2005/10/01 by Yi-Zhi Huang, YI-ZHI HUANG · 74 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Associative algebra #Associative property #Commutative property #Conformal field theory #Conformal map #Current algebra #Differential operator #Discrete mathematics #Division algebra #Graph #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Modular design #Modular invariance #Nonlinear Waves and Solitons #Operator algebra #Pure mathematics #Subalgebra #Vertex (graph theory) #Vertex operator algebra
paper · doi:10.1142/s021919970500191x
published in Communications in Contemporary Mathematics 07(05), 649-706 (World Scientific)
crossref issued 2005/10/01 · crossref published 2005/10/01 · crossref published-print 2005/10/01 · openalex publication_date 2005/10/01 · crossref created 2005/10/07 · crossref published-online 2011/11/20 · crossref deposited 2019/08/06 · openalex created_date 2025/10/10 · crossref indexed 2026/08/04 · openalex updated_date 2026/08/05
We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V (n) = 0 for n < 0 and V (0) = ℂ1, (ii) every ℕ-gradable weak V-module is completely reducible and (iii) V is C 2 -cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and others. In particular, we show that the q-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.