Cardy condition for open-closed field algebras

Abstract

Let $V$ be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that ${\mathcal{C}}_V$, the category of V-modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over certain $\C$-extension of the Swiss-cheese partial dioperad, and we obtain Ishibashi states easily in such algebras. We formulate Cardy condition algebraically in terms of the action of the modular transformation $S:\tau \mapsto -\frac{1}{\tau }$ on the space of intertwining operators. We then derive a graphical representation of S in the modular tensor category ${\mathcal{C}}_V$. This result enables us to give a categorical formulation of Cardy condition and modular invariant conformal full field algebra over $V\otimes V$. Then we incorporate the modular invariance condition for genus-one closed theory, Cardy condition and the axioms for open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called Cardy ${\mathcal{C}}_V|{\mathcal{C}}_{V\otimes V}$-algebra. We also give a categorical construction of Cardy ${\mathcal{C}}_V|{\mathcal{C}}_{V\otimes V}$-algebra in Cardy case.

70 page, 105 figures, references are updated. less typos, to appear in Comm. Math. Phys

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