438 research outputs found

    Bethe Equation of τ(2)\tau^{(2)}-model and Eigenvalues of Finite-size Transfer Matrix of Chiral Potts Model with Alternating Rapidities

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    We establish the Bethe equation of the τ(2)\tau^{(2)}-model in the NN-state chiral Potts model (including the degenerate selfdual cases) with alternating vertical rapidities. The eigenvalues of a finite-size transfer matrix of the chiral Potts model are computed by use of functional relations. The significance of the "alternating superintegrable" case of the chiral Potts model is discussed, and the degeneracy of τ(2)\tau^{(2)}-model found as in the homogeneous superintegrable chiral Potts model.Comment: Latex 25 pages; Typos corrected, Minor changes for clearer presentation, References added-Journal versio

    The Q-operator and Functional Relations of the Eight-vertex Model at Root-of-unity η=2mKN\eta = \frac{2m K}{N} for odd N

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    Following Baxter's method of producing Q_{72}-operator, we construct the Q-operator of the root-of-unity eight-vertex model for the crossing parameter η=2mKN\eta = \frac{2m K}{N} with odd NN where Q_{72} does not exist. We use this new Q-operator to study the functional relations in the Fabricius-McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N-state chiral Potts model. By the compatibility of the constructed Q-operator with the structure of Baxter's eight-vertex (solid-on-solid) SOS model, we verify the set of functional relations of the root-of-unity eight-vertex model using the explicit form of the Q-operator and fusion weights of SOS model.Comment: Latex 28 page; Typos corrected, minor changes in presentation, References added and updated-Journal versio

    Duality and Symmetry in Chiral Potts Model

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    We discover an Ising-type duality in the general NN-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N=2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small kk') to those of another chiral Potts model at a high temperature (of k1k'^{-1}). The τ(2)\tau^{(2)}-model and chiral Potts model on the dual lattice are established alongside the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl2sl_2-loop-algebra symmetry of its associated spin-N12\frac{N-1}{2} XXZ chain through the identification of their eigenstates.Comment: Latex 34 pages, 2 figures; Typos and misprints in Journal version are corrected with minor changes in expression of some formula

    On τ(2)\tau^{(2)}-model in Chiral Potts Model and Cyclic Representation of Quantum Group Uq(sl2)U_q(sl_2)

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    We identify the precise relationship between the five-parameter τ(2)\tau^{(2)}-family in the NN-state chiral Potts model and XXZ chains with Uq(sl2)U_q (sl_2)-cyclic representation. By studying the Yang-Baxter relation of the six-vertex model, we discover an one-parameter family of LL-operators in terms of the quantum group Uq(sl2)U_q (sl_2). When NN is odd, the NN-state τ(2)\tau^{(2)}-model can be regarded as the XXZ chain of Uq(sl2)U_{\sf q} (sl_2) cyclic representations with qN=1{\sf q}^N=1. The symmetry algebra of the τ(2)\tau^{(2)}-model is described by the quantum affine algebra Uq(sl^2)U_{\sf q} (\hat{sl}_2) via the canonical representation. In general for an arbitrary NN, we show that the XXZ chain with a Uq(sl2)U_q (sl_2)-cyclic representation for q2N=1q^{2N}=1 is equivalent to two copies of the same NN-state τ(2)\tau^{(2)}-model.Comment: Latex 11 pages; Typos corrected, Minor changes for clearer presentation, References added and updated-Journal versio

    A new example of N=2 supersymmetric Landau-Ginzburg theories: the two-ring case

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    The new example of N=2 supersymmetric Landau-Ginzburg theories is considered when the critical values of the superpotential w(x) form the regular two-ring configuration. It is shown that at the deformation, which does not change the form of this configuration, the vacuum state metric satisfies the equation of non-Abelian 2 x 2 Toda system.Comment: LaTeX, 13p

    The Q-operator for Root-of-Unity Symmetry in Six Vertex Model

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    We construct the explicit QQ-operator incorporated with the sl2sl_2-loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the QQ-operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable NN-state chiral Potts model. We show that the whole set of functional equations is valid for the QQ-operator. Direct calculations in certain cases are also given here for clearer illustration about the nature of the QQ-operator in the symmetry study of root-of-unity six-vertex model from the functional-relation aspect.Comment: Latex 26 Pages; Typos and small errors corrected, Some explanations added for clearer presentation, References updated-Journal version with modified labelling of sections and formula

    Quotients of E^n by A_{n+1} and Calabi-Yau manifolds

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    We give a simple construction, starting with any elliptic curve E, of an n-dimensional Calabi-Yau variety of Kummer type (for any n>1), by considering the quotient Y of the n-fold self-product of E by a natural action of the alternating group A_{n+1} (in n+1 variables). The vanishing of H^m(Y, O_Y) for 0<m<n follows from the non-existence of (non-zero) fixed points in certain representations of A_{n+1}. For n<4 we provide an explicit crepant resolution X in characteristics different from 2,3. The key point is that Y can be realized as a double cover of P^n branched along a hypersurface of degree 2(n+1).Comment: 9 page

    A Note on ODEs from Mirror Symmetry

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    We give close formulas for the counting functions of rational curves on complete intersection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the instanton corrected Yukawa coupling.Comment: 24 pages using harvma

    The Onsager Algebra Symmetry of τ(j)\tau^{(j)}-matrices in the Superintegrable Chiral Potts Model

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    We demonstrate that the τ(j)\tau^{(j)}-matrices in the superintegrable chiral Potts model possess the Onsager algebra symmetry for their degenerate eigenvalues. The Fabricius-McCoy comparison of functional relations of the eight-vertex model for roots of unity and the superintegrable chiral Potts model has been carefully analyzed by identifying equivalent terms in the corresponding equations, by which we extract the conjectured relation of QQ-operators and all fusion matrices in the eight-vertex model corresponding to the TT^T\hat{T}-relation in the chiral Potts model.Comment: Latex 21 pages; Typos added, References update
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