Two-term dilogarithms, mixed-base Nahm sums, and Fricke symmetry
Abstract
We study the arithmetic constraints that two-term Rogers dilogarithm relations impose on rank-two Nahm systems with mixed denominator steps. Monomial complement equations give a uniform conversion to quadratic data in every algebraic degree. Within a specified ten-record simplest-cubic family, positive coupling, positive definiteness, and integer-valued exponents leave, for m > 2, exactly the steps (1,3), (1,13), and (1,31). At the smallest compatible modulus 31, the required support cannot be a multiplicative subgroup coset. Quadratic examples include known product identities and two systems for which the first radial correction excludes every individual modular sum with rational linear terms. For the index-13 system, we prove a two-term Rogers identity and recover its arguments from the matrix of a previously proved two-dimensional Fricke transformation. Its cusp growth agrees with the resulting saddle action, and an independent calculation reproduces all three leading Fricke amplitudes exactly. The associated two-summand product evaluations remain conjectural; the small rank and dimension contrast with the classical rank-five, six-component Andrews-Gordon construction at the same modulus. A mixed-signature cubic family illustrates the limits of the arithmetic restriction
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BibTeX
@misc{hakimoglubrown2026twoterm,
title = {{Two-term dilogarithms, mixed-base Nahm sums, and Fricke symmetry}},
author = {Cetin Hakimoglu-Brown},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14940},
eprint = {2609.14940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14940},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-term dilogarithms, mixed-base Nahm sums, and Fricke symmetry
AU - Cetin Hakimoglu-Brown
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14940
UR - https://arxiv.org/abs/2609.14940
AB - We study the arithmetic constraints that two-term Rogers dilogarithm relations impose on rank-two Nahm systems with mixed denominator steps. Monomial complement equations give a uniform conversion to quadratic data in every algebraic degree. Within a specified ten-record simplest-cubic family, positive coupling, positive definiteness, and integer-valued exponents leave, for m > 2, exactly the steps (1,3), (1,13), and (1,31). At the smallest compatible modulus 31, the required support cannot be a multiplicative subgroup coset. Quadratic examples include known product identities and two systems for which the first radial correction excludes every individual modular sum with rational linear terms. For the index-13 system, we prove a two-term Rogers identity and recover its arguments from the matrix of a previously proved two-dimensional Fricke transformation. Its cusp growth agrees with the resulting saddle action, and an independent calculation reproduces all three leading Fricke amplitudes exactly. The associated two-summand product evaluations remain conjectural; the small rank and dimension contrast with the classical rank-five, six-component Andrews-Gordon construction at the same modulus. A mixed-signature cubic family illustrates the limits of the arithmetic restriction
ER -