发表机构
Shanghai University; East China Normal University（上海大学; 华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为任意秩的带符号广义Nahm和构造不定theta表示，确定其模变换法则，并给出相关乘积向量分量的多重和表示。
AI 中文摘要
我们为任意秩 \\(r\geq2\\) 的一族带符号的广义Nahm和构造了不定theta表示。在显式归一化和边界修正之后，一个带符号的分支给出了与签名\\((1,1)\\)的格相关联的\\((2r-1)\\)分量全纯theta向量的偶分量。Euler型多重和提供了奇分量，从而产生了在秩和分量指标上均一致的公式。利用Zwegers理论，我们确定了实解析完备化的权一模变换法则，并给出了\\(S\\)-变换的显式有限正弦矩阵。我们将非全纯修正分解为单变量theta函数与一维误差函数级数的乘积的有限和，并计算了完备化的反全纯导数。结合完备化的变换法则，这些公式给出了全纯向量的非齐次变换法则。我们还计算了两个带符号分支的线性组合，将其表示为Jacobi乘积商的差。该恒等式以及两族可对称化矩阵及其对偶的进一步求值，结合已知结果，给出了四个相关乘积向量的每个分量的多重和表示。这些表示是带有指定矩阵和对称化子的广义Nahm和的有限组合；在对偶族中，允许带符号的和以及半周期平移。
英文摘要
We construct indefinite theta representations for a family of signed generalized Nahm sums of arbitrary rank \(r\geq2\). After explicit normalization and boundary correction, one signed branch gives the even components of a \((2r-1)\)-component holomorphic theta vector associated with a lattice of signature \((1,1)\). Euler-type multiple sums provide the odd components, yielding formulas uniform in both the rank and the component index. Using Zwegers' theory, we determine the weight-one modular transformation laws of the real-analytic completion, with an explicit finite sine matrix for the \(S\)-transformation. We decompose the nonholomorphic correction into a finite sum of products of unary theta functions and one-dimensional error-function series, and compute the antiholomorphic derivative of the completion. Together with the completed transformation laws, these formulas give the inhomogeneous transformation law of the holomorphic vector. We also evaluate a linear combination of two signed branches as a difference of Jacobi product quotients. This identity and further evaluations for two families of symmetrizable matrices and their duals, combined with known results, give multiple-sum representations for every component of four associated product vectors. These representations are finite combinations of generalized Nahm sums with the prescribed matrices and symmetrizers; in the dual families, signed sums and half-period translates are allowed.
Comments47 pages
