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93
TAU FUNCTION AND HIROTA BILINEAR EQUATIONS FOR THE EXTENDED BIGRADED TODA HIERARCHY
, 906
"... Abstract. In this paper we generalize the Sato theory to the extended bigraded Toda hierarchy (EBTH). We revise the definition of the Lax eqution, give the Sato equations, wave operators and show the existence of tau function. Meanwhile we prove the validity of its Faylike identities and Hirota bil ..."
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Cited by 14 (10 self)
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Abstract. In this paper we generalize the Sato theory to the extended bigraded Toda hierarchy (EBTH). We revise the definition of the Lax eqution, give the Sato equations, wave operators and show the existence of tau function. Meanwhile we prove the validity of its Faylike identities and Hirota bilinear equations (HBEs) in terms of nonscaled vertex operators. Mathematics Subject Classifications(2000). 37K10, 37K20.
Identification of the Givental formula with the spectral curve topological recursion procedure
 Comm. Math. Phys
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BCOV theory via Givental group action on . . .
, 2008
"... In a previous paper, Losev, me, and Shneiberg constructed a full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra. This contruction extended the construction of Barannikov and Kontsevich of solution of the WDVV equation, based on the earlier paper of Bershadsky, Cecotti, O ..."
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Cited by 13 (6 self)
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In a previous paper, Losev, me, and Shneiberg constructed a full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra. This contruction extended the construction of Barannikov and Kontsevich of solution of the WDVV equation, based on the earlier paper of Bershadsky, Cecotti, Ooguri, and Vafa. In the present paper, we give an interpretation of this full descendant potential in terms of Givental group action on cohomological field theories. In particular, the fact that it satisfies all
DEFORMATIONS OF THE MONGE/RIEMANN HIERARCHY AND APPROXIMATELY INTEGRABLE SYSTEMS
, 2002
"... Abstract. Dispersive deformations of the Monge equation uu = uux are studied using ideas originating from topological quantum field theory and the deformation quantization programme. It is shown that, to a highorder, the symmetries of the Monge equation may also be appropriately deformed, and that, ..."
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Abstract. Dispersive deformations of the Monge equation uu = uux are studied using ideas originating from topological quantum field theory and the deformation quantization programme. It is shown that, to a highorder, the symmetries of the Monge equation may also be appropriately deformed, and that, if they exist at all orders, they are uniquely determined by the original deformation. This leads to either a new class of integrable systems or to a rigorous notion of an approximate integrable system. QuasiMiura transformations are also constructed for such deformed equations. Consider a general scalar evolution equation 1.
On deformations of quasiMiura transformations and the DubrovinZhang bracket
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On almost duality for Frobenius manifolds
, 2004
"... We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theo ..."
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Cited by 12 (1 self)
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We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups and Shephard groups, from the Seiberg Witten duality.
THE SPACES OF LAURENT POLYNOMIALS, GROMOVWITTEN THEORY OF P 1ORBIFOLDS, AND INTEGRABLE HIERARCHIES
, 2007
"... Abstract. Let Mk,m be the space of Laurent polynomials in one variable x k +t1x k−1 +...tk+mx −m, where k, m ≥ 1 are fixed integers and tk+m ̸ = 0. According to B. Dubrovin [11], Mk,m can be equipped with a semisimple Frobenius structure. In this paper we prove that the corresponding descendent and ..."
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Cited by 9 (3 self)
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Abstract. Let Mk,m be the space of Laurent polynomials in one variable x k +t1x k−1 +...tk+mx −m, where k, m ≥ 1 are fixed integers and tk+m ̸ = 0. According to B. Dubrovin [11], Mk,m can be equipped with a semisimple Frobenius structure. In this paper we prove that the corresponding descendent and ancestor potentials of Mk,m (defined as in [16]) satisfy Hirota quadratic equations (HQE for short). Let Ck,m be the orbifold obtained from P 1 by cutting small discs D1 ∼ = {z  ≤ ǫ} and D2 ∼ = {z −1  ≤ ǫ} around z = 0 and z = ∞ and gluing back the orbifolds D1/Zk and D2/Zm in the obvious way. We show that the orbifold quantum cohomology of Ck,m coincides with Mk,m as Frobenius manifolds. Modulo some yettobeclarified details, this implies that the descendent (respectively the ancestor) potential of Mk,m is a generating function for the descendent (respectively ancestor) orbifold Gromov–Witten invariants of Ck,m. There is a certain similarity between our HQE and the Lax operators of the Extended bigraded Toda hierarchy, introduced by G. Carlet in [7]. Therefore, it is plausible that our HQE characterize the taufunctions of this hierarchy and we expect that the Extended bigraded Toda hierarchy governs the Gromov–Witten theory of Ck,m. 1.
GromovWitten invariants of target curves via symplectic field theory. arXiv:0709.2860
"... We compute the GromovWitten potential at all genera of target smooth Riemann surfaces using Symplectic Field Theory techniques and establish differential equations for the full descendant potential. This amounts to impose (and possibly solve) different kinds of Schrödinger equations related to some ..."
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We compute the GromovWitten potential at all genera of target smooth Riemann surfaces using Symplectic Field Theory techniques and establish differential equations for the full descendant potential. This amounts to impose (and possibly solve) different kinds of Schrödinger equations related to some quantization of the dispersionless KdV hierarchy. In particular we find very explicit formulas for the GromovWitten invariants of low degree of P 1 with descendants of the Kähler class.
Wconstraints for the total descendant potential of a simple singularity
 Compositio Math
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