engleski

An analogue of modular BPZ-equation in logarithmic (super)conformal field theory

We prove a general result on the size of the largest $L(0)$-Jordan cell in the category of modules for any $C_2$-cofinite vertex algebra. Then we analyze certain null vector conditions for the triplet $\mathcal{; ; ; W}; ; ; (p)$ (and the supertriplet vertex algebra $\mathcal{; ; ; SW}; ; ; (m)$), allowing us to construct modular differential equations satisfied by its vacuum pseudotraces (i.e, {; ; ; generalized characters}; ; ; ). Consequently, the category of weak modules for $\mathcal{; ; ; W}; ; ; (p)$ (or $\mathcal{; ; ; SW}; ; ; (m)$) admits $L(0)$-Jordan cells of size at most two, while the vector space of generalized characters for $\mathcal{; ; ; W}; ; ; (p)$ and $\mathcal{; ; ; SW}; ; ; (m)$ is $(3p-1)$ and $(3m+1)$- dimensional, respectively. Closely related to our modular differential equations are certain "logarithmic" $q$-series identities for powers of the Dedekind $\eta$-function, obtained by using ideas from [M1]-[M4].

vertex operator algebras ; triplet vertex algebras ; logarithmic conformal field theory ; modular differential equations

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