THE LOOP GROUP OF E 8 AND TARGETS FOR SPACETIME

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: The M5-brane As a Framed Submanifoldmentioning

confidence: 99%

Framed M-branes, corners, and topological invariants

Journal of Mathematical Physics

2018

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“…Let us now go with the restriction of the E 8 bundle on Z 8 to its boundary Y 7 . This will give an E 8 bundle, whose reduction over the circle fiber down to M 6 leads again to an LE 8 bundle, in this case in exact analogy to the case of going from Z 12 to the M-theory boundary and then reducing on the M-theory circle to get type IIA string theory (see [MS,Sa07,Sa10a]). Therefore, we have the following statement For any exceptional Lie group G, there is a loop G bundle on the fivebrane worldvolume.…”

Section: Which Gauge Group(s)?mentioning

confidence: 99%

Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

Fiorenza

Advances in Theoretical and Mathematical Physics

Schreiber

2014

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100

The worldvolume theory of coincident M5-branes is expected to contain a nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of self-dual Yang-Mills theory. But the precise details -in particular the global moduli / instanton / magnetic charge structure -have remained elusive. Here we deduce from anomaly cancellation a natural candidate for the holographic dual of this nonabelian 2-form field, under AdS7/CFT6 duality. We find this way a 7-dimensional nonabelian Chern-Simons theory of String 2-connection fields, which, in a certain higher gauge, are given locally by non-abelian 2-forms with values in an affine Kac-Moody Lie algebra. We construct the corresponding action functional on the entire smooth moduli 2-stack of field configurations, thereby defining the theory globally, at all levels and with the full instanton structure, which is nontrivial due to the twists imposed by the quantum corrections. Along the way we explain some general phenomena of higher nonabelian gauge theory that we need.

“…Thus, in this special case, we have an F 4 bundle and an ΩF 4 bundle over X 10 . This is analogous to the case of E 8 [54].…”

Section: Compatibility With Ten-dimensional Superstring Theoriesmentioning

confidence: 65%

On the Geometry of the Supermultiplet in M-Theory

Int. J. Geom. Methods Mod. Phys.

2011

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The massless supermultiplet of eleven-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F4 into those of its maximal compact subgroup Spin(9). In an earlier paper, a dynamical Kaluza-Klein origin of this observation is proposed with internal space the Cayley plane, OP 2 , and topological aspects are explored. In this paper we consider the geometric aspects and characterize the corresponding forms which contribute to the action as well as cohomology classes, including torsion, which contribute to the partition function. This involves constructions with bilinear forms. The compatibility with various string theories are discussed, including reduction to loop bundles in ten dimensions. * 2. The identity representation of F 4 , i.e. the one corresponding to Dynkin index [0, 0, 0, 0], generates the three representations of Spin(9) [51] Id(F 4 ) −→ (44, 128, 84), the numbers on the right hand side correctly matching the number of degrees of freedoms of the massless bosonic content of eleven-dimensional supergravity with the individual summands corresponding, respectively, to the graviton, the gravitino, and the C-field (see the beginning of section 2).The main Idea of this paper, presented in section 2.1 is interpreting Ramond's triplets as arising from OP 2 bundles with structure group F 4 over our eleven-dimensional manifold Y 11 , on which M-theory is 'defined'. We first discuss in section 2.2 the geometric properties of OP 2 , including Spin(9)-structures and characteristic classes. This leads to one of the main results, theorem 2.9, that the massless fields of M-theory are encoded in the spinor bundle of OP 2 . We then relate Spin(9)-structures on the 9-dimensional vector space V 9 to the geometry of the eight-sphere S 8 which in turn, by [24], is related to Killing spinors on the cone over S 8 . This shows that the unification of the fields, as well as their supersymmetry, in M-theory can be seen from the eight-sphere over the Cayley plane (cf. proposition 2.11). We then show that fields can be given yet another interpretation via the index of the twisted (Kostant) Dirac operator of [44]. The identity representation of F 4 encoding the supergravity multiplet is the space of twisted harmonic spinors on OP 2 , which is propositon 2.12.After studying structures on OP 2 , we use that space itself as the fiber over eleven-dimensional spacetime. In section 2.4.1 we explore the consequences of this idea by relating the geometry and the characteristic classes of the base to that of the total space, using the knowledge of that of the fiber studied in section 2. If the base Y 11 has positive Ricci curvature then so does the total space M 27 . This and related matters are discussed in section 2.4.1. In section 2.5 we relate structures, such as Fivebrane structures [57] [58], as well as genera on the base space to genera on the total space. This includes elliptic genera, Witten genera, Ochanine genera and is the content of proposition 2.15. We use this to r...