1,292 research outputs found

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

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    We compute off-critical local height probabilities in regime-III restricted solid-on-solid models in a 4N4 N-quadrant spiral geometry, with periodic boundary conditions in the angular direction, and fixed boundary conditions in the radial direction, as a function of NN, the winding number of the spiral, and Ο„\tau, the departure from criticality of the model, and observe that the result depends only on the product N τN \, \tau. In the limit Nβ†’1N \rightarrow 1, Ο„β†’Ο„0\tau \rightarrow \tau_0, such that Ο„0\tau_0 is finite, we recover the off-critical local height probability on a plane, Ο„0\tau_0-away from criticality. In the limit Nβ†’βˆžN \rightarrow \infty, Ο„β†’0\tau \rightarrow 0, such that N τ=Ο„0N \, \tau = \tau_0 is finite, and following a conformal transformation, we obtain a critical partition function on a cylinder of aspect-ratio Ο„0\tau_0. We conclude that the off-critical local height probability on a plane, Ο„0\tau_0-away from criticality, is equal to a critical partition function on a cylinder of aspect-ratio Ο„0\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

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    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

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    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

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    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

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    We show, in a number of simple examples, that Macdonald-type qtqt-deformations of topological string partition functions are equivalent to topological string partition functions that are without qtqt-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

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    We consider the AGT correspondence in the context of the conformal field theory M p,pβ€²M^{\, p, p^{\prime}} βŠ—\otimes MHM^{H}, where M p,pβ€²M^{\, p, p^{\prime}} is the minimal model based on the Virasoro algebra V p,pβ€²V^{\, p, p^{\prime}} labeled by two co-prime integers {p,pβ€²}\{p, p^{\prime}\}, 1<p<pβ€²1 < 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,pβ€²M^{\, p, p^{\prime}} βŠ—\otimes MHM^{H} leads to ill-defined or incorrect expressions. Let Bn p,pβ€²,HB^{\, p, p^{\prime}, H}_n be a conformal block in M p,pβ€²M^{\, p, p^{\prime}} βŠ—\otimes MHM^{H}, with nn consecutive channels Ο‡i\chi_{i}, i=1,⋯ ,ni = 1, \cdots, n, and let Ο‡i\chi_{i} carry states from Hri,sip,pβ€²H^{p, p^{\prime}}_{r_{i}, s_{i}} βŠ—\otimes FF, where Hri,sip,pβ€²H^{p, p^{\prime}}_{r_{i}, s_{i}} is an irreducible highest-weight V p,pβ€²V^{\, p, p^{\prime}}-representation, labeled by two integers {ri,si}\{r_{i}, s_{i}\}, 0<ri<p0 < r_{i} < p, 0<si<pβ€²0 < s_{i} < p^{\prime}, and FF is the Fock space of HH. We show that restricting the states that flow in Ο‡i\chi_{i} to states labeled by a partition pair {Y1i,Y2i}\{Y_1^{i}, Y_2^{i}\} such that Y2,Riβˆ’Y1,R+siβˆ’1iβ‰₯1βˆ’riY^{i}_{2, {\tt R}} - Y^{i}_{1, {\tt R} + s_{i} - 1} \geq 1 - r_{i}, and Y1,Riβˆ’Y2,R+pβ€²βˆ’siβˆ’1iβ‰₯1βˆ’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,pβ€²,HB^{\, p, p^{\prime}, H}_n. We check the correctness of this expression for 1.{\bf 1.} Any 1-point B1 p,pβ€²,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

    Polynomial identities of the Rogers--Ramanujan type

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    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
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