arXiv subjects
Publications and source records attributed to Shlomo S. Razamat.
In this note we discuss several properties of the Schur index of N=2 superconformal theories in four dimensions. In particular, we study modular properties of this index under SL(2,Z) transformations of its parameters.
The analytic properties of the N = 2 superconformal index are given a physical interpretation in terms of certain BPS surface defects, which arise as the IR limit of supersymmetric vortices. The residue of the index at a pole in flavor fugacity is interpreted as the index of a superconformal field theory without this flavor symmetry, but endowed with an additional surface defect. The residue can be efficiently extracted by acting on the index with a difference operator of Ruijsenaars-Schneider type. By imposing the associativity constraints of S-duality, we are then able to evaluate the index of all generalized quiver theories of type A, for generic values of the three superconformal fugacities, with or without surface defects.
Recently a prescription to compute the superconformal index for all theories of class S was proposed. In this paper we discuss some of the physical information which can be extracted from this index. We derive a simple criterion for the given theory of class S to have a decoupled free component and for it to have enhanced flavor symmetry. Furthermore, we establish a criterion for the "good", the "bad", and the "ugly" trichotomy of the theories. After interpreting the prescription to compute the index with non-maximal flavor symmetry as a residue calculus we address the computation of the index of the bad theories. In particular we suggest explicit expressions for the superconformal index of higher rank theories with E_n flavor symmetry, i.e. for the Hilbert series of the multi-instanton moduli space of E_n.
The Hilbert Series (HS) of the moduli space of two G instantons on C^2, where G is a simple gauge group, is studied in detail. For a given G, the moduli space is a singular hyperKahler cone with a symmetry group U(2) \times G, where U(2) is the natural symmetry group of C^2. Holomorphic functions on the moduli space transform in irreducible representations of the symmetry group and hence the Hilbert series admits a character expansion. For cases that G is a classical group (of type A, B, C, or D), there is an ADHM construction which allows us to compute the HS explicitly using a contour integral. For cases that G is of E-type, recent index results allow for an explicit computation of the HS. The character expansion can be expressed as an infinite sum which lives on a Cartesian lattice that is generated by a small number of representations. This structure persists for all G and allows for an explicit expressions of the HS to all simple groups. For cases that G is of type G_2 or F_4, discrete symmetries are enough to evaluate the HS exactly, even though neither ADHM construction nor index is known for these cases.
We identify the 2d topological theory underlying the N=2 4d superconformal index with an explicit model: q-deformed 2d Yang-Mills. By this route we are able to evaluate the index of some strongly-coupled 4d SCFTs, such as Gaiotto's T_N theories.
We derive a holographic dual description of free quantum field theory in arbitrary dimensions, by reinterpreting the exact renormalization group, to obtain a higher spin gravity theory of the general type which had been proposed and studied as a dual theory. We show that the dual theory reproduces all correlation functions.
We evaluate the superconformal index of the Y^{p,q} quiver gauge theories using Romeslberger's prescription. For the conifold quiver Y^{1,0} we find exact agreement at large N with a previous calculation in the dual AdS_5 X T^{1,1} supergravity.
We derive an integral representation for the superconformal index of the strongly-coupled N=2 superconformal field theory with E_6 flavor symmetry. The explicit expression of the index allows highly non-trivial checks of Argyres-Seiberg duality and of a class of S-dualities conjectured by Gaiotto.
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 discuss an explicit construction of a string dual for the Gaussian matrix model. Starting from the matrix model and employing Strebel differential techniques we deduce hints about the structure of the dual string. Next, following these hints a worldheet theory is constructed. The correlators in this string theory are assumed to localize on a finite set of points in the moduli space of Riemann surfaces. To each such point one associates a Feynman diagram contributing to the correlator in the dual matrix model, and thus recasts the worldsheet expression as a sum over Feynman diagrams.
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 study correlation functions of single-cycle chiral operators in 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 study the superconformal index for the class of N=2 4d superconformal field theories recently introduced by Gaiotto. These theories are defined by compactifying the (2,0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A_1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult.
The ratio of central charges in four-dimensional CFTs has been suggested by Hofman and Maldacena to lie within an interval whose boundaries are fixed by the number of supersymmetries. We compute this ratio for a set of interacting N=1 superconformal field theories which arise as RG fixed points of supersymmetric Yang-Mills theories with adjoint and fundamental matter. We do not find violations of the proposed bounds, which appear to be saturated by free field theories.
We discuss a modification of Gopakumar's prescription for constructing string theory duals of free gauge theories. We use this prescription to construct an unconventional worldsheet dual of the Gaussian matrix model.
We discuss scattering of fundamental matter in the planar and strong coupling limit via the AdS/CFT correspondence, generalizing the recently proposed calculation for adjoint matter due to Alday and Maldacena [arXiv:0705.0303]. Color decomposition of quark amplitudes is a key property allowing to repeat the procedure in the case of fundamental matter and to derive the relation of these strong coupling amplitudes to minimal area problems. We present the results for two different D3-D7 systems, one is only conformal in the planar limit and the other is exactly conformal. Our results suggest a universal behavior of scattering amplitudes at strong coupling and planar limit (both for gluons and quarks).
We discuss fluctuations on solitons in the dilaton/graviton/tachyon system using the low energy effective field theory approach. It is shown that closed string solitons are free of tachyons in this approximation, regardless of the exact shape of the tachyon potential.
We continue to investigate properties of the worldsheet conformal field theories (CFTs) which are conjectured to be dual to free large N gauge theories, using the mapping of Feynman diagrams to the worldsheet suggested in hep-th/0504229. The modular invariance of these CFTs is shown to be built into the formalism. We show that correlation functions in these CFTs which are localized on subspaces of the moduli space may be interpreted as delta-function distributions, and that this can be consistent with a local worldsheet description given some constraints on the operator product expansion coefficients. We illustrate these features by a detailed analysis of a specific four-point function diagram. To reliably compute this correlator we use a novel perturbation scheme which involves an expansion in the large dimension of some operators.