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Gauge Enhancement of Super M-Branes Via Parametrized Stable Homotopy Theory

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Abstract

A key open problem in M-theory is to explain the mechanism of “gauge enhancement” through which M-branes exhibit the nonabelian gauge degrees of freedom seen perturbatively in the limit of 10d string theory. In fact, since only the twisted K-theory classes represented by nonabelian Chan–Paton gauge fields on D-branes have an invariant meaning, the problem is really the understanding the M-theory lift of the classification of D-brane charges by twisted K-theory. Here we show that this problem has a solution by universal constructions in rational super homotopy theory. We recall how double dimensional reduction of super M-brane charges is described by the cyclification adjunction applied to the 4-sphere, and how M-theory degrees of freedom hidden at ADE singularities are induced by the suspended Hopf action on the 4-sphere. Combining these, we demonstrate that, in the approximation of rational homotopy theory, gauge enhancement in M-theory is exhibited by lifting against the fiberwise stabilization of the unit of this cyclification adjunction on the A-type orbispace of the 4-sphere. This explains how the fundamental D6 and D8 brane cocycles can be lifted from twisted K-theory to a cohomology theory for M-brane charge, at least rationally.

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Notes

  1. At least in mathematics it is not uncommon that a theory is conjectured to exist before its actual nature is known—famous examples of this include the theory of motives, which has meanwhile been discovered, and the field with one element.

  2. In [Wi96, first line on p. 8] the argument was introduced as an “obvious guess”. Most subsequent references cite this as a fact, e.g. the review [My03, Sec. 3], despite the lack of a formal argument.

  3. [HW06]: “As it has been proposed that [this] theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committedly call it the M-theory, leaving to the future the relation of M to membranes.”

  4. While a derivation of K-theory from M-theory is suggested by the title of [DMW03], that article only checks that the behavior of the partition function of the 11d supergravity C-field is compatible with the a priori K-theory classification of D-branes. Seeking a generalized cohomology describing the M-field and M-branes was originally advocated for in [Sa05a, Sa05b, Sa06, Sa10].

  5. This torsion is in the sense of cohomology or homotopy classes. In the following paragraph we use torsion in the sense of differential (super)geometry. We hope that the distinction will be clear from the context.

  6. Here and elsewhere, “(pb)" denotes a (homotopy-)pullback square.

  7. we will always assume that all topological spaces are compactly generated, so that \(\mathrm {Maps}(G,Y)\) is the exponential object in the category of compactly generated spaces—this completely specifies the topology.

  8. The identity \(d H_7 = F_2 \wedge F_6 - \tfrac{1}{2} F_4 \wedge F_4\) for the type IIA \(\mathrm {NS5}\)-brane flux does not hold after fiberwise stabilization in (60)—indeed, the flux form \(H_7\) is part of the obstruction to completing the zig-zag truncation map \(\tau _6\) in Example 2.48 to an actual homomorphism.

  9. Models of particle physics obtained from dimensional reductions of M-theory on singular manifolds of \(G_2\)-holonomy (see [Ka17]) are among the globally supersymmetric extensions of the Standard Model of particle physics that are, so far, still consistent with experimental constraints [BGK18]. If and when supersymmetric extensions of the Standard Model are ruled out, then this will also rule out dimensional reductions of M-theory on singular fiber manifolds of \(G_2\)-holonomy as realistic models for particle physics. However, this particular type of dimensional reduction is in no way dictated by the theory, and are certainly not generic amongst all possibilities, but were motivated by the expectation of global supersymmetry in the first place. What is dictated by the theory is local supersymmetry, which is already present as soon as fermions are.

  10. From the point of view of homotopy theory, there is little difference between working over \(\mathbb {Q}\) or over \(\mathbb {R}\).

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Acknowledgements

We are grateful to Augustí Roig and Martintxo Saralegi-Aranguren for discussion of [RS00], as well as to David Corfield, Ted Erler, Domenico Fiorenza, and David Roberts for useful comments. We also thank the anonymous referee for their careful reading and helpful suggestions. VBM acknowledges partial support of SNF Grant No. 200020_172498/1. This research was partly supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation, and by the COST Action MP1405 QSPACE, supported by COST (European Cooperation in Science and Technology).

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  1. Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Vincent Braunack-Mayer

  2. Division of Science and Mathematics, New York University, Abu Dhabi, UAE

    Hisham Sati & Urs Schreiber

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Braunack-Mayer, V., Sati, H. & Schreiber, U. Gauge Enhancement of Super M-Branes Via Parametrized Stable Homotopy Theory. Commun. Math. Phys. 371, 197–265 (2019). https://doi.org/10.1007/s00220-019-03441-4

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