Minimal resolutions and other minimal models

Roig, Agustí

doi:10.5565/publmat_37293_04

Cited by 10 publications

(13 citation statements)

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“…Note that Propodition 6.8 together with Lemma 6.11 make Sullivan minimal algebras in Alg, minimal in an abstract categorical sense (c.f. [Roi94b], [Roi93], [GNPR10]). Theorem 6.13.…”

Section: Algebras Over Variable Operadsmentioning

confidence: 99%

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Sullivan minimal models of operad algebras

Cirici¹,

Roig²

2019

Publicacions Matemàtiques

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We prove the existence of Sullivan minimal models of operad algebras, for a quite wide family of operads in the category of complexes of vector spaces over a field of characteristic zero. Our construction is an adaptation of Sullivan's original step by step construction to the setting of operad algebras. The family of operads that we consider includes all operads concentrated in degree 0 as well as their minimal models. In particular, this gives Sullivan minimal models for algebras over Com, Ass and Lie, as well as over their minimal models Com∞, Ass∞ and Lie∞. Other interesting operads, such as the operad Ger encoding Gerstenhaber algebras, also fit in our study.2010 Mathematics Subject Classification. 18D50, 55P62.

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“…Note that Propodition 6.8 together with Lemma 6.11 make Sullivan minimal algebras in Alg, minimal in an abstract categorical sense (c.f. [Roi94b], [Roi93], [GNPR10]). Theorem 6.13.…”

Section: Algebras Over Variable Operadsmentioning

confidence: 99%

“…Then there exists a morphism of P -algebras g : M → A such that f g = id M .Proof. We rewrite the proof of Gómez-Tato (see Lemma 4.4 of[GT93]) for Com-algebras in the Palgebra setting (see also Theorem 14.11 of[FHT01] and[Roi94b],[Roi93] for a categorical version).…”

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confidence: 99%

Sullivan minimal models of operad algebras

Cirici¹,

Roig²

2019

Publicacions Matemàtiques

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“…Alternatively, we could have said that ρ is a morphism of M \DGM(A), the category of A-dg modules under M . So, a minimal KS-factorization is, simply, a minimal model in M \DGM(A) (see [20] for the precise statement of this). Theorem 1.3.1 Let A be a dgc algebra and let ϕ : M −→ N be an A-dg module morphism such that ϕ 0 * : Given a morphism ϕ : M → N of A-dg modules we will need to construct a model of N ⊕ϕ M .…”

Section: Models Ofmentioning

confidence: 99%

“…in the sense that the couple (A ′ , M ′ ) is unique up to isomorphism (of DGM) and the couple of quis (ρ, ϕ ) is also unique up to homotopies (of DGM): this follows from [20,Theorem 3.4], which tells us that the couple (A, M ) is minimal in DGM if and only if A is a minimal R-dgc algebra and M is an A-dg minimal module.…”

Section: Proofmentioning

confidence: 99%

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Minimal models for non-free circle actions

Roig

Saralegi-Aranguren

2000

Illinois J. Math.

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Let Φ : S 1 × M → M be a smooth action of the unit circle S 1 on a manifold M . In this work, we compute the minimal model of M in terms of the orbit space B and the fixed point set F ⊂ B, as a dg-module over the Sullivan's minimal model of B.The question we treat in this work is the following: given a smooth action Φ : S 1 × M → M , is it possible to construct a model of M using just basic data? The answer is well known when the fixed point set is empty (in particular, when the action is free). A dgc algebra model of M is given by a Hirsch extension of the dgc algebra Sullivan minimal model A(B) of the orbit space Bwhere the degree of x is 1 and dx defines the Euler form of the action (see for example [8]). This formula does not apply when the fixed point set F is not empty. Roughly speaking this happens because the Euler form does not live on B.Our answer to the above question is a minimal model of the deRham dgc algebra Ω(M ) of M , which is a dg module over the Sullivan minimal model A(B) of B. Such structure is associated to M by means of the canonical projection π : M → B. We prove that the minimal model of M , as an A(B)-dg module, is the graded cone

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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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Minimal resolutions and other minimal models

Cited by 10 publications

References 2 publications

Sullivan minimal models of operad algebras

Sullivan minimal models of operad algebras

Minimal models for non-free circle actions

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

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