scholarly journals Integration of Differential Graded Manifolds

2019 ◽  
Vol 2020 (20) ◽  
pp. 6769-6814
Author(s):  
Pavol Ševera ◽  
Michal Širaň
Keyword(s):  
Integral Transformation ◽  
Differential Forms ◽  
Gauge Condition ◽  
Courant Algebroids ◽  
Finite Dimensional ◽  
Simplicial Category ◽  
Simplicial Manifold ◽  
Graded Manifold ◽  
Graded Manifolds

Abstract We consider the problem of integration of $L_\infty $-algebroids (differential non-negatively graded manifolds) to $L_\infty $-groupoids. We first construct a “big” Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer–Cartan equation. The main analytic trick in our work is an integral transformation sending the solutions of the Maurer–Cartan equation to closed differential forms. Following the ideas of Ezra Getzler, we then impose a gauge condition that cuts out a finite-dimensional simplicial submanifold. This “smaller” simplicial manifold is (the nerve of) a local Lie $\ell $-groupoid. The gauge condition can be imposed only locally in the base of the $L_\infty $-algebroid; the resulting local $\ell $-groupoids glue up to a coherent homotopy, that is, we get a homotopy coherent diagram from the nerve of a good cover of the base to the (simplicial) category of local $\ell $-groupoids. Finally, we show that a $k$-symplectic differential non-negatively graded manifold integrates to a local $k$-symplectic Lie $\ell$-groupoid; globally, these assemble to form an $A_\infty$-functor. As a particular case for $k=2$, we obtain integration of Courant algebroids.

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.

The Hodge Operator in Fermionic Fock Space

2005 ◽  
Vol 70 (7) ◽  
pp. 979-1016 ◽  
Author(s):  
Leszek Z. Stolarczyk
Keyword(s):  
Fock Space ◽  
Differential Forms ◽  
Grassmann Algebra ◽  
Basis Set ◽  
Electron Systems ◽  
Linear Transformations ◽  
Spin Orbital ◽  
Creation And Annihilation Operators ◽  
Finite Dimensional ◽  
Hodge Operator

The Hodge operator ("star" operator) plays an important role in the theory of differential forms, where it serves as a tool for the switching between the exterior derivative and co-derivative. In the theory of many-electron systems involving a finite-dimensional fermionic Fock space, one can define the Hodge operator as a unique (i.e., invariant with respect to linear transformations of the spin-orbital basis set) antilinear operator. The similarity transformation based on the Hodge operator results in the switching between the fermion creation and annihilation operators. The present paper gives a self-contained account on the algebraic structures which are necessary for the construction of the Hodge operator: the fermionic Fock space, the corresponding Grassmann algebra, and the generalized creation and annihilation operators. The Hodge operator is then defined, and its properties are reviewed. It is shown how the notion of the Hodge operator can be employed in a construction of the electronic time-reversal operator.

scholarly journals POISSON BRACKET IN CLASSICAL FIELD THEORY AS A DERIVED BRACKET

2008 ◽  
Vol 05 (07) ◽  
pp. 1051-1055
Author(s):  
S. A. POL'SHIN
Keyword(s):  
Field Theory ◽  
Poisson Bracket ◽  
Maxwell Equations ◽  
Differential Forms ◽  
Classical Field Theory ◽  
Classical Field ◽  
Hamiltonian Form ◽  
Jet Bundle ◽  
Finite Dimensional ◽  
Derived Bracket

We construct a Leibniz bracket on the space Ω• (Jk (π)) of all differential forms over the finite-dimensional jet bundle Jk (π). As an example, we write Maxwell equations with sources in the covariant finite-dimensional hamiltonian form.

scholarly journals General Yang–Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical frameworkorFrom Bianchi identities to twisted Courant algebroids

2014 ◽  
Vol 12 (01) ◽  
pp. 1550009 ◽  
Author(s):  
Melchior Grützmann ◽  
Thomas Strobl
Keyword(s):  
Gauge Theories ◽  
Differential Forms ◽  
Coupled System ◽  
Scalar Fields ◽  
Gauge Fields ◽  
Courant Algebroids ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Derived Bracket ◽  
Form Degree

Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into what is called a Q-structure or, equivalently, an L∞-algebroid. This has many technical and conceptual advantages: complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines and is given by what is called the derived bracket construction. This paper aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form degree p, we pay particular attention to p = 2, i.e. 1- and 2-form gauge fields coupled nonlinearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g. one as a particular kind of a vector-bundle twisted Courant algebroid.

scholarly journals Quantitative towers in finite difference calculus approximating the continuum

2020 ◽  
Author(s):  
R Lawrence ◽  
N Ranade ◽  
D Sullivan
Keyword(s):  
Finite Difference ◽  
Differential Forms ◽  
Nonlinear Problems ◽  
Finite Dimensional ◽  
Inverse Systems ◽  
Dimensional Models ◽  
Composition Product ◽  
The Continuum

Abstract Multivector fields and differential forms at the continuum level have respectively two commutative associative products, a third composition product between them and various operators like $\partial$, d and ‘*’ which are used to describe many nonlinear problems. The point of this paper is to construct consistent direct and inverse systems of finite dimensional approximations to these structures and to calculate combinatorially how these finite dimensional models differ from their continuum idealizations. In a Euclidean background, there is an explicit answer which is natural statistically.

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 Theory of Connections on Graded Principal Bundles

1998 ◽  
Vol 10 (01) ◽  
pp. 47-79 ◽  
Author(s):  
T. Stavracou
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Lie Superalgebra ◽  
Principal Bundles ◽  
Graded Manifold ◽  
Definition Of ◽  
Manifold Theory ◽  
Free Actions

The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant–Berezin–Leites. We first review the basic elements of this theory establishing at the same time supplementary properties of graded Lie groups and their actions. Particular emphasis is given in introducing and studying free actions in the graded context. Next, we investigate the geometry of graded principal bundles; we prove that they have several properties analogous to those of ordinary principal bundles. In particular, we show that the sheaf of vertical derivations on a graded principal bundle coincides with the graded distribution induced by the action of the structure graded Lie group. This result leads to a natural definition of the graded connection in terms of graded distributions; its relation with Lie superalgebra-valued graded differential forms is also exhibited. Finally, we define the curvature for the graded connection and we prove that the curvature controls the involutivity of the horizontal graded distribution corresponding to the graded connection.

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