HEXAGONbeta

Lefschetz coefficients of quantum theta functions

A mix of human-written and AI-generated textHuman understanding: some partsmath.QA — Quantum Algebramath.RT — Representation Theory

Contributed by Qiyue Tang ↗

Version 1 / Oct 27, 2026 / CC BY 4.0

Abstract

We prove that the coefficients of quantum theta functions in the skew-symmetric setting are characters of finite-dimensional sl2\mathfrak{sl}_2 representations. This proves Conjectures 1.4, 2.17 and 3.13 of Davison–Mandel and the two-wall conjecture stated as Conjecture 3.12 in Davison’s ICM 2026 survey.

Provenance statement

The present result was obtained through two interactive sessions with GPT-6-Astra-Pro, totaling approximately 2.5 hours. Its output exceeded the authors' expectations, and the authors subsequently performed multiple rounds of refinement using built-in Skills. Additionally, this manuscript passed the verification of Rethlas-OSV (running in 2 parallel instances at Astra-Xhigh reasoning effort).

References

  1. S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations , arXiv:math/0104151. https://arxiv.org/abs/math/0104151.arXiv
  2. V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm , arXiv:math/0311245v7. https://arxiv.org/abs/math/0311245.arXiv
  3. M. Gross, P. Hacking and S. Keel, Mirror symmetry for log Calabi--Yau surfaces I , arXiv:1106.4977v3. https://arxiv.org/abs/1106.4977.arXiv
  4. M. Gross, P. Hacking, S. Keel and M. Kontsevich, Canonical bases for cluster algebras , arXiv:1411.1394v2. https://arxiv.org/html/1411.1394v2.arXiv
  5. A. Berenstein and A. Zelevinsky, Quantum cluster algebras , arXiv:math/0404446v2. https://arxiv.org/abs/math/0404446.arXiv
  6. T. Mandel, Scattering diagrams, theta functions, and refined tropical curve counts , J. Lond. Math. Soc. 104 (2021), 2299--2334. https://doi.org/10.1112/jlms.12498.DOI
  7. B. Davison, Positivity for quantum cluster algebras , arXiv:1601.07918v6. https://arxiv.org/abs/1601.07918.arXiv
  8. B. Davison and T. Mandel, Strong positivity for quantum theta bases of quantum cluster algebras , Invent. Math. 226 (2021), 725--843; arXiv:1910.12915v3. https://arxiv.org/html/1910.12915v3.arXiv
  9. B. Davison and S. Meinhardt, Cohomological Donaldson--Thomas theory of a quiver with potential and quantum enveloping algebras , arXiv:1601.02479v5. https://arxiv.org/pdf/1601.02479.arXiv
  10. S. Meinhardt and M. Reineke, Donaldson--Thomas invariants versus intersection cohomology of quiver moduli , arXiv:1411.4062v2. https://arxiv.org/pdf/1411.4062.arXiv
Show all 17 referencesShow fewer references
  1. T. Feng, Z. Yun and W. Zhang, Higher Siegel--Weil formula for unitary groups: the non-singular terms , arXiv:2103.11514v4, §3. https://arxiv.org/pdf/2103.11514.arXiv
  2. A. D. King, Moduli of representations of finite dimensional algebras , Quart. J. Math. 45 (1994), 515--530. https://www.math.uni-bielefeld.de/~sek/sem/stability/king.pdf.link
  3. K. Karu, Hard Lefschetz theorem for nonrational polytopes , Invent. Math. 157 (2004), 419--447. https://personal.math.ubc.ca/~karu/papers/hlef.pdf.link
  4. Q. Tang, A Note on Universal Positivity of Rank-Two Quantum Greedy Elements , arXiv:2609.37452v1 (26 September 2026). https://arxiv.org/html/2609.37452v1.arXiv
  5. B. Davison, BPS cohomology in geometry and representation theory , ICM 2026 proceedings paper, arXiv:2601.08004v1 (2026), §3.3. https://arxiv.org/html/2601.08004v1.arXiv
  6. H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, S. Liu, L. Chen, Yutong Wang, Yuefeng Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai and B. Dong, Automated Conjecture Resolution with Formal Verification , arXiv:2604.03789v2 (2026). https://arxiv.org/abs/2604.03789v2.arXiv
  7. F. Nie, H. Ye, S. Liu, Z. Wang, S. Ermon and J. Zou, One-Sided Verification Under Limited Expert Review , arXiv:2610.18089v2 (2026). https://arxiv.org/abs/2610.18089v2.arXiv

Version history

  1. v1Submitted by Qiyue TangInitial depositCurrentOct 27, 2026