Probing bad theories with the dualization algorithm I

TL;DR

This paper applies a dualization algorithm to analyze bad 3d and 4d supersymmetric theories, revealing that their indices and partition functions behave as distributions rather than regular functions, especially in the case of SQCD.

Contribution

It introduces a new method to study bad theories using dualization, providing exact indices and partition functions that exhibit distributional behavior.

Findings

Indices and partition functions of bad theories behave as distributions.

The method successfully analyzes bad SQCD with specific gauge groups.

Surprising distributional features emerge in the deformation parameters.

Abstract

Recently an algorithm to build $SL(2,\mathbb{Z})$ duals, including mirror duals, of 3d $\mathcal{N}=4$ quiver theories and their 4d $\mathcal{N}=1$ uplift has been introduced. In this work we use this new tool to study the so-called bad theories. Our approach allows us to determine exactly indices/partition functions for generic values of fugacities/real mass and FI parameters revealing their surprising feature: the 4d index/3d partition function of a bad theory behaves as a sum of distributions rather than an ordinary function of the deformation parameters. We focus on the bad SQCD, with $U(N_c)$ gauge group in 3d and $USp(2N_c)$ in 4d, while in an upcoming paper we will consider linear quivers which, in the 3d case, have both unitary and special unitary bad nodes.