标题： 非扭模块和扭模块之间交织算子的微分方程 显示英文标题

标题： Differential equations for intertwining operators among untwisted and twisted modules

摘要： 对于任何顶点算子代数$ V $和自同构$ g $，当$ W_1 $是一个未扭的$ V $-模，而$ W_2 $和$ W_3 $是$ g $-扭的$ V $-模时，我们推导出类型为$ \left( \begin{smallmatrix} W_3\\ W_1 \, W_2 \end{smallmatrix}\right) $的交织算子$ \mathcal{Y} $的雅可比恒等式。 我们称这样的交织算子为$\left(\!\begin{smallmatrix} g\\ 1 \ g \end{smallmatrix}\!\right)$类型。 利用雅可比恒等式，我们得到当$ \mathcal{Y}_j $为$\left(\!\begin{smallmatrix} g\\ 1 \ g \end{smallmatrix}\!\right)$类型且模是$ C_1 $-有限且离散分次时，多级数$ \langle w_0, \mathcal{Y}_1(w_1,z_1) \cdots \mathcal{Y}_N(w_N,z_N) w_{N+1} \rangle $所满足的齐次线性微分方程。 在$ V $是仿射顶点算子代数的特殊情况中，我们推导出“扭曲的KZ方程”，并证明当$ g $有有限阶时，其解在某些指定点处具有正则奇点。 当$ V $一般且$ g $具有有限阶时，我们使用正则奇点理论证明，当$ |z_1| > \cdots > |z_N| > 0 $时，多重级数$ \langle w_0, \mathcal{Y}_1(w_1,z_1) \cdots \mathcal{Y}_N(w_N,z_N) w_{N+1} \rangle $绝对收敛到一个多重解析函数，并在区域$ z_i, z_i - z_j \neq 0 $内解析延拓。 此外，当$ N = 2 $时，我们证明这些多重解析函数在某些指定的点具有正则奇点。 显示英文摘要

摘要： 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.