arXiv subjects

Theodore Th. Voronov

Publications and source records attributed to Theodore Th. Voronov.

10 recordsLinked to original sources

arXiv2024

Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology

We show how the tangent functor extends from ordinary smooth maps to "microformal morphisms" (also called "thick morphisms") of supermanifolds. Microformal morphisms generalize ordinary maps and correspond to formal canonical relations between the cotangent bundles specified by generating functions depending on position variables on the source manifold and momentum variables on the target manifold (as formal power expansions), regarded as part of the structure. Microformal morphisms act on functions by non-linear (in general) pullbacks. We obtain here non-linear pullbacks of (pseudo)differential forms and show that they respect the de Rham differentials as "non-linear chain maps" that can induce non-linear transformations of cohomology.

math.DG↗

arXiv2019

Graded Geometry, $Q$-Manifolds, and Microformal Geometry

We give an exposition of graded and microformal geometry, and the language of $Q$-manifolds. $Q$-manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non-linear analogue of Lie algebras (in parallel with even and odd Poisson manifolds), a basis of "non-linear homological algebra", and a powerful tool for describing algebraic and geometric structures. This language goes together with that of graded manifolds, which are supermanifolds with an extra $\mathbb{Z}$-grading in the structure sheaf. "Microformal geometry" is a new notion referring to "thick" or "microformal" morphisms, which generalize ordinary smooth maps, but whose crucial feature is that the corresponding pullbacks of functions are nonlinear. In particular, "Poisson thick morphisms" of homotopy Poisson supermanifolds induce $L_{\infty}$-morphisms of homotopy Poisson brackets. There is a quantum version based on special type Fourier integral operators and applicable to Batalin-Vilkovisky geometry. Though the text is mainly expository, some results are new or not published previously.

hep-th↗

arXiv2016

"Nonlinear pullbacks" of functions and $L_{\infty}$-morphisms for homotopy Poisson structures

We introduce mappings between spaces of functions on (super)manifolds that generalize pullbacks with respect to smooth maps but are, in general, nonlinear (actually, formal). The construction is based on canonical relations and generating functions. (The underlying structure is a formal category, which is a "thickening" of the usual category of supermanifolds; it is close to the category of symplectic micromanifolds and their micromorphisms considered recently by A. Weinstein and A. Cattaneo--B. Dherin--Weinstein.) There are two parallel settings, for even and odd functions. As an application, we show how such nonlinear pullbacks give $L_{\infty}$-morphisms for algebras of functions on homotopy Schouten or homotopy Poisson manifolds.

math.DG↗

arXiv2013

Vector fields on mapping spaces and a converse to the AKSZ construction

The well-known AKSZ construction (for Alexandrov--Kontsevich--Schwarz--Zaboronsky) gives an odd symplectic structure on a space of maps together with a functional $S$ that is automatically a solution for the classical master equation $(S,S)=0$. The input data required for the AKSZ construction consist of a volume element on the source space and a symplectic structure of suitable parity on the target space, both invariant under given homological vector fields on the source and target. In this note, we show that the AKSZ setup and their main construction can be naturally recovered from the single requirement that the `difference' vector field arising on the mapping space be gradient (or Hamiltonian). This can be seen as a converse statement for that of AKSZ. We include a discussion of properties of vector fields on mapping spaces.

math-ph↗

arXiv2012

$Q$-Manifolds and Mackenzie Theory

Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids in a supermanifold language. Namely, we show that a double Lie algebroid in Mackenzie's sense is equivalent to a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights. Our approach helps to simplify and elucidate Mackenzie's original definition; we show how it fits into a bigger picture of equivalent structures on `neighbor' double vector bundles. It also opens ways for extending the theory to multiple Lie algebroids, which we introduce here.

math-ph↗

arXiv2010

$Q$-manifolds and Higher Analogs of Lie Algebroids

We show how the relation between $Q$-manifolds and Lie algebroids extends to ``higher'' or ``non-linear'' analogs of Lie algebroids. We study the identities satisfied by a new algebraic structure that arises as a replacement of operations on sections of a Lie algebroid. When the base is a point, we obtain a generalization of Lie superalgebras.

math.DG↗

arXiv2006

Mackenzie theory and Q-manifolds

We give a simple characterization of Mackenzie's double Lie algebroids in terms of homological vector fields. Application to the `Drinfeld double' of Lie bialgebroids is given and an extension to the multiple case is suggested.

math.DG↗

arXiv2006

Differential forms and odd symplectic geometry

We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {Š}evera. We study the relationship of odd symplectic geometry with classical objects. We show that the Berezinian of a canonical transformation for an odd symplectic form is a polynomial in matrix entries and a complete square. This is a simple but fundamental fact, parallel to Liouville's theorem for an even symplectic structure. We draw attention to the fact that the de Rham complex on $M$ naturally admits an action of the supergroup of all canonical transformations of $ΠT^*M$. The infinitesimal generators of this action turn out to be the classical `Lie derivatives of differential forms along multivector fields'.

math.DG↗

arXiv2005

New Facts about Berezinians

We consider a new formula for Berezinian (superdeterminant). The Berezinian of a supermatrix $A$ is expressed as the ratio of polynomial invariants of $A$. This formula follows from recurrence relations existing for supertraces of exterior powers.

math-ph↗

arXiv2005

Higher Derived Brackets for Arbitrary Derivations

We introduce and study a construction of higher derived brackets generated by a (not necessarily inner) derivation of a Lie superalgebra. Higher derived brackets generated by an element of a Lie superalgebra were introduced in our earlier work. Examples of higher derived brackets naturally appear in geometry and mathematical physics. From a totally different viewpoint, we show that higher derived brackets arise when one wants to turn the inclusion map of a subalgebra of a differential Lie superalgebra, with a given complementary subalgebra, into a fibration. (For a non-Abelian complementary subalgebra, this leads to a generalization of $L_{\infty}$-algebras with dropped or weakened (anti)symmetry of the brackets.)

math.QA↗