On invariants of two-dimensional minimally supersymmetric field theories

TL;DR

This paper introduces a new invariant for two-dimensional $ = (0,1)$ supersymmetric quantum field theories, expanding the toolkit for classifying such theories beyond known elliptic genus invariants.

Contribution

It constructs a novel invariant for 2D $ = (0,1)$ SQFTs, complementing existing invariants and providing a systematic way to detect all invariants including the new one.

Findings

New invariant for 2D $ = (0,1)$ SQFTs introduced.

Spectral invariant of supercharge defined and used to detect invariants.

Spectral pairing systematizes the detection of known and new invariants.

Abstract

We construct a new invariant of two-dimensional $\mathcal{N}{=}(0,1)$ supersymmetric quantum field theories (SQFTs), under a couple of assumptions on the general properties of such SQFTs motivated by considerations in heterotic string theory. This new invariant is in addition to the known 'primary' invariants, i.e. the ordinary elliptic genus and the mod-2 elliptic genus, and the known 'secondary' invariant, which can be considered as an SQFT analog of the Atiyah-Patodi-Singer $\eta$-invariant when its continuous variation vanishes. We also define a spectral invariant of supercharge more generally by the integration of the expectation value of the supercharge over the fundamental region of $\mathrm{ SL}(2,\mathbb{Z})$. With this spectral invariant, we construct a spectral pairing which can be used to systematically detect all the known invariants, including the new one we introduce in the first half of this paper.