Correlator correspondences for Gaiotto-Rapčák dualities and first order formulation of coset models

We derive correspondences of correlation functions among dual conformal field theories in two dimensions by developing a “first order formulation” of coset models. We examine several examples, and the most fundamental one may be a conjectural equivalence between a coset (SL(n)k ⊗SL(n)−1)/SL(n)k−1 and $$ \mathfrak{sl}(n) $$ sl n Toda field theory with generic level k. Among others, we also complete the derivation of higher rank FZZ-duality involving a coset SL(n + 1)k /(SL(n)k ⊗ U(1)), which could be done only for n = 2, 3 in our previous paper. One obstacle in the previous work was our poor understanding of a first order formulation of coset models. In this paper, we establish such a formulation using the BRST formalism. With our better understanding, we successfully derive correlator correspondences of dual models including the examples mentioned above. The dualities may be regarded as conformal field theory realizations of some of the Gaiotto-Rapčák dualities of corner vertex operator algebras.

Black hole microstates from the worldsheet

Recently an exact worldsheet description of strings propagating in certain black hole microstate geometries was constructed in terms of null-gauged WZW models. In this paper we consider a family of such coset models, in which the currents being gauged are specified by a set of parameters that a priori take arbitrary values. We show that consistency of the spectrum of the worldsheet CFT implies a set of quantisation conditions and parity restrictions on the gauging parameters. We also derive these constraints from an independent geometrical analysis of smoothness, absence of horizons and absence of closed timelike curves. This allows us to prove that the complete set of consistent backgrounds in this class of models is precisely the general family of (NS5-decoupled) non-BPS solutions known as the JMaRT solutions, together with their various (BPS and non-BPS) limits. We clarify several aspects of these backgrounds by expressing their six-dimensional solutions explicitly in terms of five non-negative integers and a single length-scale. Finally we study non-trivial two-charge limits, and exhibit a novel set of non-BPS supergravity solutions describing bound states of NS5 branes carrying momentum charge.

Exact Modular $S$ Matrix for ${\mathbb Z}_{k}$ Parafermion Quantum Hall Islands and Measurement of Non-Abelian Anyons

Using the decomposition of rational conformal filed theory characters for the $\Z_k$ parafermion quantum Hall droplets for general $k=2,3,\dots$, we derive analytically the full modular $S$ matrix for these states, including the $\uu$ parts corresponding to the charged sector of the full conformal field theory and the neutral parafermion contributions corresponding to the diagonal affine coset models. This precise neutral-part parafermion $S$ matrix is derived from the explicit relations between the coset matrix and those for the numerator and denominator of the coset and the latter is expressed in compact form due to the level-rank duality between the affine Lie algebras $\widehat{\frak{su}(k)_2}$ and $\widehat{\frak{su}(2)_k}$. The exact results obtained for the $S$ matrix elements are expected to play an important role for identifying interference patterns of fractional quantum Hall states in Fabry-P\'erot interferometers which can be used to distinguish between Abelian and non-Abelian statistics of quasiparticles localized in the bulk of fractional quantum Hall droplets as well as for nondestructive interference measurement of Fibonacci anyons which can be used for universal topological quantum computation

Integrability in sigma-models

The following topics are covered in this chapter: (1) Homogeneous spaces, (2) Classical integrability of sigma-models in two dimensions, (3) Topological terms, (4) Background-field method and beta-function, (5) S-matrix bootstrap in the O(N) model, (6) Supersymmetric coset models and strings on AdS(d) x X.

Loop groups and noncommutative geometry

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level [Formula: see text] projective unitary positive-energy representations of any given loop group [Formula: see text]. The construction is based on certain supersymmetric conformal field theory models associated with [Formula: see text] in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.