Anton M. Zeitlin

On first-order formalism in string theory

Abstract We consider the first-order formalism in string theory, providing a new off-shell description of the non-trivial backgrounds around an “infinite metric”. The OPE of the vertex operators, corresponding to the background fields in some “twistor representation”, and conditions of conformal invariance results in the quadratic equation for the background fields, which appears to be equivalent to the Einstein equations with a Kalb–Ramond B -field and a dilaton. Using a new representation for the Einstein equations with B -field and dilaton we find a new class of solutions including the plane waves for metric (graviton) and the B -field. We discuss the properties of these background equations and main features of the BRST operator in this approach.

Homotopy Lie superalgebra in Yang-Mills theory

The Yang-Mills equations are formulated in the form of generalized Maurer-Cartan equations, such that the corresponding algebraic operations are shown to satisfy the defining relations of homotopy Lie superalgebra.

Formal Maurer-Cartan structures: from CFT to Classical Field Equations

We show how the well-known classical field equations as Einstein and Yang-Mills ones, which arise as the conformal invariance conditions of certain two-dimensional theories, expanded up to the second order in the formal parameter, can be reformulated as Generalized/formal Maurer-Cartan equations (GMC), where the differential is the BRST operator of String theory. We introduce the bilinear operations which are present in GMC, and study their properties, allowing us to find the symmetries of the resulting equations which will be naturally identified with the diffeomorphism and gauge symmetries of Einstein and Yang-Mills equations correspondingly.

Integrable structure of superconformal field theory and quantum super-KdV theory

Abstract The integrable structure of the two-dimensional superconformal field theory is considered. The classical counterpart of our constructions is based on the osp (1|2) super-KdV hierarchy. The quantum version of the monodromy matrix associated with the linear problem for the corresponding L-operator is introduced. Using the explicit form of the irreducible representations of osp q (1|2) , the so-called “fusion relations” for the transfer matrices considered in different representations of osp q (1|2) are obtained. The possible integrable perturbations of the model (primary operators, commuting with integrals of motion) are classified and the relation with the supersymmetric osp (1|2) Toda field theory is discussed.

Perturbed beta–gamma systems and complex geometry

Abstract We consider the equations, arising as the conformal invariance conditions of the perturbed curved beta–gamma system. These equations have the physical meaning of Einstein equations with a B-field and a dilaton on a Hermitian manifold, where the B-field 2-form is imaginary and proportional to the canonical form associated with Hermitian metric. We show that they decompose into linear and bilinear equations and lead to the vanishing of the first Chern class of the manifold where the system is defined. We discuss the relation of these equations to the generalized Maurer–Cartan structures related to BRST operator. Finally we describe the relations of the generalized Maurer–Cartan bilinear operation and the Courant/Dorfman brackets.

Superconformal field theory and SUSY N=1 KdV hierarchy I: vertex operators and Yang–Baxter equation

Abstract The supersymmetry invariant integrable structure of two-dimensional superconformal field theory is considered. The classical limit of the corresponding infinite family of integrals of motion (IM) coincide with the family of IM of SUSY N = 1 KdV hierarchy. The quantum version of the monodromy matrix, generating quantum IM, associated with the SUSY N = 1 KdV is constructed via vertex operator representation of the quantum R-matrix. The possible applications to the perturbed superconformal models are discussed.

Conformal Field Theory and algebraic structure of gauge theory

We consider various homotopy algebras related to Yang-Mills theory and twodimensional conformal field theory (CFT). Our main objects of study are Yang-Mills L∞ and C∞ algebras and their relation to the certain algebraic structures of Lian-Zuckerman type in CFT. We also consider several examples of algebras related to gauge theory, involving first order formulations and gauge theories with matter fields.

BRST, generalized Maurer–Cartan equations and CFT

Abstract The paper is devoted to the study of BRST charge in perturbed two-dimensional conformal field theory. The main goal is to write the operator equation expressing the conservation law of BRST charge in perturbed theory in terms of purely algebraic operations on the corresponding operator algebra, which are defined via the OPE. The corresponding equations are constructed and their symmetries are studied up to the second order in formal coupling constant. It appears that the obtained equations can be interpreted as generalized Maurer–Cartan ones. We study two concrete examples in detail: the bosonic nonlinear sigma model and perturbed first order theory. In particular, we show that the Einstein equations, which are the conformal invariance conditions for both these perturbed theories, expanded up to the second order, can be rewritten in such generalized Maurer–Cartan form.

String field theory-inspired algebraic structures in gauge theories

We consider gauge theories in a String Field Theory-inspired formalism. The constructed algebraic operations lead in particular to homotopy algebras of the related BV theories. We discuss invariant description of the gauge fixing procedure and special algebraic features of gauge theories coupled to matter fields.

Batalin-Vilkovisky Yang-Mills theory as a homotopy Chern-Simons theory via string field theory

We show explicitly how Batalin–Vilkovisky Yang–Mills action emerges as a homotopy generalization of Chern–Simons theory from the algebraic constructions arising from string field theory.We show that BV Yang-Mills action can be reformulated in the homotopy Chern-Simons form. The corresponding formalism is based on the constructions introduced in [4], where the Yang-Mills equations were rewritten as the generalized Maurer-Cartan equations for some Homotopy Lie algebra.