aa r X i v : . [ m a t h - ph ] D ec A REMARK ON DEFORMATIONS OF HURWITZFROBENIUS MANIFOLDS
A. BURYAK AND S. SHADRIN
Abstract.
In this note we use the formalism of multi-KP hi-erarchies in order to give some general formulas for infinitesimaldeformations of solutions of the Darboux-Egoroff system. As anapplication, we explain how Shramchenko’s deformations of Frobe-nius manifold structures on Hurwitz spaces fit into the general for-malism of Givental-van de Leur twisted loop group action on thespace of semi-simple Frobenius manifolds.
Contents
1. Introduction 11.1. Organization of the paper 21.2. Acknowledgements 22. Frobenius structures associated to Hurwitz spaces 22.1. Darboux-Egoroff equations 22.2. Hurwitz spaces 32.3. Shramchenko’s deformations 33. Van de Leur’s formalism for Frobenius manifolds 43.1. Basic definitions 43.2. Infenitesimal deformations 54. Special deformations 74.1. The input 74.2. Special deformations 74.3. Shramchenko’s formulas 9References 91.
Introduction
In [1], Dubrovin has associated a structure of Frobenius manifoldsto an arbitrary Hurwitz space of meromorphic functions on Riemannsurfaces of genus g with simple finite critical values and a prescribedramification indices over infinity. Shramchenko observed [10] that thestructure of Frobenius manifold associated to a Hurwitz space can beincluded into a family of Frobenius manifold structures parametrized A. B is partially supported by the grants RFBR-07-01-00593, NSh-709.2008.1.Both A. B. and S. S. are partly supported by the Vidi grant of NWO. by a symmetric g × g matrix. There is a beautiful description of thisdeformation in terms of the values of holomorphic differentials at thecritical points and their B -periods matrix.Meanwhile, Givental in [2, 3] and, independently, van de Leur in [7]have constructed an action of the twisted loop group of GL n on thespace of semi-simple Frobenius manifolds. Moreover, Givental hasshown that this group acts transitively on the space semi-simple Frobe-nius manifolds. This two constructions of the group action were iden-tified in [4] via an identification of the formulas of Y.-P. Lee for theinfinitesimal Givental action [6] with the tangent van de Leur actioncomputed in [4] in terms of twisted wave functions of multi-componentKP hierarchies.In this paper, we extend in some way the formulas for the tangent vande Leur action computed in [4]. Namely, we express infinitesimal Liealgebra action on the space of solutions of the Darboux-Egoroff systemin terms of the twisted wave functions of multi-component KP. In prin-ciple, these formulas are of independent interest. In particular, theyallow us to fit Shramchenko’s deformations into a general Givental-vande Leur scheme. In particular, it is interesting to trace a corrsepon-dence between geometric ingridients of Shramchenko’s deformation andparticular wave functions of the multi-component KP hierarchy that isassociated to an arbitrary solution of the Darboux-Egoroff system invan de Leur’s approach .1.1. Organization of the paper.
In section 2, we recall the con-structions of Hurwitz Frobenius manifolds and their deromations. Insection 3, we recall the van de Leur approach to Frobenius manifols anduse it in order to derive explicit formulas for the Givental-van de Leurinfinitesimal deformations of solutions of the Darboux-Egoroff equa-tions. In section 4 we discuss the simplest possible example of suchinfinitesimal deformations that can be integrated explicitely and showthat it is exactly the way one could obtain Shramchenko’s deformationsof Hurwitz Frobenius manifolds.1.2.
Acknowledgements.
The authors are grateful to H. Posthumafor a useful discussion.2.
Frobenius structures associated to Hurwitz spaces
Darboux-Egoroff equations.
In this paper we consider onlysemi-simple Frobenius manifolds. There is a way to encode the struc-ture of a semi-simple Frobenius manifold in canonical coordinates asa solution of a system of PDEs that is called Darboux-Egoroff equa-tions [1].
N DEFORMATIONS OF HURWITZ FROBENIUS MANIFOLDS 3
Let n ≥
1. We consider functions γ ij = γ ji , i, j = 1 , . . . , n , i = j , invariables u , . . . , u n . The Darboux-Egoroff equations read: ∂γ ij ∂u k = γ ik γ kj , i = j = k = i (1) n X k =1 ∂γ ij ∂u k = 0 i = j Is it convenient to collect γ ij into a symmetric matrix with the diag-onal terms that can be either equal to 0 or just arbitrary. We introducea special notation for that. Let M be a symmetric matrix. By n . d .M we denote the same matrix with non-specified diagonal terms.2.2. Hurwitz spaces.
We fix some integer numbers a , . . . , a m > g ≥
0. Let H be the space of the equivalence classes of the tuplesof data ( C g , { a i , b i } gi =1 , f : C g → C P ), where C g is a Riemann surfaceof genus g , { a i , b i } gi =1 is a choice of the canonical basis of cycles on C g ,and f : C g → C P is a meromorphic function of degree d := P mi =1 a m with exactly m poles of multiplicity a , . . . , a m and n := 2 g + d + m − x , . . . , x n ∈ C g . In addition, we choose localparameters z , . . . , z n at the points x , . . . , x n ∈ C g such that f = z i ina neighbourhood of x i . Two tuples of this data are equivalent if thereis a biholomorphic map between two source curves that preserves therest of the data.The critical values of meromorphic functions u i := f ( x i ), i = 1 , . . . , n ,are local coordinates on the space H .We recall the Kokotov-Korotkin construction [5] of a solution of theDarboux-Egoroff equations. Let W ( P, Q ) be the canonical meromor-phic bidifferential on a Riemann surface C g . That is, W ( P, Q ) is spec-ified by the following properties: it is symmetric, it has a quadraticpole on the diagonal P = Q with biresidue 1, and its a -periods withrespect to both variables vanish. Then the functions(2) γ ij := 12 W ( x i , x j ) := 12 W ( P, Q ) dz i ( P ) dz j ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) P = x i ,Q = x j in variables u , . . . , u n satisfy the Darboux-Egoroff equations.2.3. Shramchenko’s deformations.
Let ω i , i = 1 , . . . , n , be the ba-sis of holomorphic differentials on C g normalized by R a i ω j = δ ij . De-note by ω the matrix of the values of ω i at critical points:(3) ω ij := ω i ( x j ) := ω i ( P ) dz j ( P ) (cid:12)(cid:12)(cid:12)(cid:12) P = x j Denote by B the matrix of b -periods of these differentials divided by π √− B ij := π √− R b i ω j . Let M be an arbitrary g × g symmetric ma-trix such that B + M is non-degenerate. Shramchenko’s deformations A. BURYAK AND S. SHADRIN of Hurwitz Frobenius manifolds [10] are given by the formula(4) n . d .γ ( M ) := n . d . (cid:0) γ − ω t ( B + M ) − ω (cid:1) . Here n . d .γ is given by equation (2). Shramchenko proved that γ ij ( M )are solutions of the Darboux-Egoroff equations in the variables u , . . . , u n in the domain det( B + M ) = 0. Observe that n . d .γ ( M ) tends to n . d .γ when ( B + M ) − tends to zero.The proof that n . d .γ ( M ) is a solution of the Darboux-Egoroff equa-tions is based on Rauch variational formula and its corollaries: ∂W ( P, Q ) ∂u j = 12 W ( P, x j ) W ( Q, x j ) , (5) ∂ω i ( P ) ∂u j = 12 ω i ( x j ) W ( P, x j ) , (6) ∂B kl ∂u j = ω k ( x j ) ω l ( x j ) , (7)where evaluation of differentials at particular points is defined in (2)and (3).3. Van de Leur’s formalism for Frobenius manifolds
In this section we explain van de Leur’s construction of a Frobeniusstructure associated to a point in the isotropic semi-infinite Grassman-nian.3.1.
Basic definitions.
Let V = h e , . . . , e n i be an n -dimensional vec-tor space over C . Let z be a formal variable. We denote by V thevector space Λ ∞ / ( V ⊗ C [ z − , z ]) spanned by the semi-infinite wegdeproducts ω = ( e i z d ) ∧ ( e i z d ) ∧ ( e i z d ) ∧ . . . such that the tail of ω coinsides with the tail of vacuum vector | i := ( e z ) ∧ · · · ∧ ( e n z ) ∧ ( e z ) ∧ · · · ∧ ( e n z ) ∧ . . . . By tail of ω we call another basis vector in V that is obtained from ω by removing the first few factors in the wedge product.Consider a matrix series A ( z ) ∈ End ( V ) ⊗ C [[ z − , z ]] such that A t ( − z ) A ( z ) = Id · z (it is better to imagine it as a finite productof invertible matrix series in End ( V ) ⊗ C [[ z ]] and End ( V ) ⊗ C [[ z − ]]satisfying the same symplectic condition).Let α i be a local Lie algebra element whose action on V ⊗ C [ z − , z ]is defined by α i ( e j z d ) := (cid:26) e j z d +1 if i = j, V by the Leibnitz rule. N DEFORMATIONS OF HURWITZ FROBENIUS MANIFOLDS 5
All basic objects that we are going to consider are some matrix ele-ments of the operator A := exp( n X i =1 α i u i ) A ( z ) , where u , . . . , u n are formal variables.We denote by γ ij = γ ij ( A ), i, j = 1 , . . . , n , i = j , the following matrixelements of A : γ ij := ± (cid:10) | i (cid:12)(cid:12) A (cid:12)(cid:12) ( e i z − ) ∧ ∂ ( e j z ) | i (cid:11) h| i |A || ii . (the vector ( e i z − ) ∧ ∂ ( e j z ) | i is obtained, up to a sign, from the vacuumvector | i by the replacement of ( e j z ) by ( e i z − )).We denote by (Ψ d ) ij = (Ψ d ) ij ( A ), i, j = 1 , . . . , n , the following ma-trix elements of A :(Ψ d ) ij := (cid:10) ( e j z − ) ∧ | i (cid:12)(cid:12) A (cid:12)(cid:12) ( e i z − − d ) ∧ | i (cid:11) h| i |A || ii . These matrices are can be arranged into a generating series Ψ( z ) := P ∞ d =1 z d Ψ d that would be a wave function of multi-KP hierarchy multi-plied by A ( z ) from the right. The property A t ( − z ) A ( z ) = Id · z implythat Ψ t ( − z )Ψ( z ) = Id · z .Van de Leur has shown in [7] that a formal locally semi-simple Frobe-nius structure can expressed in terms of the matrices γ and Ψ d , d ≥ γ ij is a solution of the Darboux-Egoroff system; u , . . . , u n are canonical coordinates; Ψ t Ψ is a column of flat coordinates (here is the column of units); and (1 / · t Ψ t ( − Ψ Ψ t + Ψ Ψ t )Ψ is theprepotential of a Frobenius manifold.3.2. Infenitesimal deformations.
From the previous section we seethat there is an action of the groups of matrices A ( z ) ∈ Hom ( V, V ) ⊗ C [[ z ]], A t ( − z ) A ( z ) = Id · z , and A ( z − ) ∈ Hom ( V, V ) ⊗ C [[ z − ]], A t ( − z − ) A ( z − ) = Id · z . This group action is crucially important,see, e.g., [4] for a list of references for particular applications.We discuss the corresponding Lie algebra action. Let k ≥ ℓ >
0. Let matrices r and s be symmetric for odd ℓ and skewsymmetricfor even ℓ . It is proven in [4] that ∂∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 Ψ k ( A exp ǫ ( rz − ℓ )) =(8) Ψ ℓ + k ( A ) r − ℓ X p =1 ℓ − p X q =0 ( − ℓ − p − q Ψ q ( A ) r Ψ tℓ − p − q ( A )Ψ p + k ( A ); ∂∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 Ψ k ( A exp ǫ ( sz ℓ )) = ( Ψ k − ℓ ( A ) s, ℓ ≤ k ;0 , ℓ > k (9) A. BURYAK AND S. SHADRIN
This allows to compute the action of this Lie algebra on the prepotentialof Frobenius manifolds in flat coordinates, since both the prepotentialand the flat coordinates are expressed in terms of Ψ d , d ≥ n . d .γ ( A ). Theorem 3.1.
Let ℓ ≥ . Let matrices r and s be symmetric for odd ℓ and skewsymmetric for even ℓ . We have: ∂∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 n . d .γ ( A exp ǫ ( rz − ℓ )) = n . d . X i + j = ℓ − ( − j − Ψ i ( A ) r Ψ tj ( A )(10) ∂∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 n . d .γ ( A exp ǫ ( sz ℓ )) = 0 . (11) Proof.
This theorem is an easy consequence of formulas (8) and (9).Indeed, there is a relation between Ψ d , d ≥ n . d .γ that is provenin [7]. For any d ≥ k = 1 , . . . , n , we have:(12) ∂∂u i Ψ d = E kk Ψ d − + [ n . d .γ, E kk ]Ψ d Here and below we assume that Ψ − = 0 and we use γ with arbitrarydiagonal terms since they disappear in the commutator with E kk . By E kk we denote the matrix unit, that is ( E kk ) ij := δ ik δ jk . This formulagives an expression for all elements of n . d .γ in terms of Ψ and itsderivatives.We combine equations (8) and (12): ∂∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 ∂∂u i Ψ ( A exp ǫ ( rz − ℓ )) = [ ∂∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 γ ( A exp ǫ ( rz − ℓ )) , E kk ]Ψ ( A )+ [ γ ( A ) , E kk ] ∂∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 Ψ ( A exp ǫ ( rz − ℓ ))We denote ∂∂ǫ (cid:12)(cid:12) ǫ =0 γ ( A exp ǫ ( rz − ℓ )) by δγ . Using that Ψ t ( − z )Ψ( z ) =Id · z , we obtain the following equation: ∂∂u i Ψ ℓ r + ℓ − X q =0 ( − ℓ − q Ψ q r Ψ tℓ − q Ψ ! =[ δγ, E kk ]Ψ + [ γ, E kk ] Ψ ℓ r + ℓ − X q =0 ( − ℓ − q Ψ q r Ψ tℓ − q Ψ ! Using equation (8), we see that[ δγ, E kk ]Ψ = [ E kk , ℓ − X i =0 ( − i Ψ ℓ − − i r Ψ i ]Ψ . N DEFORMATIONS OF HURWITZ FROBENIUS MANIFOLDS 7
Since Ψ is invertible, Ψ Ψ t = 0, we obtain equation (10). Equa-tion (11) can be proven in the same way, but in fact it is obvious fromthe definition of γ . (cid:3) Example 3.2.
The simplest non-trivial deformation would be by anelement rz − , where r is an arbitrary symmetric matrix. Denote by δ r γ and δ r Ψ d the corresponding infinitesimal deformations. We havethe following system of equations: n . d .δ r γ = − n . d . Ψ r Ψ t (13) δ r Ψ = Ψ r − Ψ r Ψ t Ψ δ r Ψ = Ψ r − Ψ r Ψ t Ψ and so on.4. Special deformations
Shramchenko’s deformations of Hurwitz Frobenius manifolds dis-cussed in section 2.3 fits into a special case of example 3.2 that canbe integrated explicitely.4.1.
The input.
Consider a symmetric n × n matrix n . d .γ whose el-ements are functions in u , . . . , u n . Let n . d .γ be a solution of theDarboux-Egoroff equations (1). There are two important geometricstructures associated to the Frobenius structure corresponding to n . d .γ .First, there is a solution of the commutativity equations [8], whichis a symmetric n × n matrix C = C ( u , . . . , u n ) such that dC ∧ dC = 0.In terms of multi-KP tau-functions, C = Ψ t Ψ .Second, one can consider Ψ itself. In geometric terms Ψ is definedby the equation dC = Ψ t · diag ( du , . . . , du n ) · Ψ . An alternative wayto define Ψ is the following. Consider the system of equations (it isequivalent to equation (12)): ∂ (Ψ ) ij ∂u k = γ ik (Ψ ) kj , i = k, (14) n X k =1 ∂ (Ψ ) ij ∂u k = 0 . (15)Compatibility of this system of equations follows from the Darboux-Egoroff equations for n . d .γ . This system of equations implies that ∂ (Ψ t Ψ ) /∂u k = 0, k = 1 , . . . , n , and Ψ that we need is a particularsolution of this system of equations such that Ψ t Ψ = id.4.2. Special deformations.
We consider a distribution in the tangentbundle of the moduli space of the solutions of the Darboux-Egoroffequations. It is given by the Givental-van de Leur tangent vectors ofthe type (13) described in example 3.2. It is easy to see that, roughlyspeaking, a deformation of a particular solution of the Darboux-Egoroff
A. BURYAK AND S. SHADRIN equations is given by an ordinary differential equations of the infiniteorder.However, there is a special class of infinitesimal deformations thatcan be reduced to a finite order ODEs. We fix a positive integer g ≤ n/
2. Let D be a g × n constant matrix of rank g such that DD t = 0. Letus consider the distribution in the tangent bundle of the moduli space ofthe solutions of the Darboux-Egoroff equations given by the Givental-van de Leur tangent vectors of the type (13) described in example 3.2with the matrix r that can be represented as r = D t M D , where M isan arbitrary symmetric g × g matrix. In that case equation (13) canbe reduced to an ODE of finite order. Proposition 4.1.
Equation (13) for the matrix r = D t M D impliesthe following system of ODEs for n . d .γ , ω := D Ψ t , and B := DCD t : n . d .δ M γ = − n . d .ω t M ω ;(16) δ M ω = − BM ω ; δ M B = − BM B.
Proof.
Direct computation. (cid:3)
In order to use this proposition for a particular n . d .γ without goingback to the full multi-KP framework, we need an independent defini-tions of ω and B in terms of γ and D . We define ω as a g × n -matrix-valued solutions of the equation(17) dω = ω · [ diag ( du , . . . , du n ) , n . d .γ ]with the constant term ω | u =0 = D Ψ t | u =0 . We define B to be a g × g -matrix-valued solution of the equation(18) dB = ω · diag ( du , . . . , du n ) · ω t with the constant term B | u =0 = DCD t | u =0 .Equations (16) can be integrated explicitely in the case when M is aconstant matrix independent of n . d .γ , ω , and B . Indeed, let us define n . d .γ ( ǫ ), ω ( ǫ ), and B ( ǫ ) by the following formulas: n . d .γ ( ǫ ) := n . d .γ − n . d .ω t ǫM (1 + ǫBM ) − ω ;(19) ω ( ǫ ) = (1 + ǫBM ) − ω ; B ( ǫ ) = (1 + ǫBM ) − B. (these formulas are defined in the domain where (1 + ǫBM ) is invert-ible). Proposition 4.2.
The matrices n . d .γ ( ǫ ) , ω ( ǫ ) , and B ( ǫ ) satisfy equa-tions (17) and (18) for any ǫ ≥ . They integrate the constant vector N DEFORMATIONS OF HURWITZ FROBENIUS MANIFOLDS 9 field determined by the matrix M , that is, n . d . ∂γ ( ǫ ) ∂ǫ = − n . d .ω ( ǫ ) t M ω ( ǫ ); n . d .γ ( ǫ ) | ǫ =0 = n . d .γ ; ∂ω ( ǫ ) ∂ǫ = − B ( ǫ ) M ω ( ǫ ); ω ( ǫ ) | ǫ =0 = ω ; ∂B ( ǫ ) ∂ǫ = − B ( ǫ ) M B ( ǫ ); B ( ǫ ) | ǫ =0 = B ; Proof.
Direct computation. (cid:3)
Shramchenko’s formulas.
In this context, Shramchenko’s for-mulas are a version of formulas (16) for ǫ = 1, with some appropriatechanges. Let use n . d .γ , ω , and B defined in section 2.3. Equations (5)-(7) imply Darboux-Egoroff equations for n . d .γ and equations (17)-(18).Therefore, we are indeed have a system suitable for deformation givenby (16) (the initial conditions for ω and B depend on the choice of aparticular point of a formal expansion). Indeed, let us substitute ǫ = 1in equation (19). We have: n . d .γ ( ǫ ) | ǫ =1 = n . d .γ ( ǫ ) | ǫ =0 − n . d .ω t ( M − + B ) − ω. If we change the notations in order to replace M − with M , we obtainexactly formula (4). Remark 4.3.
One could obtain the same solution of the Darboux-Egoroff equations from a special deformation of the following triple: n . d . ˜ γ := γ − ω t Bω , ˜ ω := B − ω , and ˜ B := B − . In that case someformulas would look a bit simpler. Remark 4.4.
Deformations of “real doubles” [9] of Hurwitz Frobeniusmanifolds fit into exactly the same scheme as we discuss in section 4.2.
References [1] B. Dubrovin, Geometry of 2D topological field theories, in: Integrable systemsand quantum groups (Montecatini Terme, 1993), 120–348, Lecture Notes inMath. , Springer, Berlin, 1996.[2] A. Givental, Semisimple Frobenius structures at higher genus, Int. Math. Res.Not. , no. 23, 1265–1286.[3] A. Givental, Gromov-Witten invariants and quantization of quadratic Hamil-tonians, Mosc. Math. J. (2001), no. 4, 551–568.[4] E. Feigin, J. van de Leur, S. Shadrin, Givental symmetries of Frobenius man-ifolds and multi-component KP tau-functions, Adv. Math., to appear, arXiv:0905.0795, 25 pp.[5] A. Kokotov, D. Korotkin, A new hierarchy of integrable systems associated toHurwitz spaces, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. (2008), no. 1867, 1055–1088.[6] Y.-P. Lee, Invariance of tautological equations I: conjectures and applications,J. Eur. Math. Soc. (JEMS) (2008), no. 2, 399–413. [7] J. van de Leur, Twisted GL n loop group orbit and solutions of the WDVVequations, Int. Math. Res. Not. , no. 11, 551–573.[8] S. Shadrin, D. Zvonkine, A group action on Losev-Manin cohomological fieldtheories, arXiv: 0909.0800, 21 pp.[9] V. Shramchenko, “Real doubles” of Hurwitz Frobenius manifolds, Comm.Math. Phys. (2005), no. 3, 635–680.[10] V. Shramchenko, Deformations of Frobenius structures on Hurwitz spaces, Int.Math. Res. Not. , no. 6, 339–387. A. Buryak:Department of Mathematics, University of Amsterdam,P. O. Box 94248, 1090 GE Amsterdam, The NetherlandsandDepartment of Mathematics, Moscow State University,Leninskie gory, 19992 GSP-2 Moscow, Russia
E-mail address : [email protected], [email protected] S. Shadrin:Department of Mathematics, University of Amsterdam,P. O. Box 94248, 1090 GE Amsterdam, The NetherlandsandDepartment of Mathematics, Institute of System Research,Nakhimovsky prospekt 36-1, Moscow 117218, Russia
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