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

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

An approach to anomalies in M-theory via KSpin

The M-theory fieldstrength and its dual, given by the integral lift of the left hand side of the equation of motion, both satisfy certain cohomological properties. We study the combined fields and observe that the multiplicative structure on the product of the corresponding degree four and degree eight cohomology fits into that given by Spin K-theory. This explains some earlier results and leads naturally to the use of Spin characteristic classes. We reinterpret the one-loop term in terms of such classes and we show that it is homotopy invariant. We argue that the various anomalies have natural interpretations within Spin K-theory. In the process, mod 3 reductions play a special role.Comment: 23 page