[Paper Review] Integrable structure of melting crystal model with external potentials

Toshio Nakatsu, Kanehisa Takasaki|arXiv (Cornell University)|Jul 31, 2008

Black Holes and Theoretical Physics28 references3 citations

TL;DR

This paper establishes that the partition function of the melting crystal model with external potentials is, up to a simple factor, a tau function of the 1D Toda hierarchy. By constructing a fermionic representation via transfer matrices and identifying a hidden quantum torus Lie algebra structure with shift symmetries, the authors reformulate the partition function into the standard form of a Toda tau function, revealing an underlying integrable structure in 5D supersymmetric Yang-Mills theory and topological string theory.

ABSTRACT

This is a review of the authors' recent results on an integrable structure of the melting crystal model with external potentials. The partition function 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 ordinary 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.

Motivation & Objective

• To uncover the integrable structure underlying the melting crystal model with external potentials.
• To connect the partition function of this model to the 1D Toda hierarchy through fermionic representations.
• To identify the role of the quantum torus Lie algebra and its shift symmetries in reorganizing the partition function into standard tau function form.
• To relate the model to 5D and 4D supersymmetric gauge theories and topological string amplitudes on toric Calabi-Yau threefolds.

Proposed method

• The partition function is expressed using a complex fermion system, leveraging the method of transfer matrices to reorganize sums over 3D Young diagrams into sums over 2D partitions.
• A fermionic realization of the quantum torus Lie algebra is constructed from transfer matrices and fermion bilinear forms.
• Shift symmetry relations are derived for the transfer matrices and fermion bilinear operators, enabling a transformation to the standard Toda hierarchy form.
• The partition function is rewritten in terms of a $GL(\infty)$ element that satisfies the reduction condition to the 1D Toda hierarchy, confirming its status as a tau function.
• The connection to topological strings is established via the topological vertex, where $Z_p(t)$ corresponds to the amplitude for $\mathcal{O}\oplus\mathcal{O}(-2)\to\mathbb{CP}^1$.
• Operator identities from the quantum torus algebra are used to derive constraints on Lax and Orlov-Schulman operators, resembling $q$-deformed string equations.

Experimental results

Research questions

• RQ1 Does the partition function of the melting crystal model with external potentials exhibit integrable structure, and if so, which hierarchy does it belong to?
• RQ2 How can the fermionic representation of the partition function be transformed into the standard form of a 1D Toda hierarchy tau function?
• RQ3 What algebraic structure underlies the transfer matrices and fermion bilinear forms in this model?
• RQ4 How does the quantum torus Lie algebra and its shift symmetries facilitate the reorganization of the partition function?
• RQ5 What is the relationship between this model and 5D supersymmetric Yang-Mills theory, topological strings, and the 2D Toda hierarchy?

Key findings

• The partition function $Z_p(t)$ of the melting crystal model with external potentials is, up to a simple factor, a tau function of the 1D Toda hierarchy.
• The fermionic representation of $Z_p(t)$ is reformulated into the standard form of a Toda tau function using shift symmetry of the transfer matrices and fermion bilinear operators.
• A realization of the quantum torus Lie algebra emerges from the transfer matrix and fermion bilinear structure, providing the algebraic foundation for the integrable structure.
• The model's partition function is shown to be equivalent to a tau function of the 2D Toda hierarchy that depends only on $t - \bar{t}$, indicating a reduction to the 1D Toda hierarchy.
• The results connect the model to topological string amplitudes on $\mathcal{O}\oplus\mathcal{O}(-2)\to\mathbb{CP}^1$, with $q = e^{-g_{\mathrm{st}}}$ and $Q = e^{-a}$.
• Operator identities from the quantum torus algebra lead to constraints on the Lax and Orlov-Schulman operators that resemble $q$-deformed string equations, suggesting a $q$-deformation of $W_\infty$ algebraic structures.