5d Field Theories and M Theory

TL;DR

The paper recasts 5d N=1 field theories in terms of M-theory curves derived from IIB (p,q) 5-brane webs, using a toroidal M-theory origin to define a complex curve F(s,t)=0 that resolves vertex singularities. It develops a complex-variable framework (s,t) with w,z and shows how Coulomb-branch data are encoded in polynomial coefficients, illustrated by explicit SU(2) examples and their exponentially small corrections resembling worldline instantons. A key idea is the diagrammatic interpretation of polynomials as projections of F=0, linking brane geometry, polynomial data, and 5d dynamics. The approach provides a potential new toolkit to compare geometric corrections with field-theoretic instanton-like effects and to map between polynomials and their associated diagrams.

Abstract

5-brane configurations describing 5d field theories are promoted to an M theory description a la Witten in terms of polynomials in two complex variables. The coefficients of the polynomials are the Coulomb branch. This picture resolves apparent singularities at vertices and reveals exponentially small corrections. These corrections ask to be compared to world line instanton corrections. From a different perspective this procedure may be used to define a diagrammatic representation of polynomials in two variables.

Figures (2)

• Figure 1: A vertex configuration (a - left) is smoothed out in (b -right)
• Figure 2: A 5brane configuration resulting in an SU(2) 5d theory