Large-N limit of the two-dimensoinal Yang-Mills theory on surfaces with boundaries

TL;DR

This paper investigates the large-N limit of 2D Yang-Mills theory on surfaces with boundaries, revealing that boundary conditions influence the effective area and critical behavior similarly to boundaryless surfaces.

Contribution

It demonstrates that boundary holonomies modify the effective area in the large-N limit, extending the understanding of 2D Yang-Mills theory to surfaces with boundaries.

Findings

Critical behavior matches boundaryless case with modified area

Effective area depends only on boundary holonomies

Results apply to SU(N) and other simple groups

Abstract

The large-N limit of the two-dimensional U$(N)$ Yang-Mills theory on an arbitrary orientable compact surface with boundaries is studied. It is shown that if the holonomies of the gauge field on boundaries are near the identity, then the critical behavior of the system is the same as that of an orientable surface without boundaries with the same genus but with a modified area. The diffenece between this effective area and the real area of the surface is obtained and shown to be a function of the boundary conditions (holonomies) only. A similar result is shown to hold for the group SU$(N)$ and other simple groups.