Neveu–Schwarz 非正则顶点算子、分解定理与双线性算子
Neveu--Schwarz Irregular Vertex Operators, Decomposition Theorems, and Bilinear Operators
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中文总结 AI 辅助
该研究构造了 Neveu–Schwarz 代数的零秩非正则顶点算子，将其分解定理推广到非正则情形，导出的双线性微分算子与量子 Painlevé V、IV 的 τ 函数方程中的算子一致。
中文摘要 AI 辅助
我们构造了 Neveu–Schwarz 代数的零秩非正则顶点算子，作为相同秩非正则 Verma 模之间的线性映射，满足通常的超共形对易关系和非正则渐近条件。在非退化假设 Λ₂ₚ≠0 下，我们证明了这些算子的存在性与唯一性。随后，我们将 Neveu–Schwarz 代数的分解定理推广到非正则情形：自由费米 Fock 模与秩 p 的 Neveu–Schwarz 非正则 Verma 模的张量积，分解为两个 Virasoro 非正则 Verma 模的张量积的无限直和。我们还证明了非正则顶点算子可兼容分解为 Virasoro 非正则顶点算子的张量积。利用适当的配对和模式插入，我们导出了类型为 (0,0,1) 和 (0,2) 的 Virasoro 非正则共形块乘积加权和的双线性微分方程。在明确参数识别和 Painlevé V 情形的规范变换后，所得双线性微分算子与量子 Painlevé V 和 IV τ 函数方程中出现的算子一致。
英文摘要
We construct rank-zero irregular vertex operators for the Neveu--Schwarz algebra as linear maps between irregular Verma modules of the same rank satisfying the usual superconformal commutation relations and an irregular asymptotic condition. Under the nondegeneracy assumption $Λ_{2p}\neq0$, we prove their existence and uniqueness. We then extend the decomposition theorem for the Neveu--Schwarz algebra to the irregular setting: the tensor product of the free-fermion Fock module with a rank-$p$ Neveu--Schwarz irregular Verma module decomposes into an infinite direct sum of tensor products of two Virasoro irregular Verma modules. We also prove a compatible decomposition of the irregular vertex operators into tensor products of Virasoro irregular vertex operators. Using suitable pairings and mode insertions, we derive bilinear differential equations for weighted sums of products of Virasoro irregular conformal blocks of types $(0,0,1)$ and $(0,2)$. After explicit parameter identifications and a gauge transformation in the Painlevé V case, the resulting bilinear differential operators agree with those appearing in the quantum Painlevé V and IV tau-function equations.
