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 a certain $${\mathbb {C}}$$ -extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of the modular transformation $${S: \tau \mapsto -\frac{1}{\tau}}$$ on the space of intertwining operators of V. 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 the Cardy condition and the modular invariance condition for 1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy $${\mathcal{C}_V|\mathcal{C}_{V \otimes V}}$$ -algebra. In the end, we give a categorical construction of the Cardy $${\mathcal{C}_V|\mathcal{C}_{V \otimes V}}$$ -algebra in the Cardy case.