We develop a diagrammatic language for symmetric product orbifolds of two-dimensional conformal field theories. Correlation functions of twist operators are written as sums of diagrams: each diagram corresponds to a branched covering map from a surface where the fields are single-valued to the base sphere where twist operators are inserted. This diagrammatic language facilitates the study of the large N limit and makes more transparent the analogy between symmetric product orbifolds and free non-abelian gauge theories. We give a general algorithm to calculate the leading large N contribution to four-point correlators of twist fields.

We present a Hamiltonian Monte Carlo algorithm to sample from multivariate Gaussian distributions in which the target space is constrained by linear and quadratic inequalities or products thereof. The Hamiltonian equations of motion can be integrated exactly and there are no parameters to tune. The algorithm mixes faster and is more efficient than Gibbs sampling. The runtime depends on the number and shape of the constraints but the algorithm is highly parallelizable. In many cases, we can exploit special structure in the covariance matrices of the untruncated Gaussian to further speed up the runtime. A simple extension of the algorithm permits sampling from distributions whose log-density is piecewise quadratic, as in the "Bayesian Lasso" model.

Abstract:We compute the partition function of the supersymmetric two-dimensional Euclidean black hole geometry described by the SL(2, R)/U (1) superconformal field theory. We decompose the result in terms of characters of the N = 2 superconformal symmetry. We point out puzzling sectors of states besides finding expected discrete and continuous contributions to the partition function. By adding an N = 2 minimal model factor of the correct central charge and projecting on integral N = 2 charges we compute the partition function of the background dual to little string theory in a double scaling limit. We show the precise correspondence between this theory and the background for NS5-branes on a circle, due to an exact description of the background as a null gauging of SL(2, R) × SU (2). Finally, we discuss the interplay between GSO projection and target space geometry.

Abstract:We study D-branes in the mirror pair N = 2 Liouville/supersymmetric SL(2, R)/U (1) coset superconformal field theories. After revisiting the duality between the two models, we build D0, D1 and D2 branes, on the basis of the boundary state construction for the H + 3 conformal field theory. We also construct D0-branes in an orbifold that rotates the angular direction of the cigar. We show how the poles of correlators associated to localized states and bulk interactions naturally decouple in the one-point functions of localized and extended branes. We stress the role played in the analysis of D-brane spectra by primaries in SL(2, R)/U (1) which are descendents of the parent theory.

We study correlation functions of single-cycle chiral operators in Sym N T 4 , the symmetric product orbifold of N supersymmetric four-tori. Correlators of twist operators are evaluated on covering surfaces, generally of different genera, where fields are single-valued. We compute some simple four-point functions and study how the sum over inequivalent branched covering maps splits under OPEs. We then discuss extremal n-point correlators, i.e. correlators of n−1 chiral and one anti-chiral operators. They obey simple recursion relations involving numbers obtained from counting branched covering maps with particular properties. In most cases we are able to solve explicitly the recursion relations. Remarkably, extremal correlators turn out to be equal to Hurwitz numbers.

We consider "gauge invariant" operators in Sym N T 4 , the symmetric product orbifold of N copies of the 2d supersymmetric sigma model with T 4 target. We discuss a spin chain representation for single-cycle operators and study their two point functions at large N . We perform systematic calculations at the orbifold point ("tree level"), where non-trivial mixing is already present, and some sample calculations to first order in the blow-up mode of the orbifold ("one loop").

We carry out an exact worldsheet computation of tree level threepoint correlators of chiral operators in type IIB string theory on AdS 3 × S 3 × T 4 with NS-NS flux. We present a simple representation for the string chiral operators in the coordinate basis of the dual boundary CFT. Striking cancelations occur between the three-point functions of the H + 3 and the SU(2) WZW models which result in a simple factorized form for the final correlators. We show, by fixing a single free parameter in the H + 3 WZW model, that the fusion rules and the structure constants of the N = 2 chiral ring in the bulk are in precise agreement with earlier computations in the boundary CFT of the symmetric product of T 4 at the orbifold point in the large N limit.e-print archive: http://lanl.arXiv.org/abs/hep-th/0703022 ATISH DABHOLKAR AND ARI PAKMAN

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