Off-critical local height probabilities on a plane and critical partition functions on a cylinder

We compute off-critical local height probabilities in regime-III restricted solid-on-solid models in a $4 N$-quadrant spiral geometry, with periodic boundary conditions in the angular direction, and fixed boundary conditions in the radial direction, as a function of $N$, the winding number of the spiral, and $\tau$, the departure from criticality of the model, and observe that the result depends only on the product $N \, \tau$. In the limit $N \rightarrow 1$, $\tau \rightarrow \tau_0$, such that $\tau_0$ is finite, we recover the off-critical local height probability on a plane, $\tau_0$-away from criticality. In the limit $N \rightarrow \infty$, $\tau \rightarrow 0$, such that $N \, \tau = \tau_0$ is finite, and following a conformal transformation, we obtain a critical partition function on a cylinder of aspect-ratio $\tau_0$. We conclude that the off-critical local height probability on a plane, $\tau_0$-away from criticality, is equal to a critical partition function on a cylinder of aspect-ratio $\tau_0$, in agreement with a result of Saleur and Bauer.Comment: 28 page

An Izergin-Korepin procedure for calculating scalar products in six-vertex models

Using the framework of the algebraic Bethe Ansatz, we study the scalar product of the inhomogeneous XXZ spin-1/2 chain. Inspired by the Izergin-Korepin procedure for evaluating the domain wall partition function, we obtain a set of conditions which uniquely determine the scalar product. Assuming the Bethe equations for one set of variables within the scalar product, these conditions may be solved to produce a determinant expression originally found by Slavnov. We also consider the inhomogeneous XX spin-1/2 chain in an external magnetic field. Repeating our earlier procedure, we find a set of conditions on the scalar product of this model and solve them in the presence of the Bethe equations. The expression obtained is in factorized form.Comment: 32 pages, 24 figure

Hall-Littlewood plane partitions and KP

MacMahon's classic generating function of random plane partitions, which is related to Schur polynomials, was recently extended by Vuletic to a generating function of weighted plane partitions that is related to Hall-Littlewood polynomials, S(t), and further to one related to Macdonald polynomials, S(t,q). Using Jing's 1-parameter deformation of charged free fermions, we obtain a Fock space derivation of the Hall-Littlewood extension. Confining the plane partitions to a finite s-by-s square base, we show that the resulting generating function, S_{s-by-s}(t), is an evaluation of a tau-function of KP.Comment: 17 pages, minor changes, added a subsection and comments to clarify content, no changes made to conclusions, version to appear in IMR

Variations on Slavnov's scalar product

We consider the rational six-vertex model on an L-by-L lattice with domain wall boundary conditions and restrict N parallel-line rapidities, N < L/2, to satisfy length-L XXX spin-1/2 chain Bethe equations. We show that the partition function is an (L-2N)-parameter extension of Slavnov's scalar product of a Bethe eigenstate and a generic state, with N magnons each, on a length-L XXX spin-1/2 chain. Decoupling the extra parameters, we obtain a third determinant expression for the scalar product, where the first is due to Slavnov [1], and the second is due to Kostov and Matsuo [2]. We show that the new determinant is a discrete KP tau-function in the inhomogeneities, and consequently that tree-level N = 4 SYM structure constants that are known to be determinants, remain determinants at 1-loop level.Comment: 17 page

Macdonald topological vertices and brane condensates

We show, in a number of simple examples, that Macdonald-type $qt$-deformations of topological string partition functions are equivalent to topological string partition functions that are without $qt$-deformations but with brane condensates, and that these brane condensates lead to geometric transitions.Comment: 23 pages, 5 figures. v2: minor changes, published versio

AGT, Burge pairs and minimal models

We consider the AGT correspondence in the context of the conformal field theory $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$, where $M^{\, p, p^{\prime}}$ is the minimal model based on the Virasoro algebra $V^{\, p, p^{\prime}}$ labeled by two co-prime integers $\{p, p^{\prime}\}$, $1 < p < p^{\prime}$, and $M^{H}$ is the free boson theory based on the Heisenberg algebra $H$. Using Nekrasov's instanton partition functions without modification to compute conformal blocks in $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$ leads to ill-defined or incorrect expressions. Let $B^{\, p, p^{\prime}, H}_n$ be a conformal block in $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$, with $n$ consecutive channels $\chi_{i}$, $i = 1, \cdots, n$, and let $\chi_{i}$ carry states from $H^{p, p^{\prime}}_{r_{i}, s_{i}}$ $\otimes$ $F$, where $H^{p, p^{\prime}}_{r_{i}, s_{i}}$ is an irreducible highest-weight $V^{\, p, p^{\prime}}$-representation, labeled by two integers $\{r_{i}, s_{i}\}$, $0 < r_{i} < p$, $0 < s_{i} < p^{\prime}$, and $F$ is the Fock space of $H$. We show that restricting the states that flow in $\chi_{i}$ to states labeled by a partition pair $\{Y_1^{i}, Y_2^{i}\}$ such that $Y^{i}_{2, {\tt R}} - Y^{i}_{1, {\tt R} + s_{i} - 1} \geq 1 - r_{i}$, and $Y^{i}_{1, {\tt R}} - Y^{i}_{2, {\tt R} + p^{\prime} - s_{i} - 1} \geq 1 - p + r_{i}$, where $Y^{i}_{j, {\tt R}}$ is row-${\tt R}$ of $Y^{i}_j, j \in \{1, 2\}$, we obtain a well-defined expression that we identify with $B^{\, p, p^{\prime}, H}_n$. We check the correctness of this expression for ${\bf 1.}$ Any 1-point $B^{\, p, p^{\prime}, H}_1$ on the torus, when the operator insertion is the identity, and ${\bf 2.}$ The 6-point $B^{\, 3, 4, H}_3$ on the sphere that involves six Ising magnetic operators.Comment: 22 pages. Simplified the presentatio

Polynomial identities of the Rogers--Ramanujan type

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by establishing a graphical one-to-one correspondence between those two kinds of restricted partitions.Comment: 27 page