A Mirzakhani-style volume and intersection theory for the SU(2) character varieties of bordered surfaces with fixed boundary holonomy

Mirzakhani's 2007 theory expresses the Weil-Petersson volumes of the moduli spaces $\mathcal{M}_{g,n}(b)$ of hyperbolic surfaces with geodesic boundary as polynomials in the boundary lengths, computes them recursively by integrating a generalized McShane identity over moduli space, and identifies the coefficients with intersection numbers of tautological classes on $\overline{\mathcal{M}}_{g,n}$ . The closing paragraph of the paper's introduction ('Analogies with moduli spaces of stable bundles') states that, as in its argument for Theorem 6.1, it is possible to derive recursive formulas for intersection numbers of line bundles on the $SU(2)$ character varieties $R_{g,n}(\theta_1,\dots,\theta_n)$ of surface groups with boundary monodromy in fixed conjugacy classes, equipped with the Atiyah-Bott-Goldman symplectic form, by relating these numbers to the symplectic volume, an approach the paper attributes to Witten and to Weitsman. Since the literal target quantities - the volumes of $R_{g,n}(\theta)$ and their intersection numbers - were already obtained by independent methods (Witten's surgery, Jeffrey-Kirwan localization, Meinrenken-Woodward cobordism, Weitsman's intersection-ring recursions, and recent positive polytope formulas), the open content of that direction is the transfer itself: a derivation that relates intersection numbers of the boundary determinant line bundles on $R_{g,n}$ to the Goldman symplectic volume $\mathrm{Vol}(R_{g,n}(\theta))$ in the manner of Mirzakhani's dictionary, under the structural constraint that these volumes are character series in the angles rather than polynomials and under the constraint of the independently known evaluations.

Background

Let $S_{g,n}$ be a smooth, compact, oriented surface of genus $g$ with $n$ boundary components, and let $\mathcal{M}_{g,n}(b_1,\dots,b_n)$ be the moduli space - the quotient, by the mapping class group (the group of isotopy classes of diffeomorphisms of $S_{g,n}$ ), of the set of complete finite-area hyperbolic metrics on $S_{g,n}$ in which every boundary component is a geodesic of prescribed length $b_i$ . The Weil-Petersson (WP) symplectic form $\omega$ is the natural symplectic structure on this space - in Fenchel-Nielsen coordinates $(\ell_j,\tau_j)_j$ , the coordinates cutting the surface into pairs of pants (three-holed spheres) along simple closed curves, it reads $\omega=\sum_j d\ell_j\wedge d\tau_j$ - and $V_{g,n}(b)$ denotes the total WP volume of $\mathcal{M}_{g,n}(b)$ . Mirzakhani's 2007 JAMS paper established: (i) $V_{g,n}(b)$ is a polynomial in $b_1^2,\dots,b_n^2$ with an explicit constant term; (ii) a recursion for $V_{g,n}$ in the complexity $(g,n)$ ; and (iii) a dictionary identifying the coefficients of this polynomial with intersection numbers on the Deligne-Mumford compactification $\overline{\mathcal{M}}_{g,n}$ (the compactification by stable nodal curves) - integrals of the tautological $\psi$ classes (the $\psi_i$ are the first Chern classes of the cotangent line bundles recording the direction of a marked point) against powers of $\omega$ - from which she derived a new proof of the Witten-Kontsevich theorem (the Virasoro constraints: an infinite system of linear relations, arising from a representation of the Virasoro algebra, that determine all these intersection numbers). The engine of the proof is a generalized McShane identity: an absolutely convergent identity on each hyperbolic surface in the moduli space, expressing a boundary-anchored quantity as a sum over simple closed curves whose summands depend only on the curve length and the boundary lengths; integrating this identity over moduli space and evaluating the resulting twist integrals produces the volume recursion. Two orbifold caveats are recorded in the paper. The coefficient dictionary (Theorem 4.4) carries an exceptional factor $2^{m(g,n)|\alpha|}$ with $m(g,n)=1$ exactly when $(g,n)=(1,1)$ ( $\alpha$ indexing the $\psi$ -powers): on the once-holed torus every Riemann surface admits an order-2 automorphism (the elliptic involution), so $\int_{\mathcal{M}_{1,1}}\psi_1=\tfrac{1}{24}$ , half of what naive counting predicts. Correspondingly, the volume recursion excludes $(g,n)=(1,1)$ (and $(0,3)$ ) and acquires a compensating factor $\tfrac12$ there, coming from a half-twist in the stabilizer of a separating curve. These exceptional factors are handled case-by-case rather than by a uniform mechanism. The paper's introduction suggests an analog for a different family of moduli spaces. Let $R_{g,n}(\theta_1,\dots,\theta_n)$ be the character variety: the space of conjugacy classes of homomorphisms $\rho\colon\pi_1(S_{g,n})\to SU(2)$ whose monodromy around the $i$ -th boundary component lies in the fixed conjugacy class of $\mathrm{diag}(e^{i\theta_i},e^{-i\theta_i})$ (conjugacy classes of $SU(2)$ are parametrized by an angle $\theta_i\in[0,\pi]$ , so fixing a conjugacy class is the exact group-theoretic analog of fixing a boundary length). The smooth irreducible part of $R_{g,n}(\theta)$ carries the Atiyah-Bott-Goldman symplectic form (the natural closed 2-form on a character variety, obtained by transgressing the invariant inner product on the Lie algebra of $SU(2)$ along loops on the surface) and has finite total volume, making $\mathrm{Vol}(R_{g,n}(\theta))$ the direct analog of $V_{g,n}(b)$ . The analogy has a structural asymmetry that the paper records explicitly: $R_{g,n}(\theta)$ is analogous to the Teichmüller space of $S_{g,n}$ with fixed boundary lengths (the space of marked hyperbolic metrics before the mapping-class-group quotient), but it already has finite volume without quotienting, and the mapping-class-group action on it is ergodic (Goldman) rather than proper. The paper states that, as in its proof of the intersection-number dictionary, 'it is possible to derive recursive formulas for intersection numbers of line bundles on $R_{g,n}$ by relating these numbers to the symplectic volume of $R_{g,n}$ ', attributing the approach to Witten and to Weitsman. For the $SU(2)$ / $U(n)$ side, the corresponding quantities were computed by gauge-theoretic rather than hyperbolic-geometric methods, both before and after 2007. Witten's pants-surgery/fusion analysis expresses the volume as a series of characters of the unitary group; the gluing was made rigorous by Jeffrey-Kirwan and by Meinrenken-Woodward (via Hamiltonian cobordism); Jeffrey's extended moduli spaces and the Jeffrey-Weitsman inductive toric decompositions make the computations effective; Sengupta recovered the volume measure as a Yang-Mills limit; Mulase-Penkava recovered it by push-forward of the group-algebra measure. On the intersection side, Weitsman determined the geometry of the intersection ring of the moduli space of flat $SU(2)$ connections, verifying the Newstead-Witten vanishing predictions (the conjecture that the rational cohomology of the moduli space vanishes above half of its dimension), with recursions in $(g,n)$ ; Gamse-Weitsman extended the line to parabolic bundles. In genus $0$ , the space $R_{0,n}$ is the moduli space of spatial polygons with fixed side lengths, for which Agapito-Godinho proved a recursion in the number of sides for intersection numbers. Most recently the Lévy school produced positive, polytope-based (hive/honeycomb, in the sense of the Knutson-Tao hive calculus for eigenvalue polytopes) formulas for volumes of moduli of flat $U(n)$ connections on punctured surfaces with prescribed holonomy: the three-holed sphere (François-Tarrago 2024) and arbitrary genus (François-García-Zelada-Lévy-Tarrago 2026). None of these works uses or produces a McShane-type identity, an integration-over-the-moduli-space scheme in Mirzakhani's sense, or a coefficient-to-intersection-number dictionary with orbifold bookkeeping; and their formulas show that the volumes depend on the angle data through character (Fourier) series and polytope integrals, not polynomials - so the literal transfer of Mirzakhani's polynomiality theorem cannot hold, and any derivation on the character-variety side must instead reproduce that character-series dependence. Separately, the McShane-identity program has been extended in several directions: to hyperbolic surfaces with cusps, geodesic boundary and cone points, and to the $SL_2(\mathbb{C})$ character variety of the one-holed torus via Markoff triples (solutions of $x^2+y^2+z^2=xyz$ , which parametrize that character variety; surveyed by Tan-Wong-Zhang); to super Teichmüller space, the supersymmetric extension of Teichmüller space (Huang-Penner-Zeitlin); and to higher Teichmüller theory (higher-rank analogs of Teichmüller space obtained from special surface-group representations into semisimple Lie groups), where Huang-Sun proved identities for positive representations into real split groups via the Goncharov-Shen potential. Do-Norbury studied WP volumes for cone surfaces (surfaces with prescribed cone-angle singularities); Nguyen computed intersection-theoretic volumes of moduli spaces of flat metrics on the sphere, finding piecewise-polynomial dependence on the cone angles. None of these covers compact $SU(2)$ character varieties $R_{g,n}(\theta)$ of general surfaces, and none of them is integrated over a character variety to produce volume recursions.

Problem Statement

Carry Mirzakhani's volume-and-intersection theory over to the $SU(2)$ character varieties $R_{g,n}(\theta_1,\dots,\theta_n)$ of bordered surfaces with fixed boundary holonomy (the spaces of conjugacy classes of homomorphisms $\pi_1(S_{g,n})\to SU(2)$ whose boundary monodromy lies in the prescribed conjugacy classes, equipped with the Atiyah-Bott-Goldman symplectic form), in the direction stated in the introduction of her paper and attributed there to Witten and Weitsman: derive recursive formulas for intersection numbers of line bundles on $R_{g,n}$ by relating these numbers to the symplectic volume of $R_{g,n}(\theta)$ , as in her argument for Theorem 6.1, where the volume theory on $\mathcal{M}_{g,n}(b)$ feeds a dictionary identifying its coefficients with intersection numbers of tautological classes on $\overline{\mathcal{M}}_{g,n}$ . Concretely, the objective is such a derivation for $R_{g,n}(\theta)$ : recursive formulas, in the complexity $(g,n)$ , for the intersection numbers of the boundary determinant line bundles $L_1,\dots,L_n$ on $R_{g,n}$ - the $n$ line bundles attached to the boundary holonomies, whose first Chern classes are the analogues of the $\psi$ classes and are paired with powers of the Goldman symplectic form - obtained by relating these numbers to $\mathrm{Vol}(R_{g,n}(\theta))$ on the smooth irreducible part. The derivation has two coupled outputs: (i) an explicit dictionary, in the sense of Mirzakhani's identification of volume coefficients with intersection numbers, expressing these intersection numbers through $\mathrm{Vol}(R_{g,n}(\theta))$ ; and (ii) the recursion in the complexity $(g,n)$ that this dictionary yields for the named intersection numbers. The known gauge-theoretic evaluations show that the angle dependence enters through character (Fourier) series rather than polynomials, so the direct analog of her polynomiality theorem cannot hold and any derivation must reproduce the known character-series structure instead.

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