The long flow to freedom

TL;DR

This paper investigates the low-energy behavior of certain two-dimensional N=(2,2) supersymmetric gauge theories with non-compact target spaces, revealing an order of limits issue in their renormalization group flow and discussing dualities.

Contribution

It demonstrates that specific non-compact 2D supersymmetric gauge theories flow to free theories at low energies and clarifies the order of limits problem in their RG flow.

Findings

The U(1) gauge theory with one flavor flows to free chiral multiplets.

Pure SU(N) gauge theories also flow to free theories.

An order of limits issue affects the interpretation of the RG flow.

Abstract

Two-dimensional field theories do not have a moduli space of vacua. Instead, it is common that their low-energy behavior is a sigma model with a target space. When this target space is compact its renormalization group flow is standard. When it is non-compact the continuous spectrum of operators can change the qualitative behavior. Here we discuss two-dimensional gauge theories with N=(2,2) supersymmetry. We focus on two specific theories, for which we argue that they flow to free chiral multiplets at low energies: the U(1) gauge theory with one flavor (two chiral superfields with charges plus and minus one) and a non-zero Fayet-Iliopoulos term, and pure SU(N) gauge theories. We argue that the renormalization group flow of these theories has an interesting order of limits issue. Holding the position on the target space fixed, the space flattens out under the renormalization group. On the other hand, if we first go to infinity on the target space and then perform the renormalization group, we always have a non-trivial space, e.g. a cone with a deficit angle. We explain how to interpret low-energy dualities between theories with non-compact target spaces. We expect a similar qualitative behavior also for other non-compact sigma models, even when they do not flow to free theories.