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116
Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and GromovWitten invariants
, 2001
"... We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their ..."
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Cited by 44 (2 self)
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We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov Witten classes and their descendents.
Moduli spaces of higher spin curves and integrable hierarchies
 Compositio Math
"... Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin curves assemble into a generating function which yields a solution of the s ..."
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Cited by 37 (8 self)
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Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdVr equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r −1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity Ar−1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of GromovWitten invariants and quantum cohomology. The moduli space of stable curves of genus g with n marked points Mg,n is a fascinating object. Mumford [37] introduced tautological cohomology classes associated to the universal curve Cg,n
Complex curve of the two matrix model and its taufunction
 J. Phys. A
, 2003
"... We study the hermitian and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports. Its planar graph interpretation is given. The study reveals a general structure of the underlying analytic complex curve, different from the hyperelliptic curve of the one matr ..."
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Cited by 31 (3 self)
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We study the hermitian and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports. Its planar graph interpretation is given. The study reveals a general structure of the underlying analytic complex curve, different from the hyperelliptic curve of the one matrix model. The matrix model quantities are expressed through the periods of meromorphic generating differential on this curve and the partition function of the multiple support solution, as a function of filling numbers and coefficients of the matrix potential, is shown to be a quasiclassical taufunction. The relation to N = 1 supersymmetric YangMills theories is discussed. A general class of solvable multimatrix models with treelike interactions is considered.
A class of Einstein–Weyl spaces associated to an integrable system of hydrodynamic type
 J. Geom. Phys
, 2004
"... HyperCR Einstein–Weyl equations in 2+1 dimensions reduce to a pair of quasilinear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydro ..."
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Cited by 28 (11 self)
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HyperCR Einstein–Weyl equations in 2+1 dimensions reduce to a pair of quasilinear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions yield new Einstein–Weyl structures. 1 The Equation Let us consider a pair of quasilinear PDEs ut + wy + uwx − wux = 0, uy + wx = 0, (1.1) for two real functions u = u(x, y, t), w = w(x, y, t). This system of equation has recently attracted a lot of attention in the integrable systems literature [18, 9, 10, 17]. In [3] it arouse in a different context, as a symmetry reduction of the heavenly equation. The system (1.1) shares many properties with two more prominent dispersionless integrable equations: the dispersionless Kadomtsev–Petviashvili equation (dKP), and the SU(∞) Toda equation, but it is simpler in some ways. • Its Lax representation
Hierarchies of Isomonodromic Deformations and Hitchin Systems
, 1998
"... We investigate the classical limit of the KnizhnikZamolodchikovBernard equations, considered as a system of nonstationar Schröodinger equations on singular curves, where times are the moduli of curves. It has a form of reduced nonautonomous hamiltonian systems which include as particular examples ..."
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Cited by 22 (3 self)
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We investigate the classical limit of the KnizhnikZamolodchikovBernard equations, considered as a system of nonstationar Schröodinger equations on singular curves, where times are the moduli of curves. It has a form of reduced nonautonomous hamiltonian systems which include as particular examples the Schlesinger equations, Painlevé VI equation and their generalizations. In general case, they are defined as hierarchies of isomonodromic deformations (HID) with respect to changing the moduli of underling curves. HID are accompanying with the Whitham hierarchies. The phase space of HID is the space of flat connections of G bundles with some additional data in the marked points. HID can be derived from some free field theory by the hamiltonian reduction under the action of the gauge symmetries and subsequent factorization with respect to diffeomorphisms of curve. This approach allows to define the Lax equations associated with HID and the linear system whose isomonodromic deformations are provided by HID. In addition, it leads to description of solutions of HID by the projection method. In some special limit HID convert into the Hitchin systems. In particular, for SL(N,C) bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero Nbody system.
Differentialgeometric approach to the integrability of hydrodynamic chains: the Haantjes tensor
"... The integrability of an mcomponent system of hydrodynamic type, ut = V (u)ux, by the generalized hodograph method requires the diagonalizability of the m × m matrix V (u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to ..."
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Cited by 20 (7 self)
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The integrability of an mcomponent system of hydrodynamic type, ut = V (u)ux, by the generalized hodograph method requires the diagonalizability of the m × m matrix V (u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains — infinitecomponent systems of hydrodynamic type for which the ∞× ∞ matrix V (u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is welldefined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semiHamiltonian hydrodynamic reductions, thus providing an easytoverify necessary condition for the integrability.
On the integrable geometry of soliton equations and N=2 supersymmetric gauge theories
 hepth/9604199, J. Diff. Geom
, 1997
"... We provide a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric YangMills theories and soliton equations. Their phase spaces are Jacobiantype bundles over the leaves of a foliation in a universal configuration space. On one hand, imbedded into finit ..."
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Cited by 18 (2 self)
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We provide a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric YangMills theories and soliton equations. Their phase spaces are Jacobiantype bundles over the leaves of a foliation in a universal configuration space. On one hand, imbedded into finitegap solutions of soliton equations, these symplectic forms assume explicit expressions in terms of the auxiliary Lax pair, expressions which generalize the wellknown GardnerFaddeevZakharov bracket for KdV to a vast class of 2D integrable models; on the other hand, they determine completely the effective Lagrangian and BPS spectrum when the leaves are identified with the moduli space of vacua of an N=2 supersymmetric gauge theory. For SU(Nc) with Nf ≤ Nc + 1 flavors, the spectral curves we obtain this way agree with the ones derived by Hanany and Oz and others from physical considerations.
Extended SeibergWitten Theory and Integrable Hierarchy
"... The prepotential of the effective N = 2 superYangMills theory, perturbed in the ultraviolet by the descendents ∫ d4θ trΦk+1 of the singletrace chiral operators, is shown to be a particular taufunction of the quasiclassical Toda hierarchy. In the case of noncommutative U(1) theory (or U(N) theory ..."
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Cited by 16 (1 self)
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The prepotential of the effective N = 2 superYangMills theory, perturbed in the ultraviolet by the descendents ∫ d4θ trΦk+1 of the singletrace chiral operators, is shown to be a particular taufunction of the quasiclassical Toda hierarchy. In the case of noncommutative U(1) theory (or U(N) theory with 2N −2 fundamental hypermultiplets at the appropriate locus of the moduli space of vacua) or a theory on a single fractional D3 brane at the ADE singularity the hierarchy is the dispersionless Toda chain, and we present its explicit solution. Our results generalise the limit shape analysis of LoganSchepp and VershikKerov, support the prior work [1], which established the equivalence of these N = 2 theories with the topological A string on CP 1, and clarify the origin of the EguchiYang matrix integral. In the higher rank case we find an appropriate variant of the quasiclassical taufunction, show how the SeibergWitten curve is deformed by Toda flows, and fix the contact term ambiguity.
INSTANTONS IN NONCRITICAL STRINGS FROM THE TWOMATRIX MODEL
"... We derive the nonperturbative corrections to the free energy of the twomatrix model in terms of its algebraic curve. The eigenvalue instantons are associated with the vanishing cycles of the curve. For the (p, q) critical points our results agree with the geometrical interpretation of the instanto ..."
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Cited by 15 (1 self)
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We derive the nonperturbative corrections to the free energy of the twomatrix model in terms of its algebraic curve. The eigenvalue instantons are associated with the vanishing cycles of the curve. For the (p, q) critical points our results agree with the geometrical interpretation of the instanton effects recently discovered in the CFT approach. The form of the instanton corrections implies that the linear relation between the FZZT and ZZ disc amplitudes is a general property of the 2D string theory and holds for any classical background. We find that the agreement with the CFT results holds in the presence of infinitesimal perturbations by order operators and observe that the ambiguity in the interpretation of the eigenvalue instantons as ZZbranes (four different choices for the matter and Liouville boundary conditions lead to the same result) is not lifted by the perturbations. We find similar results to the c = 1 string theory in the presence of tachyon perturbations. ∗Membre de l’Institut Universitaire de France † Associate member of the Institute for Nuclear Research and Nuclear Energy (I�I�E),