On classification of extremal non-holomorphic conformal field theories

  title={On classification of extremal non-holomorphic conformal field theories},
  author={James E. Tener and Zhenghan Wang},
  journal={arXiv: Mathematical Physics},
Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $\mathcal{C}$ and a central charge $c$. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category $\mathcal{C}$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $\mathcal{C}_V$ is $\mathcal… 
In and around abelian anyon models
Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two
Galois symmetry induced by Hecke relations in rational conformal field theory and associated modular tensor categories
Hecke operators relate characters of rational conformal field theories (RCFTs) with different central charges, and extend the previously studied Galois symmetry of modular representations and fusion
Double-Janus linear sigma models and generalized reciprocity for Gauss sums
We study the supersymmetric partition function of a 2d linear $\sigma$-model whose target space is a torus with a complex structure that varies along one worldsheet direction and a Kahler modulus
Wronskian indices and rational conformal field theories
The classification scheme for rational conformal field theories, given by the Mathur-Mukhi-Sen (MMS) program, identifies a rational conformal field theory by two numbers: ( n, l ). n is the number of
Conformal Field Theories as Scaling Limit of Anyonic Chains
We provide a mathematical definition of a low energy scaling limit of a sequence of general non-relativistic quantum theories in any dimension, and apply our formalism to anyonic chains. We formulate
Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions
We study 3D pure Einstein quantum gravity with negative cosmological constant, in the regime where the AdS radius $l$ is of the order of the Planck scale. Specifically, when the Brown-Henneaux
A mathematical theory of gapless edges of 2d topological orders. Part I
This is the first part of a two-part work on a unified mathematical theory of gapped and gapless edges of 2d topological orders. We analyze all the possible observables on the 1+1D world sheet of a
Classifying three-character RCFTs with Wronskian Index equalling $\mathbf{0}$ or $\mathbf{2}$.
In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers
Classification of extremal vertex operator algebras with two simple modules
In recent work, Wang and the third author defined a class of 'extremal' vertex operator algebras (VOAs), consisting of those with at least two simple modules and conformal dimensions as large as


Rank-finiteness for modular categories
We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the
From Vertex Operator Algebras to Conformal Nets and Back
We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure
Modular invariance of characters of vertex operator algebras
In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain
Vertex operator algebras, the Verlinde conjecture, and modular tensor categories.
  • Yi-Zhi Huang
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 2005
A proof of the Verlinde conjecture for V is announced of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation tau |--> -1/tau on the space of characters of irreducing V- modules.
Conformal invariance of chiral edge theories
The low-energy effective quantum field theory of the edge excitations of a fully-gapped bulk topological phase corresponding to a local interaction Hamiltonian must be local and unitary. Here it is
Genera of Vertex Operator Algebras and three dimensional Topological Quantum Field Theories
The notion of the genus of a quadratic form is generalized to vertex operator algebras. We define it as the modular braided tensor category associated to a suitable vertex operator algebra together
Vector-valued modular functions for the modular group and the hypergeometric equation
A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of
The Theory of Vector-Valued Modular Forms for the Modular Group
We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we
We consider holomorphic vector-valued modular forms F of integral weight k on the full modular group Γ = SL(2, ℤ) corresponding to representations of Γ of arbitrary finite dimension p. Assuming that