Convolution symmetries of integrable hierarchies, matrix models and tau-functions

Convolution symmetries of integrab le hiera r chies, matrix mo dels and τ -functions ∗ J. Harnad 1 , 2 and A. Y u. Orlov 3 1 Centr e de r e cher ches math ´ ematiques, Universit´ e de Montr ´ eal C. P. 6128 , suc c. c entr e vil le, Montr ´ eal, Qu´ eb e c, Cana da H3C 3J7 e-mail: h arnad@crm.umont r e al.c a 2 Dep artment of Mathematics and St atistics, Conc or dia Univ ersity 1455 de Maisonneuve Blvd. W. Montr e al, Queb e c , Canada H3G 1M8 3 Nonline ar Wave Pr o c esses L ab or atory, Institute of Oc e anolo gy, 36 Nakhimovskii Pr osp e ct, Mosc ow 117851, Russia e-mail: o rlovs55@mail.r u, orlovs@o c e an.ru Abstract Generalized con v olution s ymmetries of inte grable h ierarc hies of K P and 2KP-T o da t yp e act diagonally on the Hilb ert sp ace H = L 2 ( S 1 ) in the standard monomial b asis. The indu ced transformations on the Hilb ert space Grassmann ian Gr H + ( H ) ma y b e view ed as symmetries of th ese hierarc hies, acting u p on the S ato -Segal-Wi lson τ -functions, and thereby generating new solutions of th e hierarc hies. The corresp onding transformations of the asso ciat ed fermionic F o c k sp ace are also d iagonal in th e standard orthonormal basis, la b eled by inte ger partitions. The Pl¨ uc ke r co ordinates of the image under the P l¨ uc k er map of the eleme n t W ∈ Gr H + ( H ) defining th e initial p oin t und er the commuting KP flows are the co efficien ts in the single and double S c hur fun ctio n expans ions of the asso ciated τ f unctions. These are therefore m ultiplied b y the eigen v alues of th e con volution action in the fermionic represent ation. Applying suc h transformations to standard matrix mo del inte grals, w e obtain new matrix mo dels of externally coupled t yp e wh ose partition fun ctions are th us also seen to b e KP or 2KP-T o da τ -functions. More general m u ltiple int egral representa tions of tau functions a re similarly obtained, as w ell as finite determinant al expr essions for them. ∗ W ork of J.H. supp orted by the Natural Sciences a nd Engineer ing Res earch Council of Canada (NSER C) a nd the F onds Qu´ ebe cois de la recherch e sur la na ture et les technologies (FQRNT). W or k of A.O. supp orted by RFBR g ran t 11-01-0 0 440-a and RAS Pr ogram “F undament al Metho ds in Nonlin- ear Physics”. 1 1 In tr o ducti o n: con v olutio n symmetrie s of τ -function s Solutions of inte grable hierarc hies of KP and 2 KP-T o da t yp e are determined b y their τ - functions [23, 24, 2 5]. Infinite sequences of suc h KP τ - functions { τ ( N , t ) } N ∈ Z , dep ending on the infinite set of comm uting flow parameters t = ( t 1 , t 2 , . . . ) and a n in teger lattice lab el N , ma y b e associated in a standard fashion [23, 24, 25], to elemen ts of a “ univ ersal phase” space, view ed as an infinite Grassm ann manifold or flag manif o ld. These satisfy the Hirota bilinear equations of t he KP hierar c h y and also, in certain cases (e.g. exp onen tial flo ws of matrix mo del integrals ind uced b y trace in v ariants), the e quations of the T o da lattice hierarc hy . The τ -functions may b e expanded as infinite series in a basis of Sc hur functions s λ ( t ), lab elled by in teger partitions λ = ( λ 1 ≥ λ 2 ≥ · · · ≥ 0) τ ( N , t ) = X λ π N ( λ ) s λ ( t ) . (1.1) In the a ppro ac h of Sat o and Segal-Wilson [23, 25], the co efficien ts π N ( λ ) are interpreted as Pl ¨ uc ker co ordinat es of the image P ( W ) of an elemen t W of a Hilb ert space Grassmannian Gr H + ( H ) under the Pl ¨ uc ker map P : Gr H + ( H ) → P ( F ) (1.2) in to the pro jectiv ation of the semi-infinite exterior space F := Λ H (the F ermionic F o c k space). In [25], the Hilb ert pace H is c hosen as the square in tegrable f unctions L 2 ( S 1 ) on the unit circle in the complex z -plane and the elemen t s of Gr H + ( H ) a re subspace s of H = L 2 ( S 1 ) that are “ commensurable” with the subspace H + ⊂ H of functions a dmitting a holomorphic extens ion to the in terior disk. The image P ( Gr H + ( H )) of t he Grassmannian under the Pl ¨ uc k er map consists of all decomp osable elemen ts of Λ H , whic h is the in tersection of the infinite set of quadrics defined b y the Pl ¨ uc ker relations. The latter are equiv alen t to the infinite set o f Hirota bilinear differen tial relations [15, 23, 24] for τ ( N , t ), whic h are the defining property of τ -functions. Through the Sato form ula fo r the Bak er-Akhiezer function Ψ N ( z , t ) = e P ∞ i =1 t i z i τ ( N , t − [ z − 1 ]) τ ( N , t ) , [ z − 1 ] := ( z − 1 , 2 z − 2 , 3 z − 3 , . . . ) , (1.3) these equations are equiv alen t to the K P hierarch y and their asso ciated Lax equations. The 2KP-T o da hierarc h y [15, 27] can similarly be expressed in terms of τ -functions dep ending on N , t and a further infinite sequence of flow parameters ˜ t = ( ˜ t 1 , ˜ t 2 , . . . ). 2 These admit double Sch ur function expansions [26] τ (2) ( N , t , ˜ t ) = X λ X µ B N ( λ, µ ) s λ ( t ) s µ ( ˜ t ) , (1.4) in whic h t he co efficien ts B N ( λ, µ ) hav e a similar interpretation in terms of Pl ¨ uc k er co- ordinates. They also satisfy an infinite set of bilinear differen tial Hirota-type relations in b oth sequences of flo w v ariables and difference-differen tial equations relating differen t lattice p oints. F or fixed N , they include the K P Hirota equtions of the K P hierarch y in eac h of t he tw o sets of flo w v aria bles, so w e refer to t hem as 2KP-T o da tau f unctions. Starting with an y given τ -f unction of KP-T o da or 2KP-T o da t yp e, it will b e sho wn in the following that new τ -functions can b e constructed, satisfying the same sets of bilinear relations, ha ving the follo wing Sc h ur function expansions: ˜ C ρ ( τ )( N , t ) = X λ r λ ( N ) π N ( λ ) s λ ( t ) (1.5) ˜ C (2) ρ, ˜ ρ ( τ (2) )( N , t , ˜ t ) = X λ X µ r λ ( N ) B N ( λ, µ ) ˜ r µ ( N ) s λ ( t ) s µ ( ˜ t ) , (1.6) where the factors r λ ( N ), ˜ r λ ( N ) are defined in terms of a given pair o f infinite sequences of non-v anishing constan ts { r i } i ∈ Z , { ˜ r i } i ∈ Z through the form ulae r λ ( N ) := c r ( N ) Y ( i,j ) ∈ λ r N − i + j , ˜ r µ ( N ) := c ˜ r ( N ) Y ( k, l ) ∈ µ ˜ r N − k + l . (1.7) Here the pro ducts are o v er pairs of p ositiv e in tegers ( i, j ) ∈ λ and ( k , l ) ∈ µ that lie within the matrix lo cations re presen ted by the Y o ung diagra ms of the partitions λ and µ , respectiv ely , c r ( N ) := ∞ Y i =1 ρ N − i ρ − i , (1.8) and r i = ρ i ρ i − 1 . (1.9) The seq uence of nonv anishing parameters { ρ i } may b e view ed as F ourier co efficien ts of a function ρ ( z ) on the unit circle, or a dis tribution. It will b e sho wn ( Prop osition 3.1 ) that, in terms of the elemen t s of the subspace W ⊂ L 2 ( S 1 ) corresp onding to a p oint o f the Grassmannian, the transformations (1.5 ) , (1.6) mean taking a generalized conv olution pro duct with ρ ( z ) (and similarly for ˜ r i ). These will therefore b e referred to a s (generalized) c onvolution s ymm etries . 3 With the usual 2KP-T o da flow parameters ( t , ˜ t ) fixed at some sp ecific v alues, such transformations extend to an infinite abelian group of comm uting flo ws whose parameters determine the ρ i ’s. This has b een used to generate new classes of solutio ns of in tegrable hierarc hies [5, 20, 21]. In the presen t w ork, they ar e studied r a ther as individual transfor- mations, for fixed v alues of the paramete rs ρ i whic h, when applied to a giv en KP -T o da or 2KP-T o da τ -f unction, pro duce a new one. P articular cases that implicitly use suc h transformations as symmetries ha ve fo und applications, e.g., as generating functions for top ological inv ariants related t o Riemann surfaces, suc h as Gromov-Witten in v a rian ts and Hurwitz n um b ers [17, 18]. As an immediate application, we may start with an integral o v er N × N Hermitian matrices: Z N ( t ) = Z M ∈ H N × N dµ ( M ) e tr P ∞ i =1 t i M i , (1.10) where dµ is a suitably defined U ( N ) conjugation in v ariant measure on the space H N × N of Hermitian N × N matrices,. This is kno wn to b e a KP-T o da τ -function [29]. Applying a con v o lution symmetry (1.5) with ρ ( z ) tak en esse n tially as the exponen tial function e z on the unit disc, and ev aluating at flow parameter v alues t i = 1 i tr( A i ) , t = [ A ] = ( t 1 , t 2 , . . . ) (1.11) for a fixed N × N Hermitian matrix A w e obtain, within a constant m ultiplicativ e factor , the externally coupled matrix mo del integral ( P roposit ion 4.1 ). Z N ,ext ([ A ]) := Z M ∈ H N × N dµ ( M ) e tr AM = ( N − 1 Y i =1 i !) ˜ C ρ ( Z N )([ A ]) . (1.12) Suc h integrals arise in a n um b er con texts, suc h as the K o n tsevic h- Witten generating function [7], the Brezin-Hik ami mo del [6, 3 0, 31] and the complex Wishart ensem ble [4, 28]. More general c hoices for the function ρ ( z ) are sho wn in Prop osition 4.2 to also determine KP-T o da τ - functions as externally coupled matr ix in tegrals. It is further sho wn, in P r oposition 4.3 , that these matrix mo del τ - functions can b e expresse d as finite N × N determinan ts. Similarly , Hermitian t w o-ma t rix in tegrals with exp onen tial coupling of Itzykson-Zub er t ype [10] Z (2) N ( t , ˜ t ) = Z M 1 ∈ H N × N dµ ( M 1 ) Z M 2 ∈ H N × N d ˜ µ ( M 2 ) e tr( P ∞ i =1 ( t i M i 1 + ˜ t i M i 2 )+ M 1 M 2 ) (1.13) 4 are kno wn to b e 2KP-T o da τ - functions [1 , 11, 12, 19]. Applying t he con v olution symmetry (1.6) to (1.13) g iv es an externally coupled t w o-mat r ix in tegral ( Prop osition 4.4 ). ˜ C (2) ρ, ˜ ρ ( Z (2) N )([ A ] , [ B ]) = Z M 1 ∈ H N × N dµ ( M 2 ) Z M 2 ∈ H N × N d ˜ µ ( M 2 ) τ r ( N , [ A ] , [ M 2 ]) τ ˜ r ( N , [ B ] , [ M 2 ]) e tr( M 1 M 2 ) , (1.14) where [ A ] a nd [ B ] signify the seq uences { 1 i tr( A i ) } i ∈ N + and { 1 i tr( B i ) } i ∈ N + of trace in v ari- an ts for the pair of Hermitian matrices A and B and τ r ( N , [ A ] , [ M 1 ]) = X λ r λ ( N ) s λ ([ A ]) s λ ([ M 1 ]) , (1.15) τ ˜ r ( N , [ B ] , [ M 2 ]) = X λ ˜ r λ ( N ) s λ ([ B ]) s λ ([ M 2 ]) . (1.16) This doubly externally couple d t w o-matrix mo del τ -function can also be expres sed in a finite N × N determinan tal form (eq. (4.37), Prop osition 4.5 ). This approach can also b e extended to more general 2KP-T o da τ -functions admitting m ultiple in tegra l represen tations o f the form (4.46). Applying the con volution symmetry (1.6) t hen giv es a new 2KP-T o da τ -function expressible either as a multiple integral (eq. (4.47), Prop osition 4.6 ) or as a finite determinan t (eq. ( 4 .49), Prop osition 4.7 ). The k ey to understanding these constructions, and further results follo wing from them, is the in terpretatio n of t he Sato τ - function as a v acuum state exp ectation v alue of pro ducts of exp onen tials of bilinear combinations of fermionic creation a nd annihilation op erators [23, 15, 2 7]. This w ell- known construction will b e summarized in the next section. 2 F ermionic co n struction of τ -functions W e re call here the approach to the construction o f τ -functions for integrable hierarchies of the KP and T o da types due to Sa to [23, 24], the Ky oto sc ho ol [8, 9, 15, 27] and Segal and Wilson [25]. 2.1 Hilb ert space Grassmannian and fermionic F o c k space W e b egin with the “first quan tized” Hilb ert space H , whic h will b e iden tified, as in [25], with the space o f square in tegrable functions on the unit circle H = L 2 ( S 1 ) = H + + H − , (2.1) 5 decomp osed as the direct sum o f the subspaces H + = span { z i } i ∈ N and H − = span { z − i } i ∈ N + consisting of functions tha t admit holomorphic extensions, respectiv ely , to the in terior and exterior of the unit circle S 1 in the complex z -plane, with the latter v a nishing a t z = ∞ . F or cons istency with other con ve n tions, the monomial (orthonorma l) basis ele- men ts { e i } i ∈ Z of H will b e denoted e i := z − i − 1 , i ∈ Z . (2.2) Tw o infinite ab elian groups act o n H by m ultiplication: Γ + := { γ + ( t ) := e P ∞ i =1 t i z i } , and Γ − := { γ − ( t ) := e P ∞ i =1 t i z − i } , (2.3) where t := ( t 1 , t 2 , . . . ) is an infinite sequenc e of (complex) flo w parameters corresp ond- ing to the one-parameter subgroups. More generally , we ha ve the g eneral linear gro up GL ( H ) consisting of in v ertible endomorphisms connected to the iden tity with w ell define d determinan ts. (See [25] for mor e detailed definitions o f this and what follo ws.) W e consider the G rassmannian G r H + ( H ) o f subspaces W ⊂ H that are c ommensur able with H + ⊂ H (in the sense of [25]; i.e., that orthogo nal pr o jection π + : W →H + to H + is a F redholm op erator while pro jection π − : W →H − to H − is Hilb ert-Sch midt). The connected comp onen ts of Gr H + ( H ), denoted Gr H N + ( H ), N ∈ Z , consis t of those W ∈ Gr H + ( H ) for which the F redholm index of π + : W →H + is N . These are the G L ( H ) orbits of the subspaces H N + := z − N H + ⊂ H , (2.4) whose elemen ts are denoted W g ,N = g ( H N + ) ∈ Gr H N + ( H ). The solutions to the KP hier- arc h y a re giv en b y t he τ -f unction τ N ,g ( t ) as defined b elo w, whic h determines the orbit of W g ,N in Gr H N + ( H ) under Γ + through its Pl ¨ uc k er co ordinates. In the terminolog y [25], the index N is called the “virtual dimension” o f the elemen ts W g ,N ∈ Gr H N + ( H ); i.e. their dimension r elative to the those in the comp onen t Gr H 0 + ( H ) con taining H + . The F ermionic F o ck sp ac e is the exterior space F := Λ H consisting of (a completion of ) the span of the semi-infinite w edge pro ducts: | λ, N i := e l 1 ∧ e l 2 ∧ · · · , (2.5) where { l j } j ∈ N + is a strictly decreasing sequence of in tegers that saturates, for sufficien tly large j , to a descending sequence of consecutiv e in tegers. This is equiv alen t to requiring that there b e an ass o ciated pair ( λ, N ) consisting of an in teger N and a partition λ = ( λ 1 , . . . , λ ℓ ( λ ) , 0 , 0 , . . . ) of length l ( λ ) and w eigh t | λ | = P ℓ ( λ ) i =1 λ i , where the part s λ i are a 6 w eakly decreasing sequenc e of non-negativ e integers that are positive for i ≤ ℓ ( λ ), a nd zero for i > ℓ ( λ ), suc h that the sequenc e { l j } j ∈ N + is giv en by l j := λ j − j + N . (2.6) In particular, for the t r ivial par t it io n λ = (0), w e ha ve t he “charge N v acuum” v ector | 0 , N i = e N − 1 ∧ e N − 2 ∧ . . . , (2.7) whic h will henceforth b e denoted | N i . The full F o ck space F th us a dmits a decomp osition as an o r t hogonal direct sum of the subspaces F N of states with charge N F = M N ∈ Z F N . (2.8) Denoting b y { ˜ e i } i ∈ Z the basis for H ∗ dual to the monomial basis { e i } i ∈ Z for H , w e define the F ermi creation and annihilatio n op erators ψ i and ψ † i on an arbitrary v ector v ∈ F b y exterior and interior multiplication, respectiv ely: ψ i v = e i ∧ v , ψ † i v := i ˜ e i v , v ∈ F . (2.9) These satisfy the standard canonical an ti-comm utation r elatio ns generating the Clifford algebra on H + H ∗ with respect t o the natural corresp onding quadratic form [ ψ i , ψ j ] + = [ ψ † i , ψ † j ] + = 0 , [ ψ i , ψ † j ] + = δ ij . (2.10) The basis states | λ, N i ma y b e expres sed in terms o f creation a nd annihilation op erators acting up on the c harge N v acuum ve ctor as follo ws [14] | λ, N i = ( − 1) P k i =1 β i k Y i =1 ψ N + α i ψ † N − β i − 1 | N i , (2.11) where ( α 1 , . . . α k | β 1 , . . . , β k ) is the F rob enius not a tion (see [16]) fo r the partition λ ; i.e., α i is t he n um b er of b o xes in the corresp onding Y oung diagram to the righ t o f the i th diagonal elemen t and β i the n umber b elow it. The Pl¨ uc ker map P : Gr H + ( H ) → P ( F ) t ak es the subspace W = span ( w 1 , w 2 , . . . ) into the pro jectivization of the exterior pro duct of its basis elemen ts: P : span( w 1 , w 2 , . . . ) 7→ [ w 1 ∧ w 2 ∧ · · · ] , (2.12) 7 and ma y b e lifted to a ma p fro m the bundle F r H + ( H ) of frames o n G r H + ( H ) to F ˆ P : F r H + ( H ) → F ˆ P : ( w 1 , w 2 , . . . ) 7→ w 1 ∧ w 2 ∧ · · · . (2.13) These interlace the lif t of the action of the ab elian group Γ + × H →H to F r H + ( H ) or Gr H + ( H ) with the follo wing represen tation of Γ + on F (a nd its pro jectivization) γ + ( t ) : v 7→ ˆ γ + ( t ) v , ˆ γ + ( t ) := e P ∞ i =1 t i H i , v ∈ F , (2.14) where H i := X n ∈ Z ψ n ψ † n + i , i ∈ Z , i 6 = 0 , (2.15) and t = ( t 1 , t 2 , . . . ) is the infinite sequence of flo w parameters. Similarly , the Pl ¨ uck er maps ˆ P and P interlace the lift of the a ction of the ab elian group Γ − × H →H to F r H + ( H ) o r Gr H + ( H ) with the follo wing represen tation of Γ − on F (a nd its pro jectivization). γ − ( t ) : v 7→ ˆ γ − ( t ) v , ˆ γ − ( t ) := e P ∞ i =1 t i H − i , v ∈ F , (2.16) Remark 2.1 Note that the image under the Pl ¨ uc k er map of the virtual dimension N comp o- nen t Gr H N + ( H ) of the Grassmannian Gr H + ( H ) is the GL ( H ) orbit of th e c harged v acuum state | N i , consisting of all decomp osable elemen ts of F N . The KP-T o da τ -function τ g ( N , t ) corresp o nding to the ele men t W g ,N ∈ Gr H + ( H ) is giv en, within a nonzero m ultiplicativ e constan t, by applying the gro up elemen ts γ + ( t ) to W g ,N , to obtain the Γ + orbit { W g ,N ( t ) := γ + ( t )( W g ,N ) } , and taking the linear co ordinate (within pro jectivization) of the image under the Pl ¨ uc k er map corresp onding to pro j ection along the basis elemen t | N i τ g ( N , t ) = h N | ˆ P ( W g ,N ( t )) i . (2.17) If the group eleme n t g ∈ GL ( H ) is interpreted, relativ e to the monomial basis { e i } i ∈ Z , as an infinite mat r ix exp onen tia l g = e A of an elemen t of the Lie algebra A ∈ gl ( H ) with matrix elemen ts A ij , t hen the corresp onding represen ta t ion of GL ( H ) on F is giv en by ˆ g := e P i,j ∈ Z A ij : ψ i ψ † j : , (2.18) where : : denotes norma l ordering (i.e. annihilation op erators ψ † j app earing to the rig h t when j ≥ 0 a nd creation op erators ψ i to the right when i < 0). T his give s the followin g 8 expression for τ N ,g ( t ) as a charge N v acuum state exp ectation v alue of a pro duct of exp o nen tiated bilinears in the F ermi creation and annihilation op erators τ g ( N , t ) = h N | ˆ γ + ( t ) ˆ g | N i . (2.19) The equations of the KP hierarc h y are then equiv alent to the w ell-kno wn infinite system of Hirota bilinear equations [15, 23 , 24] whic h, in turn, are just t he Pl ¨ uc k er relations for the decomposable elemen t P ( W g ,N ( t )) ∈ P ( F N ). Similarly , w e may de fine a 2 - T o da sequence of double KP τ - functions asso ciated to the group elemen t ˆ g τ (2) g ( N , t , ˜ t ) = h N | ˆ γ + ( t ) ˆ g ˆ γ − ( ˜ t ) | N i , (2.20) where ˜ t = ( ˜ t 1 , ˜ t 2 , . . . ) is a second infinite set of flow parameters. This ma y similarly b e sho wn to satisfy the Hirota bilinear relations of the 2KP-T o da hierarc h y . 2.2 Sc hur fun ction expansions Ev a luating the matrix elemen ts of ˆ γ + ( t ) and ˆ γ − ( t ) b et w een the states | N i and | λ, N i giv es the Sch ur function h N | ˆ γ + ( t ) | λ, N i = h λ , N | ˆ γ − ( t ) | N i = s λ ( t ) , (2.21) (cf. [23, 24, 1 2, 13] whic h is determined through the Jacobi-T rudy f o rm ula s λ ( t ) = det( h λ i − i + j ( t )) | 1 ≤ i,j ≤ ℓ ( λ ) (2.22) in terms o f the complete symmetric functions h i ( t ), defined b y e P ∞ i =1 t i z i = ∞ X i =0 h i ( t ) z i . (2.23) Inserting a sum o v er a complete set of in termediate stat es in eqs. (2.19), (2.20), w e obtain the single and double Sc hur functions expansions τ g ( N , t ) = X λ π N ,g ( λ ) s λ ( t ) , (2.24) τ (2) g ( N , t , ˜ t ) = X λ X µ B N ,g ( λ, µ ) s λ ( t ) s µ ( ˜ t ) . (2.25) Here the sum is ov er all partitions λ and µ and π N ,g ( λ ) = h λ, N | ˆ g | N i (2.26) 9 is the Pl ¨ uc k er c o ordinate of the imag e of the elemen t g ( H N + ) ∈ Gr H N + ( H ) under t he Pl ¨ uc ker map P along the basis direc tion | λ, N i in the c harge N sector F N of the F o c k space. Similarly , B N ,g ( λ, µ ) = h λ , N | ˆ g | µ, N i (2.27) ma y be view ed a s the | λ, N i Pl¨ uc k er co ordinate of the image of the e lemen t g ( w µ,N ) ∈ Gr H N + ( H ), where w µ,N := span { e µ i − i + N } ∈ Gr H N + ( H ) . (2.28) In particular, c ho osing g to b e the iden tit y elemen t I , and using Wic k’s theorem (or equiv alently , t he Cauc h y-Binet iden tit y in semi-infinite form), w e obtain [12] τ (2) I ( N , t , ˜ t ) = h N | ˆ γ + ( t ) ˆ γ − ( ˜ t ) | N i = X λ s λ ( t ) s λ ( ˜ t ) = e P ∞ i =1 it i ˜ t i , (2.29) where the last equalit y is the Cauc h y-Littlewoo d identit y (cf. ref. [16]). 3 Con v oluti on symmetries 3.1 Con v olution action on H and Gr H + ( H ) Consider no w a n infinite sequence of complex n um b ers { T i } i ∈ Z , and define ρ i := e T i , r i := ρ i ρ i − 1 , i ∈ Z . (3.1) In the follo wing, we will assume that the series P ∞ i =1 T − i con v erges and that lim i →∞ | r i | = r ≤ 1 , (3.2) (although, for some purp oses, the latter condition ma y b e w eak ened). It follo ws that the t w o series ρ + ( z ) = ∞ X i =0 ρ i z i , ρ − ( z ) = ∞ X i =1 ρ − i z − i , (3.3) are absolutely con vergen t in the in terior and exterior o f the unit circle | z | = 1, r esp ectiv ely , defining analytic functions ρ ± ( z ) in these r egio ns and that R ρ := ∞ Y i =1 ρ − i (3.4) 10 con v erges to a finite v alue. If the inequalit y (3.2) is strict, ρ + ( z ) extends to the unit circle, defining a function in L 2 ( S 1 ). Henceforth, w e denote the pair ( ρ + , ρ − ) by ρ , where the latter can b e view ed as a sum ρ − + ρ + in the sens e of distributiona l con v o lutio ns, as defined b elo w. If w ∈ L 2 ( S 1 ) has the F ourier series decomp osition w ( z ) = ∞ X i = −∞ w i e i = ∞ X i = −∞ w i z − i − 1 = w + ( z ) + w − ( z ) (3.5) where w + ( z ) := ∞ X i =0 w − i − 1 z i , w − ( z ) := ∞ X i =1 w i z − i − 1 , (3.6) (note the differen t labelling con ve n tions in (3.3) and (3.6)), we can define a b ounded linear map C ρ : L 2 ( S 1 ) → L 2 ( S 1 ) that has the effect of multiplyin g eac h F ourier co effic ien t w i b y the factor ρ i , a nd hence eac h basis elemen t e i b y ρ i : C ρ ( w )( z ) = ∞ X i = −∞ ρ i w i z − i − 1 = C ρ ( w ) + + C ρ ( w ) − (3.7) This can b e in terpreted as taking a con v olutio n pr o duct with the function (or distribution) ˜ ρ ( z ) = ˜ ρ + ( z ) + ˜ ρ − ( z ) (3.8) where ˜ ρ + ( z ) := z − 1 ρ − ( z − 1 ) = ∞ X i =0 ρ − i − 1 z i , (3.9) ˜ ρ − ( z ) := z − 1 ρ + ( z − 1 ) = ∞ X i =0 ρ i z − i − 1 , (3.10) C ρ ( w ) + ( z ) := lim ǫ → 0 + 1 2 π i I | ζ | =1 − ǫ ˜ ρ + ( ζ ) w + ( z /ζ ) ζ − 1 dζ , (3.11) C ρ ( w ) − ( z ) := lim ǫ → 0 + 1 2 π i I | ζ | =1+ ǫ ˜ ρ − ( ζ ) w − ( z /ζ ) ζ − 1 dζ (3.12) (with the con tour integrals tak en coun terclo c kwise). 11 If ρ − ( z ) extends analytically to S 1 , eq. (3.11) is an ordinary con v olutio n pro duct on the circle (in exp onen tial v ar ia bles ). In the examples detailed b elow , a ll but a finite num b er of the T − i v alues v a nish for i > 0, and hence the infinite pro duct (3.4) is really finite, but ρ − ( z ) is rational with a p ole at z = 1 and the con v olution pro duct (3.12) ma y b e understo o d on S 1 only in the sense of distributions. Remark 3.1 Note that the class of generalize d con v olution mappings defined by (3.7) - (3.1 2) only forms a semi-group since, although they ma y b e in v ertible, their inv erse do es not generally b elong to th e same class. It ma y b e extended t o a group by droppin g the cond ition ( 3.2), o r restricted to one by r equ iring r = 1, b ut this will not b e needed in the sequ el. The linear maps C ρ : H → H ma y nev erth ele ss b e interpreted as elemen ts of GL ( H ), and are simply repr esen ted in the m onomial basis { e i } b y the d iag onal matrix diag { ρ i } . They thus b elong to the ab elian subgroup of GL ( H ) consisting of in v ertible elemen ts that are diagonal in th e monomial b asis. Remark 3.2 Since t he Baker-Akhiez er function (1.3), ev aluated at all v alues of the param- eters t = ( t 1 , t 2 , . . . ), sp ans th e shifted elemen t z N ( W g ,N ) ∈ Gr H 0 + ( H ) in the ze ro virtual di- mension comp onen t of the Grassmannian, the con v olution action (3.11), (3.12), lifted to the Grassmannian, may b e obtained by applying its conju gat e z N ◦ C ρ ◦ z − N under the shif t map z − N : Gr H 0 + ( H ) → Gr H N + ( H ) to Ψ N ( z , t ). Bu t note that, at fixed v alues of the flo w parameters t , this do es not equal the v alue of the Bak er-Akhiezer function corresp onding to the transformed τ - function as d efined b elo w; only the subsp ac es of H that they span, v aryin g ov er the t v alues, will coincide. This fact w ill not b e u sed explicitly in the follo win g, but it un derlies the geometrical meaning of generalized conv olutions as symmetries of KP-T o da and 2KP-T o da hierarchies. 3.2 Con v olution action on F o c k sp ace W e no w consider t he action ˆ C × F →F o f t he ab elian subgroup of GL ( H ) consisting of diagonal elemen ts in the monomial basis, and asso ciate a n elemen t ˆ C ρ ∈ ˆ C to eac h sequence { ρ i } i ∈ Z defined as ab o ve , suc h that the Pl ¨ uc k er map ˆ P in tert wines the ˆ C ρ action with tha t of C ρ , lifted to the bundle F r H + ( H ) of frames o v er Gr H + ( H ), and is equiv arian t with respect t o group multiplic ation in ˆ C . T o do this, w e first in tro duce the ab elian algebra generated b y the op erators K i := : ψ i ψ † i : =  ψ i ψ † i if i ≥ 0 − ψ † i ψ i if i < 0 , (3.13) [ K i , K j ] = 0 , i, j ∈ Z . (3.14) F or { ρ i = e T i } i ∈ Z as ab o ve, define the op erator ˆ C ρ := e P ∞ i = −∞ T i K i . (3.15) 12 Definition 3.1 F or e a ch p ai r ( λ , N ) , wher e N ∈ Z , and λ is a p artition which , expr esse d in F r ob enius notation, is ( α 1 · · · α k | β 1 · · · β k ) , l e t r λ ( N ) := c r ( N ) Y ( i,j ) ∈ λ r N − i + j = c r ( N ) k Y i =1 ρ N + α i ρ N − β i − 1 ! , (3.16) wher e c r ( N ) :=      Q N − 1 i =0 ρ i if N > 0 1 if N = 0 1 Q − 1 i = N ρ i if N < 0 . (3.17) Her e the inclusion ( i, j ) ∈ λ is understo o d to me an that the matrix lo c ation ( i, j ) c orr e- sp onds to a b o x within the Y oung diagr am of the p artition λ ; i.e., 1 ≤ i ≤ ℓ ( λ ) , 1 ≤ j ≤ λ i . The se c o n d e quality in (3.1 6 ) fol lo w s fr om the definition (3.1). It follo ws t ha t ˆ C ρ acts diagonally in the basis {| λ, N i} , with eigen v alues r λ ( N ). Lemma 3.1 ˆ C ρ | λ, N i = r λ ( N ) | λ , N i . (3.18) Pro of : Since the F o c k space basis elemen t | λ, N i is an infinite w edge pro duct | λ, N i = e l 1 ∧ e l 2 ∧ · · · = ( − 1) P k i =1 β i k Y i =1 ψ N + α i ψ † N − β i − 1 | N i , ( 3.19) l j := λ j − j + N , j ∈ N + , (3.20) it follows from the definition (2.9 ) and the normal ordering in (3.13 ) that the effect of the action of e T i K i on | λ, N i is to intro duce a m ultiplicativ e factor ρ i if i ≥ 0 and e i is presen t in the w edge pro duct (3.19) or ρ − 1 i if i < 0 and it is absen t, and otherwise no factor. Therefore ˆ C ρ | λ, N i = ˆ C ρ ( − 1) P k i =1 β i k Y i =1 ψ N + α i ψ † N − β i − 1 | N i = Q ∞ i =1 ρ N − i Q ∞ i =1 ρ − i k Y i =1 ρ N + α i ρ N − β i − 1 ! | λ, N i = c r ( N ) k Y i =1 ρ N + α i ρ N − β i − 1 ! | λ, N i 13 = r λ ( N ) | λ , N i . (3.21) Q.E.D. No w let W = span { w i ( z ) ∈ L 2 ( S 1 ) } i ∈ N + ∈ Gr H + ( H ) and vie w { w i } i ∈ N + as a frame for W . Lemma 3.2 The Pl ¨ ucker map ˆ P i n tertwines the c onv olution a c tion ( 3 .7) and the ˆ C - action on F ˆ P ( { C ρ ( w i ) } i ∈ N + ) = R ρ ˆ C ρ ( ˆ P { w i } i ∈ N + ) , (3.2 2) with m ultiplic ative factor R ρ := Q ∞ i =1 ρ − i . Pro of : Applying C ρ to eac h ele men t w i ∈ L 2 ( S 1 ) defining the fra me for W ∈ Gr H + ( H ) just m ultiplies its F ourier co efficien ts by the factors ρ j as in eq. (3.7 ). It follows tha t the basis elemen t | λ, N i is m ultiplied b y the pro duct of the fa ctors ρ l j corresp onding to the terms e l j it con tains, as in (3.18). Eq. (3.22) the n follow s from the definition of the Pl ¨ uc ker map ˆ P and linearity . Q.E.D. Example 3.1 Cho ose ρ + ( z ) = e z = ∞ X i =0 z i i ! , | z | ≤ 1 (3 .23) ρ − ( z ) = 1 z − 1 = ∞ X i =1 z − i | z | > 1 , (3.24) so ρ i =  1 i ! if i ≥ 1 1 if i ≤ 0 , (3.25) r i =  1 i if i ≥ 1 1 if i ≤ 0 , (3.26) r λ ( N ) = 1 ( Q N − 1 i =1 i !)( N ) λ if ℓ ( λ ) ≤ N (3.27) where ( N ) λ := ℓ ( λ ) Y i =1 λ i Y j =1 ( N − i + j ) (3.28) is the extended Poc hhammer sym b ol. 14 Example 3.2 Cho ose ρ + ( z ) = 1 (1 − ζ z ) a = ∞ X i =0 ( a ) i ( ζ z ) i i ! , | ζ | < 1 , | z | ≤ 1 (3.29) and ρ − ( z ) again as in (3.24), so ρ i =  ( a ) i ζ i i ! if i ≥ 1 1 if i ≤ 0 , (3.30) r i =  a − 1+ i i ζ if i ≥ 1 1 if i ≤ 0 , (3.31) r λ ( N ) = N − 1 Y i =0 ( a ) i i ! ! ζ | λ | + 1 2 N ( N − 1) ( a − 1 + N ) λ ( N ) λ if ℓ ( λ ) ≤ N . (3.32) 3.3 Con v olutions and S ch ur func tion expansions of τ -fu n ctions W e now consider the KP-T o da tau function τ C ρ g ( N , t ) = h N | ˆ γ + ( t ) ˆ C ρ ˆ g | N i (3.33) obtained b y replacing the group elemen t g in ( 2 .19) by C ρ g . Such a τ -function, obtained from τ g b y applying a conv olution symmetry will b e denoted τ C ρ g =: ˜ C ρ ( τ g ) . (3.34) In tro ducing a second pair ( ˜ ρ + , ˜ ρ − ), defined as in (3.3 ) , with the F ourier co efficien ts ρ i replaced b y ˜ ρ i , we also consider t he 2-T o da tau function τ (2) C ρ g C ˜ ρ ( N , t , ˜ t ) = h N | ˆ γ + ( t ) ˆ C ρ ˆ g ˆ C ˜ ρ ˆ γ − ( ˜ t ) | N i (3.35) obtained b y replacing the g roup elemen t g in (2.20) b y C ρ g C ˜ ρ , and denote this transformed 2-T o da τ - function τ (2) C ρ ˆ g C ˜ ρ =: ˜ C (2) ( ρ, ˜ ρ ) ( τ (2) g ) . (3.36) Inserting sums ov er complete sets of intermed iate orthonormal basis states in (3 .33) and (3.35), a nd defining ˜ r λ ( N ) as in (3.16), with the factors ρ i replaced by ˜ ρ i , w e obtain the follo wing form for the Sc h ur function expansions (2.24), (2.25). 15 Prop osition 3.1 The effe ct of the c on volution action s (3. 3 4), (3.36) is to multiply the c o efficients in the Schur function exp ansions of τ C ρ g ( N , t ) and τ (2) C ρ ˆ g C ˜ ρ ( N , t , ˜ t ) by the diag- onal factors r λ ( N ) and ˜ r µ ( N ) . τ C ρ g ( N , t ) = X λ r λ ( N ) π N ,g ( λ ) s λ ( t ) , (3.37) τ (2) C ρ g C ˜ ρ ( N , t , ˜ t ) = X λ X µ r λ ( N ) B N ,g ( λ, µ ) ˜ r µ ( N ) s λ ( t ) s µ ( ˜ t ) . (3.38) The Pl¨ ucker c o or dinates for the mo di fi e d Gr assmannia n elem e n ts C ρ g ( H N + ) and C ρ g C ˜ ρ ( w µ,N ) ar e thus π N ,C ρ g ( λ ) = r λ ( N ) π N ,g ( λ ) (3.39) B N ,C ρ g C ˜ ρ ( λ, µ ) = r λ ( N ) B N ,g ( λ, µ ) ˜ r µ ( N ) . (3.40) Pro of : This follows immediately from the diagona l form (3.18) of the ˆ C action in the orthonormal basis {| λ , N i} , substituted into the expansions (2.24), (2.25), using the defi- nitions (2.26) and (2.27) of the Pl ¨ uc k er co ordinates π N ,C ρ g ( λ ) and B N ,C ρ g C ˜ ρ ( λ, µ ). Q.E.D. In particular, setting g = C ˜ ρ = I , in (3.3 8) w e obtain τ (2) C ρ ( N , t , ˜ t ) = X λ r λ ( N ) s λ ( t ) s λ ( ˜ t ) =: τ r ( N , t , ˜ t ) (3.41) where τ r ( N , t , ˜ t ) is de fined b y the second equalit y . Suc h τ - functions ha ve been studied as gene ralizations of h ypergeometric functions in [21, 20]. (Cf. also [12, 13], where the notation differs slightly due to the presence o f the nor malization factor c r ( N ) in the definition (3.16) of r λ ( N ).) In the fo llo wing, the infinite sequence of par a meters t = ( t 1 , t 2 , . . . ) will o f ten b e chose n as the trace in v aria nts of some square matrix M . The sequence so for med will b e denoted t = [ M ] =  1 i tr( M i )      i ∈ N + , [ M ] i := 1 i tr( M i ) . (3.42) If t and ˜ t in (3.41) are replaced b y [ A ] and [ B ], respectiv ely , where A and B ar e a pa ir of diagonal matrices A = diag( a 1 , . . . , a N ) , B = diag( b 1 , . . . , b N ) (3.43) 16 with distinct eigen v alues, and ∆( A ) := n Y 1 ≤ i ℓ ( λ ). Since all expressions in the s um will b e p olynomials in the para meters ( a i , b i ) there is no loss of generalit y in assuming tha t t hese lie within the unit disc. W e define F i ( z ) := ρ + ( a i z ) , G i ( z ) := (1 − b i z ) − 1 (3.51) and hence F ij = ρ i ( a j ) , G ij = ( b j ) i . (3 .5 2) F rom the character fo rm ula s λ ([ A ]) = det( a λ j − j + N i ) ∆( A ) , s λ ([ B ]) = det( b λ j − j + N i ) ∆( B ) , (3 .53) it follo ws that the determinan t factors on the RHS of (3.4 9) are det( F λ i − i + N ,j ) = det( a λ i − i + N j ρ λ i − i + N ) = N Y i =1 ρ λ i − i + N ! s λ ([ A ])∆([ A ]) , (3.54) det( G λ i − i + N ,j ) = det( b λ i − i + N j ) = s λ ([ B ])∆( B ) . (3.55) F rom the definitions (3 .16) and (3.1 7), it follow s that N Y i =1 ρ λ i − i + N ! = r λ ( N ) , (3.56) so the R HS of the Cauc h y-Binet iden tity (3.49) is just t he RHS o f eq. (3.45) m ultiplied b y ∆([ A ])∆([ B ]). On the o ther hand, from (3.48), the LHS o f (3.49) is det( F i , G j ) = det  1 2 π i I z ∈ S 1 ρ + ( a i z ) z − b j dz z  = det( ρ + ( a i b j )) , (3.57) whic h is just the expression (3 .46) multiplie d b y ∆([ A ])∆([ B ]). Q.E.D. 18 Remark 3.4 Note that, for the ca se of Example 3.1 , eq. (3.46) b ecomes the k ey iden tit y (cf. [13, 31]) X ℓ ( λ ) ≤ N 1 ( N ) λ s λ ([ A ]) s λ ([ B ]) = N − 1 Y k =1 k ! ! det( e a i b j ) | 1 ≤ i,j ≤ N ∆( A )∆( B ) , (3.58) whic h, together with the c haracter in tegral [16] d λ,N Z U ∈ U ( N ) dµ H ( U ) s λ ([ AU X U † ]) = s λ ([ A ]) s λ ([ X ]) , (3.59) (where dµ H ( U ) is the Haar measure on U ( N )), imp lies the Harish-Chand ra-Itzykso n-Zub er (HCIZ) in tegral [10] Z U ∈ U ( N ) dµ H ( U ) e tr( AU X U † ) = N − 1 Y k =1 k ! ! det( e a i x j ) ∆( A )∆( X ) . (3.60) Remark 3.5 The condition that the eigen v alues { a i } and { b i } of A and B b e distinct can b e eliminated simply by taking limits in which some or all of these are made to coincide. In the re- sulting determinanta l formulae, lik e (3.46), and those app earing in subsequent sections, in whic h a V andermonde determinan t ∆( A ) or ∆( B ) app ears in th e denominator, the only mod ificatio n is that the terms in the numerator d eterminan ts dep end ing on the a i ’s and b i ’s are replaced by their deriv ativ es with resp ect to these parameters, tak en to the same degree as the degeneracy of their v alues, wh ile th e denominator V andermond e determinants are corresp ondingly replaced b y their lo w er dim en sional analogs. This will not b e fur th er deve lop ed here, b ut will be con- sidered elsewhere, in connection with correlation k ern els for externally coupled matrix m o dels. All formulae b elo w in whic h no V andermonde determinant facto rs ∆( A ) or ∆( B ) app ear in the denominator remain v alid in th e case of degenerate eige n v alues. 4 Applicatio n s to matrix mo dels W e no w consider N × N ma t rix Hermitian in tegrals that are τ - functions, and sho w how the application of con v olutio n symmetries leads to new matrix mo dels of the externally coupled t yp e. In the follo wing, let dµ ( M ), b e a measure o n the space of N × N Hermitian matrices M ∈ H N × N that is inv ariant under conjugatio n b y unitary matrices, and suc h that the reduced measure, pro jected to the s pace of eigen v alues b y integration o v er the group U ( N ), is a pro duct of N identical measures dµ 0 on R , times the Jacobian factor ∆ 2 ( X ), Z U ∈ U ( N ) dµ ( U X U † ) = N Y a =1 dµ 0 ( x a )∆ 2 ( X ) . (4.1) where X = dia g( x 1 , . . . , x N ). 19 4.1 Con v olution symmetries, externally coup led Hermitian matrix mo dels and τ - functions as fi nite determinan ts It is w ell-know n that Hermitian matrix in tegrals of the form Z N ( t ) = Z M ∈ H N × N dµ ( M ) e tr P ∞ i =1 t i M i (4.2) = N Y a =1 Z R dµ 0 ( x a ) e P ∞ i =1 t i x i a ∆ 2 ( X ) , (4.3) are KP-T o da τ -functions [29]. The Sc h ur function expansion is Z N ( t ) = X ℓ ( λ ) ≤ N π N ,dµ ( λ ) s λ ( t ) , (4.4 ) where the c o efficien ts π N ,dµ ( λ ) are expressible as determinants in terms of the matrix of momen ts [11, 12, 13] π N ,dµ ( λ ) = N Y a =1  Z R dµ 0 ( x a )  ∆ 2 ( X ) s λ ([ X ]) (4.5) = ( − 1) 1 2 N ( N − 1) N ! det( M λ i + N − i,j − 1 ) | 1 ≤ i,j ≤ N (4.6) M ij := Z R dµ 0 ( x ) x i + j . (4.7) No w consider t he externally coupled matrix model integral (cf. refs. [6, 28, 30, 31]) Z N ,ext ( A ) := Z M ∈ H N × N dµ ( M ) e tr( AM ) , (4.8) where A ∈ H N × N is a fi xed N × N Hermitian matrix. This can be obtained b y simply applying a conv olution symmetry transformation of the t ype give n in E xamp le 3.1 to the τ - f unction defined by the matrix inte gral (4.3). Prop osition 4.1 Applying the c o nvolution symmetry ˜ C ρ to the τ -function Z N ( t ) , wher e ρ + ( z ) a nd ρ − ( z ) ar e define d as in (3. 2 3 ), ( 3 .24), and cho o s ing the KP flow p ar ameters as t = [ A ] gives, within a multiplic ative c onstant, the external ly c ouple d m atrix inte gr al (4.8) ˜ C ρ ( Z N )([ A ]) = N − 1 Y i =1 i ! ! − 1 Z N ,ext ( A ) . (4.9) 20 Pro of: Substituting the expansion [13] e tr AM = X ℓ ( λ ) ≤ N d λ,N ( N ) λ s λ ([ AM ]) , (4.10) in to (4.8), where d λ,N = s λ ( I N ) (4.11) is the dimension of the irreducible GL ( N ) t ensor represen tation of symmetry ty p e λ , and expressing M in diagonalized form as M = U X U † , (4.12) where U ∈ U ( N ) and X = diag ( x 1 , . . . x N ), giv es Z N ,ext ( A ) = X ℓ ( λ ) ≤ N Z U ∈ U ( N ) dµ H ( U ) N Y a =1 Z R dµ 0 ( x a ) e P ∞ i =1 t i x i a ∆ 2 ( X ) d λ,N ( N ) λ s λ ([ AU X U † ]) . (4.13) Ev a luating the character integral ( 3.59) and using (4.5), it follows that Z N ,ext ( A ) = X ℓ ( λ ) ≤ N N Y a =1 Z R dµ 0 ( x a ) e P ∞ i =1 t i x i a ∆ 2 ( X ) 1 ( N ) λ s λ ([ A ]) s λ ([ X ]) = X ℓ ( λ ) ≤ N 1 ( N ) λ π N ,dµ ( λ ) s λ ([ A ]) = X ℓ ( λ ) ≤ N ( N − 1 Y i =1 i !) r λ ( N ) s λ ([ A ]) = ( N − 1 Y i =1 i !) ˜ C ρ ( Z N ) | t =[ A ] . (4.14) where the third line follows from the expression ( 3 .27) for r λ ( N ) in example 3.1 and the last from Propo sition 4.1 , eq. (3.1). Q.E.D. More generally , giv en an arbitrary f unction ρ + ( z ), analytic on the interior o f S 1 and c ho osing ρ − ( z ) as in (3 .24), we may define a new externally coupled matrix in tegral Z N ,ρ ( A ) := Z M ∈ H N × N dµ ( M ) τ r ( N , [ AM ]) , (4.15) 21 in whic h e tr AM is replaced b y τ r ( N , [ M ]) := τ r ( N , [ I N ] , [ M ]) = X ℓ ( λ ) ≤ N d λ,N r λ ( N ) s λ ([ M ]) . (4.16) Then b y the same calculation as ab o ve, it follows that Z N ,ρ ( A ) is again just the τ -function obtained b y applying the conv olution symmetry ˜ C ρ to Z N , ev aluated at the parameter v alues t = [ A ]. Prop osition 4.2 Applying the c on volution symmetry ˜ C ρ to Z N gives ˜ C ρ ( Z N )([ A ]) = Z N ,ρ ( A ) . (4.17) In particular, if we tak e ( ρ + , ρ − ) as in Example 3.2 ab ov e, w e obtain (cf. [13 ]) Z N ,ρ ( A ) = N − 1 Y i =0 ( a ) i i ! ! ζ 1 2 N ( N − 1) Z M ∈ H N × N dµ ( M ) det(1 − ζ AM ) − a − N +1 , (4.18) sho wing that t his also is a KP-T o da τ -function ev aluated at parameter v alues t = [ A ]. Returning t o t he general case, a finite determinantal fo r m ula for Z N ,ρ ( A ) is give n b y the follo wing. Prop osition 4.3 Z N ,ρ ( A ) = ( − 1) 1 2 N ( N − 1) N ! ∆( A ) det( G ij ( ρ, A )) | 1 ≤ i,j ≤ N , (4.19) wher e G ij ( ρ, A ) := Z R dµ 0 ( x ) x i − 1 ρ + ( a j x ) . (4.20) Pro of: Applying the c haracter in tegral iden tity (3.59) to (4.15) giv es Z N ,ρ ( A ) = Z M ∈ H N × N dµ ( M ) X ℓ ( λ ) ≤ N r λ ( N ) s λ ([ A ]) s λ ([ M ]) (4.21) = 1 ∆( A ) Z dµ 0 ( X )∆( X )det( ρ + ( a i x j )) | 1 ≤ i,j ≤ N (4.22) = ( − 1) 1 2 N ( N − 1) N ! ∆( A ) det( G ij ( ρ, A )) | 1 ≤ i,j ≤ N , (4.23) 22 with G ij ( ρ, A ) defined b y (4.20). Here, the in tegration o v er the U ( N ) gro up has b een p erformed a nd Lemma 3.3 has b een us ed in eq. (4.22). Eq. (4.23) follow s from (4.22) b y applying the Andr ´ eief iden tit y [3] in the form N Y m =1 Z dµ 0 ( x m ) ! det( φ i ( x j ))det( ψ k ( x l ))    1 ≤ i,j ≤ N 1 ≤ k,l ≤ N = N !det  Z φ i ( x ) ψ j ( x )     1 ≤ i,j ≤ N (4.24) with φ i ( x ) = x N − i , ψ j ( x ) := ρ + ( a j x ) , (4.25) since ∆( X ) = det( φ i ( x j )) . (4.26) Q.E.D. 4.2 Externally c oupled t w o-matrix mo dels W e now turn to the case of t w o- matrix mo dels. F or simplicit y , w e only consider the Itzykson-Zub er exp onen tia l coupling [10 ], althoug h the same double con v olution trans- formations ma y b e applied to all the couplings considered in ref. [13]. Using the HCIZ iden tity (3.60) to ev aluat e the in tegrals o ve r the unitary groups U ( N ), w e obtain Z (2) N ( t , ˜ t ) = Z M 1 ∈ H N × N dµ ( M 1 ) Z M 2 ∈ H N × N d ˜ µ ( M 2 ) e tr( P ∞ i =1 ( t i M i 1 + ˜ t i M i 2 )+ M 1 M 2 ) (4.27) = N Y k =1 k ! N Y a =1  Z R dµ 0 ( x a ) Z R d ˜ µ 0 ( y a ) e P ∞ i =1 ( t i x i a + ˜ t i y i a + x a y a )  ∆( X )∆( Y ) , where Y = dia g( y 1 , . . . , y N ). This is kno wn to b e a 2KP-T o da τ -f unction [1, 2, 11, 12, 1 3, 22], with do uble Sc h ur function expansion Z (2) N ( t , ˜ t ) = X λ X µ B N ,dµ,d ˜ µ ( λ, µ ) s λ ( t ) s µ ( ˜ t ) , (4.28) where the co efficien ts B N ,dµ,d ˜ µ ( λ, µ ) are N × N determinan ts of submatrices in terms o f the matrix of bimoments B N ,dµ,d ˜ µ ( λ, µ ) = N Y k =1 k ! N Y a =1  Z R dµ 0 ( x a ) Z R d ˜ µ 0 ( y a ) e x a y a  ∆( X )∆( Y ) s λ ([ X ]) s µ ([ Y ]) 23 = ( N !) N Y k =1 k ! det( B λ i − i + N , µ j − j + N ) | 1 ≤ i,j ≤ N (4.29) B ij := Z R dµ 0 ( x a ) Z R d ˜ µ 0 ( y a ) e x a y a x i y j . (4.30) No w, c ho osing a pair of elemen ts ( ρ, ˜ ρ ), with b oth ρ − and ˜ ρ − as in (3.24), w e ma y define a family o f externally coupled tw o-matrix mo dels, b y Z (2) N ,ρ, ˜ ρ ( A, B ) := Z M 1 ∈ H N × N dµ ( M 1 ) Z M 2 ∈ H N × N d ˜ µ ( M 2 ) τ r ( N , [ A ] , [ M 1 ]) τ ˜ r ( N , [ B ] , [ M 2 ]) e tr( M 1 M 2 ) . (4.31) where A, B are a pair of hermitian N × N matrices. This class may b e obtained as t he 2KP-T o da τ - function resulting from applying the con v o lut io n symme try ˜ C ρ, ˜ ρ to Z (2) N . Prop osition 4.4 Applying the c onv o lution symmetry ˜ C ρ, ˜ ρ to Z (2) N and evaluating at the p ar ameter values t = [ A ] , ˜ t = [ B ] gives the external ly c ouple d matrix inte gr al (4.31) ˜ C (2) ρ, ˜ ρ ( Z (2) N )([ A ] , [ B ])) = Z (2) N ,ρ, ˜ ρ ( A, B ) . (4.32) Pro of: Because of the U ( N ) × U ( N ) in v ariance of the measures d µ and d ˜ µ in (4.3 1) and all factors in the in tegrand, except for the coupling term e tr( M 1 M 2 ) , we may carry out the t w o U ( N ) in tegrations, using the HCIZ identit y (3.6 0), to obtain a reduced in tegr a l ov er the diagonal matrices X = diag( x 1 , . . . , x N ) , Y = diag( y 1 , . . . , y N ) o f eigen v a lues of M 1 and M 2 , Z (2) N ,ρ, ˜ ρ ( A, B ) = N Y k =1 k ! N Y a =1  Z R dµ 0 ( x a ) Z R d ˜ µ 0 ( y a ) e x a y a  ∆( X )∆( Y ) (4.33) × τ C ρ ( N , [ A ] , [ X ]) τ C ˜ ρ ( N , [ B ] , [ Y ]) = X ℓ ( λ ) ≤ N X ℓ ( µ ) ≤ N r λ ( N ) B N ,dµ,d ˜ µ ( λ, µ ) ˜ r λ ( N ) s λ ([ A ]) s µ ([ B ]) (4 .34) = ˜ C (2) ρ, ˜ ρ ( Z (2) N )([ A ] , [ B ])) . (4.35) where the s econd equality f ollo ws from eq. (3.41) and the last from Prop osition 3 .1 , eq. (3.38). Q.E.D. Since the dep endence on A and B is U ( N ) × U ( N ) conjugation inv ariant we may c ho ose, without loss of generalit y , A and B to b e diago nal matrices A = diag( a 1 , . . . , a N ) , B = diag( b 1 , . . . , b N ) , (4.36) 24 W e then obtain, as in the one-mat r ix case, a finite determinan tal form ula f o r the 2KP- T o da τ -function Z (2) N ,ρ, ˜ ρ ( A, B ). Prop osition 4.5 Z (2) N ,ρ, ˜ ρ ( A, B ) = N !( Q N k =1 k !) ∆( A )∆( B ) det( G ij ( ρ, ˜ ρ, A, B ) | 1 ≤ i,j ≤ N , (4.37) wher e G ij ( ρ, ˜ ρ, A, B ) := Z R dµ 0 ( x ) Z R d ˜ µ 0 ( y ) e xy ρ + ( a i x ) ˜ ρ + ( b j y ) . (4.38) Pro of : Z (2) N ,ρ, ˜ ρ ( A, B ) = Z M 1 ∈ H N × N dµ ( M 1 ) Z M 2 ∈ H N × N d ˜ µ ( M 2 ) e tr( M 1 M 2 ) × X ℓ ( λ ) ≤ N r λ ( N ) s λ ([ A ]) s λ ([ M 1 ]) X ℓ ( µ ) ≤ N ˜ r µ ( N ) s µ ([ B ]) s µ ([ M 2 ]) (4.39) = ( Q N k =1 k !) ∆( A )∆( B ) Z dµ ( X ) Z d ˜ µ ( Y ) e P N i =1 x i y i × det( ρ + ( a k x l )) | 1 ≤ k ,l ≤ N det( ˜ ρ + ( b m y n )) | 1 ≤ m,n ≤ N (4.40) = N !( Q N k =1 k !) ∆( A )∆( B )) det( G ij ( ρ, ˜ ρ, A, B ) | 1 ≤ i,j ≤ N . (4.41) In (4.40), w e hav e used the HCIZ iden tit y (3.60), an tisymmetry of the dete rminan ts in the integrand with r esp ect to p ermu tations in the in tegration v a riables ( x 1 , . . . , x N ) and ( y 1 , . . . , y N ) and Lem ma 3.3 tw ice, while in (4.41), w e hav e use d the Andr´ eief ide n tit y [3] in t he form N Y m =1 Z dµ ( x m , y m ) ! det( φ i ( x j ))det( ψ k ( y l ))    1 ≤ i,j ≤ N 1 ≤ k,l ≤ N = N !det  Z dµ ( x, y ) φ i ( x ) ψ j ( y )     1 ≤ i,j ≤ N (4.42) Q.E.D. As t he simplest example of a 2KP-T o da τ -function obtained through Propositions 4.4 and 4.5 , consider the case when the measures d µ 0 ( x ) a nd d µ 0 ( y ) are bo t h Gaussian, and ρ + and ˜ ρ + are b oth taken as the exp onen tial function. Example 4.1 dµ 0 ( x ) = e − σx 2 dx, dµ 0 ( y ) = e − σy 2 dy , ρ + ( x ) = e x , ˜ ρ + ( y ) = e y . (4.43) 25 Ev aluating the Gaussian integral s giv es G ij = 2 π √ 1 + 4 σ 2 e σ ( a 2 i + b 2 j ) − a i b j 4 σ 2 − 1 , (4.44) and hence Z N ,ρ ( A ) = (2 π ) N N ! Q N k =1 k ! (1 + 4 σ 2 ) N 2 ∆( A )∆( B ) e σ 4 σ 2 − 1 P N i =1 ( a 2 i + b 2 i ) det( e σ a i b j 1 − 4 σ 2 ) . (4.45) The fa ctor e σ 4 σ 2 − 1 P N i =1 ( a 2 i + b 2 i ) is a linear e xp onen tial in terms of the 2KP flo w v ariable t 2 and ˜ t 2 and hence, through the Sato formula (1.3 ), pro duces ju st a gauge f actor multiplying the Bak er-Akhiezer fun ction [25]. Therefore (4.45) is just a rescaled, gauge transformed v ersion of the 2KP τ -function of h yp ergeometric type app earing in t he integrand of the Itzykson-Zub er coupled t w o-matrix mo del [10]. 4.3 More general 2KP-T o da τ -functions as m ultiple in tegrals W e ma y extend the ab o v e results to mo r e general 2KP-T o da τ -functions expressed as m ultiple in tegra ls and finite determinan ts. T o b egin with, the follo wing multiple integral τ (2) dµ ( N , t , ˜ t ) = N Y a =1  Z Γ Z ˜ Γ dµ ( x a , y a ) e P ∞ i =1 ( t i x i a + ˜ t i y i a )  ∆( X )∆( Y ) , (4.46) where Γ, ˜ Γ are curv es in t he complex x - and y - planes and dµ ( x, y ) is a measure on Γ × ˜ Γ, is a 2KP-T o da τ -function [13] for a large class of measures d µ 0 ( x, y ) . Applying a double con v olution symme try ˜ C ρ, ˜ ρ , with ρ − and ˜ ρ − the same as in ( 3 .24), gives a new 2KP-T o da τ -function, also ha ving a multiple in tegral represen tation. Prop osition 4.6 ˜ C (2) ρ, ˜ ρ ( τ (2) dµ )( N , t , ˜ t ) = N Y a =1  Z Γ Z ˜ Γ dµ ( x a , y a )  ∆( X )∆( Y ) τ r ( N , t , [ X ]) τ ˜ r ( N , ˜ t , [ Y ]) . (4.47) Pro of: This is prov ed similarly to Prop osition 4.4 , using the Cauc hy-Littlew o o d iden- tit y (2.29) t wice in the form N Y a =1 e P ∞ i =1 ( t i x i a + ˜ t i y i a ) = X ℓ ( λ ) ≤ N s λ ( t ) s λ ([ X ]) X ℓ ( µ ) ≤ N s µ ( ˜ t ) s µ ([ Y ]) . (4.48) Q.E.D. Ev a luating at parameter v alues t = [ A ] and ˜ t = [ B ], and applying again Lemma 3. 3 giv es the τ - function o f eq. (4.47) in N × N determinan tal form. 26 Prop osition 4.7 ˜ C (2) ρ, ˜ ρ ( τ (2) dµ )([ A ] , [ B ]) = N ! ∆( A )∆( B ) det( G ij ( ρ, ˜ ρ, A, B ) | 1 ≤ i,j ≤ N , (4.49) wher e G ij ( ρ, ˜ ρ, A, B ) := Z Γ Z ˜ Γ dµ ( x, y ) ρ + ( a j x ) ˜ ρ + ( b j y ) . (4.50) Pro of: ˜ C (2) ρ, ˜ ρ ( τ (2) dµ )([ A ] , [ B ]) = 1 ∆( A )∆( B ) N Y a =1  Z Γ Z ˜ Γ dµ ( x a , y a )  × det( ρ + ( a k x l )) | 1 ≤ k ,l ≤ N det( ˜ ρ + ( b m y n )) | 1 ≤ m,n ≤ N = N ! ∆( A )∆( B ) det( G ij ( ρ, ˜ ρ, A, B )) | 1 ≤ i,j ≤ N , (4.51) where ag ain, w e ha v e used the Lemma 3.3 t wice and the Andr ´ eief iden tit y in the f o rm (4.42). Q.E.D. This therefore provides a new class of 2KP-T o da τ -functions express ible in suc h a finite determinan tal form, asso ciated to an y pair o f curv es Γ, ˜ Γ, together with a measure dµ on their pro duct, a nd a pair of functions ρ + ( x ) and ˜ ρ + ( y ), suc h t ha t the in tegrals in (4.50) are well defined a nd conv ergen t. A cknow le dgemen ts. The authors would lik e to thank D. W ang for helpful discussions relating to this work. References [1] M. Adler and P . v an Mo erb ek e, “The sp ectrum of coupled ra ndom matrices”, Ann. Math. 149 , 92 1-976 (199 9). [2] M. Adler and P . v an M o erb ek e, “Virasoro a ction on S c h ur function expansions, sk ew Y oung t a bleaux and random w alks”, Com mun. Pur e. Appl. Math.. 58 , 362- 408 (2 004). [3] C. Andr´ eief, “Note sur une relation p our les int ´ egra les d ´ efinies des pro duits des fonctions”, M ´ em. So c. Sci. Bor de aux , 2 , 1–14 (1883). 27 [4] Z. Bai and J. Silv erstein, “On the empirical distribution of eigen v a lues of a class of large dimensional random matrices.”, J. Multivariate Anal. 54 , 175- 192 (1995) . [5] E. Bettelheim, A.G. Abanov and P . Wiegmann. “Nonlinear dynamics o f quan tum systems and solition theory”, J. Phys. A 40 , F193-F 207 (2 007). [6] E. Br ´ ezin a nd S. Hik ami, “Correlations of nearb y lev els induced b y a random p oten- tial”, Nucl. Phys. B 479 , 697- 706 (1996). [7] M. Kon tsevic h, “Interse ction theory on the mo duli space of curv es and the matrix Airy function”, Commun. Math. Phys. 147 , 1-23 ( 1 992). [8] E. D ate, M. Jim b o, M. Kashiw ara a nd T. Miwa, “T ransformatio n groups for soliton equations, IV A new hierarc h y of soliton equations of KP-t ype”, Physic a D 4 , 343- 365 (1982). [9] E. D ate, M. Jim b o, M. Kashiw ara a nd T. Miwa, “T ransformatio n groups for soliton equations”, in: Nonlin e ar inte gr a ble systems-clas s i c al the ory and quantum the ory , eds. M. Jimbo, and T. Miw a, W o rld Scien tific, pp. 39-12 0 , ( 1 983). [10] C. Itzykson and J.B. Zub er, “ The planar appro ximation (I I)”, J. Math. Phys. 21 , 411-421 (1980). [11] J. Harnad and A. Y u. Orlo v, “Matr ix in tegrals as Borel sums of Sc h ur func tion expansions”, In: Symmetries and P erturbation theory SPT2002, eds. S. Ab enda a nd G. Gaeta, W orld Scien tific, Singap ore (2003). [12] J. Harnad and A.Y u. Orlo v, “Scalar pro ducts of symmetric functions and matrix in tegrals”, The or. Math. Phys. 137 , 1676-169 0 (2003). [13] J. Harnad and A. Y u. Orlo v, “F ermionic construction of par t ition functions for tw o- matrix mo dels and p erturbative Sch ur function expansions ”, J. Phys. A 39 , 8 783- 8809 (2006). [14] J. Harnad and A. Y u. Orlov, “F ermionic construction of tau functions and random pro cesse s”, Physic a D 235 , 168-206 (2007). [15] M. Jim b o and T. Miw a “Solitons and Infinite Dimensional Lie Algebras”, Publ. RIMS Kyoto Univ. 19 , 943-1 001 (1 983). 28 [16] I.G. Macdonald, Symmetric F unctions and Hal l Polynom i a ls , Clarendon Press, Ox- ford, (1995). [17] A. Ok ounk o v, “T o da eq uations for Hurwitz n um b ers”, Math, R es. L ett. 7 , 447-453 (2000). [18] A. Okounk o v and R. Pandharipande, “Gromov-Witten theory , Hurw itz theory , a nd completed cycles ” Ann. Math. 163 , 517-560 (2006). [19] A. Y u. Orlo v, “New Solv able Matrix In tegrals”, In tern. J. Mo d. Phys. A 19 , Supple- men t, 27 6-293 (20 0 4). [20] A. Y u. Orlov, “Hyp ergeometric functions as infi nite-soliton tau functions ”, The or. Math. Phys. 146 , 183-20 6 (2006). [21] A. Y u. O rlo v and D . M. Sche rbin, “Multiv ariate h yp ergeometric functions as t a u- functions of T o da lattice a nd Kadom tsev-P etviash vili equation” Physic a D 152 , 51-65 (2001). [22] A. Y u. Orlov and T. Shiota, “Sch ur function expansion for normal matrix mo del and asso ciated discrete matrix mo dels”, Phys. L e tt. A 343 , 384-39 6 (2005). [23] M. Sa t o , “Soliton equations as dynamical systems on infinite dimensional Gra ss mann manifolds”, RIMS, Ky oto Univ. Kokyur oku 439 , 30-46 (1981) . [24] M. Sato and Y. Sato. “Soliton equations as dynamical systems on infinite dimensional Grassmann manifold”, in: Nonlinear PDE in Applied Science, Pro c. U. S.- Japan Seminar, T oky o 1982 , K ino kuniya, T oky o, 1983, pp. 259-27 1. [25] G. Segal and G . Wilson, “Lo op groups a nd e quations of KdV type ”, Public a tions Mathmatiques de l’IH ´ ES 6 , 5 -65 (1985 ) . [26] K. T ak asaki, “Initial v alue problem for the T o da la t tice hierarch y”, A dv. Stud. Pur e Math. 4 , 13 9-163 (198 4). [27] K.Ueno and K.T ak asaki, “T o da lat t ice hierarch y”, A dv. Stud. Pur e Math. 4 , 1-95 (1984). [28] Dong W ang, “R andom matrices with external source and KP τ -functions”, J. Math. Phys. 50 0 73506 (2009 ) 29 [29] A. Zabrodin, S. K ha rc hev, A. Mironov , A.Marshak o v and A. Orlo v, “Matrix Mo d- els among In tegrable Theories: F orced Hierarchie s and Op erator F ormalism”, Nucl. Phys. B 366 , 569-601 (1991). [30] P . Zinn-Justin, Univ ersalit y o f correlation functions of hermitian random matrices in an external field, Co m mun. Math. Phys. 194 , 631-65 0 (1998). [31] P . Zinn-Justin, “HCIZ integral and 2D T o da lattice hierarch y”, Nu cl. Phys. B 634 , 417-432 (2002). 30