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Register a new userThe character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes
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The celebrated Green-Schwarz mechanism in heterotic string theory has been suggested to secretly underly a higher gauge theoretic phenomenon, embodying a higher Bianchi identity for a higher-degree analog of a curvature form of a higher gauge field. Here we prove that the non-perturbative Horava-Witten Green-Schwarz mechanism for heterotic line bundles in heterotic M-theory with M5-branes parallel to MO9-planes on $A_1$-singularities is accurately encoded in the higher gauge theory for higher gauge group of the equivariant homotopy type of the Z/2-equivariant A-infinity-loop group of twistor space. In this formulation, the flux forms of the heterotic gauge field, the B-field on the M5-brane, and of the C-field in the M-theory bulk are all unified into the character image of a single cocycle in equivariant twistorial Cohomotopy theory; and that cocycle enforces the quantization condition on all fluxes: the integrality of the gauge flux, the half-shifted integrality of the C-field flux and the integrality of the dual C-field flux (i.e., of the Page charge in the bulk and of the Hopf-WZ term on the M5-brane). This result is in line with the Hypothesis H that M-brane charges are quantized in J-twisted Cohomotopy theory. The mathematical essence of our proof is, first, the construction of the equivariant twisted non-abelian character map via an equivariant twisted non-abelian de Rham theorem, which we prove; and, second, the computation of the equivariant relative minimal model of the Z/2-equivariant Sp(1)-parametrized twistor fibration. We lay out the relevant background in equivariant rational homotopy theory and explain how this brings about the subtle flux quantization relations in heterotic M-theory.
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We characterize the integral cohomology and the rational homotopy type of the maximal Borel-equivariantization of the combined Hopf/twistor fibration, and find that subtle relations satisfied by the cohomology generators are just those that govern Horava-Wittens proposal for the extension of the Green-Schwarz mechanism from heterotic string theory to heterotic M-theory. We discuss how this squares with the Hypothesis H that the elusive mathematical foundation of M-theory is based on charge quantization in J-twisted Cohomotopy theory.
We show that charge-quantization of the M-theory C-field in J-twisted Cohomotopy implies emergence of a higher Sp(1)-gauge field on single heterotic M5-branes, which exhibits worldvolume twisted String structure.
We highlight what seems to be a remaining subtlety in the argument for the cancellation of the total anomaly associated with the M5-brane in M-theory. Then we prove that this subtlety is resolved under the hypothesis that the C-field flux is charge-quantized in the generalized cohomology theory called J-twisted Cohomotopy.
There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments,lacking a microscopic underpinning in M-brane charge quantization. We provide a framework for answering these questions, crucial not only for mathematical consistency but also for phenomenological accuracy of string theory, by formulating the M-theory C-field on flat M-orientifolds in the generalized cohomology theory called Equivariant Cohomotopy. This builds on our previous results for smooth but curved spacetimes, showing in that setting that charge quantization in twisted Cohomotopy rigorously implies a list of expected anomaly cancellation conditions. Here we further expand this list by proving that brane charge quantization in unstable equivariant Cohomotopy implies the anomaly cancellation conditions for M-branes and D-branes on flat orbi-orientifolds. For this we (a) use an unstable refinement of the equivariant Hopf-tom Dieck theorem to derive local/twisted tadpole cancellation, and (b) the lift to super-differential cohomology to establish global/untwisted tadpole cancellation. Throughout, we use (c) the unstable Pontrjagin-Thom theorem to identify the brane/O-plane configurations encoded in equivariant Cohomotopy and (d) the Boardman homomorphism to equivariant K-theory to identify Chan-Paton representations of D-brane charge. We find that unstable equivariant Cohomotopy, but not its image in K-theory, distinguishes D-brane charge from the finite set of types of O-plane charges.
In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining techniques from homotopy theory and from supergeometry to what we call super-exceptional geometry within super-homotopy theory, we present an elegant joint solution to all three problems. This leads to a unified description of the Nambu-Goto, Perry-Schwarz, and topological Yang-Mills Lagrangians in the topologically nontrivial setting. After explaining how charge quantization of the C-field in Cohomotopy reveals DAuria-Fres hidden supergroup of 11d supergravity as the super-exceptional target space, in the sense of Bandos, for M5-brane sigma-models, we prove, in exceptional generalization of the doubly-supersymmetric super-embedding formalism, that a Perry-Schwarz-type Lagrangian for single (abelian) M5-branes emerges as the super-exceptional trivialization of the M5-brane cocycle along the super-exceptional embedding of the half M5-brane locus, super-exceptionally compactified on the Horava-Witten circle fiber. From inspection of the resulting 5d super Yang-Mills Lagrangian we find that the extra fermion field appearing in super-exceptional M-geometry, whose physical interpretation had remained open, is the M-theoretic avatar of the gaugino field.
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