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GromovWitten theory of CP 1 and integrable hierarchies, arxiv: mathph/0605001
"... Abstract. The ancestor Gromov–Witten invariants of a compact Kähler manifold X can be organized in a generating function called the total ancestor potential of X. In this paper, we construct Hirota Quadratic Equations (HQE shortly) for the total ancestor potential of CP 1. The idea is to adopt the f ..."
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Abstract. The ancestor Gromov–Witten invariants of a compact Kähler manifold X can be organized in a generating function called the total ancestor potential of X. In this paper, we construct Hirota Quadratic Equations (HQE shortly) for the total ancestor potential of CP 1. The idea is to adopt the formalism developed in [G1, GM] to the mirror model of CP 1. We hope that the ideas presented here can be generalized to other manifolds as well. As a corollary, using the twisted loop group formalism from [G3], we obtain a new proof of the following version of the Toda conjecture: the total descendant potential of CP 1 (known also as the partition function of the CP 1 topological sigma model) is a taufunction of the Extended Toda Hierarchy. 1.
Zhang Y.: Bihamiltonian Cohomologies and Integrable Hierarchies II: The General Case
 In preparation
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GromovWitten invariants of CP 1 and integrable hierarchies, available at arXiv:math.AG/0501336
"... Abstract. The paper [CDZ] gives a Lax type presentation of the flows of the Extended Toda Hierarchy (shortly ETH). Our first result is a description of the ETH in terms of Hirota Quadratic Equations (shortly HQEs), which can be viewed as flows on a certain infinite dimensional manifold of functions, ..."
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Abstract. The paper [CDZ] gives a Lax type presentation of the flows of the Extended Toda Hierarchy (shortly ETH). Our first result is a description of the ETH in terms of Hirota Quadratic Equations (shortly HQEs), which can be viewed as flows on a certain infinite dimensional manifold of functions, called taufunctions of the ETH. A new feature here is that the Hirota equations are given in terms of vertex operators taking values in the algebra of differential operators on the affine line. On the other hand, in [G1], the author constructs vertex operators in terms of period mappings associated with isolated singularities of holomorphic functions. In the second part of this paper we apply the methods from [G1] to the mirror of CP 1 to construct certain deformations of the HQEs of the ETH, equivalent to the original ones and parameterized by the semisimple points in the cohomology algebra H ∗ (CP 1; C). In particular, applying Givental’s formula which describes the total descendent potential of CP 1 in terms of taufunctions of the KdV hierarchy, we obtain a new proof of the Toda conjecture: the Gromov–Witten invariants of CP 1 are governed by the flows of the ETH. 1.
THE 2TODA HIERARCHY AND THE EQUIVARIANT GROMOV–WITTEN THEORY OF CP 1
, 2005
"... Abstract. The equivariant Toda conjecture says that the equivariant Gromov– Witten invariants of CP 1 are governed by the flows of the 2Toda hierarchy. The 2Toda flows can be presented on the bosonic Fock space C[[yi, y j  i, j ≥ 0]] via vertex operators and Hirota quadratic equations (shortly HQ ..."
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Abstract. The equivariant Toda conjecture says that the equivariant Gromov– Witten invariants of CP 1 are governed by the flows of the 2Toda hierarchy. The 2Toda flows can be presented on the bosonic Fock space C[[yi, y j  i, j ≥ 0]] via vertex operators and Hirota quadratic equations (shortly HQE). The key result in this paper is a formula expressing the vertex operators in terms of the equivariant mirror model of CP 1. In particular we give a new proof of the equivariant Toda conjecture. 1.
COHOMOLOGY, PERIODS AND THE HODGE STRUCTURE OF TORIC HYPERSURFACES
"... Let f be a Laurent polynomial considered as a regular function on a ddimensional algebraic torus T d. The aim of these notes is to explain some ideas in the study of cohomology groups H ∗ (Zf) of nondegenerate affine toric hypersurfaces Zf defined by the equation f = 0. The central role is played b ..."
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Let f be a Laurent polynomial considered as a regular function on a ddimensional algebraic torus T d. The aim of these notes is to explain some ideas in the study of cohomology groups H ∗ (Zf) of nondegenerate affine toric hypersurfaces Zf defined by the equation f = 0. The central role is played by the differential equations for periods of Zf. We explain the relation of periods of Zf to the GKZhypergeometric functions and discuss their applications
Symmetry, Integrability and Geometry: Methods and Applications An Exactly Solvable Spin Chain Related to Hahn Polynomials
"... doi:10.3842/SIGMA.2011.033 Abstract. We study a linear spin chain which was originally introduced by Shi et al. [Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter α and depends on the parity of the chain site. Extending the model by a second parameter β, ..."
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doi:10.3842/SIGMA.2011.033 Abstract. We study a linear spin chain which was originally introduced by Shi et al. [Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter α and depends on the parity of the chain site. Extending the model by a second parameter β, it is shown that the single fermion eigenstates of the Hamiltonian can be computed in explicit form. The components of these eigenvectors turn out to be Hahn polynomials with parameters (α, β) and (α + 1, β − 1). The construction of the eigenvectors relies on two new difference equations for Hahn polynomials. The explicit knowledge of the eigenstates leads to a closed form expression for the correlation function of the spin chain. We also discuss
The (n, 1)Reduced DKP Hierarchy, the String Equation and W Constraints?
"... Abstract. The total descendent potential of a simple singularity satisfies the Kac–Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding Walgebra. This was used by Liu, Yang and Zhang to prove its uniqueness. We construct th ..."
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Abstract. The total descendent potential of a simple singularity satisfies the Kac–Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding Walgebra. This was used by Liu, Yang and Zhang to prove its uniqueness. We construct this principal hierarchy of type D in a different way, viz. as a reduction of some DKP hierarchy. This gives a Lax type and a Grassmannian formulation of this hierarchy. We show in particular that the string equation induces a large part of the W constraints of Bakalov and Milanov. These constraints are not only given on the tau function, but also in terms of the Lax and Orlov–Schulman operators. Key words: affine Kac–Moody algebra; loop group orbit; Kac–Wakimoto hierarchy; isotropic Grassmannian; total descendent potential; W constraints