A comparison of two models of orbispaces

Körschgen, Alexander

doi:10.4310/hha.2018.v20.n1.a19

Cited by 11 publications

(24 citation statements)

References 7 publications

(25 reference statements)

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“…In Section 3, we recall two definitions of model categories of orbispaces: the category L-T as defined by Schwede [37] and the category T Orb which is due to Gepner and Henriques [14]. Moreover, we recall a result by Körschgen [19] establishing a zig-zag of Quillen equivalences between these two model categories. We establish two constructions for associating an orbispace to an orbifold: In Section 4.1, we use effective orbifolds and the the category L-T for a non-functorial but easily computable construction.…”

Section: Organization Of the Papermentioning

confidence: 99%

“…We discuss two definitions: L-spaces as defined by Schwede [37] and Orb-spaces as defined by Gepner and Henriques [14]. Moreover, we explain how to compare these two constructions by recalling a result from Körschgen's paper [19].…”

Section: Definition 221 (Essential Equivalencementioning

confidence: 99%

“…Körschgen's paper [19] provides a zig-zag of Quillen equivalences between T O gl and T Orb . We briefly recall some important definitions and results.…”

Section: Comparison Between L-spaces and Orb-spacesmentioning

confidence: 99%

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Orbifolds, Orbispaces and Global Homotopy Theory

Juran

2020

Preprint

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Given an orbifold, we construct an orthogonal spectrum representing its stable global homotopy type. Orthogonal spectra now represent orbifold cohomology theories which automatically satisfy certain properties as additivity and the existence of Mayer-Vietoris sequences. Moreover, the value at a global quotient orbifold M G can be identified with the G-equivariant cohomology of the manifold M . Examples of orbifold cohomology theories which are represented by an orthogonal spectrum include Borel and Bredon cohomology theories and orbifold K-theory. This also implies that these cohomology groups are independent of the presentation of an orbifold as a global quotient orbifold.

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Section: Organization Of the Papermentioning

confidence: 99%

Section: Definition 221 (Essential Equivalencementioning

confidence: 99%

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Orbifolds, Orbispaces and Global Homotopy Theory

Juran

2020

Preprint

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“…If the G-action on X is free, the comparison map X/ /G → X/G is a weak homotopy equivalence (see e.g. [Kö18]).…”

Section: The Ext/cyc-adjunctionmentioning

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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“…The approach that most easily feeds into our present context are the notions of topological stacks respectively orbispaces as developed by Gepner and Henriques in [10]. The present paper and the article [12] by Körschgen together identify the Gepner-Henriques model with the global homotopy theory of orthogonal spaces as established by the author in [21,Ch. 1].…”

Section: Introductionmentioning

confidence: 97%

Orbispaces, orthogonal spaces, and the universal compact Lie group

Schwede

2019

Math. Z.

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This paper identifies the homotopy theories of topological stacks and orbispaces with unstable global homotopy theory. At the same time, we provide a new perspective by interpreting it as the homotopy theory of 'spaces with an action of the universal compact Lie group'. The upshot is a novel way to construct and study genuine cohomology theories on stacks, orbifolds, and orbispaces, defined from stable global homotopy types represented by orthogonal spectra.The universal compact Lie group (which is neither compact nor a Lie group) is a wellknown object, namely the topological monoid L of linear isometric self-embeddings of R ∞ . The underlying space of L is contractible, and the homotopy theory of L-spaces with respect to underlying weak equivalences is just another model for the homotopy theory of spaces. However, the monoid L contains copies of all compact Lie groups in a specific way, and we define global equivalences of L-spaces by testing on corresponding fixed points. We establish a global model structure on the category of L-spaces and prove it to be Quillen equivalent to the global model category of orthogonal spaces, and to the category of orbispaces, i.e., presheaves of spaces on the global orbit category.

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A comparison of two models of orbispaces

Cited by 11 publications

References 7 publications

Orbifolds, Orbispaces and Global Homotopy Theory

Orbifolds, Orbispaces and Global Homotopy Theory

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

Orbispaces, orthogonal spaces, and the universal compact Lie group

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