Indefinite theta vectors and generalized Nahm sums of arbitrary rank
Abstract
We construct indefinite theta representations for a family of signed generalized Nahm sums of arbitrary rank r≥2. 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.
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BibTeX
@misc{chen2026indefinite,
title = {{Indefinite theta vectors and generalized Nahm sums of arbitrary rank}},
author = {Dandan Chen and Siyu Yin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00967},
eprint = {2610.00967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00967},
}
RIS
TY - UNPB
TI - Indefinite theta vectors and generalized Nahm sums of arbitrary rank
AU - Dandan Chen
AU - Siyu Yin
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.00967
UR - https://arxiv.org/abs/2610.00967
AB - We construct indefinite theta representations for a family of signed generalized Nahm sums of arbitrary rank r≥2. 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.
ER -