Signatures in TQFT : Asymptotics and Modularity
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914351481356288 |
|---|---|
| author | Marché, Julien Masbaum, Gregor |
| author_facet | Marché, Julien Masbaum, Gregor |
| contents | We study the signature $σ_g(\frac q p)$ of $\mathrm{SU}_2$-TQFT vector spaces associated to surfaces of genus $g$, as a function of the defining root of unity $ζ=e^{iπq/p}$. We prove that $\frac{1}{p^2}σ_2(\frac{q}{p})$ converges to $Λ(θ)=\frac{16}{π^3}\sum\limits_{n\ge 1, \textrm{ odd}}\frac{1}{n^3\sin(nπθ)}$ when $\frac{q}{p}$ goes to an irrational number $θ\in [0,1]$ under certain conditions. We also observe that the function $Λ(θ)$ is the boundary value of an Eichler integral of a level $2$ modular form of weight $4$, and use this to propose a conjectural transformation law for the signature function in genus 2 similar to the reciprocity formula for classical Dedekind sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13450 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Signatures in TQFT : Asymptotics and Modularity Marché, Julien Masbaum, Gregor Geometric Topology Number Theory Quantum Algebra 57K16, 11F11, 11K50 We study the signature $σ_g(\frac q p)$ of $\mathrm{SU}_2$-TQFT vector spaces associated to surfaces of genus $g$, as a function of the defining root of unity $ζ=e^{iπq/p}$. We prove that $\frac{1}{p^2}σ_2(\frac{q}{p})$ converges to $Λ(θ)=\frac{16}{π^3}\sum\limits_{n\ge 1, \textrm{ odd}}\frac{1}{n^3\sin(nπθ)}$ when $\frac{q}{p}$ goes to an irrational number $θ\in [0,1]$ under certain conditions. We also observe that the function $Λ(θ)$ is the boundary value of an Eichler integral of a level $2$ modular form of weight $4$, and use this to propose a conjectural transformation law for the signature function in genus 2 similar to the reciprocity formula for classical Dedekind sums. |
| title | Signatures in TQFT : Asymptotics and Modularity |
| topic | Geometric Topology Number Theory Quantum Algebra 57K16, 11F11, 11K50 |
| url | https://arxiv.org/abs/2512.13450 |