Instantons and 2d Superconformal field theory

A recently proposed correspondence between 4-dimensional N=2 SUSY SU(k) gauge theories on R^4/Z_m and SU(k) Toda-like theories with Z_m parafermionic symmetry is used to construct four-point N=1 super Liouville conformal block, which corresponds to the particular case k=m=2. The construction is based on the conjectural relation between moduli spaces of SU(2) instantons on R^4/Z_2 and algebras like \hat{gl}(2)_2\times NSR. This conjecture is confirmed by checking the coincidence of number of fixed points on such instanton moduli space with given instanton number N and dimension of subspace degree N in the representation of such algebra.Comment: 13 pages, exposition improved, references adde

Low-Lying Dirac Eigenmodes, Topological Charge Fluctuations and the Instanton Liquid Model

The local structure of low-lying eigenmodes of the overlap Dirac operator is studied. It is found that these modes cannot be described as linear combinations of 't Hooft "would-be" zeromodes associated with instanton excitations that underly the Instanton Liquid Model. This implies that the instanton liquid scenario for spontaneous chiral symmetry breaking in QCD is not accurate. More generally, our data suggests that the vacuum fluctuations of topological charge are not effectively dominated by localized lumps of unit charge with which the topological "would-be" zeromodes could be associated.Comment: Presented by I. Horvath at the NATO Advanced Research Workshop "Confinement, Topology, and other Non-Perturbative Aspects of QCD", January 21-27, 2002, Stara Lesna, Slovakia. 12 pages, 6 figures, uses crckapb.st

Vertex operator algebras and operads

Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, $n$-ary operations for all $n$ greater than or equal to $0$, not just binary products. In this paper, a reformulation of the notion of vertex operator algebra in terms of operads is presented. This reformulation shows that the rich geometric structure revealed in the study of conformal field theory and the rich algebraic structure of the theory of vertex operator algebras share a precise common foundation in basic operations associated with a certain kind of (two-dimensional) ``complex'' geometric object, in the sense in which classical algebraic structures (groups, algebras, Lie algebras and the like) are always implicitly based on (one-dimensional) ``real'' geometric objects. In effect, the standard analogy between point-particle theory and string theory is being shown to manifest itself at a more fundamental mathematical level.Comment: 16 pages. Only the definitions of "partial operad" and of "rescaling group" have been improve

Conformal Invariance in Percolation, Self-Avoiding Walks and Related Problems

Over the years, problems like percolation and self-avoiding walks have provided important testing grounds for our understanding of the nature of the critical state. I describe some very recent ideas, as well as some older ones, which cast light both on these problems themselves and on the quantum field theories to which they correspond. These ideas come from conformal field theory, Coulomb gas mappings, and stochastic Loewner evolution.Comment: Plenary talk given at TH-2002, Paris. 21 pages, 9 figure

Field theory of scaling lattice models. The Potts antiferromagnet

In contrast to what happens for ferromagnets, the lattice structure participates in a crucial way to determine existence and type of critical behaviour in antiferromagnetic systems. It is an interesting question to investigate how the memory of the lattice survives in the field theory describing a scaling antiferromagnet. We discuss this issue for the square lattice three-state Potts model, whose scaling limit as T->0 is argued to be described exactly by the sine-Gordon field theory at a specific value of the coupling. The solution of the scaling ferromagnetic case is recalled for comparison. The field theory describing the crossover from antiferromagnetic to ferromagnetic behaviour is also introduced.Comment: 11 pages, to appear in the proceedings of the NATO Advanced Research Workshop on Statistical Field Theories, Como 18-23 June 200

Conformal Toda theory with a boundary

We investigate sl(n) conformal Toda theory with maximally symmetric boundaries. There are two types of maximally symmetric boundary conditions, due to the existence of an order two automorphism of the W(n>2) algebra. In one of the two cases, we find that there exist D-branes of all possible dimensions 0 =< d =< n-1, which correspond to partly degenerate representations of the W(n) algebra. We perform classical and conformal bootstrap analyses of such D-branes, and relate these two approaches by using the semi-classical light asymptotic limit. In particular we determine the bulk one-point functions. We observe remarkably severe divergences in the annulus partition functions, and attribute their origin to the existence of infinite multiplicities in the fusion of representations of the W(n>2) algebra. We also comment on the issue of the existence of a boundary action, using the calculus of constrained functional forms, and derive the generating function of the B"acklund transformation for sl(3) Toda classical mechanics, using the minisuperspace limit of the bulk one-point function.Comment: 42 pages; version 4: added clarifications in section 2.2 and footnotes 1 and