A Toy Model of the M5-brane: Anomalies of Monopole Strings in Five Dimensions

TL;DR

This paper analyzes a five-dimensional SU(2) gauge theory with an adjoint Higgs and fermions, which supports a monopole-string solution hosting chiral zero modes on its worldsheet. By integrating out massive bulk fermions, the authors derive parity-odd Chern-Simons terms in the low-energy effective action, and show that anomaly inflow from these terms cancels the worldsheet gauge and gravitational anomalies for various fermion representations. The analysis extends to arbitrary isospin representations, providing explicit inflow and zero-mode anomaly expressions that cancel under suitable SO(3)_{diag} embeddings, while highlighting an $N$-dependence distinction for monopole charge $N>1$ and drawing cautions about direct M-theory extrapolations. The work strengthens the analogy to M-theory’s M5-brane inflow mechanism, clarifies the role of topological backgrounds in defining CS terms, and identifies open questions about chiral dyon modes and the precise high-energy origin of CS-like couplings.

Abstract

We study a five-dimensional field theory which contains a monopole (string) solution with chiral fermion zero modes. This monostring solution is a close analog of the fivebrane solution of M-theory. The cancellation of normal bundle anomalies parallels that for the M-theory fivebrane, in particular, the presence of a Chern-Simons term in the low-energy effective U(1) gauge theory plays a central role. We comment on the relationship between the the microscopic analysis of the world-volume theory and the low-energy analysis and draw some cautionary lessons for M-theory.