Relative cohomology spaces for some osp(n|2)-modules

Didier Arnal; Mabrouk Ben Ammar; Wafa Mtaouaa; Zeineb Selmi

doi:10.1063/1.5026103

The affine cohomology spaces and its applications

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988781650016x ◽

2016 ◽

Vol 13 (02) ◽

pp. 1650016 ◽

Cited By ~ 1

Author(s):

Nizar Ben Fraj ◽

Ismail Laraiedh

Keyword(s):

Large Class ◽

Differential Operators ◽

Lie Superalgebra ◽

Invariant Differential Operators ◽

Linear Differential Operators ◽

Relative Cohomology ◽

Linear Formula ◽

Invariant Differential ◽

Linear Differential ◽

Cohomology Spaces

We compute the [Formula: see text] cohomology space of the affine Lie superalgebra [Formula: see text] on the (1,1)-dimensional real superspace with coefficient in a large class of [Formula: see text]-modules [Formula: see text]. We apply our results to the module of weight densities and the module of linear differential operators acting on a superspace of weighted densities. This work is the generalization of a result by Basdouri et al. [The linear [Formula: see text]-invariant differential operators on weighted densities on the superspace [Formula: see text] and [Formula: see text]-relative cohomology, Int. J. Geom. Meth. Mod. Phys. 10 (2013), Article ID: 1320004, 9 pp.]

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On the 𝔞𝔣𝔣(m|1)-relative cohomology of the orthosymplectic superalgebra 𝔬𝔰𝔭(m|2) and 𝔞𝔣𝔣(m|1)-trivial deformation

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988781750027x ◽

2017 ◽

Vol 14 (02) ◽

pp. 1750027

Author(s):

Hafedh Khalfoun ◽

Thamer Faidi

Keyword(s):

Lie Superalgebra ◽

Lie Derivative ◽

Relative Cohomology ◽

Orthosymplectic Superalgebra ◽

Orthosymplectic Lie Superalgebra ◽

Cohomology Spaces

Over the [Formula: see text]-dimensional supercircle [Formula: see text], we consider the action of the orthosymplectic Lie superalgebra [Formula: see text], by the Lie derivative on the superpseudodifferential operators [Formula: see text]. We compute the [Formula: see text]-relative cohomology spaces [Formula: see text], where [Formula: see text] is the affine Lie superalgebra on [Formula: see text]. We explicitly give cocycles spanning these cohomology spaces. We study the [Formula: see text]-trivial deformations of the structure of the [Formula: see text]-modules [Formula: see text].

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THE LINEAR $\mathfrak{aff}(n|1)$-INVARIANT DIFFERENTIAL OPERATORS ON WEIGHTED DENSITIES ON THE SUPERSPACE ℝ1|n AND $\mathfrak{aff}(n|1)$-RELATIVE COHOMOLOGY

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887813200041 ◽

2013 ◽

Vol 10 (04) ◽

pp. 1320004 ◽

Cited By ~ 10

Author(s):

IMED BASDOURI ◽

ISMAIL LARAIEDH ◽

OTHMEN NCIB

Keyword(s):

Vector Fields ◽

Differential Operators ◽

Lie Superalgebra ◽

Differential Cohomology ◽

Invariant Differential Operators ◽

Linear Differential Operators ◽

Relative Cohomology ◽

Invariant Differential ◽

Linear Differential ◽

Cohomology Spaces

Over the (1, n)-dimensional real superspace, we classify [Formula: see text]-invariant linear differential operators acting on the superspaces of weighted densities, where [Formula: see text] is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of [Formula: see text] with coefficients in the superspace of weighted densities, vanishing on the Lie superalgebra [Formula: see text]. We explicitly give 1-cocycles spanning these cohomology spaces.

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𝔞𝔣𝔣(1|1)-Relative cohomology on ℝ1|1

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817501742 ◽

2017 ◽

Vol 14 (12) ◽

pp. 1750174

Author(s):

Hafedh Khalfoun

Keyword(s):

Vector Fields ◽

Differential Operators ◽

Lie Superalgebra ◽

Poisson Superalgebra ◽

Relative Cohomology ◽

Cohomology Space ◽

Cohomology Spaces

Over the [Formula: see text]-dimensional real superspace [Formula: see text], we classify [Formula: see text]-invariant bilinear differential operators acting on the superspaces of weighted densities. We compute the second [Formula: see text]-relative cohomology space of [Formula: see text] with coefficients in the module of [Formula: see text]-densities [Formula: see text] on [Formula: see text], where [Formula: see text] is the Lie superalgebra of contact vector fields on [Formula: see text] and [Formula: see text] is the affine Lie superalgebra. This result allows us to compute the second [Formula: see text]-relative cohomology space of [Formula: see text] with coefficients in the Poisson superalgebra [Formula: see text]. We explicitly give 2-cocycles spanning these cohomology spaces.

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Coarse cohomology with twisted coefficients

Mathematica Slovaca ◽

10.1515/ms-2017-0440 ◽

2020 ◽

Vol 70 (6) ◽

pp. 1413-1444

Author(s):

Elisa Hartmann

Keyword(s):

Sheaf Cohomology ◽

Grothendieck Topology ◽

Coarse Structure ◽

Relative Cohomology ◽

Number Of Ends ◽

Constant Coefficients ◽

Coarse Space ◽

Grothendieck Topologies

AbstractTo a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

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WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000232 ◽

2021 ◽

pp. 1-38

Author(s):

Xianzhe Dai ◽

Junrong Yan

Keyword(s):

Essential Role ◽

Morse Function ◽

Noncompact Manifold ◽

Morse Inequalities ◽

Bounded Geometry ◽

Noncompact Manifolds ◽

Witten Laplacian ◽

Relative Cohomology ◽

Witten Deformation ◽

Small Eigenvalues

Abstract Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

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