M-Theory with Framed Corners and Tertiary Index Invariants

Journal of Mathematical Physics

2018

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We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q, Z and Z 2 , the Kervaire invariant, the f -invariant, and the ν-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 M5brane 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 f -invariant, which is an invariant at chromatic level two. Finally, considering M-theory on manifolds with G 2 holonomy we show that the canonical G 2 structure minimizes the topological part of the M5-brane action. This is done via the ν-invariant and a variant that we introduce related to the one-loop polynomial. *

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Stable Framingmentioning

confidence: 92%

See 1 more Smart Citation

Framed M-branes, corners, and topological invariants

Journal of Mathematical Physics

2018

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“…(ii) As the structure of the functional in the proposition involves a product of two Chern-Simons forms, this suggests a formulation where X 10 is viewed as a manifold of corners of codimension two, in the sense of the setting in [Sa11] [Sa14]. We hope to take up this point of view elsewhere.…”

Section: Fiber Integration Of Massey Products and Anomaly Line Bundlesmentioning

confidence: 99%

Massey products in differential cohomology via stacks

Grady

J. Homotopy Relat. Struct.

2017

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We extend Massey products from cohomology to differential cohomology via stacks, organizing and generalizing existing constructions in Deligne cohomology. We study the properties and show how they are related to more classical Massey products in de Rham, singular, and Deligne cohomology. The setting and the algebraic machinery via stacks allow for computations and make the construction well-suited for applications. We illustrate with several examples from differential geometry and mathematical physics.Comment: 50 pages, improved presentation and many corrections, published versio

“…The expression for CS p1q pηq is a sum of higher product abelian Chern-Simons theories, in the sense of [FSS13]. The next terms (not explicitly recorded for brevity) correspond to tertiary and higher structures, in the sense of [FSS13] [Sa14].…”

mentioning

confidence: 99%

Higher-twisted periodic smooth Deligne cohomology

Grady

Homology, Homotopy and Applications

2019

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Degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology, was established in previous work. Here we consider higher degree twists. The Rham complex, hence de Rham cohomology, admits twists of any odd degree. However, in order to consider twists of integral cohomology we need a periodic version. Combining the periodic versions of both ingredients leads us to introduce a periodic form of Deligne cohomology. We demonstrate that this theory indeed admits a twist by a gerbe of any odd degree. We present the main properties of the new theory and illustrate its use with examples and computations, mainly via a corresponding twisted differential Atiyah-Hirzebruch spectral sequence.1 This is sometimes also denoted Z 8 D pnq or Zpnq 8 D . We are in the smooth setting throughout, so we will not need extra decorations. 2 This would be H n pM ; Dpnqq if we use the opposite convention. However, the one we use is positively graded, hence better adapted for stacks.where L is the stackification functor. 11 Let φ : R n Ñ M be a local chart. For a convex open subset U Ă R n , this stack can be evaluated on the corresponding open subset V " φpU q via MappV, B 2 U p1q ∇ q » DK`C 8 pV, U p1qq d log Ý ÝÝ Ñ Ω 1 pV q d ÝÑ Ω 2 pV q˘. More generally, descent for the stack B 2 U p1q ∇ implies that, for any choice of good open cover tU α u of M , the space of maps MappM, B 2 U p1q ∇ q can be identified by replacing M with theČech nerveČptU α uq of tU α u and considering instead the space of maps Map´ČptU α uq, DK`U p1q d logBy the basic properties of the Dold-Kan correspondence we have an isomorphism 12By [Br93, Theorem 5.3.11], the elements on the right parametrize the homotopy classes of the gerbes with connective structure and curving considered in Example 6.11 This is a functor which turns a prestack into a stack, analogously to the way a sheafification functor turns a presheaf into a sheaf. See [Lu09, Sec. 6.5.3] for details. 12 The shift in degree occurs because on the left we consider the complex U p1q d log Ý ÝÝ Ñ Ω 1 d Ý Ý Ñ Ω 2 as being shifted up two degrees relative to the complex appearing on the right. 15