Grand Partition Functions of Little Matrix Models with ABCD

TL;DR

This paper explores grand partition functions of little matrix models with ABCD gauge groups, analyzing their differential equations, singularities, and implications for brane configurations and monopoles in string theory.

Contribution

It introduces the computation of grand partition functions for USp, SO, and SU matrix models, revealing their differential equations and pole structures related to brane and orientifold configurations.

Findings

Partition functions follow second-order differential equations.

Singularities occur at q=0 and q=∞, allowing analytic continuation.

Different pole choices correspond to distinct brane and orientifold setups.

Abstract

Itoyama-Tokura type USp matrix model is discussed. Non-Abelian Berry's phases in a T-dualized model of IT model were reconsidered. These phases describe the higher dimensional monopoles; Yang monopole and nine-dimensional monopole. They are described by the connections of the BPST instanton on S^4 and the Tchrakian-GKS instanton on S^8, respectively. As a preparation to understand their effect in original zero-dimensional model, we consider partition function of simplified matrix models. We compute partition functions of SU, SO and USp reduced matrix models. Groups SO and USp appear in low energy effective theories of string against orientifold background. In this evaluation we chose different poles from that of Moore-Nekrasov-Shatashvili and our previous result. The position of poles explain branes' and the orientifold's configurations. There is a brane which is sitting on the orientifold in the SO(2N) model, while in USp(2N) and SO(2N+1) model there are no branes on the orientifold. The grand partition functions of these models are considered. They follow to linear second order ordinary differential equations and their singularities are q=0,\infty. Their solutions can be analytically continued to whole q plane. We show the expectation values of the number N of A and C cases as examples. There is an ambiguity coming from the problem on sign. Grand partition functions with minus sign give effective actions which have cusp singularities.