Integrable structure of melting crystal model with external potentials

arXiv:0807.4970v4 [math-ph] 23 Feb 2010 Integrable structure of melting crystal model with external potentials Toshio Nakatsu1 and Kanehisa Takasaki2∗ 1Faculty of Engineering, Mathematics and Physics,Setsunan University Ikedanakamachi, Neyagawa, Osaka 572-8508, Japan 2Graduate School of Human and Environmental Studies, Kyoto University Yoshida, Sakyo, Kyoto 606-8501, Japan Abstract This is a review of the authors’ recent results on an integrable structure of the melting crystal model with external potentials. The partition func- tion of this model is a sum over all plane partitions (3D Young diagrams). By the method of transfer matrices, this sum turns into a sum over ordi- nary partitions (Young diagrams), which may be thought of as a model of q -deformed random partitions. This model can be further translated to the language of a complex fermion system. A fermionic realization of the quantum torus Lie algebra is shown to underlie therein. With the aid of hidden symmetry of this Lie algebra, the partition function of the melting crystal model turns out to coincide, up to a simple factor, with a tau function of the 1D Toda hierarchy. Some related issues on 4D and 5D supersymmetric Yang-Mills theories, topological strings and the 2D Toda hierarchy are briefly discussed. ∗E-mail: takasaki@math.h.kyoto-u.ac.jp 1 Figure 1: Melting crystal corner 1 Introduction The melting crystal model is a model of statistical mechanics that describes a melting corner of a semi-infinite crystal (Figure1). The crystal is made of unit cubes, which are initially placed at regular positions and fills the positive octant x, y, z ≥0 of the three dimensional Euclidean space. As the crystal melts, a finite number of cubes are removed from the corner. The present model excludes such crystals that have “overhangs” viewed from the (1, 1, 1) direction. In other words, the complement of the crystal in the positive octant is assumed to be a 3D analogue of Young diagrams (Figure2). Since 3D Young diagrams are represented by “plane partitions”, the melting crystal model is also referred to as a model of “random plane partitions”. Though combinatorics of plane partitions has a rather long history [1], Ok- ounkov and Reshetikhin [2] proposed an entirely new approach in the course of their study on a kind of stochastic process of random partitions (the Schur pro- cess). Their approach was based on “diagonal slices” of 3D Young diagrams and “transfer matrices” between those slices. As a byproduct, they could re-derive a classical result of MacMahon [1] on the generating function of the numbers of plane partitions. Actually, this generating function is nothing but the par- tition function of the aforementioned melting crystal model. The method of Okounkov and Reshetikhin was soon generalized [3] to deal with the topological vertex [4, 5] of A-model topological strings on toric Calabi-Yau threefolds. The melting crystal model is also closely related to supersymmetric gauge theories. Namely, with slightest modification, the partition function can be interpreted as the instanton sum of 5D N = 1 supersymmetric (SUSY) U(1) Yang-Mills theory on partially compactified space-time R4 × S1 [6]. This in- 2 Figure 2: 3D Young diagram as complement of crystal corner stanton sum is a 5D analogue of Nekrasov’s instanton sum for 4D N = 2 SUSY gauge theories [7, 8]. The 4D instanton sum is a statistical sum over ordinary partitions (or “colored” partitions in the case of SU(N) theory), hence a model of random partitions. Nekrasov and Okounkov [9] used such models of random partitions to re-derive the Seiberg-Witten solutions [10] of 4D N = 2 SUSY gauge theories. Actually, by the aforementioned method of transfer matrices, the statistical sum over plane partitions can be reorganized to a sum over par- titions. This is a kind of q-deformations of 4D instanton sums. A 5D analogue of the Seiberg-Witten solution can be derived from this q-deformed instanton sum [9, 11]. In this paper, we review our recent results [12] on an integrable structure of the melting crystal model (and the 5D U(1) instanton sum) with external potentials. The partition function Zp(t) of this model is a function of the cou- pling constants t = (t1, t2, . . .) of the external potentials. A main conclusion of these results is that Zp(t) is, up to a simple factor, a tau function of the 1D Toda hierarchy, in other words, a tau function τp(t, ¯t) of the 2D Toda hierarchy [13] that depends only on the difference t −¯t of the two sets t, ¯t of time vari- ables. To derive this conclusion, we first rewrite Zp(t) in terms of a complex fermion system. In the case of 4D instanton sum, such a fermionic representa- tion was proposed by Nekrasov et al. [14, 9]. In the present case, we can use the aforementioned transfer matrices [2] to construct a fermionic representation. This fermionic representation, however, does not take the form of a standard fermionic representation of the (1D or 2D) Toda hierarchy [15, 16].

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