Related papers: Mapping class group representations and Conformal …
Rational conformal field theories produce a tower of finite-dimensional representations of surface mapping class groups, acting on the conformal blocks of the theory. We review this formalism. We show that many recent mathematical…
It is shown that for the modular representations associated to Rational Conformal Field Theories, the kernel is a congruence subgroup whose level equals the order of the Dehn-twist. An explicit algebraic characterization of the kernel is…
The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of…
The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters on the one hand and its interpretation of modules as objects in a modular tensor category on the other one. Overarching…
A general procedure is presented to determine, given any suitable representation of the modular group, the characters of all possible Rational Conformal Field Theories whose associated modular representation is the given one. The relevant…
Unitary representations of centrally extended mapping class groups $\tilde M_{g,1}, g\geq 1$ are given in terms of a rational Hopf algebra $H$, and a related generalization of the Verlinde formula is presented. Formulae expressing the…
The notion of Galois currents in Rational Conformal Field Theory is introduced and illustrated on simple examples. This leads to a natural partition of all theories into two classes, depending on the existence of a non-trivial Galois…
We examine general aspects of parity functions arising in rational conformal field theories, as a result of Galois theoretic properties of modular transformations. We focus more specifically on parity functions associated with affine Lie…
Explicit formulae describing the genus one characters and modular transformation properties of permutation orbifolds of arbitrary Rational Conformal Field Theories are presented, and their relation to the theory of covering surfaces is…
We describe the construction which takes as input a profinite group, which when applied the the absolute Galois group of a geometric field F agrees in some cases with the algebraic K-theory of F. We prove that it agrees in the case of a…
We discuss some basic problems and conjectures in a program to construct general orbifold conformal field theories using the representation theory of vertex operator algebras. We first review a program to construct conformal field theories.…
This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes…
The generalized Verlinde formulae expressing traces of mapping classes corresponding to automorphisms of certain Riemann surfaces, and the congruence relations on allowed modular representations following from them are presented. The…
It is shown that traces of mapping classes of finite order may be expressed by Verlinde-like formulae. The 3D topological argument is explained, and the resulting trace identities for modular matrix elements are presented.
We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each $k\in\{12,16,18,20,22,26\}$, we give explicit rational primes $\l$…
We show that any complex (respectively real) representation of finite group naturally generates a open-closed (respectively Klein) topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in…
In conformal field theory the understanding of correlation functions can be divided into two distinct conceptual levels: The analytic properties of the correlators endow the representation categories of the underlying chiral symmetry…
This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with a pure Virasoro example, critical percolation, then continues with a detailed exposition of symplectic fermions,…
After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RCQFTs) we focus on pairs (A,F) of conformal field theories, where F has a finite group G of global symmetries…
We study an analogue of Serre's modularity conjecture for projective representations $\overline{\rho}: \operatorname{Gal}(\overline{K} / K) \rightarrow \operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases…