CRM-1844 (1993) Dual Isomonodromic Deformations and Moment Maps to Loop Algebras † J. Harnad‡ Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montr´eal, Canada H4B 1R6, and 3 Centre de recherches math´ematiques, Universit´e de Montr´eal 9 9 C. P. 6128-A, Montr´eal, Canada H3C 3J7 1 n Abstract a J TheHamiltonianstructureofthemonodromypreservingdeformationequationsofJimboetal [JMMS] 9 1 isexplainedintermsofparameterdependentpairsofmomentmapsfromasymplecticvectorspacetothedual 2 spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations v areobtainedbypullingbackspectralinvariantsonPoissonsubspacesconsistingofelementsthatarerationalin 6 7 theloopparameterandidentifyingthedeformationparameterswiththosedeterminingthemomentmaps. This 0 1 construction is shown to lead to “dual” pairs of matrix differential operators whose monodromy is preserved 0 underthesamefamilyofdeformations. Asillustrativeexamples,involvingdiscreteandcontinuousreductions,a 3 9 higherrankgeneralizationoftheHamiltonianequationsgoverningthecorrelationfunctionsforanimpenetrable / h Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painleve t - p transcendents PV and PVI. e h : v 1. Monodromy Preserving Hamiltonian Systems i X The following integrable Pfaffian system was studied by Jimbo, Miwa, Mˆori and Sato in r a [JMMS]: n dN = [N ,N ]dlog(α α ) [N ,d(α Y)+Θ]. (1.1) i i j i j i i − − − j=1 Xj6=i Here N (α ,...,α , y ,...,y ) is a set of r r matrix functions of n + r (real or i 1 n 1 r i=1,...n { } × complex) variables α ,y , Y is the diagonal r r matrix Y = diag(y ,...,y ) and i a i=1,...n 1 r { }a=1,...r × the matrix differential form Θ is defined by: n Θ = (1 δ )( N ) dlog(y y ). (1.2) ab ab i ab a b − − i=1 X † Research supported in part by the Natural Sciences and Engineering Research Council of Canada. ‡ e-mail address: [email protected] or [email protected] 1 This system determines deformations of the differential operator: ∂ := (λ), (1.3) λ D ∂λ −N where n N i (λ) := Y + , (1.4) N λ α i i=1 − X thatpreserve itsmonodromy aroundtheregularsingularpoints λ = α andat λ = . i i=1,...n { } ∞ It was observed in [JMMS] (appendix 5) that, expressing the N ’s as i N = GTF , (1.5) i i i where F ,G Mki×r are pairs of maximal rank k r rectangular matrices (k r), i i i=1,...n i i { ∈ } × ≤ with F GT = L gl(k ) constant matrices related to the monodromy of at { i i i ∈ i }i=1,...n Dλ α , eq. (1.1) may be expressed as a set of compatible nonautonomous Hamiltonian i i=1,...n { } systems: dF = F ,ω (1.6a) i i { } dG = G ,ω . (1.6b) i i { } Here d denotes the total differential with respect to the variables α ,y , the 1 form i a i=1,...n { }a=1,...r − ω is defined as: n r ω := H dα + K dy , (1.7) i i a a i=1 a=1 X X with n tr(N N ) i j H := tr(YN )+ , i = 1,...,n (1.8a) i i α α i j j=1 − Xj6=i n r ( n N ) ( n N ) i=1 i ab j=1 j ba K := α (N ) + , a = 1,...,r (1.8b). a i i aa y y i=1 b=1 P a −Pb X Xb6=a and the Poisson brackets in the space of (F ,G )’s are defined to be such that the matrix i i elements of F ,G are canonically conjugate: i i i=1,...n { } (F ) ,(G ) = δ δ δ (1.9) i a a j b b ij a b ab { i i } i j i,j = 1,...,n, a,b = 1,...,r, a ,b = 1,...,k . i i i 2 The Frobenius integrability of the differential system (1.1) follows from the fact that the Hamiltonians H ,K Poisson commute. It was also noted in [JMMS] that, with i a i=1,...n { }a=1,...r respect to the Poisson brackets (1.9), the matrices N satisfy: i i=1,...,n { } (N ) ,(N ) = δ (δ (N ) δ (N ) ), (1.10) i ab j cd ij bc i ad ad i cb { } − which is the Lie Poisson bracket on the space ( n gl(r))∗ dual to the direct sum of n copies ⊕i=1 of the Lie algebra gl(r) with itself. The 1-form ω is exact on the parameter space and may be interpreted as the logarithmic differential of the τ-function, ω = dlogτ. (1.11) Numerous applications of such systems exist; in particular, in the calculation of correla- tionfunctionsforintegrablemodelsinstatisticalmechanicsandquantumfieldtheory[JMMS, IIKS], in matrix models of two dimensional quantum gravity [M] and in the computation of level spacing distribution functions for random matrix ensembles [TW1, TW2]. In the next section these systems will be examined within the context of loop algebras, using an approach originally developed for the autonomous case, involving isospectral flows, in [AHP, AHH1]. This is based on “dual” pairs of parameter dependent moment maps from n symplectic vector spaces to two different loop algebras gl(r) and gl(N), where N = k . i=1 i The nonautonomousHamiltoniansystems (1.6)–(1.8)willbe generatedby pulling back certain P e e spectral invariants, viewed as polynomial functions on rational coadjoint orbits, under these moment maps, and identifying the parameters determining the maps with the deformation parameters of the system. This construction leads to a pair of “dual” first order matrix differ- ential operators with regular singular points at finite values of the spectral parameter, both of whose monodromy data are invariant under the deformations generated by these Hamil- tonian systems. In section 3, the generic systems so obtained will be reduced under various discrete and continuous Hamiltonian symmetry groups. A rank r = 2s generalization of the systems determining the correlation functions for an impenetrable Bose gas (or, equivalently, the generating function for the level spacing distribution functions for random matrix ensem- bles [TW1]) will be derived by reduction to the symplectic loop algebra sp(2s). The “dual” isomonodromy representations of the equations for the Painlev´e transcendents P and P V VI will also be derived, and their Hamiltonian structure deduced through redeuctions under con- tinuous groups. A brief discussion of generalizations to systems with irregular singular points is given in section 4. 2. Loop Algebra Moment Maps and Spectral Invariants 3 In [AHP, AHH1], an approach to the embedding of finite dimensional integrable sys- tems into rational coadjoint orbits of loop algebras was developed, based on a parametric family of equivariant moment maps J : M gl(r)∗ from the space M = F,G MN×r A −→ + { ∈ } of pairs of N r rectangular matrices, with canonical symplectic structure × e e ω = tr(dF dGT) (2.1) ∧ to the dual of the positivehalf of the loop algebragl(r). The maps J , which are parametrized A by the choice of an N N matrix A MN×N with generalized eigenspaces of dimension × ∈ n e e k , k = N and eigenvalues α , are defined by: { i}i=1,...n i=1 i { i}i=1,...n P J : (F,G) GT(A λI )−1F (2.2) A N 7−→ − where IN denotes the N Neidentity matrix. The conventions here are such that all the × eigenvalues α are interior to a circle S1 in the complex λ-plane on which the loop i i=1,...n { } algebra elements X(λ) gl(r) are defined. The two subalgebras gl(r) ,gl(r) consist of + − ∈ elements X gl(r) , X gl(r) that admit holomorphic extensions, respectively, to the + + − − ∈ ∈ e e e interior and exterior regions, with X ( ) = 0. The space gl(r) is identified as a dense − ∞ subspace of its deual space gl(r)e∗, through the pairing e e< X ,X > = tr(X (λ)X (λ))dλ (2.3) 1 2 1 2 IS1 X gl(r)∗, X gl(r). 1 2 ∈ ∈ This also gives identifications of the dual speaces gl(r)∗± weith the opposite subalgebras gl(r)∓. Taking the simplest case, when A is diagonal, the image of the moment map is e e n N (λ) = GT(A λI )−1F = i , (2.4a) 0 N N − λ α i i=1 − X N := GTF , (2.4b) i i i − where (F ,G ) are the k r blocks in (F,G) corresponding to the eigenspaces of A with i i i × eigenvalues α . The set of all gl(r) having the pole structure given ineq. (2.4a) i i=1,...n 0 − { } N ∈ forms a Poisson subspace of gl(r) , which we denote g . The coadjoint action of the loop − A e group Gl(r) corresponding to the algebra gl(r) , restricted to the subspace g , is given by: + + A e f Ad∗(Gl(r)+) :egA gA −→ n N n g N g−1 Ad∗(g)f: i i i i (2.5) λ α −→ λ α i i i=1 − i=1 − X X g := g(α ), i = 1,...,n. i i 4 We see that g could equally have been identified with the dual space ( n gl(r))∗ of the A ⊕j=1 direct sum of n copies of gl(r) with itself, and the Lie Poisson bracket on gl(r)∗ gl(r) : + ∼ − f, g =< ,[df,dg] >, e e (2.6) N 0 N { }| 0 N | 0 reduces on the Poisson subspace g to that for ( n gl(r))∗, as given in eq. (1.10). A ⊕j=1 Inthe approach developed in[AHP, AHH1], one studies commuting Hamiltonianflows on spaces of type g (in general, rational Poisson subspaces involving higher order poles if A the matrix A is nondiagonalizable), generated by elements of the Poisson commuting spectral ring Y of polynomials on gl(r)∗ invariant under the Ad∗Gl(r)–action (conjugation by loop IA group elements), restricted to the affine subspace Y + g , where Y gl(r) is some fixed A ∈ e f r r matrix. The pullback of such Hamiltonians under J generates commuting flows in A × M that project to the quotient of M by the Hamiltonian action of the stability subgroup e G := Stab(A) Gl(N). The Adler-Kostant-Symes (AKS) theorem then tells us that: A ⊂ (i) Any two elements of Y Poisson commute (and hence, so do their pullbacks under the IA Poisson map J ). A (ii) Hamilton’s equations for H Y have the Lax pair form: e ∈ IA d N = [(dH) , ] = [(dH) , ], (2.7) + − dt N − N where (λ,t) := Y + (λ,t), (2.8) 0 N N with gl(r) of the form (2.4a), dH viewed as an element of (gl(r)∗)∗ gl(r), 0 − N N ∈ | ∼ and the subscripts denoting projections to the subspace gl(r) . ± ± e e e The coefficients of the spectral curve of (λ), determined by theecharacteristic equation N det(Y +GT(A λI )−1 zI ) = 0, (2.9) N r − − are the generators of the ring Y. IA In particular, choosing Y := diag(y ,...,y ) (2.10) 1 r and defining H Y by: { i ∈ IA}i=1,...n n 1 tr(N N ) H ( ) := tr(( (λ))2dλ = tr(YN )+ i j (2.11) i i N 4πi N α α Iλ=αi j=1 i − j Xj6=i 5 (where denotes integration around a small loop containing only this pole), we see that λ=αi these coincide with the H ’s defined in eq. (1.8a). Thus, the Poisson commutativity of the H i H ’s follows from the AKS theorem. The Lax form of Hamilton’s equations is: i ∂ N = [(dH ) , ] (2.12) i − ∂t − N i where N i (dH ) = gl(r) . (2.13) i − − λ α ∈ i − Evaluating residues at each λ = αi, we see that this iseequivalent to: ∂N [N ,N ] j j i = , j = i, i,j = 1,...n. (2.14a) ∂t α α 6 i j i − n ∂N N i j = [Y + ,N ]. (2.14b) i ∂t α α i i j j=1 − Xj6=i If we now identify the flow parameters t with the eigenvalues α , we i i=1,...n i i=1,...n { } { } obtain the nonautonomous Hamiltonian systems ∂N [N ,N ] j j i = , j = i, i,j = 1,...n, (2.15a) ∂α α α 6 i j i − n ∂N N i j = [Y + ,N ], (2.15b) i ∂α α α i i j j=1 − Xj6=i which are the α components of the system (1.1). Viewing the N ’s as functions on the i i fixed phase space M, eqs. (2.15a,b) are induced by the nonautonomous Hamiltonian sys- tems generated by the pullback of the H ’s under the parameter dependent moment map J . i A Eqs. (2.15a,b) are equivalent to replacing the Lax equations (2.12) by the system: e ∂ ∂(dH ) i − N = [(dH ) , ] , (2.16) i − ∂α − N − ∂λ i which is just the condition of commutativity of the system of operators , , where λ i i=1,...n {D D } is given by (1.3) and λ D ∂ ∂ N i := +(dH ) (λ) = + . (2.17) i i − D ∂α ∂α λ α i i i − Remark. The system (2.16) could also be viewed as a Lax equation defined on the dual of the centrally extended loop algebra gl(r)∧, in which the Ad∗ action is given by gauge e 6 transformations rather than conjugation [RS]. The analogue of the spectral ring Y is the IA ring of monodromy invariants, restricted to a suitable Poisson subspace with respect to a modified (R–matrix) Lie Poisson bracket structure. This viewpoint will not be developed here, but is essential to deriving such systems through reductions of autonomous Hamiltonian systems of PDE’s. The fact that the matrices F GT = L gl(k ) are constant under the defor- { i i i ∈ i }i=1,...n mations generated by the eqs. (2.15a,b), (2.16) follows from the fact that J (F,G) := diag(F GT,...,F GT) gl(N) (2.18) GA 1 1 n n ∈ is the moment map generating the Hamiltonian action of the stabilizer of A in Gl(N): N G := Gl(k ) = Stab(A) Gl(N), (2.19) A i ⊂ i=1 Y this action being given by G : M M A −→ K : (F,G) (KF,(KT)−1G) (2.20) −→ K = diag(K ,...,K ) G , K Gl(k ). 1 n A i i ∈ ∈ The orbits of G are just the fibres of J , so (J ,J ) form a “dual pair” of moment maps A A A G A [W]. Evidently, thepullback J∗(H)isG –invariant forallH Y, and hence J isconstant A A ∈ IA GA e e under the H flows. i e So far, we have only considered the part of the system (1.1) relating to the parameters α . What about the Hamiltonians K that generate the y components? As i i=1,...n a a=1,...r a { } { } shown in [AHH1], besides J there is, for each Y gl(r), another moment map A ∈ e J : M gl(N)∗ gl(N) Y −→ + ∼ − J : (F,G) F(Y zI )−1GT, (2.21) Y r e 7−→ −e − e where z denotes the loop paraemeter for the loop algebra gl(N), whose elements are defined on a circle S1 in the complex z-plane containing the eigenvalues of Y in its interior. The pairing identifying gl(N) as a dense subspace of gl(N)∗ is defiened similarly to (2.3), for elements X gl(N)∗, X gl(N). The subalgebras gl(N) are similarly defined with respect to this 1 2 ± ∈ ∈ circle, and teheir dual spaces gl(N)∗ are ideentified analogously with gl(N) . ± ∓ e e e The moment map J is also “dual” to J , but in a different sense than J – one that Y e A e GA is relevant for the construction of the remaining Hamiltonians K . The space g a a=1,...r A { } e e 7 may be identified with the quotient Poisson manifold M/G , with symplectic leaves given by A the level sets of the symmetric invariants formed from each F GT, since these are the Casimir i i invariants on g . Since the Hamiltonians in Y are all also invariant under the action of the A IA stabilizer G = Stab(Y) Gl(r), where Gl(r) Gl(r) is the subgoup of constant loops, Y ⊂ ⊂ we may also quotient by this action to obtain g /G = M/(G G ). Doing this in the A Y Y A × opposite order, we may define g gl(N)∗ gl(N)f as the Poisson subspace consisting of Y ⊂ + ∼ − elements of the form: e e r M (z) = F(Y zI )−1GT = a , (2.22a) 0 N M − − z y a a=1 − X (M ) := F G , i,j = 1,...,n, a = 1,...r (2.22b) a ij ia ja (where, if the y are distinct, the residue matrices are all of rank 1), and identify g a a=1,...r Y { } with M/G . Denoting by A the ring of Ad∗-invariant polynomials on gl(N)∗, restricted to Y IY the affine subspace A+g consisting of elements of the form Y − e = A+ , g , (2.23) 0 0 Y M − M M ∈ the pullback of the ring A under the moment map J also gives a Poisson commuting ring IY Y whose elements are both G and G invariant, and hence project to M/(G G ). In fact, Y A Y A × the two rings J∗( Y) and J∗( A) coincide (cf. [AHHe1]), because of the identity: A IA Y IY det(A λI )det(Y +GT(A λI )−1F zI ) = det(Y zI )det(A+F(Y zI )−1GT λI ), N e e N r r r N − − − − − − (2.24) which shows that the spectral curve of (z), defined by M det(A+F(Y zI )−1GT λI ) = 0 (2.25) r N − − and that of (λ), given by eq. (2.9), are birationally equivalent (after reducing out the trivial N factors det(A λI ) and det(Y zI )). N r − − Now, similarly to the definition of the elements H Y , we may define K { i ∈ IA}i=1,...n { a ∈ A as: IY }a=1,...r r 1 tr(M M ) K := tr( (z))2dz = tr(AM )+ a b (2.26) a a 4πi M − y y Iz=ya b=1 a − b Xb6=a To verify that these coincide with the K ’s defined in eq. (1.8b), we use eqs. (2.4a), (2.8), a (2.22a) and (2.23) to express K as: a 1 1 K = dz dλλtr( (z)(A λI )−1)2 (2.27a) a N 4πi 2πi M − Iz=ya IS1 1 1 = dλλ dz tr((Y zI )−1 (λ))2 2tr (Y zI )−1 (λ) .(2.27) r r 4πi 2πi − N − − N IS1 Iz=ya (cid:2) (cid:0) (cid:1)(cid:3) 8 Evaluating residues at y in z and at in λ gives (1.8b). The Poisson commutativity a a=1,...r { } ∞ of the K ’s again follows from the AKS theorem, and the commutativity with the H ’s follows a i from the equality of the two rings J∗ Y = J∗ A. To compute the Lax form of the equations AIA YIY of motion generated by the K ’s, we evaluate their differentials, viewing them as functions of a e e (λ) defined by eq. (2.27). Evaluating the z integral, this gives: N r E (λ)E +E (λ)E a b b a dK (λ) = λE + N N gl(r) (2.28) a a y y ∈ a b b=1 − Xb6=a e where E denotes the elementary diagonal r r matrix with (aa) entry equal to 1 and zeroes a × elsewhere. Taking the projection to gl(r) gives: + r n e E N E +E N E a i b b i a (dK ) (λ) = λE + gl(r) , (2.29) a + a + y y ∈ a b b=1 i=1 − Xb6=a X e and hence r (dK ) dy = λY +Θ, (2.30) a + a a=1 X where Θ is defined in eq. (1.2). By the AKS theorem, the autonomous form of the equations of motion is ∂ N = [(dK ) , ], (2.31) a + ∂τ N a while the nonautonomous version is ∂ ∂(dK ) a + N = [(dK ) , ]+ = [(dK ) , ]+E . (2.32) a + a + a ∂y N ∂λ N a Evaluating residues at λ = α gives the equations i ∂N i = [(dK ) (α ),N ], i = 1,...,n (2.33) a + i i ∂τ a for the autonomous case and ∂N i = [(dK ) (α ),N ], i = 1,...,n (2.34) a + i i ∂y a for the nonautonomous one. Eqs. (2.34) are just the y components of the system (1.1). a Eq. (2.32) is equivalent to the commutativity of the operators , ∗ , where λ a a=1,...r {D D } ∂ ∗ := (dK ) (λ), (2.35) Da ∂y − a + a 9 and implies that the monodromy of is invariant under the y deformations. In fact, it fol- λ a D lows from the AKS theorem that the complete system of operators , , ∗ λ i a i=1,...n,a=1,...r {D D D } commutes. Turning now to the dual system, it follows from the AKS theorem that the Lax form of the equations of motion induced by the K ’s on gl(N) , viewed now as functions of , in a − M the autonomous case is ∂ e M = [(dK ) , ], (2.36) a − ∂τ − M a where M a (dK ) (z) = gl(N) . (2.37) a − − z y ∈ a − (Note that the differential dK entering in eqs. (2.36),e(2.37) and below has a different signif- a icance from that appearing in eqs. (2.28)–(2.35).) Evaluating residues at z = y shows that a (2.36) is equivalent to the system ∂M [M ,M ] b b a = , b = a, a,b = 1,...r, (2.38a) ∂τ y y 6 a b a − n ∂M M a b = [ A+ ,M ]. (2.38b) a ∂τ − y y a a b b=1 − Xb6=a Identifying the flow parameters τ now with the eigenvalues y of Y gives a a=1,...r a a=1,...r { } { } the nonautonomous Hamiltonian system ∂M [M ,M ] b b a = , b = a, a,b = 1,...r, (2.39a) ∂y y y 6 a b a − n ∂M M a b = [ A+ ,M ]. (2.39b) a ∂y − y y a a b b=1 − Xb6=a or, equivalently, ∂ ∂(dK ) a − M = [(dK ) , ] . (2.40) a − ∂y − M − ∂z a Equations (2.39a,b), (2.40) are equivalent to the commutativity of the system of operators , defined by: z a a=1,...r {D D } ∂ := (z), (2.41a) z D ∂z −M ∂ ∂ M a := +(dK ) (z) = + , a = 1,...,r. (2.41b) a a − D ∂y ∂y z y a a a − Thus the monodromy of the “dual” operator , is also preserved under the y deformations. z a D 10