Differential equations, duality and modular invariance

Differential equations, duality and modular invariance

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0 and V(0) = ℂ1, (ii) every ℕ-gradable weak V-module is completely reducible and (iii) V is C2-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and others. In particular, we show that the q-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.

我们解决了从与满足以下条件的顶点算子代数相关的亏格零手征关联函数构造所有亏格一手征关联函数的问题：（i）当\(n<0\)时\(V(n)=0\)且\(V(0)=\mathbb{C}1\)；（ii）每个\(\mathbb{N}\) - 可分次弱\(V\) - 模是完全可约的；（iii）\(V\)是\(C_2\) - 余有限的。我们确立了这些函数的基本性质，包括适当表述的交换性、结合性和模不变性。我们在此发展和使用的方法与朱和其他人先前使用的方法完全不同。特别地，我们表明某些几何修正的交织算子的乘积的\(q\) - 迹满足模不变的微分方程组，对于任何固定的模参数，这些方程组可归结为仅有正则奇点的双周期系统。连同作者在亏格零情形下得到的结果，本文的结果从本质上解决了从满足上述条件的顶点算子代数的表示构造亏格一手征弱共形场论的问题。