Year: 2026
Type: preprint
Abstract: 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, half-period translates are absorbed into signs and constant phase factors.
Source: arXiv (Cornell University)
Authors: Chen, Dandan, Yin, Siyu
Institutions: Shanghai University, East China Normal University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215897272