The Orbifold Transform and its Applications

We discuss the notion of the orbifold transform, and illustrate it on simple examples. The basic properties of the transform are presented, including transitivity and the exponential formula for symmetric products. The connection with the theory of permutation orbifolds is addressed, and the general results illustrated on the example of torus partition functions. Work supported by grants OTKA T047041, T043582, the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and EC Marie Curie MRTN-CT-2004-512194.     

Introduction to Conformal Field Theory

An elementary introduction to conformal field theory is given. Topics include free bosons and fermions, orbifolds, affine Lie algebras, coset conformal field theories, superconformal theories, correlation functions on the sphere, partition functions and modular invariance.     

From 4d superconformal indices to 3d partition functions

An exact formula for partition functions in 3d field theories was recently suggested by Jafferis, and Hama, Hosomichi, and Lee. These functions are expressed in terms of specific q-hypergeometric integrals whose key building block is the double sine function (or the hyperbolic gamma function). Elliptic hypergeometric integrals, discovered by the second author, define 4d superconformal indices. Using their reduction to the hyperbolic level, we describe a general scheme of reducing 4d superconformal indices to 3d partition functions which imply an efficient way of getting 3dN=2 supersymmetric dualities for both SYM and CS theories from the “parent” 4dN=1 dualities for SYM theories. As an example, we consider explicitly the duality pattern for 3dN=2 SYM and CS theories with SP(2N) gauge group with the antisymmetric tensor matter.     

Construction ofN = 2 superconformal algebra from affine algebras with extended symmetry: I

The purpose of this Letter is to use the idea of the Sugawara-Ka-Todorov construction of theN = 0 andN = 1 superconformal algebras to construct a very simple free-field realization of theN = 2 superconformal algebra.     

Remarks on the thermodynamics and the vacuum energy of a quantum Maxwell gas on compact and closed manifolds

The quantum Maxwell theory at finite temperature at equilibrium is studied on compact and closed manifolds in both the functional integral and Hamiltonian formalism. The aim is to shed some light onto the interrelation between the topology of the spatial background and the thermodynamic properties of the system. The quantization is not unique and gives rise to inequivalent quantum theories which are classified by θ-vacua. Based on explicit parametrizations of the gauge orbit space in the functional integral approach and of the physical phase space in the canonical quantization scheme, the Gribov problem is resolved and the equivalence of both quantization schemes is elucidated. Using zeta-function regularization the free energy is determined and the effect of the topology of the spatial manifold on the vacuum energy and on the thermal gauge field excitations is clarified. The general results are then applied to a quantum Maxwell gas on an n-dimensional torus providing explicit formulae for the main thermodynamic functions in the low- and high-temperature regimes, respectively.     

Reduction of 4-dim self-dual super Yang-Mills equations onto super Riemann surfaces

Recently, a self-dual super Yang-Mills equation over a super Reimann surface was obtained as the zero set of a moment map on the space of superconnections to the dual of the super Lie algebra of gauge transformations. We present a new formulation of the 4-dim Euclidean self-dual super Yang-Mills equations in terms of constraints on the supercurvature. By dimensional reduction, we obtain the same set of superconformal field equations which define self-dual connections on a super Rieman surface.     

Geometry of superconformal manifolds

The main facts about complex curves are generalized to superconformal manifolds. The results thus obtained are relevant to the fermion string theory and, in particular, they are useful for computation of determinants of super laplacians which enter the string partition function.To the memory of our friend and colleague Vadim Knizhnik     

Four-dimensional heterotic strings and conformal field theory

The techniques of (super) conformal field theory are applied to 4-dimensional heterotic string theories. We discuss certain aspects of 4-dimensional strings in the framework of the bosonic lattice approach such as the realization of superconformal symmetry, character valued partition functions, construction of vertex operators and ghost picture changing. As an application we compute all possible 3- and 4-point tree amplitudes of the massless fields and derive from them the low energy effective action of the massless modes. Some effects for the massless spectrum due to one-loop string effects are also mentioned.     

Modular invariance in N=2 superconformal field theories

The modular invariance properties of two-dimensional N=2 superconformal field theories are studied. It is shown that the character formulae of the central charge c<3 unitary highest weight representation for the untwisted algebras can be written in terms of the string functions and the theta functions of the affine su(2) Kac-Moody algebra. Deriving the modular transformation of the characters we construct the modular invariant partition functions on a torus. The character relation corresponding to the coset space construction of the unitary discrete series in the N=2 algebra is also obtained.     

G/H conformal field theory from gauged WZW model

We show that the coset construction for affine algebras ĝ ⊃ ĥ can be realized by coupling a group G WZW model to a gauge field taking values in the Lie algebra h. The partition function of the coset models is computed exactly in terms of the branching functions of ĝ⊃ĥ. Correlation functions may be expressed in terms of those of the G-valued WZW model and of the H_scc/H-valued one, also exactly soluble. The special cases include unitary, superconformal, paramefermionic and other discrete series.     

Modular geometry of superconformal field theory

Two-dimensional superconformal field theory is discussed as geometry on universal supermoduli space, generalizing the conformal case. The compactification divisor of supermoduli space consists of super-Riemann surfaces with spin and super nodes, which are expressed algebraically. The local behavior of the super partition function on universal supermoduli space is described.     

Self-dual Yang-Mills fields in d=7,8, octonions and ward equations

The Yang-Mills theories in d=7 and d=8 with the arbitrary gauge group G are considered. Generalized self-duality-type relations for gauge fields are reduced to systems of nonlinear differential equations on functions of one variable (Ward equations). Ward equations may be reduced to equations which follow from Yang-Baxter equations. This permits us to obtain new classes of explicit solutions of the Yang-Mills equations in d=7 and d=8.     

Finite <Emphasis Type="Italic">N</Emphasis> index and angular momentum bound from gravity

We exactly compute the finite N index and BPS partition functions for SYM theory in a newly proposed maximal angular momentum limit. The new limit is not predicted from the superconformal algebra, but naturally arises from the supergravity dual. We show that the index does not receive any finite N corrections while the free BPS partition function does.     

C-disorder fields and Γ(2)-invariant partition functions in parafermionic conformal field theories

The character formulae for the C-disorder fields corresponding to the charge conjugation symmetry in the parafermionic conformal field theories of Zamolodchikov and Fateev are obtained. These characters span a unitary representation of the level 2 subgroup Γ(2) of the full modular group. The corresponding Γ(2)-invariant partition functions are calculated. Derivation of the character formula is based on the relation between the parafermionic theories and the twisted N = 2 superconformal algebra. A similar idea is applied to explicitly construct the highest weight modules of the twisted SU(2) Kac-Moody algebra.     

A note on gauge symmetry breaking

We show that the breaking of Abelian gauge symmetry implies the existence of dipole singularities in the correlation functions of the (Abelian) Higgs model. We also show that the noninvariance of the Wightman functions does not preclude the implementability of the global gauge symmetry. An explicit example of gauge symmetry breaking (Ferrari's model) is discussed.     

Action functionals for strings in four dimensions

All possible action functionals on the space of surfaces in ⁴ that depend only on first and second derivatives of the functions, entering the equation of the surface, and satisfy the condition of invariance with respect to rigid motions, are described.     

Superconformal algebras and their relation to integrable nonlinear systems

We relate the manifold of periodic functions on a circle with values in the Grassmann algebra to extended superconformal algebras. The graded Poisson brackets of these functions give the classical realization of the corresponding superconformal algebras and determine the hamiltonian structure for a class of integrable nonlinear equations. A super-generalization of the Korteweg-de Vries equation is found among these equations. In this way an important step in the program of the quantization of the Liouville equation is realized for the supersymmetric cases which are crucial in constructing a consistent quantum string theory. The construction of Miura transformations is outlined and the results for the N = 1,2 supersymmetric cases are presented.     

The gravitational anomalies in de Sitter space

All solutions to the consistency equations are determined which have to be satisfied by anomalies in gravitational theories with a de Sitter-invariant groundstate. They turn out to be identical with the solutions for a Poincaré-invariant groundstate.