268 research outputs found

    Hall-Littlewood plane partitions and KP

    Full text link
    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

    Get PDF
    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

    Get PDF
    We consider the AGT correspondence in the context of the conformal field theory M p,p′M^{\, p, p^{\prime}} ⊗\otimes MHM^{H}, where M p,p′M^{\, p, p^{\prime}} is the minimal model based on the Virasoro algebra V p,p′V^{\, p, p^{\prime}} labeled by two co-prime integers {p,p′}\{p, p^{\prime}\}, 1<p<p′1 < p < p^{\prime}, and MHM^{H} is the free boson theory based on the Heisenberg algebra HH. Using Nekrasov's instanton partition functions without modification to compute conformal blocks in M p,p′M^{\, p, p^{\prime}} ⊗\otimes MHM^{H} leads to ill-defined or incorrect expressions. Let Bn p,p′,HB^{\, p, p^{\prime}, H}_n be a conformal block in M p,p′M^{\, p, p^{\prime}} ⊗\otimes MHM^{H}, with nn consecutive channels χi\chi_{i}, i=1,⋯ ,ni = 1, \cdots, n, and let χi\chi_{i} carry states from Hri,sip,p′H^{p, p^{\prime}}_{r_{i}, s_{i}} ⊗\otimes FF, where Hri,sip,p′H^{p, p^{\prime}}_{r_{i}, s_{i}} is an irreducible highest-weight V p,p′V^{\, p, p^{\prime}}-representation, labeled by two integers {ri,si}\{r_{i}, s_{i}\}, 0<ri<p0 < r_{i} < p, 0<si<p′0 < s_{i} < p^{\prime}, and FF is the Fock space of HH. We show that restricting the states that flow in χi\chi_{i} to states labeled by a partition pair {Y1i,Y2i}\{Y_1^{i}, Y_2^{i}\} such that Y2,Ri−Y1,R+si−1i≥1−riY^{i}_{2, {\tt R}} - Y^{i}_{1, {\tt R} + s_{i} - 1} \geq 1 - r_{i}, and Y1,Ri−Y2,R+p′−si−1i≥1−p+riY^{i}_{1, {\tt R}} - Y^{i}_{2, {\tt R} + p^{\prime} - s_{i} - 1} \geq 1 - p + r_{i}, where Yj,RiY^{i}_{j, {\tt R}} is row-R{\tt R} of Yji,j∈{1,2}Y^{i}_j, j \in \{1, 2\}, we obtain a well-defined expression that we identify with Bn p,p′,HB^{\, p, p^{\prime}, H}_n. We check the correctness of this expression for 1.{\bf 1.} Any 1-point B1 p,p′,HB^{\, p, p^{\prime}, H}_1 on the torus, when the operator insertion is the identity, and 2.{\bf 2.} The 6-point B3 3,4,HB^{\, 3, 4, H}_3 on the sphere that involves six Ising magnetic operators.Comment: 22 pages. Simplified the presentatio

    Partial domain wall partition functions

    Full text link
    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

    Full text link
    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

    Get PDF
    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
    • …
    corecore