开闭域代数的Cardy条件
Cardy condition for open-closed field algebras
摘要：
设$V$是一个顶点算子代数，满足一定的约化性和有限性条件，使得V-模范畴$\mathcal{C}_V$是一个模张量范畴.本文研究了V上开闭域代数的开和闭扇区都具有非退化不变双线性型。我们证明了它们给出了Swiss-cheese部分偶算符的某些$\C$ -扩张上的代数，并且在这样的代数中我们很容易地得到了Ishibashi态.我们用模变换$S: τ\mapsto -\frac{1}τ$在交织算子空间上的作用代数地表示Cardy条件。然后，我们推导出S在模张量范畴$\mathcal{C}_V$中的图形表示。这个结果使我们能够给出$V\otimes V$上的Cardy条件和模不变共形满域代数的范畴形式。然后，我们将亏格1闭理论的模不变性条件、Cardy条件和V上具有非退化不变双线性型的开闭域代数的公理结合到一个张量范畴概念中，称为Cardy $\mathcal{C}_V|\mathcal{C}_{V\otimes V}$ -代数.给出了Cardy情形下Cardy $\mathcal{C}_V|\mathcal{C}_{V\otimes V}$ -代数的范畴结构.
英文摘要：
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 $S: τ\mapsto -\frac{1}τ$ 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.
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