TY - JOUR
TI - Logarithmic intertwining operators and genus-one correlation functions.
DO - https://doi.org/doi:10.7282/T3Z60QW6
PY - 2015
AB - We develop the theory of modular invariance for logarithmic intertwining operators. We construct and study genus-one correlation functions for logarithmic intertwining operators between generalized modules over a quasi-rational vertex operator algebra V . We consider generalized V -modules which admit a right action of some associative algebra P, and intertwining operators between modules in this class which commute with the action of P (P -intertwining operators). We obtain duality properties, i.e., suitable associativity and commutativity properties, for P -intertwining operators. Using the concept of pseudotrace introduced by Miyamoto, we define formal q-traces of products of P -intertwining operators, and obtain certain identities for these formal series. This allows us to show that the formal q-traces satisfy a system of differential equations with regular singular points, and therefore are absolutely convergent in a suitable region and can be extended to yield multivalued analytic functions, called genus-one correlation functions. Furthermore, we show that the space of solutions of these differential equations is invariant under the action of the modular group. We obtain a characterization of symmetric functions on bimodules over associative algebras in terms of pseudotraces of certain “bimodule actions”. We conclude by sketching the steps by which these results can be used to obtain a full modular invariance theorem for the genus-one correlation functions at least when the central charge is not 0. This modular invariance generalizes the full modular invariance theorem by Huang in the rational case. Miyamoto was the first to obtain a partial result that does not involve logarithmic intertwining operators or even intertwining operators. This modular invariance has been a conjecture for many years.
KW - Mathematics
KW - Correlation (Statistics)
KW - Operator algebras
LA - eng
ER -