Searcharxiv⌕ Search

arXiv subjects

Theodore Voronov

Publications and source records attributed to Theodore Voronov.

At least 19 recordsLinked to original sources
arXiv2025

On graded and shifted notions, and thick morphisms

We consider the notions of $L_{\infty}$-, $P_{\infty}$-, and $S_{\infty}$-algebras (including "shifted" versions) in the $\mathbb{Z}_2 \times \mathbb{Z}$-graded setting. We also consider thick (microformal) morphisms and show how they work in such graded context. In particular, we show that a "shifted $S_{\infty}$-thick morphism" (which we introduce here) induces an $L_{\infty}$-morphism of shifted $S_{\infty}$-structures. The same holds for "shifted $P_{\infty}$-thick morphisms" and shifted $P_{\infty}$-structures, respectively.

math.RA↗
arXiv2024

The weighted projective superspace with weights $+1, -1$ and an analog of the Fubini--Study form

As a by-product of our work on super Plücker embedding, we came to the notion of a weighted projective superspace $P_{+1,-1}(V\oplus W)$ with weights $+1,-1$. The construction is not in itself super and makes sense in ordinary (purely even) framework. Unlike the familiar weighted projective spaces with positive weights, the (super)space $P_{+1,-1}(V\oplus W)$ is a smooth (super)manifold. We describe its structure and show that it possesses an analog of the Fubini--Study form.

math.DG↗
arXiv2021

On super Plücker embedding and cluster algebras

We define a super analog of the classical Plücker embedding of the Grassmannian into a projective space. One of the difficulties of the problem is rooted in the fact that super exterior powers $Λ^{r|s}(V)$ are not a simple generalization from the completely even case (this works only for $r|0$ when it is possible to use $Λ^r(V)$). To construct the embedding we need to non-trivially combine a super vector space $V$ and its parity-reversion $ΠV$. Our "super Plücker map" takes the Grassmann supermanifold $G_{r|s}(V)$ to a "weighted projective space" $P\left(Λ^{r|s}(V)\oplus Λ^{s|r}(ΠV)\right)$ with weights $+1,-1$. A simpler map $G_{r|0}(V)\to P(Λ^r(V))$ works for the case $s=0$. We construct a super analog of Plücker coordinates, prove that our map is an embedding, and obtain "super Plücker relations". We analyze another type of relations (due to Khudaverdian) and show their equivalence with the super Plücker relations for $r|s=2|0$. We discuss application to much sought-after super cluster algebras and construct a super cluster structure for $G_2(\mathbb{R}^{4|1})$ and $G_2(\mathbb{R}^{5|1})$.

math.DG↗
arXiv2019

L-infinity bialgebroids and homotopy Poisson structures on supermanifolds

We generalize to the homotopy case a result of K. Mackenzie and P. Xu on relation between Lie bialgebroids and Poisson geometry. For a homotopy Poisson structure on a supermanifold $M$, we show that $(TM, T^*M)$ has a canonical structure of an $L_{\infty}$-bialgebroid. (Higher Koszul brackets on forms introduced earlier by H. Khudaverdian and the author are part of one of its manifestations.) The underlying general construction is that of a "(quasi)triangular" $L_{\infty}$-bialgebroid, which is a specialization of a "(quasi)triangular" homotopy Poisson structure. We define both here.

math.DG↗
arXiv2019

Thick morphisms of supermanifolds, quantum mechanics, and spinor representation

"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 questions here. We establish relations of thick morphisms with fundamental concepts of quantum mechanics. We also show that in the linear setup quantum thick morphisms with quadratic action give (a version of) the spinor representation for a certain category of canonical linear relations, which is an analog of the Berezin--Neretin representation and a generalization of the metaplectic representation (and ordinary spinor representation).

math-ph↗
arXiv2018

Thick morphisms, higher Koszul brackets, and $L_{\infty}$-algebroids

It is a classical fact in Poisson geometry that the cotangent bundle of a Poisson manifold has the structure of a Lie algebroid. Manifestations of this structure are the Lichnerowicz differential on multivector fields (calculating Poisson cohomology) and the Koszul bracket of differential forms. "Raising indices" by the Poisson tensor maps the de Rham differential to the Lichnerowicz differential and the Koszul bracket to the Schouten bracket. In this paper, we present a homotopy analog of the above results. When an ordinary Poisson structure is replaced by a homotopy one, instead of a single Koszul bracket there arises an infinite sequence of "higher Koszul brackets" defining an $L_{\infty}$-algebra structure on forms (Khudaverdian--Voronov arXiv:0808.3406). We show how to construct a non-linear transformation, which is an $L_{\infty}$-morphism, from this $L_{\infty}$-algebra to the Lie superalgebra of multivector fields with the canonical Schouten bracket. This is done by using the new notion of "thick morphisms" of supermanifolds recently introduced (see arXiv:1409.6475 and arXiv:1411.6720).

math.DG↗
arXiv2018

Microformal geometry and homotopy algebras

We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. The result is a formal category so that its composition law is also specified by a formal power series. A microformal morphism acts on functions by an operation of pullback, which is in general a nonlinear transformation. More precisely, it is a formal mapping of formal manifolds of even functions (bosonic fields), which has the property that its derivative for every function is a ring homomorphism. This suggests an abstract notion of a "nonlinear algebra homomorphism" and the corresponding extension of the classical "algebraic-functional" duality. There is a parallel fermionic version. The obtained formalism provides a general construction of $L_{\infty}$-morphisms for functions on homotopy Poisson ($P_{\infty}$-) or homotopy Schouten ($S_{\infty}$-) manifolds as pullbacks by Poisson microformal morphisms. We also show that the notion of the adjoint can be generalized to nonlinear operators as a microformal morphism. By applying this to $L_{\infty}$-algebroids, we show that an $L_{\infty}$-morphism of $L_{\infty}$-algebroids induces an $L_{\infty}$-morphism of the "homotopy Lie--Poisson" brackets for functions on the dual vector bundles. We apply this construction to higher Koszul brackets on differential forms and to triangular $L_{\infty}$-bialgebroids. We also develop a quantum version (for the bosonic case), whose relation with the classical version is like that of the Schrödinger equation with the Hamilton--Jacobi equation. We show that the nonlinear pullbacks by microformal morphisms are the limits at $\hbar\to 0$ of certain "quantum pullbacks", which are defined as special form Fourier integral operators.

math.DG↗
arXiv2018

Differential operators on the algebra of densities and factorization of the generalized Sturm-Liouville operator

We consider factorization problem for differential operators on the commutative algebra of densities (defined either algebraically or in terms of an auxiliary extended manifold) introduced in 2004 by Khudaverdian and Voronov in connection with Batalin-Vilkovisky geometry. We consider the case of the line, where unlike the familiar setting (where operators act on functions) there are obstructions for factorization. We analyze these obstructions. In particular, we study the "generalized Sturm-Liouville" operators acting on the algebra of densities on the line. This in a certain sense is in between the 1D and 2D cases. We establish a criterion of factorizabily for the generalized Sturm-Liouville operator in terms of solution of the classical Sturm-Liouville equation. We also establish the possibility of an incomplete factorization.

math-ph↗
arXiv2017

Differential operators on the superline, Berezinians, and Darboux transformations

We consider differential operators on a supermanifold of dimension $1|1$. We define non-degenerate operators as those with an invertible top coefficient in the expansion in the "superderivative" $D$ (which is the square root of the shift generator, the partial derivative in an even variable, with the help of an odd indeterminate). They are remarkably similar to ordinary differential operators. We show that every non-degenerate operator can be written in terms of `super Wronskians' (which are certain Berezinians). We apply this to Darboux transformations (DTs), proving that every DT of an arbitrary non-degenerate operator is the composition of elementary first order transformations. Hence every DT corresponds to an invariant subspace of the source operator and, upon a choice of basis in this subspace, is expressed by a super-Wronskian formula. We consider also dressing transformations, i.e., the effect of a DT on the coefficients of the non-degenerate operator. We calculate these transformations in examples and make some general statements.

math.DG↗
arXiv2015

Quantum microformal morphisms of supermanifolds: an explicit formula and further properties

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 these nonlinear transformations of functions in search of $L_{\infty}$-morphisms for homotopy Poisson structures.

math-ph↗
arXiv2015

On volumes of classical supermanifolds

We consider the volumes of classical supermanifolds such as the supersphere, complex projective superspace, and Stiefel and Grassmann supermanifolds, with respect to the natural metrics or symplectic structures. We show that the formulas for the volumes, upon certain universal normalization, can be obtained by an analytic continuation from the formulas for the volumes of the corresponding ordinary manifolds. Volumes of nontrivial supermanifolds may identically vanish. In 1970s, Berezin discovered that the total Haar measure of the unitary supergroup $\un(n|m)$ vanishes unless $m=0$ or $n=0$, i.e., unless it reduces to the ordinary unitary group $\un(n)$ or $\un(m)$. Witten recently suggested that the (Liouville) volume of a compact even symplectic supermanifold should always be zero if it is not an ordinary manifold. Our calculations provide counterexamples to this conjecture. On the other hand, we give a simple explanation of Berezin's statement and generalize it to the Stiefel supermanifold $\st_{r|s}(\C{n|m})$. There are also possible connections with the recent works by Mkrtchyan and Veselov on `universal formulas' in Lie algebra theory.

math.DG↗
arXiv2015

Darboux transformations for differential operators on the superline

We give a full description of Darboux transformations of any order for arbitrary (nondegenerate) differential operators on the superline. We show that every Darboux transformation of such operators factorizes into elementary Darboux transformations of order one. Similar statement holds for operators on the ordinary line.

math-ph↗
arXiv2011

On a non-Abelian Poincaré lemma

We show that a well-known result on solutions of the Maurer--Cartan equation extends to arbitrary (inhomogeneous) odd forms: any such form with values in a Lie superalgebra satisfying $dø+ø^2=0$ is gauge-equivalent to a constant, $$ø=gCg^{-1}-dg\,g^{-1}\,.$$ This follows from a non-Abelian version of a chain homotopy formula making use of multiplicative integrals. An application to Lie algebroids and their non-linear analogs is given. Constructions presented here generalize to an abstract setting of differential Lie superalgebras where we arrive at the statement that odd elements (not necessarily satisfying the Maurer--Cartan equation) are homotopic\,---\,in a certain particular sense\,---\,if and only if they are gauge-equivalent.

math.DG↗
arXiv2007

Q-manifolds and Mackenzie theory: an overview

This text is meant to be a brief overview of the topics announced in the title and is based on my talk in Vienna (August/September 2007). It does not contain new results (except probably for a remark concerning Q-manifold homology, which I wish to elaborate elsewhere). "Mackenzie theory" stands for the rich circle of notions that have been put forward by Kirill Mackenzie (solo or in collaboration): double structures such as double Lie groupoids and double Lie algebroids, Lie bialgebroids and their doubles, nontrivial dualities for double and multiple vector bundles, etc. "Q-manifolds" are (super)manifolds with a homological vector field, i.e., a self-commuting odd vector field. They may have an extra Z-grading (called weight) not necessarily linked with the Z_2-grading (parity). I discuss double Lie algebroids (discovered by Mackenzie) and explain how this quite complicated fundamental notion is equivalent to a very simple one if the language of Q-manifolds is used. In particular, it shows how the two seemingly different notions of a "Drinfeld double" of a Lie bialgebroid due to Mackenzie and Roytenberg respectively, turn out to be the same thing if properly understood.

math.DG↗
arXiv2004

Higher derived brackets and homotopy algebras

We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element $Δ$. Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of $Δ^2$. This allows to control higher Jacobi identities in terms of the ``order'' of $Δ^2$. Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin--Vilkovisky algebras. There is a generalization with $\D$ replaced by an arbitrary odd derivation. We discuss applications and links with other constructions.

math.QA↗