438 research outputs found

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

We establish the Bethe equation of the $\tau^{(2)}$-model in the $N$-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 $\tau^{(2)}$-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 $\eta = \frac{2m K}{N}$ for odd N

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
$\eta = \frac{2m K}{N}$ with odd $N$ 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

We discover an Ising-type duality in the general $N$-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 $k'$) to those of another chiral Potts model at a
high temperature (of $k'^{-1}$). The $\tau^{(2)}$-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 $sl_2$-loop-algebra symmetry of its associated
spin-$\frac{N-1}{2}$ 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 $\tau^{(2)}$-model in Chiral Potts Model and Cyclic Representation of Quantum Group $U_q(sl_2)$

We identify the precise relationship between the five-parameter
$\tau^{(2)}$-family in the $N$-state chiral Potts model and XXZ chains with
$U_q (sl_2)$-cyclic representation. By studying the Yang-Baxter relation of the
six-vertex model, we discover an one-parameter family of $L$-operators in terms
of the quantum group $U_q (sl_2)$. When $N$ is odd, the $N$-state
$\tau^{(2)}$-model can be regarded as the XXZ chain of $U_{\sf q} (sl_2)$
cyclic representations with ${\sf q}^N=1$. The symmetry algebra of the
$\tau^{(2)}$-model is described by the quantum affine algebra $U_{\sf q}
(\hat{sl}_2)$ via the canonical representation. In general for an arbitrary
$N$, we show that the XXZ chain with a $U_q (sl_2)$-cyclic representation for
$q^{2N}=1$ is equivalent to two copies of the same $N$-state
$\tau^{(2)}$-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

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

We construct the explicit $Q$-operator incorporated with the
$sl_2$-loop-algebra symmetry of the six-vertex model at roots of unity. The
functional relations involving the $Q$-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 $N$-state chiral
Potts model. We show that the whole set of functional equations is valid for
the $Q$-operator. Direct calculations in certain cases are also given here for
clearer illustration about the nature of the $Q$-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

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

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 $\tau^{(j)}$-matrices in the Superintegrable Chiral Potts Model

We demonstrate that the $\tau^{(j)}$-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
$Q$-operators and all fusion matrices in the eight-vertex model corresponding
to the $T\hat{T}$-relation in the chiral Potts model.Comment: Latex 21 pages; Typos added, References update

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