q-Analogues of Mirzakhani's Recursion

• The q-analogues of Mirzakhani’s recursion are a q-deformation of the classical recursion, replacing integration kernels with q-series to compute refined Weil–Petersson volumes.
• They unify combinatorial, geometric, and representation-theoretic aspects by incorporating q-deformed kernels and spectral curve modifications, revealing quantum geometric insights.
• The framework connects with topological recursion and matrix models, providing algorithmic and analytic tools to explore quantized moduli spaces and refined enumerative invariants.

The q-analogues of Mirzakhani's recursion comprise a class of deformations of Mirzakhani's original formula for determining the Weil–Petersson (WP) volumes of moduli spaces of bordered hyperbolic surfaces. These deformations introduce a parameter $q$, interpolating between the classical theory ($q\to1$) and a refined, “quantum” regime, with deep connections to enumerative geometry, topological recursion, q-series, and matrix models. The q-analogues unify the combinatorial, geometric, and representation-theoretic aspects of moduli space invariants, yielding both new computational tools and structural insights.

1. Background: Mirzakhani’s Recursion and Its Context

Mirzakhani’s recursion computes the WP volumes $V_{g,n}(L_1, \ldots, L_n)$ of moduli spaces of hyperbolic surfaces of genus $g$ with $n$ geodesic boundaries of lengths $L_i$. The volumes are polynomial functions of the boundary lengths (in fact, in the $L_i^2$), whose top-degree coefficients encode intersection numbers of $\psi$-classes on $\overline{\mathcal{M}}_{g,n}$. The recursion arises via a decomposition of moduli space based on the McShane identity, with the structure

$$\frac{\partial}{\partial L_1} \left( 2 L_1 V_{g,n}(L_1, \ldots, L_n) \right) = \text{integrals over lower-dimensional moduli spaces involving kernel } H(x, s)$$

where $$H(x, s) = \frac{1}{1 + \exp(\frac{x+s}{2})} + \frac{1}{1 + \exp(\frac{x-s}{2})}$$.

Mirzakhani’s recursion, after Laplace transform, is shown to be equivalent to the topological recursion governing correlators in the Kontsevich matrix model, with a direct identification between WP volumes and correlators of a specific spectral curve (e.g., $x(z) = z / \sin(2\pi z)$, $y(z) = (2\pi) / \sin(2\pi z)$) (0705.3600).

This identification immediately suggests that one may consider deformations of the spectral curve or kernel, especially q-deformations, to obtain q-analogues of the geometry, with potential relevance for quantized moduli spaces, quantum Teichmüller theory, and refined enumerative invariants (Huang, 2015).

2. Definition and Structure of the q-Analogue Recursion

A rigorous q-analogue of Mirzakhani’s recursion is constructed by replacing classical integration kernels and associated WP volume polynomials with q-deformed analogues depending on a deformation parameter $q$ ($|q|<1$), leading to polynomials with $q$-series coefficients (Do et al., 14 Oct 2025):

• The classical kernel $H(x, y)$ is replaced by a q-kernel $H_q(x, y)$:

$$H_q(x, y) = \frac{1}{2} \sum_{m=1}^\infty (-1)^{m-1} q^{m^2/2} (q^{m/2}+q^{-m/2}) [ e^{\frac{1}{2}(x+y)(q^{m/2}-q^{-m/2})} + e^{\frac{1}{2}(x-y)(q^{m/2}-q^{-m/2})} ]$$

• The recursion is realized via q-deformed kernels $D_q$ and $R_q$ defined by

$$\partial_x D_q(x,y,z) = H_q(y+z, x), \quad D_q(0,y,z)=0$$

$$\partial_x R_q(x,y,z) = \frac{1}{2}[H_q(z, x+y) + H_q(z, x-y)],\quad R_q(0,y,z)=0$$

• The q-deformed volume polynomials $V_{g,n}(L_1,\ldots,L_n)$ are defined recursively by

$$\begin{aligned} L_1 V_{g,n}(L_1,\ldots,L_n) = & \sum_{j=2}^n \int_0^\infty x R_q(L_1, L_j, x) V_{g,n-1}(x, L_{K\setminus\{j\}})\, dx\ & + \frac{1}{2} \int_0^\infty \int_0^\infty x y D_q(L_1, x, y) \times\ & \qquad [ V_{g-1,n+1}(x, y, L_K) + \sum_{g_1+g_2=g,\, I \sqcup J = K} V_{g_1,|I|+1}(x, L_I) V_{g_2,|J|+1}(y, L_J) ]\,dx\,dy \end{aligned}$$

with appropriate base cases ($V_{0,3} = 1$, $V_{1,1}$ computed via $D_q$).

The $q$-series such as $q$-zeta values

$$\zeta_q(2k) = \sum_{m=1}^\infty \frac{q^{m k}}{(1 - q^m)^{2k}}$$

arise naturally in the coefficients of these polynomials; e.g., $$V_{1,1}(L) = \frac{1}{48} L^2 + \frac{1}{2} \zeta_q(2)$$ (Do et al., 14 Oct 2025).

This construction recovers the classical WP volumes in the $q\rightarrow 1$ scaling limit: $$\lim_{q\to 1} (1-q)^{6g-6+2n} V_{g,n}\left( \frac{L_1}{1-q}, \ldots, \frac{L_n}{1-q} \right) = V_{g,n}^{\mathrm{WP}}(L_1, \ldots, L_n)$$ while the kernel limits become

$$\lim_{q \to 1} (1-q) D_q\left( \frac{x}{1-q}, \frac{y}{1-q},\frac{z}{1-q} \right) = D(x, y, z)$$

and similarly for $R_q$.

3. Algebraic and Combinatorial Features

The q-recursion produces polynomials in $L_i^2$ with $q$-series coefficients, organizing refined intersection-theoretic data or “quantized” geometric information.

A key aspect is the relationship between the top-degree terms of these q-polynomials and certain quasi-polynomials $N^q_{g,n}(b_1,\ldots,b_n)$ arising from Okuyama’s q-deformation of the Gaussian Hermitian matrix model (Do et al., 14 Oct 2025). These quasi-polynomials, defined on variables $b_i$ (integers), in a rescaled $q\rightarrow 1$ limit yield the classical WP volumes and are now seen to coincide in their top-degree terms with the volumes $V_{g,n}(L_1,\ldots,L_n)$ generated via the q-analogue recursion.

This correspondence ties the geometric q-analogue approach to the algebraic and combinatorial matrix model perspective and suggests the possibility of discrete volume interpretations compatible with the q-recursion framework.

Moreover, the extension to the super setting (for super WP volumes as in the recursion of Stanford–Witten) involves restricting relevant sums to odd positive integers and incorporating additional factors from theta functions (e.g., Ramanujan’s $\psi(q)$), leading to “odd q-zeta functions” in the coefficients: $$\zeta_q^{\mathrm{odd}}(2k) = \sum_{\text{m odd}} \frac{q^{m k}}{(1-q^m)^{2k}}.$$

4. Connections with Topological Recursion and Matrix Models

The Laplace transform of Mirzakhani’s recursion for WP volumes yields the Chekhov–Eynard–Orantin (CEO) topological recursion on a particular spectral curve (0705.3600). The $q$-analogue arises naturally by deforming the spectral curve; specifically, classical functions (e.g., sine) are replaced by q-deformed functions or $q$-analogue kernel ingredients.

The recursive structure of the q-deformed volumes thus mirrors the solution to loop equations in q-deformed or quantum matrix models, leading to $q$-Virasoro or $q$-difference versions of the algebraic constraints. This correspondence motivates the approach—by importing tools from integrable systems, random matrix theory, and quantum groups, one deduces that the q-recursion captures a “quantum” enumerative structure over moduli space, with translational invariance in $q$ reflected in these algebraic relations.

Notably, the same philosophy has been applied to Masur–Veech volumes and other enumerative invariants by introducing “twists” in the topological recursion, which can be interpreted as further $q$- or parameter-dependent deformations (Fuji et al., 2023).

5. Geometric and Physical Interpretations

The q-analogue reflects a quantization of the moduli space geometry. In quantum Teichmüller theory, the classical Fenchel–Nielsen coordinates are promoted to non-commutative variables with $q$-dependent commutation relations. Consequently, a plausible implication is that the q-recursion describes volumes or functions analogous to Weil–Petersson volumes for quantized moduli spaces, with the $q$-parameter acting as a deformation or Planck-type scale (Huang, 2015).

These recursions also appear in the context of 2D quantum gravity and matrix model–gravity dualities: for example, the partition functions of Jackiw–Teitelboim (JT) gravity localize to WP volume calculations, and introducing defects, matter fields, or super-geometry gives rise to further deformations that are interpretable in this $q$-deformed framework (Fuji et al., 2023, Budd et al., 2023, Fırat et al., 4 Sep 2024).

Furthermore, in the context of hyperbolic string field theory, analogues of Mirzakhani's recursion for objects defined via systolic regions of moduli space have been developed (Fırat et al., 4 Sep 2024), with the possibility of generalizing these constructions to q-analogues suggested by the underlying combinatorics.

6. Algorithmic and Analytic Aspects

As part of a broader theme in q-special functions, hypergeometric theory, and recurrence, the q-analogue recursions draw on techniques similar to those used in the Wilf–Zeilberger (WZ) theory of q-hypergeometric summation (Au, 7 Mar 2024, Wang et al., 2018). For many classical recurrences, creative telescoping admits a parallel q-framework in which classical functions (e.g., Gamma functions, Pochhammer symbols) are replaced by their q-analogues (e.g., q-Gamma, q-Pochhammer), preserving the telescopic property and providing q-identities that reduce to the classical case as $q \to 1$.

A conceptual implication is that for geometric recursions expressible in hypergeometric or summation forms, a systematic recipe exists for lifting them to the $q$-world: replace the underlying combinatorial or analytic building blocks with their $q$-deformed analogues, verify the compatibility of limiting behavior, and paper the resulting recursion (Au, 7 Mar 2024). This principle aligns with the algebraic structure underlying the q-analogue of Mirzakhani’s recursion, uniting enumerative geometry and quantum special function theory.

7. Extensions and Outlook

Multiple generalizations exist, including:

• Modifications encoding additional geometry, such as tight boundaries (minimal geodesic length representatives), conical defects, or crosscaps, with the resulting recursions involving further parameter-dependent kernel deformations (Budd et al., 2023, Stanford, 2023).
• Super-extensions capturing volumes in the supermoduli space, relevant for string theory and supergeometry (Do et al., 14 Oct 2025).
• Deformations interpreted as twist insertions in the topological recursion, leading to families of “twisted” volume polynomials interpolating between WP and Masur–Veech volumes or between combinatorial and geometric regimes (Fuji et al., 2023).

A plausible implication is that the q-analogue framework will facilitate new computations of refined enumerative invariants, allow for the systematic quantization of geometric structures on moduli spaces, and yield new insights into quantum geometry, random matrix models, and integrable systems. The link between the q-deformed kernel recursion and matrix model quasi-polynomials further indicates potential for algorithmic and symbolic manipulation of WP-type volumes in both mathematics and theoretical physics.