W $$ \mathcal{W} $$ symmetry in six dimensions

Journal of High Energy Physics, 2004

We introduce the analytic superspace formalism for six-dimensional (N, 0) superconformal field theories. Concentrating on the (2, 0) theory we write down the Ward identities for correlation functions in the theory and show how to solve them. We then consider the four-point function of four energy momentum multiplets in detail, explicitly solving the Ward identities in this case. We expand the four-point function using both Schur polynomials, which lead to a simple formula in terms of a single function of two variables, and (a supersymmetric generalisation of) Jack polynomials, which allow a conformal partial wave expansion. We then perform a complete conformal partial wave analysis of both the free theory four-point function and the AdS dual four-point function. We also discuss certain operators at the threshold of the series a) unitary bound, and prove that some such operators can not develop anomalous dimensions, by finding selection rules for certain three-point functions. For those operators which are not protected, we find representations with which they may combine to become long.

Physics Reports, 1993

We review various aspects of V algebra symmetry in two-dimensional conformal field theory and string theory. We pay particular attention to the construction of V algebras through the quantum Drinfeld-Sokolov reduction and through the coset construction.

1994

Quantum W-algebras are defined and their relevance for conformal field theories is outlined. We describe direct constructions of W-algebras using associativity requirements. With this approach one explicitly obtains the first members of series of W-algebras, including in particular 'Casimir algebras' (related to simple Lie algebras) and extended symmetry algebras corresponding to special Virasoro-minimal models. We also describe methods for the study of highest weight representations of W-algebras. In some cases consistency of the corresponding quantum field theory already imposes severe restrictions on the admitted representations, i.e. allows to determine the field content. We conclude by reviewing known results on Walgebras and RCFTs and show that most known rational conformal fields theories can be described in terms of Casimir algebras although on the level of W-algebras exotic phenomena occur.

Nuclear Physics B, 2019

we study the holographic description of N = 2 Super Conformal Field Theories in four dimensions first given by Gaiotto and Maldacena. We present new expressions that holographically calculate characteristic numbers of the CFT and associated Hanany-Witten setups , or more dynamical observables, like the central charge. A number of examples of varying complexity are studied and some proofs for these new expressions are presented. We repeat this treatment for the case of the marginally deformed Gaiotto-Maldacena theories, presenting an infinite family of new solutions and compute some of its observables. These new backgrounds rely on the solution of a Laplace equation and a boundary condition, encoding the kinematics of the original conformal field theory.

QuantumW-algebras are defined and their relevance for conformal field theories is outlined. We describe direct constructions ofW-algebras using associativity re- quirements. With this approach one explicitly obtains the first members of series of W-algebras, including in particular 'Casimir algebras' (related to simple Lie alge- bras) and extended symmetry algebras corresponding to special Virasoro-minimal models. We also describe methods for the study of highest weight representations ofW-algebras. In some cases consistency of the corresponding quantum field the- ory already imposes severe restrictions on the admitted representations, i.e. allows to determine the field content. We conclude by reviewing known results on W- algebras and RCFTs and show that most known rational conformal fields theories can be described in terms of Casimir algebras although on the level ofW-algebras exotic phenomena occur.

Using a unified and systematic scheme, the free field realization of irreducible representations of osp(2|2) is constructed. By using these realization, the correlation functions of N = 2 super-conformal model based on osp(2|2) symmetry and free field representation of osp(2|2) generators are calculated. Free field representation of currents are used to determine the stress-energy tensor and the central charge of the model. < φ(z 1)φ(z 2)φ(z 3) >∼ z 2α 12 z 2β 13 z 2γ 23 < φ(z 1)f (z 2)f(z 3) >∼ 1 2 γz 2α 12 z 2β 13 z 2γ−1 23 < f (z 1)φ(z 2)f(z 3) >∼ 1 2 βz 2α 12 z 2β−1 13 z 2γ 23 < f (z 1)f (z 2)φ(z 3) >∼ 1 2 αz 2α−1 12 z 2β 13 z 2γ 23 < φ(z 1)K(z 2)K(z 3) >∼ 1 2 γ(γ − 1)z 2α 12 z 2β 13 z 2γ−2 23 < K(z 1)φ(z 2)K(z 3) >∼ 1 2 β(β − 1)z 2α 12 z 2β−2 13 z 2γ 23 < K(z 1)K(z 2)φ(z 3) >∼ 1 2 α(α − 1)z 2α−2 12 z 2β 13 z 2γ 23 < f (z 1)f (z 2)K(z 3) >∼ 1 2 βγz 2α 12 z 2β−1 13 z 2γ−2 23 < K(z 1)f (z 2)f (z 3) >∼ 1 2 αβz 2α−1 12 z 2β−1 13 z 2γ 23 < f (z 1)K(z 2)f (z 3) >∼ 1 2 αγz 2α−1 12 z 2β 13 z 2γ−1 23 < K(z 1)K(z 2)K(z 3) >∼ αβγz 2α−1 12 z 2β−1 13 z 2γ−1

Physics Letters B, 1999

We use the AdS/CFT correspondence to calculate three point functions of chiral primary operators at large N in d = 3, N = 8 and d = 6, N = (2, 0) superconformal field theories. These theories are related to the infrared fixed points of world-volume descriptions of N coincident M2 and M5 branes, respectively.

Journal of High Energy Physics, 2020

This is a long-overdue companion paper to [1]. We study the relation between sl(3|2) Chern-Simons supergravity on AdS3 and two-dimensional CFT’s with $$ \mathcal{N} $$ N = 2 super-$$ {\mathcal{W}}_3 $$ W 3 symmetry. Specifically, we carry out a complete analysis of asymptotic symmetries in a basis that makes the superconformal structure transparent, allowing us to establish the precise dictionary between currents and transformation parameters in the bulk and their boundary counterparts. We also discuss the incorporation of sources and display in full detail the corresponding holographic Ward identities. By imposing suitable hermiticity conditions on the CFT currents, we identify the superalgebra su(2, 1|1, 1) as the appropriate real form of sl(3|2) in Lorentzian signature. We take the opportunity to review some of the properties of the $$ \mathcal{N} $$ N = 2 super-$$ {\mathcal{W}}_3 $$ W 3 conformal algebra, including its multiplet structure, OPE’s and spectral flow invariance, cor...

Physics Letters B, 1992

W-algebras are extensions of the Virasoro algebra describing chiral subalgebras of conformal quantum field theories. Careful analysis of the four-point functions and consideration of the invariant spaces under a subgroup of the modular group SL(2, Z) allow one to find all representations of new classes of fermionic W-algebras constructed recently. For each of these W-algebras there exists only a finite number of representations. The corresponding fusion rules are calculated.

1991

We show that various actions of topological conformal theories that were suggested recentely are particular cases of a general action. We prove the invariance of these models under transformations generated by nilpotent fermionic generators of arbitrary conformal dimension, Q (n) and G (n) . The later are shown to be the n th covariant derivative with respect to "flat abelian gauge field" of the fermionic fields of those models. We derive the bosonic counterparts W (n) and R (n) which together with Q (n) and G (n) form a special N = 2 super W ∞ algebra. The algebraic structure is discussed and it is shown that it generalizes the so called "topological algebra".