Matrix Models, Integral Polyhedra and Toric Geometry

TL;DR

This paper introduces a novel matrix model approach to studying integral polyhedra, connecting combinatorics, toric geometry, and integrability, with potential applications to mirror symmetry and toric varieties.

Contribution

It develops a new matrix and tensor model framework for integral polyhedra, revealing Virasoro constraints and extending to higher dimensions, linking combinatorics with toric geometry.

Findings

Constructed matrix models for integral polygons and polytopes.

Derived Virasoro constraints in the matrix models.

Explored applications to Batyrev's mirror pairs.

Abstract

We propose to take a look at a new approach to the study of integral polyhedra. The main idea is to give an integral representation, or matrix model representation, for the key combinatorial characteristics of integral polytopes. Based on the well-known geometric interpretations of matrix model digram techniques, we construct a new model that enumerates triangulations, subdivisions, and numbers of integral points of integral polygons. This approach allows us to look at their combinatorics from a new perspective, motivated by knowledge about matrix models and their integrability. We show how analogs of Virasoro constraints appear in the resulting model. Moreover, we make a natural generalization of this matrix model to the case of polytopes of an arbitrary dimension, considering already a tensor model. We also obtain an analogue of Virasoro constraints for it and discuss their role in the solvability of these models. The deep connection between the geometry of convex polyhedra and toric geometry is the main reference point in the construction of these models. We present considerations on specific ways of applying this approach to the description of Batyrev's mirror pairs. All this allows us to formulate many interesting directions in the study of the connection between matrix/tensor models and the geometry of toric varieties.