---
author:
- 'Davide Gaiotto and Theo Johnson-Freyd'
title: Mock modularity and a secondary elliptic genus
---

Introduction
============

In [@WittenTMF] the following puzzle was posed; the goal of this paper is to propose a solution. Let us say that a (1+1)-dimensional quantum field theory with minimal, aka ${\mathcal N}=(0,1)$, supersymmetry is [*null*]{} if supersymmetry is spontaneously broken and [*nullhomotopic*]{} if it can be connected to a null theory by a sequence of deformations, including deformations that may zig-zag up and down along RG flow lines.

\[mainpuzzle\] Show that the supersymmetric quantum field theory of three free antichiral fermions $\bar\psi_1,\bar\psi_2,\bar\psi_3$ and supersymmetry generated by $G = \sqrt{-1}\, {:}\bar\psi_1\bar\psi_2\bar\psi_3{:}$ is not nullhomotopic.

For the remainder of this paper we will write simply “SQFT” for “(1+1)-dimensional quantum field theory with ${\mathcal N}=(0,1)$ supersymmetry”, and “SCFT” for an SQFT which is furthermore superconformal. The SQFT in Puzzle \[mainpuzzle\] is an SCFT, and is the (conjectured) limit under RG flow of the ${\mathcal N}=(0,1)$ sigma model with target the round $S^3$ and minimal nonzero B-field. The puzzle is difficult because, as is shown in [@WittenTMF], the direct sum of 24 copies of the SQFT in Puzzle \[mainpuzzle\] *is* nullhomotopic (as is the ${\mathcal N}=(0,1)$ 