268 research outputs found

### 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

### 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

### Partial domain wall partition functions

We consider six-vertex model configurations on an n-by-N lattice, n =< N,
that satisfy a variation on domain wall boundary conditions that we define and
call "partial domain wall boundary conditions". We obtain two expressions for
the corresponding "partial domain wall partition function", as an
(N-by-N)-determinant and as an (n-by-n)-determinant. The latter was first
obtained by I Kostov. We show that the two determinants are equal, as expected
from the fact that they are partition functions of the same object, that each
is a discrete KP tau-function, and, recalling that these determinants represent
tree-level structure constants in N=4 SYM, we show that introducing 1-loop
corrections, as proposed by N Gromov and P Vieira, preserves the determinant
structure.Comment: 30 pages, LaTeX. This version, which appeared in JHEP, has an
abbreviated abstract and some minor stylistic change

### Particles in RSOS paths

We introduce a new representation of the paths of the Forrester-Baxter RSOS
models which represents the states of the irreducible modules of the minimal
models M(p',p). This representation is obtained by transforming the RSOS paths,
for the cases p> 2p'-2, to new paths for which horizontal edges are allowed at
certain heights. These new paths are much simpler in that their weight is
nothing but the sum of the position of the peaks. This description paves the
way for the interpretation of the RSOS paths in terms of fermi-type charged
particles out of which the fermionic characters could be obtained
constructively. The derivation of the fermionic character for p'=2 and p=kp'+/-
1 is outlined. Finally, the particles of the RSOS paths are put in relation
with the kinks and the breathers of the restricted sine-Gordon model.Comment: 15 pages, few typos corrected, version publishe

### XXZ scalar products and KP

Using a Jacobi-Trudi-type identity, we show that the scalar product of a
general state and a Bethe eigenstate in a finite-length XXZ spin-1/2 chain is
(a restriction of) a KP tau function. This leads to a correspondence between
the eigenstates and points on Sato's Grassmannian. Each of these points is a
function of the rapidities of the corresponding eigenstate, the inhomogeneity
variables of the spin chain and the crossing parameter.Comment: 14 pages, LaTeX2

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