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GromovWitten theory of Fano orbifold curves, Gamma integral structures and ADEToda Hierarchies
"... Abstract. We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov–Witten (GW) invariants of the Fano orbifold projective curve P1a1,a2,a3 with positive orbifold Euler characteristic. We also identify our HQEs with an appropriate Kac–Wakimoto hiera ..."
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Abstract. We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov–Witten (GW) invariants of the Fano orbifold projective curve P1a1,a2,a3 with positive orbifold Euler characteristic. We also identify our HQEs with an appropriate Kac–Wakimoto hierarchy of ADE type. In particular, we obtain a generalization of the famous Toda conjecture about the GW invariants of P1. Contents
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