Differential equations for twisted intertwining operators in abelian orbifold conformal field theory

Let V be a vertex operator algebra with two commuting automorphisms g₁ and g₂ (possibly of infinite order and non-semisimple). We derive a Jacobi identity for a twisted intertwining operator Y of type ( smallmatrix W₃\\ W₁ \, W₂ smallmatrix) when W₁ is a g₁-twisted V-module, W₂ is a g₂-twisted V-module, and W₃ is a g₁ g₂-twisted V-module. For any abelian group G of automorphisms of a CFT-type vertex operator algebra, under certain Noetherian assumptions, we derive differential equations satisfied by the multi-series w₀, Y₁(w₁,z₁) ⋯ YN(wN,zN) wN+1 when Yⱼ are intertwining operators among C₂-cofinite and discretely graded V-modules twisted by elements in G. For modules twisted by automorphisms with only real positive eigenvalues, it suffices to assume C₁-cofiniteness and discrete grading. When all but at most two external modules are twisted by automorphisms having only positive real eigenvalues, it suffices to assume all in-state and internal modules are C₁-cofinite and discretely graded. When G is a finite abelian group, we use the theory of regular singular points to prove that the multi-series w₀, Y₁(w₁,z₁) ⋯ YN(wN,zN) wN+1 converges absolutely to a multivalued analytic function when |z₁| > ⋯ > |zN| > 0 and analytically extends to the region zᵢ ≠ 0, zⱼ - zₖ ≠ 0, for 1 ≤ i ≤ N and 1 ≤ j < k ≤ N. The vertex operator algebra V is not assumed to be C₂-cofinite or simple.