8d gauge anomalies and the topological Green-Schwarz mechanism

TL;DR

This paper investigates gauge anomalies in 8-dimensional supersymmetric gauge theories with various Lie algebras, identifying anomalies in some cases and proposing a topological Green-Schwarz mechanism for their cancellation.

Contribution

It identifies specific global gauge anomalies in 8d theories for certain Lie algebras and introduces a topological Green-Schwarz mechanism as a potential anomaly cancellation method.

Findings

Anomalies found for _4 and _{2N+1} gauge theories.

_4 and _{2N+1} theories have 8 gauge anomalies.

Green-Schwarz mechanism proposed for anomaly cancellation in _4, _{2N+1}, and _{N} theories.

Abstract

String theory provides us with 8d supersymmetric gauge theory with gauge algebras $\mathfrak{su}(N)$, $\mathfrak{so}(2N)$, $\mathfrak{sp}(N)$, $\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$, but no construction for $\mathfrak{so}(2N{+}1)$, $\mathfrak{f}_4$ and $\mathfrak{g}_2$ is known. In this paper, we show that the theories for $\mathfrak{f}_4$ and $\mathfrak{so}(2N{+}1)$ have a global gauge anomaly associated to $\pi_{d=8}$, while $\mathfrak{g}_2$ does not have it. We argue that the anomaly associated to $\pi_d$ in $d$-dimensional gauge theories cannot be canceled by topological degrees of freedom in general. We also show that the theories for $\mathfrak{sp}(N)$ have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green-Schwarz mechanism.