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M-Theory with Framed Corners and Tertiary Index Invariants
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
Ninebrane structures
String structures in degree four are associated with cancellation of
anomalies of string theory in ten dimensions. Fivebrane structures in degree
eight have recently been shown to be associated with cancellation of anomalies
associated to the NS5-brane in string theory as well as the M5-brane in
M-theory. We introduce and describe "Ninebrane structures" in degree twelve and
demonstrate how they capture some anomaly cancellation phenomena in M-theory.
Along the way we also define certain variants, considered as intermediate cases
in degree nine and ten, which we call "2-Orientation" and "2-Spin structures",
respectively. As in the lower degree cases, we also discuss the natural twists
of these structures and characterize the corresponding topological groups
associated to each of the structures, which likewise admit refinements to
differential cohomology.Comment: 22 pages, discussion on generators for 2-orientations and 2-Spin
structures corrected, presentation improved, final versio
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