Perturbation theory in differential homological algebra I

Gugenheim, V. K. A. M.; Lambe, Larry A.

doi:10.1215/ijm/1255988571

Cited by 77 publications

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“…Then all intersections of cells of X in M ⊂ M + and cells of X ∨+ in M + are transversal. Note that with this regularization, for e i any cell and e j a boundary cell of X, we have e i · κ ∂ (e j ) = 0 12 We can again use the construction of Remark 2.1. Cells κ(e) are then defined exactly as in Intersection pairing defined as above induces a non-degenerate pairing between absolute and relative chains: Note that unlike the case of closed manifolds, where the operation X → X ∨ is an involution on cellular decompositions, for manifolds with boundary X ∨+ always has more cells than X and X → X ∨+ cannot be an involution.…”

Section: 2mentioning

confidence: 99%

“…Lemma 4.2 (Homological perturbation lemma [12]). If (i, p, K) are induction data from C • , d to C • , d and δ : C • → C •+1 is such that (d + δ) 2 = 0, then the complex…”

Section: Reminder: Homological Perturbation Theorymentioning

confidence: 99%

“…Consider M an interval viewed as a cobordism between an in-point and an out-point, with X a cellular decomposition with N ≥ 1 1-cells which we denote [01],[12], . .…”

mentioning

confidence: 99%

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A Cellular Topological Field Theory

Cattaneo

Mnëv

Reshetikhin

2020

Commun. Math. Phys.

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We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect to composition of cobordisms.3 Besides casting the model into the BV-BFV setting, with cobordisms and Segal-like gluing, some of the important advancements over [20,21] are the following: general ball CW complexes are allowed (as opposed to simplicial and cubic complexes); the new construction of the cellular action which is intrinsically finite-dimensional and in particular does not use regularized infinitedimensional super-traces; a systematic, intrinsically finite-dimensional, treatment of the behavior w.r.t. moves of CW complexes -elementary collapses and cellular aggregations; understanding the constant part of the partition function (leading to the contribution of the Reidemeister torsion and the mod 16 phase); incorporating the twist by a nontrivial local system. 4 In this paper we use the convention that the polarization is linked to the designation of boundaries as in/out. Thus we link out-boundaries to "A-polarization" and in-boundaries with "B-polarization". This convention is entirely optional. On the other hand, the link between polarization (68) and the notion of the dual CW complex (Section 2.3) is essential for the construction.1 2 is an appropriately normalized half-density on B ∂ ; K is a the chain homotopy part of the retraction of cochains of X relative to the out-boundary to the cohomology relative to 6 Superscripts pertain to the polarization p : F ∂ → B ∂ (field A fixed on the out-boundary and field B fixed on the in-boundary) used to quantize the in/out-boundary.

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Section: 2mentioning

confidence: 99%

“…Lemma 4.2 (Homological perturbation lemma [12]). If (i, p, K) are induction data from C • , d to C • , d and δ : C • → C •+1 is such that (d + δ) 2 = 0, then the complex…”

Section: Reminder: Homological Perturbation Theorymentioning

confidence: 99%

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A Cellular Topological Field Theory

Cattaneo

Mnëv

Reshetikhin

2020

Commun. Math. Phys.

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“…Our G will additionally satisfy the side conditions G 2 = 0, Gi = 0, and pG = 0. As observed in [6], initial SDR-data can always be altered to satisfy these conditions, but they are corollaries of our construction. We define p = φ.…”

Section: A ∞ -Algebra Structures From Resolutionsmentioning

confidence: 98%

A∞-algebra structures associated to K2 algebras

Conner

Goetz

2011

Journal of Algebra

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“…[33]). With any SDR data (5.1) we can associate a new SDR(T (V ),d V ) differentialsd V andd W are defined by the rule (5.2), p = ∞ n=1 p ⊗n ,î = ∞ n=1…”

mentioning

confidence: 99%

On deformations of $A_\infty$ -algebras

Sharapov¹,

Skvortsov

2019

J. Phys. A: Math. Theor.

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A simple method is proposed for deforming A∞-algebras by means of the resolution technique. The method is then applied to the associative algebras of polynomial functions on quantum superspaces. Specifically, by introducing suitable resolutions, we construct explicit deformations of these algebras in the category of minimal A∞-algebras. The relation of these deformations to higher spin gravities is briefly discussed.2010 Mathematics Subject Classification. Primary 16S80; Secondary 16E40, 53D55. Key words and phrases. strong homotopy algebras, algebraic deformation theory, quantum superspace, higher spin algebras.

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Perturbation theory in differential homological algebra I

Cited by 77 publications

References 0 publications

A Cellular Topological Field Theory

A Cellular Topological Field Theory

A∞-algebra structures associated to K2 algebras

On deformations of $A_\infty$ -algebras

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