), where (122) p a − 1 p − 1 d ≤ j < p a +1 − 1 p − 1 d, 0 ≤ a < δ ( p, m ) . 26 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN F o r future reference, we will call j critic al if equality arises for j in (122). Now in each path, pro cee d one ﬁeld up a ll the wa y to row k − a − 1 : (0 , j ) , (1 , j ) , ... ( k − a − 1 , j ) . In the same row, now, pro ceed by p a +1 − 1 p − 1 d rows to the right, ( k − a − 1 , j ) , ( k − a − 1 , j + p a +1 − 1 p − 1 d ) , ... un til w e run out of co lumns in the table T . The lengths of the paths are now the exp onents of the p -p owers which a re o rders of the summands of K τ 0 ( Y ( ℓ, 1 ) ( p ) . Let us in tro duce so me terminolo g y: ﬁrst, introduce an order ing on the ﬁelds of T : ( i ′ , j ′ ) < ( i, j ) if ( i ′ , j ′ ) pr e c edes ( i, j ) on a pa th (the paths are disjoint so this creates no ambiguit y). Next, b y the divisibility of a ﬁeld ( i, j ) we shall mean the nu mber o f ﬁelds lesser that ( i, j ) in our order (clearly , this repr esents the exp onent of the p ow er of p by w hich the element the ﬁeld represents is divisible). T o describ e K τ 0 ( L H P ℓ ) ( p ) , imagine co pies T n of the table T , n = 0 , 1 , 2 , .... . W e will la be l the ( i, j )-ﬁeld in T n by ( n, i, j ), and it will r epresent the element p i a j t n in the asso cia ted gr aded abelia n group given by Theorem 3. W e will star t with the disjoint union o f the tables T n with their own paths. Ho wev er, now the paths (and the corresp onding o rder) will b e c orrected as follows: inductively in n , so me ﬁelds in T n (all in column ℓ ) will b e deleted fro m paths in T n and app ended to paths in T n − 1 . The proce dur e is this. Let p c || r , r = g cd ( m, ℓ + 1). F or a critical generator ( n, 0 , j ) (re c a ll (122)), let α j be the num b er of ﬁelds in its path in the highest row of T n the pa th reaches whic h are no t taken ov er by paths in T n − 1 , min us 1, plus c . Then app end to the path of ( n, 0 , j ), in increasing r ow o rder, all ﬁelds ( n + 1 , c + α + d, ℓ ) whose divisibility in T n +1 is ≤ α + d + k − a, d = 0 , 1 , ..., (recall (1 22) for the deﬁnition of a ), and which were not already a ppe nded to the paths of cr itical gener ators ( n, 0 , j ′ ) with j ′ < j . Note that only one set o f corrections arises for each n , so it is easy to determine the length of the corrected paths (which a re the orders o f the generators of direct summands of K τ 0 ( L H P ℓ ) ( p ) . How ever, note tha t a num b er of scenario s can occ ur, and we know of no s imple for mula describing the p ossible results in one step. Comment: It is interesting to note that by the additive extensions we ca lculated, K τ ∗ ( L H P ℓ ) with its string pro duct cannot b e a mo dule over K τ ∗ (Ω H P ℓ ) with its lo op pro duct structure. This exhibits the subtlet y of the structure s inv olved here. How- ever, as in Prop ositio n 3 .4 of [4], there is a ring map K τ ∗ ( LM ) → K τ ∗− dim M (Ω M ) given by in tersection with the subspace of base d lo o ps. Hence K τ ∗ (Ω H P ℓ ) is a mo dule ov er K τ ∗ ( L H P ℓ ). THE SYMPLE CTIC VERLINDE ALGEBRAS 27 6. The loop coproduct The twisted string K -theory ring K τ ∗− d ( LM ) a ls o admits a c o algebra ic structure by reversing the r ole of c onc at and ˜ ∆ (and the asso c ia ted Pon tr jagin-Thom maps) in the deﬁnition of the lo op pr o duct. 6.1. The copro duct. No tice that one may regard concat as an embedding of a ﬁnite-co dimension s ubma nifold, just as for ˜ ∆. Spec iﬁcally , there is a car tesian diagram LM × M LM concat / / ev ∞   LM ev 1 , − 1   M ∆ / / M × M where ev 1 , − 1 ev aluates a a lo op b oth at the ba sep oint 1 ∈ S 1 , as well as the midp oint − 1 ∈ S 1 . The pullback over the diagonal is the subspace of loops that agree at ± 1. By repa rameterizatio n, this space may b e iden tiﬁed with LM × M LM , and the inclus ion with the concatena tion of lo ops. Consequently concat admits a shr iek map in t w is ted K - theo ry , as well. Again, using the fact that concat ∗ ( τ ) = ˜ ∆ ∗ ( τ , τ ), we may consider the c omp o site K τ n + d ( LM ) concat ! / / K ˜ ∆ ∗ ( τ ,τ ) n ( LM × M LM ) ˜ ∆ ∗ / / K ( τ ,τ ) n ( LM × LM ) F o r many purp oses this map suﬃce s . How ever, to prop erly deﬁne a copro duct, we m ust as sume that the exterior cross pro duct map × is an isomorphism; the preferred wa y to do this (using the K¨ unneth Theorem) is to take our co eﬃcients for K - theo ry to be in a ﬁeld F . Deﬁne ν := × − 1 ◦ ˜ ∆ ∗ ◦ concat ! : K τ ∗ + d ( LM ; F ) → K τ ∗ ( LM ; F ) ⊗ K ∗ K τ ∗ ( LM ; F ) Theorem 20. The map ν deﬁnes t he structur e of a c o asso ciative c o algebr a on K τ ∗− d ( LM ; F ) . W e do not ex p ect ν to be counital. W er e that the case, K τ ∗ ( LM ; F ) w ould b e equipp e d with a nondegener ate trace, and thus ﬁnite dimensio nal. But, as we hav e seen in Theor em 2, this is no t g enerally the case . 6.2. The IHX relation. The comp os ite ν ◦ m of the lo o p pro duct and copro duct satisﬁes the same relations as in a F r ob enius alg ebra: Prop ositio n 21. If we write left or right multiplic ation of K τ ∗ ( LM ) on K τ ∗ ( LM ) ⊗ K τ ∗ ( LM ) by · , then ν ( xy ) = x · ν ( y ) = ν ( x ) · y Pr o of. Cons ide r the diagram: LM × L M LM × ( L M × M LM ) 1 × concat o o 1 × ˜ ∆ / / LM × L M × L M LM × M LM ˜ ∆ O O concat   LM × M LM × M LM 1 × concat o o 1 × ˜ ∆ / / ˜ ∆ × 1 O O concat × 1   ( LM × M LM ) × L M ˜ ∆ × 1 O O concat × 1   LM LM × M LM concat o o ∆ / / LM × L M 28 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN Going a round the top a nd r ight o f the diag r am, r eplacing w r ong wa y maps by the asso ciated umkehr ma p (i.e., “push-pull”) g ives x · ν ( y ). Push-pull alo ng the left and b ottom gives ν ( xy ). All of the squares except the low er le ft are ca rtesian, and the low er left is homotopy cartesian. Co nsequently the tw o push-pull s e quences are equal. A similar diag r am prov es that ν ( x · y ) = ν ( x ) · y .  6.3. Stabilization ov er gen us. The “1-lo o p transla tion op e rator” o f Theorem 11 is of cons iderable in terest. In the context o f K τ ∗ ( LM ), T is the other comp osite o f the co pro duct a nd pro duct: T = m ◦ ν : K τ ∗ ( LM ) → K τ ∗− 2 d ( LM ) In the full ﬁeld theor etic languag e o f string top olo g y (whic h, to o ur knowledge, has not b een co nstructed in the twisted K -theory setting), this is the op era tion induced by the class o f a p oint in B Γ 1 , 1+1 , the mo duli o f Riemann surfac e s of genus 1 with 1 inco ming and 1 outoing b ounda ry . Notice that, since we hav e assumed that M is K -orientable, the tangent bundle gives an element T M ∈ K 0 ( M ). Deﬁnition 22. L et E ∈ K 0 ( M ) b e the K - t he or etic Euler class of T M : E := d X k =0 ( − 1) k Λ k ( T M ) W e r ecall that K τ ∗ ( LM ) is a module over K 0 ( M ) via cap pro ducts along c onstant lo ops (P rop osition 1 6). Theorem 23. In K τ ∗ ( LM ) , T is given by c ap pr o duct with t he squar e of E : T = E 2 Pr o of. This is ess e n tially a consequence o f Atiy a h-Singer’s computation of shr iek maps in K -theory [1]. Recall that if i : Y → X is an embedding of ﬁnite co dimension with K - oriented nor mal bundle N , then the comp osite i ∗ i ! : K ∗ ( Y ) → K ∗ ( Y ) is given by multip lication by P k ( − 1) k Λ k ( N ) ∈ K 0 ( Y ). F urther, if N c a n b e written N ∼ = i ∗ N ′ , w he r e N ′ is a K -o riented bundle on X , then the reverse comp osite i ! i ∗ : K ∗ ( X ) → K ∗ ( X ) is m ultiplication by P k ( − 1) k Λ k ( N ′ ) ∈ K 0 ( X ). The same phenomena are tr ue in t wisted K -theor y via the mo dule s tructure over unt w is ted K - theory . W e now apply this to concat a nd ˜ ∆. T he norma l bundles to each of these embeddings a r e iso morphic to ev ∗ ∞ ( T M ) → LM × M LM where ev ∞ ev aluates at the co mmon p o int of the tw o lo o ps. In bo th cases, ev ∗ ∞ ( T M ) is a pullback: ev ∗ ∞ ( T M ) ∼ = concat ∗ ( ev ∗ ( T M )) and ev ∗ ∞ ( T M ) ∼ = ˜ ∆ ∗ ( ev ∗ ( T M ) × 0 ) W r ite E ′ = X k ( − 1) k Λ k ( ev ∗ ∞ ( T M )) ∈ K 0 ( LM × M LM ) THE SYMPLE CTIC VERLINDE ALGEBRAS 29 and notice that conca t ∗ ( ev ∗ E ) = E ′ . Then, for x ∈ K ∗ τ ( LM ), the dual ma p to T is T ∗ ( x ) = ν ∗ ( m ∗ ( x )) = concat ! ˜ ∆ ∗ ( ˜ ∆ ! concat ∗ ( x )) = concat ! ( E ′ · conc a t ∗ ( x )) = concat ! ( concat ∗ ( ev ∗ E · x )) = ( ev ∗ E ) 2 · x T ranslating this into K -homo lo gy turns the cup pro duct into a cap pro duct.  Unfortunately , how ever, we hav e the following Lemma 24. We always have E 2 = 0 . Pro of: E 2 is the Euler class o f the Whitney s um of tw o copies of the ta ngent bundle o f M . How ever, in a ny bundle o f dimension > dimM , the 0 -section can b e mov ed oﬀ itself b y genera l pos ition, and hence the Euler class is 0. (Clea rly , the same argument shows that the pro duct of the Euler class of the ta ng ent bundle with any Euler c lass o f any po sitive-dimensional vector bundle is 0 in a ny generalized cohomolog y theor y with res p ect to which M is or ient ed.)  Therefore, we hav e a Corollary 2 5. We have T = 0 .  6.4. The copro duct on K τ ∗ ( L H P ℓ ) . Despite these “nega tive results”, we can use the method o f Theorem 23 to obtain non-trivial information ab out the copr o d- uct. Let us consider the str ing copro duct for M = H P ℓ with co eﬃcients in Z /p . Concretely , we use the fact that the comp osition (123) K τ 0 ( LM × M LM , Z /p ) concat ∗ / / K τ 0 ( LM , Z /p ) concat ! / / K τ 0 ( LM × M LM , Z /p ) is given b y cap pro duct with the pullback of the Euler class of M . Using the notation of Theorem 18, 1 , t ∈ K τ 0 ( LM , Z /p ) are in the image of concat ♯ , s o we hav e (124) concat ! (1) = ( ℓ + 1) y ℓ , (125) c onc at ! ( t ) = ( ℓ + 1) y ℓ t (since the Euler clas s is ( ℓ + 1) y ℓ – reca ll a lso that the cup product in K ∗ ( M ) is Poincare dual to the string pro duct on consta nt lo o ps). F r om (124), using Poincare duality , we immediately conclude that (126) ν (1) = ( ℓ + 1) y ℓ ⊗ y ℓ . Regarding (1 25), recall that the right hand side is equal to σ m − 1 ( y ) . Consider the following condition: (127) c i = coe f f y ℓ ( σ m − 1 ( y )) is not divisible by p for i = ℓ , and is divisible by p for all i < ℓ . 30 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN If the condition (127) fails, then either p | ( σ m − 1 ( y ) , y ℓ +1 ) in which case the rela tion on K τ 0 ( LM , Z /p ) r eads ( ℓ + 1) y ℓ t = 0 , and (128) ν ( t ) = 0 , or ther e ex ists an i < ℓ such that p do es not div ide c i . Then, how ever, y ℓ − i σ m − 1 ( y ) = y ℓ mo d p , so y ℓ = 0, so (128) also o ccurs . If co ndition (12 7) is satisﬁed, then the r elation o f Theorem 18 with co eﬃcients in Z /p rea ds ( ℓ + 1) y ℓ t = c ℓ y ℓ , so the pro duct formula implies (129) ν ( t ) = c ℓ = y ℓ × y ℓ . Prop ositio n 2 6 . The c ondition (127) is e quivalent to (130) ℓ = δ ( p, m ) . Thus, if (130) holds, we have (129), else we have (128). Pro of: The ﬁr st statement follows from our calculation of K τ 0 ( Y ( ℓ, 1)) in Section 1. The seco nd statement is prov ed in the a b ove discussio n.  7. Completions and comp arison with the Gruher-Sal v a tore pr ospectra All of the manifolds that we hav e co nsidered as examples – s ymplectic Gra ssman- nians – come in families asso ciated to par ticular compact Lie groups . Indeed, our computation o f K τ ∗ ( L H P ℓ ) dep ends very muc h up on o ur co mputation of the com- pletion of the V erlinde algebra for S p (1 ). In this sectio n, we make that connection more explicit. There is a technical diﬃculty involv ed: while the manifolds we consider come in a seq uence, e.g., H P 1 → H P 2 → H P 3 → · · · , it is not the case that the induced ma ps on free lo op spaces pres erve str ing top olog y op erations (since intersection theo ry is c o nt rav ar iant, not cov ar iant). Nonetheless, one would like to study s tring top olo gy o pe rations on this system. 7.1. Adjoin t bundles. Gruher-Salv ator e achieve this g oal using an interesting construction that appr oximates the sequenc e L H P 1 → L H P 2 → L H P 3 → · · · , and furthermo re pr e serves their mo diﬁcation of the string top ology pr o duct. T o b e mo re speciﬁc , we let G denote a co mpact Lie g roup, and c ho ose a mo del for B G . W e take as given an inﬁnite sequence of even dimensional, closed, K - oriented manifolds B ℓ G , eq uipped with a commutativ e diagr am of embeddings · · · i ℓ − 1 / / B ℓ G j ℓ   i ℓ / / B ℓ +1 G j ℓ +1 z z u u u u u u u u u i ℓ +1 / / · · · B G THE SYMPLE CTIC VERLINDE ALGEBRAS 31 with the prop erty that the connectivity of j ℓ increases with ℓ . Consequen tly the induced ma p j : lim − → B ℓ G → B G is a homotop y equiv a lence. Let E ℓ G = j ∗ ℓ ( E G ) be the pullback of the universal principal G -bundle ov e r B G , with quotien t B ℓ G = E ℓ G/G . Lastly , deﬁne the adjoint bun d le Ad ( E ℓ G ) := G × G E ℓ G, where G ac ts on itself b y co njuga tion; this is a G -bundle ov er B ℓ G . F o r insta nce, if G = U (1), we may take E ℓ G = S 2 ℓ +1 ⊆ C ℓ +1 \ { 0 } and B ℓ G = C P ℓ In this case, since G is abelian, Ad ( E ℓ G ) = C P ℓ × U (1). Alternatively , if G = S p ( n ), we may pr o ceed via the manifolds des crib ed in sec- tions 1 and 2. That is , we may ta ke B ℓ ( S p ( n )) to b e the sy mplectic Grassmannia n B ℓ ( S p ( n )) = G S p ( ℓ, n ) of n - dimensional H -submo dules o f H ℓ + n . Then E ℓ ( S p ( n )) is the Stiefel-manifold W ( ℓ, n ) of symplectic ℓ -frames in H ℓ + n , a nd Ad ( E ℓ ( S p ( n )) = W ( ℓ, n ) × S p ( n ) S p ( n ) = Y ( ℓ, n ) which is a nontrivial bundle ov er G S p ( ℓ, n ). W e are in terested in Ad ( E ℓ G ) because it pr ovides an approximation to LB ℓ G and LB G . Consider the pair of ﬁbratio ns Ω( B ℓ G ) / / h   LB ℓ G / / ˜ h   B ℓ G =   G / / Ad ( E ℓ G ) / / B ℓ G where the vertical ma p h is the holonomy of a lo op; in homoto p y theo retic la nguage, it is given by the ﬁr st map in the ﬁbration sequence Ω( B ℓ G ) h / / G / / E ℓ G / / B ℓ G As ℓ tends to inﬁnity in this s equence, h tends to a ho motopy eq uiv a lence, since E ℓ G bec omes increasingly connected. Th us, as ℓ tends to inﬁnity , ˜ h : LB ℓ G → Ad ( E ℓ G ) tends to a homotopy equiv alence. 7.2. The pro duct on K τ ∗ ( Ad ( E ℓ G )) . Gruher-Sa lv atore deﬁne a ring m ultiplica- tion o n h ∗ ( Ad ( E ℓ G )) for any co homology theor y with res pe c t to which the vertical tangent bundle of Ad ( E G ) → B G is or ient ed. The multiplication is a n intermediary betw een the string top olog y pro duct on LB ℓ G a nd the fusion pr o duct o n G K ∗ τ ( G ) – it mixes int ersection theory on B ℓ G with multiplication in G . Gruher extends these res ults in [18] to show that in fact h ∗ ( Ad ( E ℓ G )) is a F ro benius algebra over h ∗ when h ∗ is a gra ded ﬁeld. One may in tr o duce t wistings to this story . Ass ume now that G is simply con- nected, so that every τ ∈ H 3 ( G ) is primitiv e. Denote also b y τ the t wistings in H 3 ( LB ℓ G ) and H 3 ( Ad ( E ℓ G )) asso ciated to τ by the Ser re spe c tral sequence. Then Gruher shows that K τ ∗ ( Ad ( E ℓ G )) a dmits the structure o f a r ing via the same m ultiplication a s in [17]. 32 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN Spec iﬁc a lly , one may deﬁne the pro duct on K τ ∗ ( Ad ( E ℓ G )) via a push- pull co n- struction. Ther e is a commutativ e diagr am of ﬁbr a tions G × G   G × G   = o o µ / / G   Ad ( E ℓ G ) × Ad ( E ℓ G )   Ad ( E ℓ G ) × B ℓ G Ad ( E ℓ G )   ˜ ∆ o o ˜ µ / / Ad ( E ℓ G )   B ℓ G × B ℓ G B ℓ G = / / ∆ o o B ℓ G Here, ˜ µ is ﬁbrewise multiplication in G . Then the pr o duct is deﬁned a s m := ˜ µ ∗ ◦ ˜ ∆ ! ◦ × . The proo f that t wistings b ehave appro priately is the same as in the construction of the lo op pro duct in section 5. Prop ositio n 2 7 . The map ˜ h ∗ : K τ ∗ ( LB ℓ G ) → K τ ∗ ( Ad ( E ℓ G )) is a ring homomorphism. Pr o of. In the diagram LB ℓ G × LB ℓ G ˜ h × ˜ h   LB ℓ G × B ℓ G LB ℓ G h ′   ˜ ∆ o o concat / / LB ℓ G ˜ h   Ad ( E ℓ G ) × Ad ( E ℓ G ) Ad ( E ℓ G ) × B ℓ G Ad ( E ℓ G ) ˜ ∆ o o ˜ µ / / Ad ( E ℓ G ) the right s quare homo topy commutes, since h may b e taken to b e an H -map. The left sq uare is in fact Cartesian, so ˜ h ∗ ( x · y ) = ˜ h ∗ concat ∗ ˜ ∆ ! ( x × y ) = ˜ µ ∗ h ′ ∗ ˜ ∆ ! ( x × y ) = ˜ µ ∗ ˜ ∆ ! ( ˜ h ∗ ( x ) × ˜ h ∗ ( y )) = ˜ h ∗ ( x ) · ˜ h ∗ ( y )  This result, co mbin ed with the high c o nnectivity o f ˜ h (for lar ge ℓ ) can b e taken as an indication that K τ ∗ ( Ad ( E ℓ G )) is an increasingly go o d approximation for the string topo logy multiplication o n K τ ∗ ( LB l G ) . 7.3. Limits. The inclusio ns E ℓ G → E ℓ +1 G a r e G -equiv aria nt, so induce inclu- sions Ad ( E ℓ G ) → Ad ( E ℓ +1 G ). A Pont rjagin-Tho m colla pse fo r this embedding (o r equiv alently , Poincar´ e duality in twisted K - theory) deﬁnes a map K τ ∗ ( Ad ( E ℓ +1 G )) → K τ ∗ ( Ad ( E ℓ G )) which do es not s hift degrees, since codim ( B ℓ G ⊆ B ℓ +1 G ) is ev en. It is a conse- quence of [1 7] that this is a r ing homomo r phism. So this giv e s r ise to an in verse system o f r ing s · · · → K τ ∗ ( Ad ( E ℓ +1 G )) → K τ ∗ ( Ad ( E ℓ G )) → · · · → K τ ∗ ( Ad ( E 0 G )) THE SYMPLE CTIC VERLINDE ALGEBRAS 33 Theorem 28. L et h ( G ) b e the dual Coxeter numb er of G . Ther e is a ring isomor- phism lim ← − K τ ∗ ( Ad ( E ℓ G )) ∼ = V ( τ − h ( G ) , G ) ∧ I wher e the left side is the inverse limit of t he string top olo gy ring structur es on K τ ∗ ( Ad ( E ℓ G )) , and the right side is the c ompletion of the V erlinde algebr a (with fusion pr o duct) at t he augmentation ide al. Pr o of. The inv ers e system of Pontrjagin-Thom co llapse ma ps · · · → Ad ( E ℓ +1 G ) − T B ℓ +1 G → Ad ( E ℓ G ) − T B ℓ G → · · · → Ad ( E 0 G ) − T B 0 G is Spanier -Whitehead dual to the direct system of inclusio ns · · · ⊇ Ad ( E ℓ +1 G ) ⊇ Ad ( E ℓ G ) ⊇ · · · ⊇ Ad ( E 0 G ) and so there is an is omorphism (131) lim ← − K τ ∗ ( Ad ( E ℓ G )) ∼ = lim ← − K −∗ τ ( Ad ( E ℓ G )) The main result of [18] is that for un twisted homolo gy theories, this duality throws the string top olo g y pr o duct onto the “fusion pro duct.” The same is true in the twisted setting: as we hav e seen, the multiplication o n K τ ∗ ( Ad ( E ℓ G )) is given by the formula m := ˜ µ ∗ ◦ ˜ ∆ ! ◦ × . This is evidently Poincar´ e dual to the pro duct p : K ∗ τ ( Ad ( E ℓ G )) ⊗ K ∗ τ ( Ad ( E ℓ G )) → K ∗ τ ( Ad ( E ℓ G )) deﬁned a s the comp osite p = ˜ µ ! ◦ ˜ ∆ ∗ ◦ × . Therefore (131) is a ring isomorphism. F ur thermore, since lim − → Ad ( E ℓ G ) = Ad ( E G ) = G × G E G , a lim 1 argument implies that lim ← − K ∗ τ ( Ad ( E ℓ G )) ∼ = K ∗ τ ( Ad ( E G )) ∼ = G K ∗ τ ( G × E G ) The last is the G -e q uiv a riant twisted K -theo ry of G × E G . By the twisted version of the A tiyah-Segal completion theorem [11, 21], this is isomorphic to G K ∗ τ ( G ) ∧ I . Combining thes e results gives a ring isomorphism lim ← − K τ ∗ ( Ad ( E ℓ G )) ∼ = G K ∗ τ ( G ) ∧ I where the m ultiplicatio n on the rig h t side of the isomo rphism is giv en by the G - equiv ariant transfer µ ! to the principal G -bundle µ : G × G → G . The main theorem in [14] then g ives the des ir ed is omorphism. That this isomorphism pres erves the ring s tr ucture is immediate, as we no te that [14] show that the fusio n pr o duct o n V ( τ − h ( G ) , G ) is carr ied to the pro duct o n G K ∗ τ ( G ) deﬁned as the tra nsfer to µ .  W e no te tha t a completion is a form o f inv erse limit; namely , V ( τ − h, G ) ∧ I = lim ← − V ( τ − h, G ) /I l W e ar e thu s lead to wonder if the isomorphism o f Theorem 28 is in fact re a lized o n a ge ometric level: Conjecture 29 . Ther e is an incr e asing function N : N → N and a c ol le ction of isomorphi sms f ℓ : K τ ∗ ( Ad ( E ℓ G )) ∼ = V ( τ − h, G ) /I N ( ℓ ) which ar e c oher en t acr oss the inverse system – that is, they induc e the isomorphism of The or em 28 up on p assage to t he limit. This is true in the case of G = S p (1) (wher e N ( ℓ ) = ℓ + 1), as evidenced by Theorem 18. 34 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN 7.4. The copro duct. Gruher also deﬁnes a copro duct on the twisted K - theo ry K τ ∗ ( Ad ( E ℓ G ); F ) with ﬁeld co eﬃcients; it is given by ν = × − 1 ◦ ˜ ∆ ∗ ◦ ˜ µ ! : K τ ∗ ( Ad ( E ℓ G ); F ) → K τ ∗ ( Ad ( E ℓ G ); F ) ⊗ K τ ∗ ( Ad ( E ℓ G ); F ) F o r unt wisted ho mology theories, this map is the Poincar ´ e dual to the pr o duct, and in fact for ms part of a F rob enius a lgebra str ucture. It is unclear whether the same is true in the twisted setting, since the c o nstant map Ad ( E ℓ G ) → pt do es no t induce a map in twisted K -homolog y . Nonetheless, the Pon trjagin-Thom map K τ ∗ ( Ad ( E ℓ +1 G )) → K τ ∗ ( Ad ( E ℓ G )) preserves the copr o duct; conse q uent ly the inv e r se limit lim ← − K τ ∗ ( Ad ( E ℓ G )) ∼ = V ( τ − h ( G ) , G ) ∧ I admits a copro duct induced by ν . It is not at all obvious whether this is r elated to the copro duct derived from the F ro be nius a lgebra structure on the V erlinde alge bra. How ever, it is not true that the map ˜ h ∗ : K τ ∗ ( LB ℓ G ; F ) → K τ ∗ ( Ad ( E ℓ G ); F ) is a homomorphism o f coa lgebras. O ne can try to mimic the pro of of Pr op osition 27, but the a rgument fails, since the right s q uare is not in fac t Cartesia n. Indeed, concat ! is given by intersection theory , whereas µ ! is given by a G -transfer . 8. Concluding remarks W e hav e examined several diﬀeren t ﬁeld theo ries in this pap er. The V er linde algebra V ( m, G ) is a Poincar´ e a lgebra, or equiv alently , a top ologica l quantum ﬁeld theory (TQFT). W e hav e constructed a pro duct and copro duct on the twisted string K -theory of a manifold K τ ∗ ( LM ). By v ir tue o f the IHX relation (Prop osition 2 1, these combine in such a wa y a s to give K τ ∗ ( LM ) the structure o f a “TQ FT without trace” or “p ositive b oundar y TQFT,” a s was done in the homologica l setting by [5]. The adjoint bundle K -theo ries K τ ∗ ( Ad ( E ℓ G )) se rve as a n imp e r fect bridge b etw ee n these, pres erving pro ducts, but not necessar ily copro ducts. Despite this co nnection, when it comes to higher -genus op era tio ns, V ( m, G ) and K τ ∗ ( LM ) display ma rkedly diﬀerent b ehavior, a s evidenced b y the v anishing of T in K τ ∗ ( LM )), and its r ational inv ertbility in V ( m, S p ( n )). It is natura l to ask for the reasons behind these diﬀerences. T o approach this question, consider yet another form of ﬁeld theo r y: Gromov-Witten theo ry . W e would like to think of the V erlinde algebra a s a twisted K -theor etic analo gue of Gromov-Witten theory for the stack [ ∗ /G ]. String top olo gy a nd Gromov-Witten theo ry share some ideas in their c onstruc- tion, at least on a schematic level. They b oth inv olve a push-pull diagra m of the form: (132) ( LX ) m ← M ap (Σ , X ) → ( LX ) n where Σ is a surface with m + n b oundar y c omp o nents, the ma ps are restrictions along b oundaries , and the v a rious function spaces ar e to b e int erpreted in the right categorie s. The shriek map in Gr omov-Witten theory applies a type o f intersection theory , using the deep fac t o f the existence of a virtual fundamental cla ss on the compact- iﬁcation of the moduli of maps Σ → X . In con trast, the one in string topo lo gy THE SYMPLE CTIC VERLINDE ALGEBRAS 35 applies a Beck er- Gottlieb type tra nsfer using the fairly stra ightforw a rd fact that the maps in (132) are ﬁbra tions with compa c t ﬁbre when X = B G . Said a nother wa y , in str ing top olo g y , the ﬁber s of the maps in (132) are compact, wherea s in Gromov- Witten theory , the spaces themselves a re compact (to the e yes of coho mology , at least). As hin ted in the Introduction, in this pap er, we were interested in K -theory in- formation. In conformal ﬁeld theory , an example of such information is the V erlinde algebra, or more pr e cisely mo dular functor o f a CFT. According to mo dern string theory , the physical asp ects of strings, such as br a nes or ev en ph ysical partition function, a re also based on K -theory . If we lo o k at homolog y with characteristic 0 co eﬃcients ins tead of K - theo ry , top olo gical quantum ﬁeld theo r ies should come out of the “ state space” of the co nfo r mal ﬁeld theory itself. In eﬀect, when N = (2 , 2) sup e rsymmetry is present, we can co nstruct s uch mo dels, na mely the A -mo del a nd the B -mo del. This a lso ﬁts into the Gromov-Witten picture: N = (2 , 2) sup ersym- metric confor mal ﬁeld theories are exp ected to a rise not from ordinary manifolds, but fro m Ca labi-Y au v a rieties. While a rigo rous direct constructio n o f s uc h mo del is not known, F an-Jar vis-Ruan [12] constructed the topo logical A -mo del, alo ng with TQ FT s tr ucture, and coupling to compactiﬁed gravit y , in the rela ted case of Landau-Ginzburg o rbifold, via applying Gromov-Witt en theo ry to the Witten equation. One ca n then a sk if one can somehow extend these methods to obtain K -theory ra ther than characteristic 0 homolog y infor mation. 9. Appendix: Some found a tions It is not the purp ose of the present pap er to discuss in detail the foundations of twisted K -theory . Many of the r esults needed here ca n b e read oﬀ from the approach o f [1 ]. More sy stemic a pproaches, needed for so me o f the mo re complicated assertions , us e the setup of parametric sp ectra, which is dev elop ed for example in [22], or [19] (the latter approa ch is simpler, but requir es some corr ections). The main po int is to note that the requir e d foundations exist, even tho ugh they ar e somewhat scattered throughout the liter ature. L et us recapitulate here so me key po int s. Let us work in the ca tegory of para metric K - mo dules ov er a space X ,and let us denote this categor y by K − mod/X . Then the pro jection p : X → ∗ (actually , the po int can b e repla ced by any spa ce) has a pullback map p ∗ : K − mod/ ∗ → K − mod X , which has a left adjoint p ♯ and a r ight adjo int p ∗ . The category K − mod/X has an in ternal (“ﬁber wise”) smash pro duct ∧ K/X and function spec tr um F K/X . Twistings τ are ob jects of K − mod/X which are inv ertible under ∧ K/X . Ther e ar e the usual “exp onential” adjunctions. Now let us co nsider the universal co eﬃcient theor em. It is better to dea l with K -mo dules than with co eﬃcients. The τ -twisted K - homology module is p ♯ τ , the τ -t wis ted K - c ohomology mo dule is p ∗ τ . W e have (133) F K/ ∗ ( p ♯ τ , K ) = p ∗ F K/ ∗ ( τ , p ∗ K ) = p ∗ F K/ ∗ ( τ , 1) = p ∗ F K/ ∗ (1 , τ − 1 ) = p ∗ τ − 1 . 36 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN So this is the usual universal co eﬃcient theo rem, the contribution o f the twisting is that it ge ts in verted. Note, howev er, that the complex conjugatio n automorphism of K - theory r everses the sign of twisting, so K -(co)-ho mo logy gr o ups with opp osite t wistings a re isomor phic. Let us no w discuss Poincare duality: When X is a closed (ﬁnite-dimensio nal) manifold, its K -dua lizing ob ject ω is K ﬁbe r wise smashed with its s table normal bundle (considered as a par ametric spectrum over X ). Up to suspension by the dimension d of X , ω is a t wisting. When ω = 1[ − d ], X is called K -orientable. (so when for exa mple H 3 ( X, Z ) = 0, X is K - orientable). Poincare duality sta tes that for a para metric K -mo dule ov er X , (134) p ∗ ? = p ♯ ( ω ∧ K/X ?) . So indeed, when the manifold is orientable, its K τ homology and cohomolog y is the same up to dimensions shift. Using these foundations, usua l results on K -theo ry extend immediately to the t wisted context. F or example, there is a twisted Serre (and hence At iyah-Hirzebruch) sp ectral s e quence con verging to twisted K - ho mology or cohomolo gy . Also, the ab ov e foundations can be ex tended to the eq uiv a riant ca se, in whic h ca se there is a completion theore m, asserting that for G compac t Lie, and a ﬁnite G -CW complex X , the non-equiv ariant t wisted K -co homology o f X × G E G is is o morphic to the completion o f the equiv aria n t twisted K -theory of X , completed at the aug men ta- tion ideal of R ( G ) = K ∗ G ( ∗ ). F o r a detailed pro of of this result, we re fer the rea der to [11, 21]. Let us make one more remark on notation. W e will be using b o th the induced map and the transfer for a closed inclus ion of manifo lds f : X → Y . Another form of duality then states that (135) f ! ≃ f ∗ . Here f ! is f ♯ comp osed with smashing with the spher e bundle which is the ﬁb erwise 1-p oint co mpa ctiﬁcation o f the nor mal bundle of X in Y (alter nately , take the co r- resp onding inv e rtible para metrized K -mo dule, a nd smash o ver K X ). Our in ter est in this is tha t we nee d to c o nsider the induced map and transfer map of a smo o th inclusion f . Fir st of all, for a twisting τ o n Y , if we denote by [? , ?] homotopy classes of maps of para metrized K -mo dules, a nd denote by 1 Y the trivia l K -mo dule on Y , and by τ a twisting on Y , clearly we hav e a map (136) f ∗ : [1 Y , τ ] → [1 X , f ∗ τ ] , as 1 X = f ∗ 1 Y . Assuming now (say) that the normal bundle of X in Y is K - orientable o f r eal co dimension k , we can pro duce a map in the other direction (137) f ! : [1 X , f ∗ τ ] → [1 Y , τ [ k ]] by taking a map 1 X = f ∗ 1 Y → f ∗ τ , taking the a djoin t 1 Y → f ∗ f ∗ 1 Y , us ing (135), and comp os ing with the counit o f the a djunction ? ∗ , ? ♯ . W e see now that the notation (136), (13 7) is not r e ally justiﬁed in terms of parametrized K - mo dules: in eﬀect, it is “ one lev el be low in terms of 2-ca teg ory theory”, and ﬁts mor e with the base change functor s applied to vector bundles which repres ent classes in the cohomology groups in volved in (136), (137). This THE SYMPLE CTIC VERLINDE ALGEBRAS 37 makes selecting notation for the corresp onding maps in co homology precar ious. As a compromise b etw e e n p oss ibly contradicting allusions, we use f ∗ for the induce d map in homology , and f ! for the transfer. Thus, f ∗ (in t w is ted K -homolo gy) is induced by the adjunction counit f ♯ f ∗ τ → τ , f ! (in ho mology) is induced by the adjunction unit τ → f ∗ f ∗ τ , together with (135). W e sho uld note that in this pap er, we a lso use inﬁnite ex tensions of these duality results, a llow ed by the work of Cohen and K lein [8]. References [1] Atiy ah, Michae l; Segal, Graeme. Twisted K -theory and cohomology . In Inspir e d by S. S. Chern , 5–43, Nank ai T racts Math., 11, W orld Sci. Publ., Hack ensac k, NJ, 2006. [2] Atiy ah, M. F.; Singer, I. M. The index of elliptic op erators. I. Ann. of Math. (2) , 87, 1968, 484–530. [3] Atiy ah, Michae l; Segal, Graeme. Twisted K -theory . Ukr. M at. Visn. 1 (2004), no. 3, 287–330; translation in Uk r. Math. Bul l. 1, (2004), no. 3, 291–334. [4] Chas, M.; Sulliv an D., String topology . pr e print: math.GT/991115 9 , 1999. [5] Cohen, Ralph L.; Godin, V´ eronique. A p olarized view of string top ology . In T op olo gy, ge ome- try and quantum ﬁe ld the ory , 127–154, L ondon Math. So c. L e ctur e Note Ser. , 308, Cambridge Univ. Press, Cambridge, 2004. [6] Cohen, Ralph L.; Jones, John D. S. A homotopy theoretic realization of stri ng topology . Math. Ann. 324 (2002), no. 4, 773–798. [7] Cohen, Ralph L.; Jones, John D. S.; Y an, Jun The lo op homology algebra of spheres and pro j ectiv e spaces. In Cate goric al de c omp osition techniques in algebra ic top olo g y (Isle of Skye, 2001) , 77–92, P r ogr. Math., 215, Birkh¨ auser, Basel, 2004. [8] Cohen, Ralph L., Klein, John R. Umkehr Maps, pr eprint , [9] Cummins, C.J.: S U ( n ) and S p (2 n ) WZW fusion rul es, J. Phys. A: Math. Gen. 24 (1991) 391-400 [10] Douglas, C hr istopher L. On the twisted K -homology of simple Lie groups. T op olo g y 45 (2006), no. 6, 955–988. [11] Dwye r, C. Twisted K-theory and completion. T o app ear. [12] F an,H., T yler, J.J. , Ruan, Y .: The Witten equation and its virtual fundamental cycle, [13] F ulton, Willi am; Harris, Joe. R epr esent ation the ory. A ﬁrst c ourse. In Gr aduate T exts in Mathematics , 129. Readings in M athematics. Spri nger-V erlag, New Y ork, 1991. xvi+551 pp. ISBN: 0- 387-97527-6; 0-387-97495-4 [14] F reed D.S.; Hopkins, M. J.; T eleman, C. Loop groups and t wi sted K-theory I. pr eprint: arXiv:0711.190 6 , 2007. [15] Gepner, D.: F usion rings and geometry , Comm. Math. Phys. 141 (1991), no. 2, 381-411 [16] Godin, V. Higher string top ology op erations. pr eprint: arXiv:0711.4859 , 2007. [17] Gruher, Kate; Salv atore, Paolo. Generalized string top ology operations. Pr o c. L ond. Math. So c. (3) 96, (2008), no. 1, 78–106. [18] Gruher, Kate. A duality betw een stri ng topology and the fusi on pro duct in equiv ari an t K - theory . Math. R es. L e tt. 14, (2007), no. 2, 303–313. [19] Hu, Po. Duality for smo oth fami lies in equiv ari an t stable homotop y theory . Ast?risque No. 285, (2003), v+108 pp. [20] J.T.Levin: On the Symplectic V erl inde Algebra and the function d ( m, n ), REU r e p ort , Univ. of M ich., 2008 [21] Lah tinen, A. The At iyah -Segal completion theorem for t wisted K-theory . pr eprint: arXiv:0809.127 3 , 2008. 38 I.KRIZ AND C.WE STERLAND, CONTRIBUTIONS BY J.T.LEVIN [22] May , J. P .; Sigurdsson, J. Par ametrize d homotop y the ory . In Mathematic al Surveys and Mono gr aphs , 132. American M athematica l So ciety , Pro vi dence, R I, 2006. x+441 pp. ISBN: 978-0-8218-3922-5; 0-8218-3922-5 [23] Tillmann, Ul rike. Higher gen us surface operad detects inﬁnite lo op spaces. Math. Ann. 317, (2000), no. 3, 613–628. [24] W esterland, Craig. D yer-Lashof oper ations in the stri ng top ology of spheres and pro jective spaces. Math. Z. 250, (2005), no. 3, 711–727 .

The symplectic Verlinde algebras and string K-theory

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment