Setting the quantum integrand of M-theory

TL;DR

This paper uses geometric index theory to define the quantum integrand in M-theory, extending its applicability to manifolds with boundaries and generalizing heterotic M-theory constructions.

Contribution

It introduces a method to set the quantum integrand of M-theory using index theory, including new results on pfaffians of Dirac operators in specific dimensions.

Findings

Quantum integrand defined for closed manifolds.

Extension to manifolds with temporal and spatial boundaries.

Generalization of heterotic M-theory on cylinders.

Abstract

In anomaly-free quantum field theories the integrand in the bosonic functional integral--the exponential of the effective action after integrating out fermions--is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand''. In the low-energy approximation to M-theory the E(8)-model for the C-field allows us to set the quantum integrand using geometric index theory. We derive mathematical results of independent interest about pfaffians of Dirac operators in 8k+3 dimensions, both on closed manifolds and manifolds with boundary. These theorems are used to set the quantum integrand of M-theory for closed manifolds and for compact manifolds with either temporal (global) or spatial (local) boundary conditions. In particular, we show that M-theory makes sense on arbitrary 11-manifolds with spatial boundary, generalizing the construction of heterotic M-theory on cylinders.