1,292 research outputs found

### 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

- β¦