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Alexei Kotov

Publications and source records attributed to Alexei Kotov.

At least 19 recordsLinked to original sources
arXiv2026

The functor between two categories of $\mathbb{Z}-$graded manifolds

This paper examines $\mathbb{Z}$-graded manifolds as semiformal homogeneity structures, comparing two polynomial filtrations from their local models. In finite dimensions, these are componentwise equivalent, yielding isomorphic graded completions; generally, one induces a finer topology. By the Batchelor-Gawedzki-type theorem (Kotov--Salnikov), every $\mathbb{Z}$-graded manifold over base $M$ is noncanonically isomorphic to one associated with its canonical $\mathbb{Z}$-graded bundle (Batchelor-Gawedzki bundle). In finite dimensions, this is the formal neighborhood of the zero section with the induced homogeneity structure. Kotov-Salnikov's graded Borel lemma extends weight-$k$ functions from the formal neighborhood to smooth ones of the same weight. Here, this generalizes to a Borel--Whitney theorem: homogeneity morphisms of formal neighborhoods lift to smooth homogeneity maps between Batchelor-Gawedzki bundles. Categorically, let $\mathsf{B}_{\mathbb{Z}}$ be the category of finite-dimensional $\mathbb{Z}$-graded vector bundles with homogeneity morphisms, and $\mathsf{Man}_{\mathbb{Z}}$ the category of finite-dimensional $\mathbb{Z}$-graded manifolds. The functor $\mathsf{F}\colon \mathsf{B}_{\mathbb{Z}} \to \mathsf{Man}_{\mathbb{Z}}$ sends bundles to formal neighborhoods of their zero sections. The graded Batchelor-Gawedzki and Borel-Whitney theorems imply $\mathsf{F}$ is full and surjective on objects.

math.DG↗
arXiv2026

Integration \`a la Harish-Chandra for bi-graded Lie algebras

We study $\mathbb Z_2\times\mathbb Z_2$ bi-graded Lie algebras. We describe their properties in relation to Lie superalgebras with some compatible structures. Then we focus on the approach to the Lie group--algebra correspondence based on Harish-Chandra pairs and provide some examples of application of it in the bi-graded setting.

math.DG↗
arXiv2022

Normal forms of $\mathbb Z$-graded $Q$-manifolds

Following recent results of A.K. and V.S. on $\mathbb Z$-graded manifolds, we give several local and global normal forms results for $Q$-structures on those, i.e. for differential graded manifolds. In particular, we explain in which sense their relevant structures are concentrated along the zero-locus of their curvatures, especially when the negative part is of Koszul--Tate type. We also give a local splitting theorem.

math.DG↗
arXiv2022

Various instances of Harish-Chandra pairs

In this paper we address several algebraic constructions in the context of groupoids, algebroids and $\mathbb Z$-graded manifolds. We generalize the results of integration of $\mathbb N$-graded Lie algebras to the honest $\mathbb Z$-graded case and provide some examples of application of the technique based on Harish-Chandra pairs. We extend the construction to the algebroids setting, the main example being the action Lie algebroid.

math.DG↗
arXiv2021

The category of $\mathbb{Z}$-graded manifolds: what happens if you do not stay positive

In this paper we discuss the categorical properties of $\mathbb{Z}$-graded manifolds. We start by describing the local model paying special attention to the differences in comparison to the $\mathbb{N}$-graded case. In particular we explain the origin of formality for the functional space and spell-out the structure of the power series. Then we make this construction intrinsic using filtrations. This sums up to proper definitions of objects and morphisms in the category. We also formulate the analogue of Batchelor's theorem for the global structure of $\mathbb{Z}$-graded manifolds.

math.DG↗
arXiv2021

Presymplectic AKSZ formulation of Einstein gravity

Any local gauge theory can be represented as an AKSZ sigma model (upon parameterization if necessary). However, for non-topological models in dimension higher than 1 the target space is necessarily infinite-dimensional. The interesting alternative known for some time is to allow for degenerate presymplectic structure in the target space. This leads to a very concise AKSZ-like representation for frame-like Lagrangians of gauge systems. In this work we concentrate on Einstein gravity and show that not only the Lagrangian but also the full-scale Batalin--Vilkovisky formulation is naturally encoded in the presymplectic AKSZ formulation, giving an elegant supergeometrical construction of BV for Cartan-Weyl action. The same applies to the main structures of the respective Hamiltonian BFV formulation.

hep-th↗
arXiv2019

The Embedding Tensor, Leibniz-Loday Algebras, and Their Higher Gauge Theories

We show that the data needed for the method of the embedding tensor employed in gauging supergravity theories are precisely those of a Leibniz algebra (with one of its induced quotient Lie algebras embedded into a rigid symmetry Lie algebra that provides an additional "represtentation constraint"). Every Leibniz algebra gives rise to a Lie n-algebra in a canonical way (for every $n\in\mathbb{N}\cup \{ \infty \}$). It is the gauging of this $L_\infty$-algebra that explains the tensor hierarchy of the bosonic sector of gauged supergravity theories. The tower of p-from gauge fields corresponds to Lyndon words of the universal enveloping algebra of the free Lie algebra of an odd vector space in this construction. Truncation to some $n$ yields the reduced field content needed in a concrete spacetime dimension.

hep-th↗
arXiv2019

Differential graded Lie groups and their differential graded Lie algebras

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differential graded Lie groups and study its properties. We show how to associate a differential graded Lie algebra to every differential graded Lie group and vice-versa. For the DGLA $\to$ DGLG direction, the main ``tools'' are graded Hopf algebras and Harish-Chandra pairs (HCP) -- we define the category of graded and differential graded HCPs and explain how those are related to the desired construction. We describe some near at hand examples and mention possible generalizations.

math.DG↗
arXiv2019

Gauge PDE and AKSZ-type Sigma Models

A gauge PDE is a natural notion which arises by abstracting what physicists call a local gauge field theory defined in terms of BV-BRST differential (not necessarily Lagrangian). We study supergeometry of gauge PDEs paying particular attention to globally well-defined definitions and equivalences of such objects. We demonstrate that a natural geometrical language to work with gauge PDEs is that of $Q$-bundles. In particular, we demonstrate that any gauge PDE can be embedded into a super-jet bundle of the $Q$-bundle. This gives a globally well-defined version of the so-called parent formulation. In the case of reparameterization-invariant systems, the parent formulation takes the form of an AKSZ-type sigma model with an infinite-dimensional target space.

hep-th↗
arXiv2019

Lie algebroids, gauge theories, and compatible geometrical structures

The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper analyzes these compatibilities from a mathematical perspective. In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the (possibly singular) foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove furthermore that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized metric structure.

math.DG↗
arXiv2019

Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries

Consider an anchored bundle $(E,ρ)$, i.e. a vector bundle $E\to M$ equipped with a bundle map $ρ\colon E \to TM$ covering the identity. M.~Kapranov showed in the context of Lie-Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid $FR(E)\supset E$. We adapt his construction to the case of an anchored bundle equipped with an arbitrary connection, $(E,\nabla)$, and show that it gives rise to a unique connection $\tilde \nabla$ on $FR(E)$ which is compatible with its Lie algebroid structure, thus turning $(FR(E), \tilde \nabla)$ into a Cartan-Lie algebroid. Moreover, this construction is universal: any connection-preserving vector bundle morphism from $(E,\nabla)$ to a Cartan-Lie Algebroid $(A,\bar \nabla)$ factors through a unique Cartan-Lie algebroid morphism from $(FR(E), \tilde \nabla)$ to $(A,\bar \nabla)$. Suppose that, in addition, $M$ is equipped with a geometrical structure defined by some tensor field $t$ which is compatible with $(E,ρ,\nabla)$ in the sense of being annihilated by a natural $E$-connection that one can associate to these data. For example, for a Riemannian base $(M,g)$ of an involutive anchored bundle $(E,ρ)$, this condition implies that $M$ carries a Riemannian foliation. %In general, the compatibility of a tensor $t$ with $(E,ρ,\nabla)$ implies its adequate invariance transversal to $ρ(E)$. It is shown that every $E$-compatible tensor field $t$ becomes invariant with respect to the Lie algebroid representation associated canonically to the Cartan-Lie algebroid $(FR(E), \tilde \nabla)$.

math.DG↗
arXiv2018

Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids

Cartan-Lie algebroids, i.e. Lie algebroids equipped with a compatible connection, permit the definition of an adjoint representation, on the fiber as well as on the tangent of the base. We call (positive) quadratic Lie algebroids, Cartan-Lie algebroids with ad-invariant (Riemannian) metrics on their fibers and base $κ$ and $g$, respectively. We determine the necessary and sufficient conditions for a positive quadratic Lie algebroid to integrate to a Riemmanian Cartan-Lie groupoid. Here we mean a Cartan-Lie groupoid $\mathcal{G}$ equipped with a bi-invariant and inversion invariant metric $η$ on $T\mathcal{G}$ such that it induces by submersion the metric $g$ on its base and its restriction to the $t$-fibers coincides with $κ$.

math.DG↗
arXiv2015

Curving Yang-Mills-Higgs Gauge Theories

Established fundamental physics can be described by fields, which are maps. The source of such a map is space-time, which can be curved due to gravity. The map itself needs to be curved in its gauge field part so as to describe interaction forces like those mediated by photons and gluons. In the present article, we permit non-zero curvature also on the internal space, the target of the field map. The action functional and the symmetries are constructed in such a way that they reduce to those of standard Yang-Mills-Higgs (YMH) gauge theories precisely when the curvature on the target of the fields is turned off. For curved targets one obtains a new theory, a curved YMH gauge theory. It realizes in a mathematically consistent manner an old wish in the community: replacing structures constants by functions depending on the scalars of the theory. In addition, we provide a simple 4d toy model, where the gauge symmetry is abelian, but turning off the gauge fields, no rigid symmetry remains---another possible manifestation of target curvature. It now remains to be seen, if internal curvature in the above sense is realized in nature. Curvature of space-time is proven, but still negligible in particle physics, except for the very early universe where quantum gravity must have played an essential role. An important question therefore is, if glimpses of target curvature can be visible in accelerator physics. We know that at contemporary energy scales, the usual (flat) standard model describes nature to a very high accuracy. Could it be that the alleged deviations in the B to D-star-tau-nu decay reported by BaBar in 2012 and recently also by LHCb are already a manifestation of target curvature? What kind of effects does target curvature have on a YMH theory in general, for what kind of effects do we need to look out for so as to detect it?

hep-th↗
arXiv2014

2d Gauge Theories and Generalized Geometry

We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra $\mathfrak{g}$ leads naturally to the appearance of the "generalized tangent bundle" $\mathbb{T}M \equiv TM \oplus T^*M$ by means of composite fields. Gauge transformations of the composite fields follow the Courant bracket, closing upon the choice of a Dirac structure $D \subset \mathbb{T}M$ (or, more generally, the choide of a "small Dirac-Rinehart sheaf" $\cal{D}$), in which the fields as well as the symmetry parameters are to take values. In these new variables, the gauge theory takes the form of a (non-topological) Dirac sigma model, which is applicable in a more general context and proves to be universal in two space-time dimensions: A gauging of $\mathfrak{g}$ of a standard sigma model with Wess-Zumino term exists, \emph{iff} there is a prolongation of the rigid symmetry to a Lie algebroid morphism from the action Lie algebroid $M \times \mathfrak{g}\to M$ into $D\to M$ (or the algebraic analogue of the morphism in the case of $\cal{D}$). The gauged sigma model results from a pullback by this morphism from the Dirac sigma model, which proves to be universal in two-spacetime dimensions in this sense.

hep-th↗
arXiv2014

Gauging without Initial Symmetry

The gauge principle is at the heart of a good part of fundamental physics: Starting with a group G of so-called rigid symmetries of a functional defined over space-time Sigma, the original functional is extended appropriately by additional Lie(G)-valued 1-form gauge fields so as to lift the symmetry to Maps(Sigma,G). Physically relevant quantities are then to be obtained as the quotient of the solutions to the Euler-Lagrange equations by these gauge symmetries. In this article we show that one can construct a gauge theory for a standard sigma model in arbitrary space-time dimensions where the target metric is not invariant with respect to any rigid symmetry group, but satisfies a much weaker condition: It is sufficient to find a collection of vector fields v_a on the target M satisfying the extended Killing equation v_{a(i;j)}=0 for some connection acting on the index a. For regular foliations this is equivalent to requiring the conormal bundle to the leaves with its induced metric to be invariant under leaf-preserving diffeomorphisms of M, which in turn generalizes Riemannian submersions to which the notion reduces for smooth leaf spaces M/~. The resulting gauge theory has the usual quotient effect with respect to the original ungauged theory: in this way, much more general orbits can be factored out than usually considered. In some cases these are orbits that do not correspond to an initial symmetry, but still can be generated by a finite dimensional Lie group G. Then the presented gauging procedure leads to an ordinary gauge theory with Lie algebra valued 1-form gauge fields, but showing an unconventional transformation law. In general, however, one finds that the notion of an ordinary structural Lie group is too restrictive and should be replaced by the much more general notion of a structural Lie groupoid.

hep-th↗
arXiv2009

Geometric structures encoded in the Lie structure of an Atiyah algebroid

We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffeomorphic. Further, we give two characterizations of the isomorphisms of the Lie algebras of sections for Atiyah algebroids associated to principle bundles with semisimple structure groups. For instance we prove that in the semisimple case the Lie algebras of sections are isomorphic if and only if the corresponding Lie algebroids are, or, as well, if and only if the integrating principal bundles are locally diffeomorphic. Finally, we apply these results to describe the isomorphisms of sections in the case of reductive structure groups -- surprisingly enough they are no longer determined by vector bundle isomorphisms and involve divergences on the base manifolds.

math.DG↗
arXiv2009

Lie algebaic characterization of supercommutative space

During the last decades algebraization of space turned out to be a promising tool at the interface between Mathematics and Theoretical Physics. Starting with works by Gel'fand-Kolmogoroff and Gel'fand-Naimark, this branch developed as from the fortieth in two directions: algebraic characterization of usual geometric space on the one hand, and algebraically defined noncommutative space, which is known to be tightly related with e.g. quantum gravity and super string theory, on the other hand. In this note, we combine both aspects, prove a superversion of Shanks and Pursell's classical result stating that any isomorphism of the Lie algebras of compactly supported vector fields is implemented by a diffeomorphism of underlying manifolds. We thus provide a super Lie algebraic characterization of super and graded spaces and describe explicitly isomorphisms of the super Lie algebras of super vector fields.

math.RA↗