scholarly journals Characteristic classes associated to Q-bundles

2014 ◽  
Vol 12 (01) ◽  
pp. 1550006 ◽  
Author(s):  
Alexei Kotov ◽  
Thomas Strobl
Keyword(s):  
Cohomology Class ◽  
Equivariant Cohomology ◽  
Principal Bundle ◽  
Characteristic Classes ◽  
De Rham ◽  
Arbitrary Dimensions ◽  
Characteristic 3 ◽  
Graded Manifold ◽  
Graded Manifolds ◽  
Exact Sequences

A Q-manifold is a graded manifold endowed with a vector field of degree 1 squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of "gauge fields" (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. As any principal bundle yields canonically a Q-bundle, this construction generalizes Chern–Weil classes. Novel examples include cohomology classes that are locally de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate the framework to equivariant cohomology and Lecomte's characteristic classes of exact sequences of Lie algebras.

scholarly journals Differential Calculus onN-Graded Manifolds

2017 ◽  
Vol 2017 ◽  
pp. 1-19
Author(s):  
G. Sardanashvily ◽  
W. Wachowski
Keyword(s):  
Differential Calculus ◽  
Differential Operators ◽  
Differential Forms ◽  
Commutative Rings ◽  
Rham Complex ◽  
Grassmann Algebras ◽  
De Rham ◽  
Linear Differential ◽  
Graded Manifold ◽  
Graded Manifolds

The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, overN-graded commutative rings and onN-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and onZ2-graded manifolds. We follow the notion of anN-graded manifold as a local-ringed space whose body is a smooth manifoldZ. A key point is that the graded derivation module of the structure ring of graded functions on anN-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its bodyZ. Accordingly, the Chevalley–Eilenberg differential calculus on anN-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus onN-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds ofN-graded bundles.

scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

Homotopy Quotients and Equivariant Cohomology

2020 ◽  
pp. 29-38
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Group ◽  
Fiber Bundle ◽  
Equivariant Cohomology ◽  
Principal Bundle ◽  
Total Space

This chapter investigates two candidates for equivariant cohomology and explains why it settles on the Borel construction, also called Cartan's mixing construction. Let G be a topological group and M a left G-space. The Borel construction mixes the weakly contractible total space of a principal bundle with the G-space M to produce a homotopy quotient of M. Equivariant cohomology is the cohomology of the homotopy quotient. More generally, given a G-space M, Cartan's mixing construction turns a principal bundle with fiber G into a fiber bundle with fiber M. Cartan's mixing construction fits into the Cartan's mixing diagram, a powerful tool for dealing with equivariant cohomology.

scholarly journals NONTRIVIAL CLASSES IN H*(Imb(S1,ℝn)) FROM NONTRIVALENT GRAPH COCYCLES

2004 ◽  
Vol 01 (05) ◽  
pp. 639-650 ◽  
Author(s):  
RICCARDO LONGONI
Keyword(s):  
Linear Combination ◽  
Cohomology Class ◽  
Feynman Diagrams ◽  
De Rham Cohomology ◽  
De Rham

We construct nontrivial cohomology classes of the space Imb (S1,ℝn) of imbeddings of the circle into ℝn by means of Feynman diagrams. More precisely, starting from a suitable linear combination of nontrivalent diagrams, we construct, for every even number n≥4, a de Rham cohomology class on Imb (S1,ℝn). We prove nontriviality of these classes by evaluation on the dual cycles.

GENERALIZED CONNECTIONS AND CHARACTERISTIC CLASSES

1991 ◽  
Vol 02 (05) ◽  
pp. 515-524
Author(s):  
HONG-JONG KIM
Keyword(s):  
Smooth Manifold ◽  
Vector Bundles ◽  
Characteristic Classes ◽  
De Rham Cohomology ◽  
De Rham ◽  
Twisted De Rham Cohomology

We study derivations on a smooth manifold, its twisted de Rham cohomology, generalized connections on vector bundles and their characteristic classes.

scholarly journals A Study of Doubly Warped Product Immersions in a Nearly Trans-Sasakian Manifold with Slant Factor

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ali H. Alkhaldi ◽  
Aliya Naaz Siddiqui ◽  
Kamran Ahmad ◽  
Akram Ali
Keyword(s):  
Cohomology Class ◽  
Warped Product ◽  
Sasakian Manifold ◽  
Sasakian Manifolds ◽  
De Rham Cohomology ◽  
De Rham

In this article, we discuss the de Rham cohomology class for bislant submanifolds in nearly trans-Sasakian manifolds. Moreover, we give a classification of warped product bislant submanifolds in nearly trans-Sasakian manifolds with some nontrivial examples in the support. Next, it is of great interest to prove that there does not exist any doubly warped product bislant submanifolds other than warped product bislant submanifolds in nearly trans-Sasakian manifolds. Some immediate consequences are also obtained.

scholarly journals Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

2017 ◽  
Vol 28 (09) ◽  
pp. 1740004 ◽  
Author(s):  
Antonio Alarcón ◽  
Finnur Lárusson
Keyword(s):  
Riemann Surface ◽  
Convex Hull ◽  
Minimal Surface ◽  
Cohomology Class ◽  
Holomorphic Maps ◽  
De Rham Cohomology ◽  
Surface Theory ◽  
Open Riemann Surface ◽  
De Rham ◽  
Smooth Locus

Let [Formula: see text] be a connected open Riemann surface. Let [Formula: see text] be an Oka domain in the smooth locus of an analytic subvariety of [Formula: see text], [Formula: see text], such that the convex hull of [Formula: see text] is all of [Formula: see text]. Let [Formula: see text] be the space of nondegenerate holomorphic maps [Formula: see text]. Take a holomorphic 1-form [Formula: see text] on [Formula: see text], not identically zero, and let [Formula: see text] send a map [Formula: see text] to the cohomology class of [Formula: see text]. Our main theorem states that [Formula: see text] is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on [Formula: see text] can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

VERTICAL TANGENTIAL INVARIANTS ON SOME FOLIATED LAGRANGE SPACES

2013 ◽  
Vol 10 (04) ◽  
pp. 1320002
Author(s):  
CRISTIAN IDA
Keyword(s):  
Tangent Bundle ◽  
Differential Forms ◽  
Characteristic Classes ◽  
Exterior Derivative ◽  
Cohomology Groups ◽  
Foliated Manifold ◽  
Smooth Complex ◽  
De Rham ◽  
Poincaré Lemma ◽  
Natural Decomposition

In this paper we consider a decomposition of tangentially differential forms with respect to the lifted foliation [Formula: see text] to the tangent bundle of a Lagrange space [Formula: see text] endowed with a regular foliation [Formula: see text]. First, starting from a natural decomposition of the tangential exterior derivative along the leaves of [Formula: see text], we define some vertical tangential cohomology groups of the foliated manifold [Formula: see text], we prove a Poincaré lemma for the vertical tangential derivative and we obtain a de Rham theorem for this cohomology. Next, in a classical way, we construct vertical tangential characteristic classes of tangentially smooth complex bundles over the foliated manifold [Formula: see text].

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