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INFINITEDIMENSIONAL FROBENIUS MANIFOLDS FOR 2 + 1 INTEGRABLE SYSTEMS
, 902
"... Abstract. We introduce a structure of an infinitedimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/ ∞ respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy assoc ..."
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Cited by 17 (4 self)
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Abstract. We introduce a structure of an infinitedimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/ ∞ respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold. 1.
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.
Multiple orthogonal polynomials of mixed type: GaussBorel factorization and the multicomponent 2D Toda hierarchy
, 2011
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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.
Dispersionless bigraded Toda hierarchy and its additional symmetry
 Reviews in Mathematical Physics
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EQUIVARIANT ORBIFOLD STRUCTURES ON THE PROJECTIVE LINE AND INTEGRABLE HIERARCHIES
, 707
"... Abstract. Let CP 1 k,m be the orbifold structure on CP 1 obtained via uniformizing the neighborhoods of 0 and ∞ respectively by z ↦ → z k and w ↦ → w m. The diagonal action of the torus T = ( S 1) 2 on CP 1 induces naturally an action on the orbifold CP 1 k,m. In this paper we prove that if k and m ..."
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Cited by 6 (1 self)
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Abstract. Let CP 1 k,m be the orbifold structure on CP 1 obtained via uniformizing the neighborhoods of 0 and ∞ respectively by z ↦ → z k and w ↦ → w m. The diagonal action of the torus T = ( S 1) 2 on CP 1 induces naturally an action on the orbifold CP 1 k,m. In this paper we prove that if k and m are coprime then Givental’s prediction of the equivariant total descendent GromovWitten potential of CP 1 k,m satisfies certain Hirota Quadratic Equations (HQE for short). We also show that after an appropriate change of the variables, similar to Getzler’s change in the equivariant GromovWitten theory of CP 1, the HQE turn into the HQE of the 2Toda hierarchy, i.e., the GromovWitten potential of CP 1 k,m is a taufunction of the 2Toda hierarchy. More precisely, we obtain a sequence of taufunctions of the 2Toda hierarchy from the descendent potential via some translations. The later condition, that all taufunctions in the sequence are obtained from a single one via translations, imposes a serious constraint on the solution of the 2Toda hierarchy. Our theorem leads to the discovery of a new integrable hierarchy (we suggest to be called the Equivariant Bigraded Toda Hierarchy), obtained from the 2Toda hierarchy via a reduction similar to the one in [13]. We conjecture that this new hierarchy governs, i.e., uniquely determines, the equivariant GromovWitten invariants of CP 1 k,m.