Andrei Losev

Issues in Topological Gauge Theory

We discuss topological theories, arising from the general N = 2 twisted gauge theories. We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various p-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme which uniquely fixes all the contact terms. We derive a formula for the correlation functions of p-observables on the manifolds of generalized simple type for 0 ⪕ p ⪕ 4 and on some manifolds with b2+ = 1. We study the theories with matter and explore the properties of universal instantons. We also discuss the compactifications of higher dimensional theories. Some relations to sigma models of type A and B are pointed out and exploited.

Four-Dimensional Avatars of Two-Dimensional RCFT

We investigate a 4D analog of 2D WZW theory. The theory turns out to have surprising finiteness properties and an infinite-dimensional current algebra symmetry. Some correlation functions are determined by this symmetry. One way to define the theory systematically proceeds by the quantization of moduli spaces of holomorphic vector bundles over algebraic surfaces. We outline how one can define vertex operators in the theories. Finally, we define four-dimensional “conformal blocks” and present an analog of the Verlinde formula.

Chern-Simons and twisted supersymmetry in various dimensions

Abstract We introduce special supersymmetric gauge theories in three, five, seven and nine dimensions, whose compactification on two-, four-, six- and eight folds produces a supersymmetric quantum mechanics on moduli spaces of holomorphic bundles and/or solutions to the analogues of instanton equations in higher dimensions. The theories may occur on the world-volumes of D-branes wrapping manifolds of a special holonomy. We also discuss the theories with matter.We describe special supersymmetric gauge theories in three, five, seven and nine dimensions, whose compactification on two-, four-, six- and eight-folds produces a supersymmetric quantum mechanics on moduli spaces of holomorphic bundles and/or solutions to the analogues of instanton equations in higher dimensions. The theories may occur on the worldvolumes of D-branes wrapping manifolds of special holonomy. We also discuss the theories with matter.

Chiral Lagrangians, anomalies, supersymmetry, and holomorphy

Abstract We investigate higher-dimensional analogues of the bc systems of 2D RCFT. When coupled to gauge fields and Beltrami differentials defining integrable holomorphic structures, the bc partition functions can be explicitly evaluated using anomalies and holomorphy. The resulting induced actions generalize the chiral algebras of 2D RCFT to 2 n dimensions. Moreover, bc systems in four and six dimensions are closely related to supersymmetric matter. In particular, we show that d = 4, N = 2 hypermultiplets induce a theory of self-dual Yang-Mills fields coupled to self-dual gravity. In this way the bc systems fermionize both the algebraic sector of the WZW 4 theory, as defined by Losev et al., and the classical open N WS = 2 string.

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.

THE FRECKLED INSTANTONS

We study instanton-corrected renormalization group flow in the two dimensional sigma models and four dimensional gauge theory. In two dimensions we do that by replacing the non-linear supersymmetric

Freckled instantons in two-dimensions and four-dimensions

{\IC\IP}^{N-1}

Hodge Strings and Elements of K. Saito’s Theory of Primitive Form

model by the gauged linear sigma model which is in the same universality class. We compare the moduli spaces of the instantons in the non-linear model and that of BPS field configurations in the linear model. We reduce the problem of matching of the parameters of the two systems to the intersection theory on the compact moduli space of the latter model. Both cases (2d and 4d) are unified by the notion of the {\it freckled instantons}. We also put an end to the discussion of the nature of the superpotentials

SMALL INSTANTONS, LITTLE STRINGS AND FREE FERMIONS

W \sim {\s} {\rm log} {\s}

Central extensions of gauge groups revisited

in 2d and 4d and discover the surprising disconnectnessness of the effective target space.We study instanton-corrected renormalization group flow in the two dimensional sigma models and four dimensional gauge theory. In two dimensions we do that by replacing the non-linear supersymmetric CP N −1 model by the gauged linear sigma model which is in the same universality class. We compare the moduli spaces of the instantons in the non-linear model and that of BPS field configurations in the linear model. We reduce the problem of matching of the parameters of the two systems to the intersection theory on the compact moduli space of the latter model. In four dimensions we find that the analogue of the linear sigma model is the gauge theory on the non-commutative space-time. Its BPS moduli space is the space of torsion free sheaves. Both cases (2d and 4d) are unified by the notion of the freckled instantons. We also put an end to the discussion of the nature of the superpotentials W ∼ σlogσ in 2d and 4d and discover the surprising disconnectnessness of the effective target space.