Differential equations for intertwining operators among untwisted and twisted modules
Daniel Tan
TL;DR
This work develops a rigorous twisted-operator framework for orbifold-like settings in vertex operator algebras by deriving a Jacobi identity for $(\!g1\ g\!)$-type intertwining operators (untwisted on the left, twisted by $g$ on the right) and establishing differential equations for their multi-point chiral correlation functions under $C_1$-cofiniteness and discrete grading. It extends Huang’s differential-operator approach to the twisted setting, deriving a twisted KZ system in the affine case and proving regularity of its solutions when $g$ has finite order. The results yield convergence of the multi-series in the standard region $|z_1|>\cdots>|z_N|>0$ and holomorphic/analytic extension to $M^N$, with regular singularities at prescribed points in the $N=2$ case; this lays groundwork for constructing a $G$-crossed tensor category structure for twisted modules without requiring $V^G$ to be $C_2$-cofinite. Overall, the paper generalizes chiral orbifold CFT techniques to a broad twisted-intertwining-operator setting, enabling rigorous control over convergence, regularity, and the analytic structure of twisted correlators.
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{smallmatrix} W_3\\ W_1 \, W_2 \end{smallmatrix}\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 $\left(\!\begin{smallmatrix} g\\ 1 \ g \end{smallmatrix}\!\right)$-type. Using the Jacobi identity, we obtain homogeneous linear differential equations satisfied by the multi-series $ \langle w_0, \mathcal{Y}_1(w_1,z_1) \cdots \mathcal{Y}_N(w_N,z_N) w_{N+1} \rangle $ when $ \mathcal{Y}_j $ are of $\left(\!\begin{smallmatrix} g\\ 1 \ g \end{smallmatrix}\!\right)$-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 $ \langle w_0, \mathcal{Y}_1(w_1,z_1) \cdots \mathcal{Y}_N(w_N,z_N) w_{N+1} \rangle $ 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 \neq 0 $. Furthermore, when $ N = 2 $, we show that these multivalued functions have regular singularities at certain prescribed points.