N = 4 superconformal coset theories

We study implications of $$ \mathcal{N}=4 $$ superconformal symmetry in three dimensions, thus extending our earlier results in [1] devoted to the $$ \mathcal{N}\le 3 $$ cases. We show that the three-point function of the supercurrent in $$ \mathcal{N}=4 $$ superconformal field theories contains two linearly independent forms. However, only one of these structures contributes to the three-point function of the energy-momentum tensor and the other one is present in those $$ \mathcal{N}=4 $$ superconformal theories which are not invariant under the mirror map. We point out that general $$ \mathcal{N}=4 $$ superconformal field theories admit two inequivalent flavour current multiplets and show that the three-point function of each of them is determined by one tensor structure. As an example, we compute the two- and three-point functions of the conserved currents in $$ \mathcal{N}=4 $$ superconformal models of free hypermultiplets. We also derive the universal relations between the coefficients appearing in the two- and threepoint correlators of the supercurrent and flavour current multiplets in all superconformal theories with $$ \mathcal{N}\le 4 $$ supersymmetry. Our derivation is based on the use of Ward identities in conjunction with superspace reduction techniques.

We analyze the relation between a topological coset model based on super SL(2, R)/U(1) coset and non-critical string theory by using free field realization. We show that the twisted N=2 algebra of the coset model can be naturally transformed into that of non-critical string. The screening operators of the coset models can be identified either with those of the minimal matters or with the cosmological constant operator. We also find that another screening operator, which is intrinsic in our approach, becomes the BRST non-trivial state of ghost number 0 (generator of the ground ring for c=1 gravity). The relation between non-critical strings and topological field theories is the subject of current interest. It has long been suggested that the latter theories describe the unbroken phase of gravity,1 but their precise relation has not been clear. It has been known that the twisting of N=2 superconformal field theory gives rise to topological theory.1,2 This suggests that any non-critical string theories may have hidden N=2 superconformal symmetry. Indeed, several authors have observed that the BRST current and the antighost field b(z) generate an algebra that is quite similar but apparently not identical to the N=2 superconformal algebra.3 It turns out that the BRST current can be modified by total derivative terms so that the antighost and the physical BRST current exactly generate a topologically twisted N=2 superconformal algebra.4,5 This does not identify, however, the structure of the models with N=2 symmetry. Recently, rather non-trivial correspondence between super SL(2, R)k/U(1) coset model6 and c=1 string has been analyzed through twisted N=2 structure. Mukhi and Vafa7 have revealed an amazing correspondence between these two models for k=3. In this letter, we discuss the relation of these models and the generalization of the correspondence to the minimal models coupled to gravity by means of the free field realization. We find that there is another interesting correspondence for k=1. Super SL(2, R)k/U(1) model is described by the bosonic coset model of SL(2, R)k×U(1)/U(1).8 For a representation of SL(2, R)k, we use the following

We study the (2,0) superconformal theories in six dimensions, which arise from the low-energy limit of k coincident 5-branes, using their discrete light-cone formulation as a superconformal quantum mechanical sigma model. We analyze the realization of the superconformal symmetry in the quantum mechanics, and the realization of primary operators. As an example we compute the spectrum of chiral primary states in symmetric Spin(5)_R representations. To facilitate the analysis we introduce and briefly discuss a new class of Lorentz non-invariant theories, which flow in the IR to the (2,0) superconformal field theories but differ from them in the UV.

In this paper, the method of point-splitting regularization is applied on the N=2 superconformal field theory, specifically the superconformal coset model formulated by Kazama and Suzuki. We obtain the correct central extensions for the N=2 superconformal algebra after many nontrivial cancellations among the various singular expressions. This shows the consistency of the point-splitting method in an N=2 superconformal system. In the process, we arrive at the Kazama–Suzuki conditions which govern the existence of an N=2 superconformal coset model in the N=1 coset model. In addition, we obtain a number of mathematical relations between the structure constants and the complex structure of the model, which allow us to simplify the U(1) current of the N=2 superconformal algebra. In the course of our analysis, we found that, at least in a two dimension conformal field theory, the operator product expansion of a composite current must be written in a way which conveys all the information of the commutator equations.

We consider conformal and ’t Hooft anomalies in six-dimensional mathcal{N} = (1, 0) superconformal field theories, focusing on those conformal anomalies that determine the two- and three-point functions of conserved flavor and SU(2)R currents, as well as stress tensors. By analyzing these correlators in superspace, we explain why the number of independent conformal anomalies is reduced in supersymmetric theories. For instance, non- supersymmetric CFTs in six dimensions have three independent conformal c-anomalies, which determine the stress-tensor two- and three-point functions, but in superconformal theories the three c-anomalies are subject to a linear constraint. We also describe anomaly multiplet relations, which express the conformal anomalies of a superconformal theory in terms of its ’t Hooft anomalies. Following earlier work on the conformal a-anomaly, we argue for these relations by considering the supersymmetric dilaton effective action on the tensor branch of such a theory. We illustrate the utility of these anomaly multiplet relations by presenting exact results for conformal anomalies, and hence current and stress-tensor correlators, in several interacting examples.

We systematically analyze the effective action on the moduli space of (2, 0) superconformal field theories in six dimensions, as well as their toroidal compactification to maximally supersymmetric Yang-Mills theories in five and four dimensions. We present a streamlined approach to non-renormalization theorems that constrain this effective action. The first several orders in its derivative expansion are determined by a one-loop calculation in five-dimensional Yang-Mills theory. This fixes the leading higher-derivative operators that describe the renormalization group flow into theories residing at singular points on the moduli space of the compactified (2, 0) theories. This understanding allows us to compute the a-type Weyl anomaly for all (2, 0) superconformal theories. We show that it decreases along every renormalization group flow that preserves (2, 0) supersymmetry, thereby establishing the a-theorem for this class of theories. Along the way, we encounter various field-theoretic arguments for the ADE classification of (2, 0) theories.

Non-hermitian symmetric N = 2 coset models, Poincaré polynomials, and string compactification

The near-horizon geometry of parallel NS5-branes is described by an exact superconformal 2d field theory based on K^10 = W(k)^4 x M^6, with M^6 a flat 5+1 space time and W(k)^4 =U(1) x SU(2)(k), a four dimensional background with non-trivial dilaton. The ten-dimensional ``BULK'' spectrum of excitations of K^10 can be derived combining unitary representations of the N=4 superconformal theory of W(k)^4 and M^6 in a modular-invariant way. All Bulk states are massive and belong to the long representations of the N=2 six dimensional space-time supersymmetry. The NS5-brane states, propagating on M^6, belong to the short representations of N=2. Both the bulk and the brane spectrum is derived using the powerful worldsheet technics of the N=4 superconformal theories. We claim that both bulk and brane states are necessary to well define the theory. The non abelian U(n) or SO(2n) structure of the 5-brane fields follows from the fusion coefficients appearing in the correlation functions involving SU(2)(k) conformal fields.

The geometry of supersymmetric coset models and superconformal algebras

Chiral generations, Poincaré polynomials and coset models with N = 2 supersymmetry

In this paper we study the massless representations of the conformal superalgebras u(2,2/N), su(2,2/N), pu(2,2/4), and psu(2,2/4). It is argued that they are the only unitary representations of conformal supersymmetry that have a realistic (i.e., finite) particle spectrum. We show that the massless unitary irreducible representations (UIR's) of an N-extended conformal supersymmetry symmetry restrict to irreducible representations of the N-extended super-Poincar\'e algebra and prove that the massless UIR's of the N-extended super-Poincar\'e algebra have unique extensions to massless UIR's of su(2,2/N) if N\ensuremath{\ne}4, or to massless UIR's of pu(2,2/4) if N=4. These facts suggest that the hidden U(N) symmetries found in N-extended supersymmetric gauge theories may be, in fact, only a subalgebra of an overall N-extended conformal superalgebra. An explicit construction of all the massless representations of su(2,2/N\ensuremath{\ne}4) and pu(2,2/4) is given. It is seen that the adjoint representation of the internal su(N) subalgebra never appears in a massless representation of N-extended conformal supersymmetry.

We show that the N = 2 superconformal coset (G/H) models, based on the hermitean symmetric spaces, are characterized by two group theoretic identities. These identities may be used as the defining relations for this class of theory.

Superstrings propagating on backgrounds of the formAdS3G=H are studied using the coset CFT approach. We focus on seven dimensional cosets which have a semiclassical limit, and which give rise to N = 3 superconformal symmetry in the dual CFT. This is realized for the two casesAdS3SU(3)=U(1) and AdS3SO(5)=SO(3), for which we present an explicit construction. The spectrum of the two coset models is analyzed and compared. We also briefly comment on the geometrical interpretation of our results.

Quaternionic superconformal field theory

We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi-Yau threefolds. Field-theoretically, these 5d SCFTs descend from 6d mathcal{N} = (1, 0) SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi-Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, these graphs manifestly capture strongly coupled data of the 5d SCFTs, such as the superconformal flavor symmetry, BPS states, and mass deformations. The capabilities of this approach are demonstrated by deriving all rank one and rank two 5d SCFTs. The full potential, how- ever, becomes apparent when applied to theories with higher rank. Starting with the higher rank conformal matter theories in 6d, we are led to the discovery of previously unknown flavor symmetry enhancements and new 5d SCFTs.