Differential equations for intertwining operators among untwisted and twisted modules

Abstract

Given any vertex operator algebra $V$ with an automorphism $g$, we derive a Jacobi identity for an intertwining operator $\mathcal{Y}$ of type $\left(\begin{array}{c}{W}_3\\ W_1{W}_2\end{array}\right)$ when $W_1$ is an untwisted $V$-module, and $W_2$ and $W_3$ are $g$-twisted $V$-modules. We say such an intertwining operator is of $\left(\begin{array}{c}g\\ 1g\end{array}\right)$-type. Using the Jacobi identity, we obtain homogeneous linear differential equations satisfied by the multi-series $⟨w_0,{\mathcal{Y}}_1(w_1,z_1)\cdots {\mathcal{Y}}_N(w_N,z_N)w_{N+1}⟩$ when ${\mathcal{Y}}_j$ are of $\left(\begin{array}{c}g\\ 1g\end{array}\right)$-type and the modules are $C_1$-cofinite and discretely graded. In the special case that $V$ is an affine vertex operator algebra, we derive the ``twisted KZ equations" and show that its solutions have regular singularities at certain prescribed points when $g$ has finite order. When $V$ is general and $g$ has finite order, we use the theory of regular singular points to prove that the multi-series $⟨w_0,{\mathcal{Y}}_1(w_1,z_1)\cdots {\mathcal{Y}}_N(w_N,z_N)w_{N+1}⟩$ converges absolutely to a multivalued analytic function when $|z_1|>\cdots >|z_N|>0$ and analytically extends to the region $z_i,z_i-z_j\ne 0$. Furthermore, when $N=2$, we show that these multivalued functions have regular singularities at certain prescribed points.

47 pages, 1 figure, minor changes to intro