arXiv subjects

Andrey Losev

Publications and source records attributed to Andrey Losev.

18 recordsLinked to original sources

arXiv2026

Cohomological beta function

We propose a cohomological approach to computing the conformal anomaly. Using the example of current-current deformations of two-dimensional conformal field theories, we reproduce the well-known Cardy formula for the leading contribution to the perturbative beta function as the coefficient of the cocycle that realizes the obstruction to deforming the conformal algebra module structure on the state space. In addition to offering a novel conceptual perspective on the conformal anomaly, the proposed approach is anticipated to provide an efficient tool for computing higher-order coefficients of perturbative beta functions.

math-ph↗

arXiv2026

Perturbative anomalies in quantum mechanics

In this work, we propose a cohomological approach to studying perturbative anomalies in quantum mechanics. The Hamiltonian $\hat{H}$ together with the symmetry generator $\hat{S}$ forms a unitary representation of the two-dimensional Abelian Lie algebra $\mathfrak{g}\cong \mathbb{R}^{2}$ on the Hilbert space $V$. We show that perturbations of such a system are related to the first Chevalley-Eilenberg cohomology group $H^{1}_{CE}(\mathbb{R}^{2},\mathfrak{u}(V))$. In turn, the perturbative anomalies of the symmetry $\hat{S}$ are related to the second cohomology group $H^{2}_{CE}(\mathbb{R}^{2},\mathfrak{u}(V))$.

math-ph↗

arXiv2025

Perturbations vs deformations

In the first part of the paper we define a perturbative (pre-formal) geometry and formulate a theorem on the relation between the construction of a perturbative neighborhood of affine varieties and the higher tangent bundles. In the second part of the paper, we discuss perturbative vector fields and related structures, which are finite-dimensional analogs of perturbation theory characteristics arising in quantum field theory.

math-ph↗

arXiv2025

Beta function without UV divergences

In this paper, we construct the beta function in the functorial formulation of two-dimensional quantum field theories (FQFT). A key feature of this approach is the absence of ultraviolet divergences. We show that, nevertheless, in the FQFT perturbation theory, the local observables of deformed theories acquire logarithmic dimension, leading to a conformal anomaly. The beta function arises in the functorial approach as an infinitesimal transformation of the partition function under the variation of the metric's conformal factor, without ultraviolet divergences, UV cutoff, or the traditional renormalization procedure.

math-ph↗

arXiv2024

On induced L-infinity action of diffeomorphisms on Cochains

One of the approaches to quantum gravity is to formulate it in terms of De Rham algebra, choose a triangulation of space-time, and replace differential forms by cochains (that form a finite dimensional vector space). The key issue of general relativity is the action of diffeomorphisms of space-time on fields. In this paper, we induce the action of diffeomorphisms on cochains by homotopy transfer (or, equivalently, BV integral) that leads to a $L_{\infty}$ action. We explicitly compute this action for the space-time being an interval, a circle, and a square.

math-ph↗

arXiv2024

Combinatorial 2d higher topological quantum field theory from a local cyclic $A_\infty$ algebra

We construct combinatorial analogs of 2d higher topological quantum field theories. We consider triangulations as vertices of a certain CW complex $Ξ$. In the "flip theory," cells of $Ξ_\mathrm{flip}$ correspond to polygonal decompositions obtained by erasing the edges in a triangulation. These theories assign to a cobordism $Σ$ a cochain $Z$ on $Ξ_\mathrm{flip}$ constructed as a contraction of structure tensors of a cyclic $A_\infty$ algebra $V$ assigned to polygons. The cyclic $A_\infty$ equations imply the closedness equation $(δ+Q)Z=0$. In this context we define combinatorial BV operators and give examples with coefficients in $\mathbb{Z}_2$. In the "secondary polytope theory," $Ξ_\mathrm{sp}$ is the secondary polytope (due to Gelfand-Kapranov-Zelevinsky) and the cyclic $A_\infty$ algebra has to be replaced by an appropriate refinement that we call an $\widehat{A}_\infty$ algebra. We conjecture the existence of a good Pachner CW complex $Ξ$ for any cobordism, whose local combinatorics is descibed by secondary polytopes and the homotopy type is that of Zwiebach's moduli space of complex structures. Depending on this conjecture, one has an "ideal model" of combinatorial 2d HTQFT determined by a local $\widehat{A}_\infty$ algebra.

math-ph↗

arXiv2024

On enumerative problems for maps and quasimaps: freckles and scars

We address the question of counting maps between projective spaces such that images of cycles on the source intersect cycles on the target. In this paper we do it by embedding maps into quasimaps that form a projective space of their own. When a quasimap is not a map, it contains freckles (studied earlier) and/or scars, appearing when the complex dimension of the source is greater than one. We consider a lot of examples showing that freckle/scar calculus (using excess intersection theory) works. We also propose the "smooth conjecture" that may lead to computation of the number of maps by an integral over the space of quasimaps.

math-ph↗

arXiv2024

Maurer-Cartan methods in perturbative quantum mechanics

We reformulate the time-independent Schrödinger equation as a Maurer-Cartan equation on the superspace of eigensystems of the former equation. We then twist the differential so that its cohomology becomes the space of solutions with a set energy. A perturbation of the Hamiltonian corresponds to a deformation of the twisted differential, leading to a simple recursive relation for the eigenvalue and eigenfunction corrections.

math-ph↗

arXiv2023

BV-refinement of the on-shell supersymmetry and localization

We generalize the BV formalism for the physical theories on supermanifolds with graded symmetry algebras realized off-shell and on-shell. An application of such generalization to supersymmetric theories allows us to formulate the new classification which refines the usual off-shell/on-shell classification. Our new classification is based on the type of higher order antifield terms in the corresponding BV action. We provide explicit examples for each class of the refined classification. We show that the supersymmetric localization for the off-shell theories is a BV integral over particular Lagrangian submanifold. We generalize the supersymmetric localization to the on-shell supersymmetric theories of quadratic type in our classification. The partition function for such theories becomes Gaussian integral for a particular choice of Lagrangian submanifold.

hep-th↗

arXiv2023

Tropical Mirror Symmetry: Correlation functions

We formulate the mirror symmetry for correlation functions of tropical observables. We prove the tropical mirror correspondence for correlation functions of evaluation observables on toric space. The key point of the proof is the localization of correlation functions for mirror states in type-B higher topological quantum mechanics on trees. The correlation functions localize to the correlation functions of holomorphic functions, defined recursively in Landau-Ginzburg-Saito theory with exponential mirror superpotential and tropical good section.

hep-th↗

arXiv2023

Tropical mirror for toric surfaces

We describe the tropical mirror for complex toric surfaces. In particular we provide an explicit expression for the mirror states and show that they can be written in enumerative form. Their holomorphic germs give an explicit form of good section for Landau-Ginzburg-Saito theory. We use an explicit form of holomorphic germs to derive the divisor relation for tropical Gromov-Witten invariants. We interpret the deformation of the theory by a point observable as a blow up of a point on the toric surface. We describe the implication of such interpretation for the tropical Gromov-Witten invariants.

hep-th↗

arXiv2023

Feynman geometry

In this paper we introduce a notion of Feynman geometry on which quantum field theories could be properly defined. A strong Feynman geometry is a geometry when the vector space of $A_\infty$ structures is finite dimensional. A weak Feynman geometry is a geometry when the vector space of $A_\infty$ structures is infinite dimensional while the relevant operators are of trace-class. We construct families of Feynman geometries with "continuum" as their limit.

hep-th↗

arXiv2023

TQFT, Homological Algebra and elements of K.Saito's Theory of Primitive Form: an attempt of mathematical text written by mathematical physicist

The text is devoted to explanation of the concept of Topological Quantum Field Theory (TQFT), its application to homological algebra and to the relation with the theory of good section from K.Saito's theory of Primitive forms. TQFT is explained in Dirac-Segal framework, one-dimensional examples are explained in detail. As a first application we show how it can be used in explicit construction of reduction of infinity-structure after contraction of a subcomplex. Then we explain Associativity and Commutativity equations using this language. We use these results to construct solutions to Commutativity equations and find a new proof of for the fact that tree level BCOV theory solved Oriented Associativity equations.

math-ph↗

arXiv2022

Two field-theoretic viewpoints on the Fukaya-Morse $A_\infty$ category

We study an enhanced version of the Morse degeneration of Fukaya $A_\infty$ category with higher compositions given by counts of gradient flow trees. The enhancement consists in allowing morphisms from an object to itself to be chains on the manifold. Higher compositions correspond to counting Morse trees passing through a given set of chains. We provide two viewpoints on the construction and on the proof of the $A_\infty$ relations for the composition maps. One viewpoint is via an effective action for the $BF$ theory computed in a special gauge. The other is via higher topological quantum mechanics.

hep-th↗

arXiv2022

Tropical Mirror

We describe the tropical curves in toric varieties and define the tropical Gromov-Witten invariants. We introduce amplitudes for the higher topological quantum mechanics (HTQM) on special trees and show that the amplitudes are equal to the tropical Gromov-Witten invariants. We show that the sum over the amplitudes in $A$-model HTQM equals the total amplitude in B-model HTQM, defined as a deformation of the $A$-model HTQM by the mirror superpotential. We derived the mirror superpotentials for the toric varieties and showed that they coincide with the superpotentials in the mirror Landau-Ginzburg theory. We construct the mirror dual states to the evaluation observables in the tropical Gromov-Witten theory.

hep-th↗

arXiv2018

Ultraviolet Properties of the Self-Dual Yang-Mills Theory

We compute the ultraviolet divergences in the self-dual Yang-Mills theory, both in the purely perturbative (zero instanton charge) and topologically non-trivial sectors. It is shown in particular that the instanton measure is precisely the same as the one-loop result in the standard Yang-Mills theory.

hep-th↗

arXiv2007

On Pure Spinor Superfield Formalism

We show that a certain superfield formalism can be used to find an off-shell supersymmetric description for some supersymmetric field theories where conventional superfield formalism does not work. This "new" formalism contains even auxiliary variables in addition to conventional odd super-coordinates. The idea of this construction is similar to the pure spinor formalism developed by N.Berkovits. It is demonstrated that using this formalism it is possible to prove that the certain Chern-Simons-like (Witten's OSFT-like) theory can be considered as an off-shell version for some on-shell supersymmetric field theories. We use the simplest non-trivial model found in [2] to illustrate the power of this pure spinor superfield formalism. Then we redo all the calculations for the case of 10-dimensional Super-Yang-Mills theory. The construction of off-shell description for this theory is more subtle in comparison with the model of [2] and requires additional Z_2 projection. We discover experimentally (through a direct explicit calculation) a non-trivial Z_2 duality at the level of Feynman diagrams. The nature of this duality requires a better investigation.

hep-th↗

arXiv2004

Riemann-Roch-Hirzebruch theorem and Topological Quantum Mechanics

In the present paper we discuss an independent on the Grothendieck-Sato isomorphism approach to the Riemann-Roch-Hirzebruch formula for an arbitrary differential operator. Instead of the Grothendieck-Sato isomorphism, we use the Topological Quantum Mechanics (more or less equivalent to the well-known constructions with the Massey operations from [KS], [P], [Me]). The statement that the Massey operations can "produce" the integral in some set-up, has an independent from the RRH theorem interest. We finish the paper by some open questions arising when the main construction is applied to the cyclic homology (instead of the Hochschild homology).

math.QA↗