Anomalies of Duality Groups and Extended Conformal Manifolds

TL;DR

This paper explores how anomalies in duality groups affect the structure of extended conformal manifolds in quantum field theories, revealing nontrivial fiberings and providing methods to determine these anomalies from higher-dimensional theories.

Contribution

It introduces a simple method to compute anomalies of duality groups via anomaly polynomials of 6d theories and clarifies the global structure of the conformal manifold in supersymmetric theories.

Findings

Extended coupling space forms a fiber bundle influenced by anomalies.

The method relates 4d anomalies to 6d anomaly polynomials.

Resolves contradictions in the global structure of the Kahler potential.

Abstract

A self-duality group $\cal G$ in quantum field theory can have anomalies. In that case, the space of ordinary coupling constants $\cal M$ can be extended to include the space $\cal F$ of coefficients of counterterms in background fields. The extended space $\cal N$ forms a bundle over $\cal M$ with fiber $\cal F$, and the topology of the bundle is determined by the anomaly. For example, the ${\cal G}=SL(2,\mathbb{Z})$ duality of the 4d Maxwell theory has an anomaly, and the space ${\cal F}=S^1$ for the gravitational theta-angle is nontrivially fibered over ${\cal M}=\mathbb{H}/SL(2,\mathbb{Z})$. We will explain a simple method to determine the anomaly when the 4d theory is obtained by compactifying a 6d theory on a Riemann surface in terms of the anomaly polynomial of the parent 6d theory. Our observations resolve an apparent contradiction associated with the global structure of the Kahler potential on the space of exactly marginal couplings of supersymmetric theories.