Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 024, 28 pages      arXiv:1203.4179      https://doi.org/10.3842/SIGMA.2014.024

M-Theory with Framed Corners and Tertiary Index Invariants

Hisham Sati
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Received March 19, 2013, in final form March 01, 2014; Published online March 14, 2014

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.

Key words: anomalies; manifolds with corners; tertiary index invariants; M-theory; elliptic genera; partition functions; eta-forms.

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