M-Theory with Framed Corners and Tertiary Index Invariants

TL;DR

This paper advances the understanding of M-theory partition functions by employing tertiary index invariants on framed manifolds with corners, revealing new insights into anomalies and constraints in heterotic and type IIA theories.

Contribution

It introduces a refined formulation of the M-theory partition function using the f-invariant, a tertiary index, for framed manifolds with corners, connecting index theory with physical anomalies.

Findings

Connection between corners and anomalies in M-theory.

Identification of the f-invariant as a key index in the partition function.

Constraints in heterotic corner and eta-form components in type IIA.

Abstract

The study of the partition function in M-theory involves the use of index theory on a twelve-dimensional bounding manifold. In eleven dimensions, viewed as a boundary, this is given by secondary index invariants such as the Atiyah-Patodi-Singer eta-invariant, the Chern-Simons invariant, or the Adams e-invariant. If the eleven-dimensional manifold itself has a boundary, the resulting ten-dimensional manifold can be viewed as a codimension two corner. The partition function in this context has been studied by the author in relation to index theory for manifolds with corners, essentially on the product of two intervals. In this paper, we focus on the case of framed manifolds (which are automatically Spin) and provide a formulation of the refined partition function using a tertiary index invariant, namely the f-invariant introduced by Laures within elliptic cohomology. We describe the context globally, connecting the various spaces and theories around M-theory, and providing a physical realization and interpretation of some ingredients appearing in the constructions due to Bunke-Naumann and Bodecker. The formulation leads to a natural interpretation of anomalies using corners and uncovers some resulting constraints in the heterotic corner. The analysis for type IIA leads to a physical identification of various components of eta-forms appearing in the formula for the phase of the partition function.