Signatures in TQFT : Asymptotics and Modularity

1. Problem Statement

The paper investigates the signature $\sigma_g(q/p)$ of the Hermitian form associated with the $SU(2)$ Topological Quantum Field Theory (TQFT) vector spaces $V_p(S_g)$ on a surface of genus $g$.

• Context: For an odd integer $p$ and a coprime odd integer $q$ ($0 < q < p$), the TQFT assigns a vector space over the cyclotomic field $\mathbb{Q}(\zeta)$ where $\zeta = e^{i\pi q/p}$. This space carries a Hermitian form $h_g$.
• The Object of Study: The signature $\sigma_g(q/p) = \text{Sign}(V_p(S_g) \otimes_{\zeta} \mathbb{C}, h_g)$.
• The Gap: While the dimension of these spaces (given by the Verlinde formula) is well-understood and related to the volume of character varieties, the behavior of the signature as a function of the root of unity parameter $q/p$ is less explored.
• The Goal: Analyze the asymptotic behavior of $\sigma_g(q/p)$ as $q/p \to \theta$ (where $\theta$ is an irrational number) and investigate its modularity properties (transformation laws under the action of the modular group).

2. Methodology

The authors employ a multi-faceted approach combining algebraic topology, combinatorics, and analytic number theory:

1. Frobenius Algebra Formalism:

   • They utilize the "signed Verlinde algebra" $V^q_p$, a commutative, semi-simple Frobenius algebra over $\mathbb{Q}$ constructed from the TQFT data.
   • The signature is expressed as a trace in this algebra: $\sigma_g(q/p) = \varepsilon(\Omega^g)$, where $\Omega$ is the inverse of the bilinear form and $\varepsilon$ is the co-unit.
   • For genus 2, this reduces to a combinatorial sum over lattice points in a polytope (Formula 1 in the paper).

2. Trigonometric Transformation:

   • Using finite Fourier transforms, the combinatorial sum for genus 2 is converted into a trigonometric sum involving roots of unity.
   • The authors apply Brion's theorem (a result from convex geometry regarding the summation of rational functions over lattice points in polytopes) to evaluate these sums explicitly.
   • They utilize a simplification provided by Y. Murakami to derive a compact trigonometric formula for $\sigma_2(q/p)$ (Corollary 3.2).

3. Asymptotic Analysis:

   • The authors analyze the limit of the trigonometric formula as $q_k/p_k \to \theta$, where $q_k/p_k$ are convergents of the continued fraction expansion of $\theta$.
   • They use Diophantine approximation properties (specifically the irrationality exponent $\mu(\theta)$) to control the convergence of the resulting infinite series.

4. Modular Forms and Eichler Integrals:

   • The limiting function is identified as the boundary value of an Eichler integral of a modular form of weight 4 for the congruence subgroup $\Gamma(2)$.
   • They draw an analogy with Dedekind sums, which are known to be boundary values of weight 1/2 modular forms, to formulate conjectures about transformation laws.

3. Key Contributions and Results

A. Genus 2 Asymptotics (Main Theorem)

The paper proves that for almost all irrational $\theta \in [0, 1]$, the normalized signature converges to a specific function $\Lambda(\theta)$:
$$\lim_{k \to \infty} \frac{\sigma_2(q_k/p_k)}{p_k^2} = \Lambda(\theta) = \frac{16}{\pi^3} \sum_{n \ge 1, \text{odd}} \frac{\sin(n\pi\theta)}{n^3}$$

• Significance: This establishes a precise link between the discrete TQFT signature and a continuous function defined by a Fourier series. The growth rate is $O(p^2)$, distinct from the dimension's growth rate of $O(p^{3g-3})$.

B. Modularity and Transformation Laws

The function $\Lambda(\theta)$ is shown to be the boundary value of an Eichler integral $G(\tau)$ associated with a weight 4 modular form $E_4^{\text{odd}}$ for $\Gamma(2)$.

• Functional Equation: The authors derive the transformation law for $\Lambda(\theta)$:
  $$\Lambda\left(\frac{\theta}{2\theta+1}\right)(2\theta+1)^2 - \Lambda(\theta) = 2\theta^2 + 2\theta + 1$$
• Conjecture (Proven by Murakami): Motivated by the modular behavior of $\Lambda(\theta)$ and the reciprocity of Dedekind sums, the authors conjectured an arithmetic transformation law for the discrete signature:
  $$\sigma_2\left(\frac{q}{2q+p}\right) - \sigma_2\left(\frac{q}{p}\right) = 2q^2 + 2pq + p^2 - 1$$
  • Update: The paper notes that Y. Murakami subsequently proved this conjecture using the trigonometric formula derived in the paper.

C. Genus 0 and Higher Genus

• Genus 0: The signature for the 4-punctured sphere is shown to be a "2-smoothed" version of the Dedekind sum $s(q, p)$, satisfying a transformation law similar to the classical Dedekind reciprocity.
• Higher Genus ($g \ge 3$): The authors propose that $\sigma_g(q/p)$ scales as $p^{2g-2}$. While they cannot prove the general asymptotic formula due to the complexity of the sign patterns in higher genus, numerical experiments support the existence of a limiting function $F_{g;\lambda}(\theta)$.

4. Significance

1. New Asymptotic Regime: The paper identifies a new scaling regime for TQFT invariants. While dimensions scale with the volume of the character variety ($p^{3g-3}$), signatures scale with $p^{2g-2}$, suggesting a different geometric or topological origin (potentially related to the boundary of the moduli space).
2. Quantum Modularity: The work provides a concrete example of "quantum modularity" where discrete quantum invariants (signatures) exhibit modular transformation properties in the limit, governed by Eichler integrals.
3. Connection to Number Theory: It bridges TQFT with classical number theory, specifically relating TQFT signatures to Dedekind sums and modular forms of half-integral and integral weights.
4. Computational Verification: The paper includes SageMath code and numerical experiments that validate the theoretical predictions, particularly the rapid convergence of the signature function to the limit $\Lambda(\theta)$.

5. Conclusion

Marché and Masbaum successfully characterize the asymptotic behavior of TQFT signatures for genus 2, revealing a deep connection to modular forms and Eichler integrals. They establish a transformation law for the limiting function and conjecture (later proven) a discrete reciprocity formula for the signatures themselves. This work opens new avenues for understanding the arithmetic and geometric properties of quantum invariants beyond their dimensions.