Theodore Voronov
We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of `quantum thick morphisms' defined here as particular oscillatory integral operators on functions.
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We give an explicit formula, as a formal differential operator, for quantum microformal morphisms of (super)manifolds that we introduced earlier. Such quantum microformal morphisms are essentially oscillatory integral operators or Fourier integral operators of a particular kind. They act on oscillatory wave functions, whose algebra extends the algebra of formal power series in Planck's constant. In the classical limit, quantum microformal morphisms reproduce, as the main term of the asymptotic, the nonlinear pullbacks of functions with respect to `classical' microformal morphisms. We found the…
The aim of this work is the construction of a "supermanifold of morphisms $X \rightarrow Y$", given two finite-dimensional supermanifolds $X$ and $Y$. More precisely, we will define an object $\underline{SC}^\infty(X,Y)$ in the category of supermanifolds proposed by Molotkov and Sachse. Initially, it is given by the set-valued functor characterised by the adjunction formula $\mathrm{Hom}(P \times X,Y) \cong \mathrm{Hom}(P,\underline{SC}^\infty(X,Y))$ where $P$ ranges over all superpoints. We determine the structure of this functor in purely geometric terms: We show that it takes values in the …
"Thick" or "microformal" morphisms of supermanifolds generalize ordinary maps. They were discovered as a tool for homotopy algebras. Namely, the corresponding pullbacks provide $L_{\infty}$-morphisms for $S_{\infty}$ or Batalin--Vilkovisky algebras. It was clear from the start that constructions used for thick morphisms closely resemble some fundamental notions in quantum mechanics and their classical limits (such as action, Schrödinger and Hamilton--Jacobi equations, etc.) There was also a natural question about any connection of thick morphisms with spinor representation. We answer both ques…
We recall the notion of a differential operator over a smooth map (in linear and non-linear settings) and consider its versions such as formal $\hbar$-differential operators over a map. We study constructions and examples of such operators, which include pullbacks by thick morphisms and quantization of symplectic micromorphisms.
We give a new and self-contained proof of the existence and unicity of the flow for an arbitrary (not necessarily homogeneous) smooth vector field on a real supermanifold, and extend these results to the case of holomorphic vector fields on complex supermanifolds. Furthermore we discuss local actions associated to super vector fields, and give several examples and applications, as, e.g., the construction of an exponential morphism for an arbitrary finite dimensional Lie supergroup. In the 2nd version we corrected typos, presentation of figures and other small points. We added references in Sec…
In this paper we analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces (i.e. following Berezin, Leites, Manin, etc.). We establish a representation formula for all morphisms from the algebra of functions on an ordinary manifolds to the superalgebra of functions on an open subset of R^{p|q}. We then derive two consequences of this result. The first one is that we can integrate the data associated with a morphism in order to get a (non unique) map defined on an ordinary space (and uniqueness can achieved…
We generalize the Donagi and Witten construction of a first obstruction class for splitting of a supermanifold via differential operators using the theory of $n$-fold vector bundles and graded manifolds. Applying the generalized Donagi--Witten construction we obtain a family of embeddings of the category of supermanifolds into the category of $n$-fold vector bundles and into the category of graded manifolds. This leads to a realization of any non-split supermanifold in terms of a collection of vector bundles and some morphism between them. Further we study the images of these embeddings into t…
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 "thic…
We introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds, referred to as holomorphic supercurves. More precisely, supercurves are morphisms from a Riemann surface, endowed with the structure of a supermanifold which is induced by a holomorphic line bundle, to an ordinary almost complex manifold. They are called holomorphic if a generalised Cauchy-Riemann condition is satisfied. We show, by means of an action identity, that holomorphic supercurves are special extrema of a supersymmetric action functional.
In this note we define Chern-Simons classes of a superconnection $D+L$ on a complex supervector bundle $E$ such that $D$ is flat and preserves the grading, and $L$ is an odd endomorphism of $E$ on a supermanifold. As an application we obtain a definition of Chern-Simons classes of a (not necessarily flat) morphism between flat vector bundles on a smooth manifold. We extend Reznikov's theorem on triviality of these classes when the manifold is a compact Kähler manifold or a smooth complex quasi--projective variety, in degrees > 1.
In this paper we develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable notion of super wavefront set for superdistributions which generalizes Dencker's polarization sets for vector-valued distributions to supergeometry. In particular, our super wavefront sets detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criter…
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,…
In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n \geq 2$, do appear quite frequently. The corresponding sign rules are determined by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The present paper is the first of a series on $\mathbb{Z}_2^n$-Supergeometry. The new theory exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise (the parity is the parity of the total degree). It is based on the hierarchy: ` $\mathbb{Z}_2^0$-Supe…
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