A note on theta dependence

TL;DR

This paper explores the theta dependence in quantum field theories, showing that in certain strongly coupled models, the usual 2 pi periodicity can be broken while periodicity under shifts by multiples of 2 pi is maintained, with implications for supersymmetric theories.

Contribution

It provides a simple 2D example demonstrating loss of 2 pi periodicity in theta dependence, extending understanding beyond instanton calculus assumptions.

Findings

2 pi periodicity can be broken in certain models

Observables remain periodic under theta -> theta + 2 k pi for k >= 2

Implications discussed for 4D N=2 supersymmetric gauge theories

Abstract

The dependence on the topological theta angle term in quantum field theory is usually discussed in the context of instanton calculus. There the observables are 2 pi periodic, analytic functions of theta. However, in strongly coupled theories, the semi-classical instanton approximation can break down due to infrared divergences. Instances are indeed known where analyticity in theta can be lost, while the 2 pi periodicity is preserved. In this short note we exhibit a simple two dimensional example where the 2 pi periodicity is lost. The observables remain periodic under the transformation theta -> theta + 2 k pi for some k >= 2. We also briefly discuss the case of four dimensional N=2 supersymmetric gauge theories.