Examples of transformation groups

Montgomery, David B.; Zippin, Leo

doi:10.1090/s0002-9939-1954-0062436-2

Cited by 30 publications

(8 citation statements)

References 7 publications

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“…For a compact connected topological group G acting non-trivially on the 4-sphere, requiring orbits to be of dimension ≤ 1 immediately forces G to be the circle group. However, the action may not be equivalent to a differentiable one [MZ54]. Furthermore, there are infinitely many nonlinear circle actions on S 4 [Pa78].…”

Section: M-theoretic Near Horizon Spacetime Of Black M5-branementioning

confidence: 99%

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

Braunack-Mayer

Sati

Schreiber

2019

Commun. Math. Phys.

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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. dimensional reduction should exhibit an equivalence ("duality") between (strongly coupled) type IIA string theory and M-theory, in particular between the full non-perturbative theory of D-branes and their M-brane pre-images. Hence if M-theory exists, then gauge enhancement on coincident D-branes must correspond to, and is potentially explained by, a corresponding phenomenon on M-branes. The most immediate incarnation of the problem of gauge enhancement in M-theory can, therefore, be succinctly phrased as: Open Problem, version 1:What is the lift to M-theory of the non-abelian Chan-Paton gauge field degrees of freedom on coincident D-branes?Since the string theory literature tends to blur the distinction between what is known and what is conjectured, we briefly highlight what the folklore on this problem (e.g. [IU12, Sec. 6.3.3], [AG04]) does and does not achieve:• A celebrated recent result (see [BLM + 13] for a review) shows the existence of a class of non-abelian gauge field theories that are plausible candidates for the worldvolume theories expected to live on black M2-branes sitting at ADE singularities (the "BLG-model" [BL08, Gus09] and, more generally, the "ABJM-model" [ABJM08]). However, a derivation of these field theories from M-theoretic degrees of freedom is missing; the argument works just by consistency checks.• On the other hand, a conjectural sketch of an explicit derivation does exist for the M-brane species MK6, whose image in the low-energy approximation provided by 11-dimensional supergravity is supposed to be the Kaluza-Klein monopole spacetime and which becomes the black D6-brane under dimensional reduction [To95a], [Se97, Sec. 1]. For an abelian gauge field on the D6-brane, a straightforward analysis shows that it is sourced by doubly dimensionally reduced M2-...

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Section: M-theoretic Near Horizon Spacetime Of Black M5-branementioning

confidence: 99%

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

Braunack-Mayer

Sati

Schreiber

2019

Commun. Math. Phys.

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“…Montgomery and Zippin described [15] an involution of E3 about a wild line. It was shown in [3] that there are such non-orthogonal rotations of other periods.…”

Section: Introductionmentioning

confidence: 99%

Inequivalent Families of Periodic Homeomorphisms of E 3

Bing¹

1964

The Annals of Mathematics

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“…As a matter of fact, examples of Bing [1] and Montgomery and Zippin [8] show that X/G is not even always locally triangulable. It may be seen by example that the space X/G of orbits is not necessarily a manifold.…”

Section: Introductionmentioning

confidence: 99%

Orbit Spaces of Circle Groups of Transformations

Conner¹,

Floyd²

1958

The Annals of Mathematics

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Examples of transformation groups

Cited by 30 publications

References 7 publications

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

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

Inequivalent Families of Periodic Homeomorphisms of E 3

Orbit Spaces of Circle Groups of Transformations

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