27 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
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
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
Factorized domain wall partition functions in trigonometric vertex models
We obtain factorized domain wall partition functions for two sets of
trigonometric vertex models: 1. The N-state Deguchi-Akutsu models, for N = {2,
3, 4} (and conjecture the result for all N >= 5), and 2. The sl(r+1|s+1)
Perk-Schultz models, for {r, s = \N}, where (given the symmetries of these
models) the result is independent of {r, s}.Comment: 12 page
Three-point function of semiclassical states at weak coupling
We give the derivation of the previously announced analytic expression for
the correlation function of three heavy non-BPS operators in N=4
super-Yang-Mills theory at weak coupling. The three operators belong to three
different su(2) sectors and are dual to three classical strings moving on the
sphere. Our computation is based on the reformulation of the problem in terms
of the Bethe Ansatz for periodic XXX spin-1/2 chains. In these terms the three
operators are described by long-wave-length excitations over the ferromagnetic
vacuum, for which the number of the overturned spins is a finite fraction of
the length of the chain, and the classical limit is known as the Sutherland
limit. Technically our main result is a factorized operator expression for the
scalar product of two Bethe states. The derivation is based on a fermionic
representation of Slavnov's determinant formula, and a subsequent bosonisation.Comment: 28 pages, 5 figures, cosmetic changes and more typos corrected in v
Domain wall partition functions and KP
We observe that the partition function of the six vertex model on a finite
square lattice with domain wall boundary conditions is (a restriction of) a KP
tau function and express it as an expectation value of charged free fermions
(up to an overall normalization).Comment: 16 pages, LaTeX2