paper

Framed M-branes, corners, and topological invariants

arXiv:1310.1060 · doi:10.1063/1.5007185

Abstract

We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over , and , the Kervaire invariant, the -invariant, and the -invariant. This requires either a framing or a corner structure. The canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies. We characterize the flux quantization condition on the C-field and the topological action of the M5-brane via the Hopf invariant, and the dual of the C-field as (a refinement of) an element of Hopf invariant two. In the signature formulation, the contribution to the M-brane effective action is given by the Maslov index of the corner. The Kervaire invariant implies that the effective action of the M5-brane is quadratic. Our study leads to viewing the self-dual string, which is the boundary of the M2-brane on the M5-brane worldvolume, as a string theory in the sense of cobordism of manifolds with corners. We show that the dynamics of the C-field and its dual are encoded in unified way in the 4-sphere, which suggests the corresponding spectrum as the generalized cohomology theory describing the fields. The effective action of the corner is captured by the -invariant, which is an invariant at chromatic level two. Finally, considering M-theory on manifolds with G_2 holonomy we show that the canonical structure minimizes the topological part of the M5-brane action. This is done via the -invariant and a variant that we introduce related to the one-loop polynomial.

33 pages, minor revisions, updated context and references, published version

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