Pure ψ-Class Intersection Numbers

• Pure ψ-class intersection numbers are fundamental invariants capturing the intersection theory of tautological cotangent line classes in the Deligne–Mumford moduli space.
• The methodology employs matrix–resolvent formalism and asymptotic analysis to derive explicit generating series and polynomial corrections for these invariants.
• Applications extend to computing Weil–Petersson volumes and bridging algebraic geometry with mathematical physics through topological recursion and integrable systems.

Pure $\psi$-class intersection numbers are fundamental invariants associated with the geometry of the Deligne–Mumford moduli space of stable curves $\overline{\mathcal{M}}_{g,n}$. They encode the intersection theory of tautological cotangent line classes at marked points, providing the primary coefficients of the KdV hierarchy’s solution and furnishing key data for enumerative geometry, mathematical physics, and the study of large-genus moduli spaces. The explicit computation, asymptotic analysis, and combinatorial structure of these intersection numbers reveal deep connections with integrable systems and topological recursion.

1. Definition and Basic Properties

For $g, n \geq 0$ with $2g-2+n > 0$, consider the Deligne–Mumford stack $\overline{\mathcal{M}}_{g,n}$ of stable n-pointed genus-$g$ curves. At each marked point $i$ is the tautological cotangent line bundle $L_i \to \overline{\mathcal{M}}_{g,n}$, with first Chern class $$\psi_i = c_1(L_i) \in H^2(\overline{\mathcal{M}}_{g,n})$$. Given nonnegative integers $d_1, \ldots, d_n$ summing to $3g-3+n$, the pure $\psi$-class intersection number is

$$\langle\tau_{d_1}\cdots\tau_{d_n}\rangle_g = \int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{d_1} \cdots \psi_n^{d_n} \in \mathbb{Q}.$$

These numbers vanish unless the degree-dimension constraint $d_1 + \cdots + d_n = 3g-3+n$ holds. They form the central coefficients of the Witten–Kontsevich tau-function and obey key constraints—namely, the string equation, dilaton equation, and Virasoro (DVV) recursion—which uniquely determine them starting from the lowest-genus cases (Guo et al., 2021).

2. Explicit Generating Series: Matrix–Resolvent Formalism

For $n \geq 2$, the $n$-point generating function is given by

$$C_n(X_1, \ldots, X_n) = \sum_{d_1,\dots,d_n \geq 0} \langle\tau_{d_1}\cdots\tau_{d_n}\rangle_g\, X_1^{d_1} \cdots X_n^{d_n},$$

which can be written in the explicit matrix–resolvent form: $$C_n(X_1, \ldots, X_n) = -\frac{1}{n} \sum_{\sigma \in S_n} \frac{ \mathrm{tr}(M(X_{\sigma(1)}) \cdots M(X_{\sigma(n)})) } { (X_{\sigma(1)}-X_{\sigma(2)}) \cdots (X_{\sigma(n)}-X_{\sigma(1)}) } + \delta_{n,2}\frac{X_1+X_2}{(X_1-X_2)^2}$$ where $M(X)\in \mathfrak{sl}_2(\mathbb{Q}((X^{-1})))$ is a particular Laurent series whose coefficients encode the recursively defined intersection numbers, and $S_n$ is the symmetric group (Guo et al., 2021). This formula generalizes the Airy matrix model’s correlators and realizes the generating series of intersection numbers as traces of products of explicit matrices.

3. Large Genus Asymptotics and Uniform Formulae

An explicit asymptotic regime governs pure $\psi$-class intersection numbers in the large-genus limit $g \to \infty$. For each fixed $C>0$, the normalized ratio

$$G_{d_1,...,d_n}(g) := \frac{(2g)! \; \langle\tau_{d_1}\cdots\tau_{d_n}\rangle_g } { (6g)^{2g-2+n} \prod_{i=1}^n (2d_i+1)!! }$$

satisfies

$$\lim_{g\to\infty} \max_{1\leq n \leq C\log g} \max_{d_1+\cdots+d_n=3g-3+n} | G_{d_1,...,d_n}(g) - 1 | = 0,$$

so that

$\langle\tau_{d_1}\cdots\tau_{d_n}\rangle_g \sim \prod_{i=1}^n (2d_i+1)!! \cdot \frac{ (6g)^{2g-2+n} }{ (2g)! } \quad \text{as %%%%20%%%%}.$

This result, originally conjectured by Delecroix–Goujard–Zograf–Zorich and proved by Aggarwal and others, demonstrates a factorial-exponential growth with explicit coefficients for fixed $n = O(\log g)$, and extends to uniform lower bounds via an explicit function $\lambda(g, L)$ (Guo et al., 2021, Delecroix et al., 2020).

4. Polynomiality and the Structure of Subleading Corrections

Subleading corrections in the large-genus expansion display a polynomial structure. Fix $n$ and nonnegative $d_1,\ldots,d_{n-1}$, set $d_n = 3g-3+n - \sum_{i=1}^{n-1} d_i$. The normalized intersection admits a full asymptotic expansion: $$G_{d_1,...,d_n}(g) = \sum_{k=0}^\infty G_k(n, p_0, ..., p_{\lfloor 3k/2\rfloor-1}) \; g^{-k},$$ where $p_i$ counts the multiplicity of $i$ among the exponents, and each $G_k$ is a polynomial in $n$, $p_j$ ($j\leq\lfloor3k/2\rfloor-1$) of degree at most $2k$. For instance,

$G_1(n,p_0) = \frac{(n-1)(n-6) + p_0(5-p_0)}{24}.$

This polynomiality phenomenon reveals universality, with the $1/g^k$ corrections governed by low-degree data, independent of the full sequence $d_1,\ldots,d_n$ (Guo et al., 2021).

5. Closed-Form and Combinatorial Formulae

Recent advances produce non-recursive, combinatorial closed-form formulae for the intersection numbers. For $g,n$ and partition $\lambda$ with $|\lambda|=3g-3+n$,

$$\langle \tau_{\lambda_1} \cdots \tau_{\lambda_n} \rangle_g = \frac{1}{24^g} \sum_{r=0}^{\min(g, \frac{(n-1)(n-2)}2)} \,12^r \; \sum_{|\nu|=d_{r,n}} \sum_{\substack{|\mu|=3g-3+n\ \mu\geq \lambda}} D_{r,n}(\nu) Q_{\nu,\mu} K_{\mu,\lambda}$$

where $D_{r,n}(\nu)$, $Q_{\nu,\mu}$, and (normalized) Kostka numbers $K_{\mu,\lambda}$ are explicitly defined, involving evaluation of Schur polynomial inner products, combinatorial factors, and Gamma-function ratios. This method bypasses the KdV and Virasoro recursions, relying purely on finite algebraic constructions and combinatorics (Eynard et al., 2022).

A natural basis for generating series $$A_{g,n}(x_1,\ldots,x_n) = \int_{\overline{\mathcal{M}}_{g,n}} \prod_{i=1}^n \frac{1}{1-x_i\psi_i}$$ is in terms of elementary symmetric polynomials $e_k(x)$. The key vanishing conjecture and its proof establish that only $g$ or fewer $e_k$ factors with $k>1$ can appear in the expansion, reflecting a strong constraint on the structure of intersection numbers (Eynard et al., 2021, Eynard et al., 2022).

6. Connections to Moduli of Curves, Combinatorics, and Topological Recursion

Combinatorial and geometric interpretations are intrinsic to the structure of pure $\psi$-class intersection numbers. The matrix–resolvent formalism expresses them via traces of $\mathfrak{sl}_2$-valued series; the enumeration of stable graphs and the action of the Virasoro constraints are naturally encoded in these expressions (Guo et al., 2021). In genus zero and for tropical settings, explicit combinatorial and inclusion–exclusion–type formulas (multinomial or signed multinomial) relate these numbers to the geometry of tropical and weighted Hassett moduli spaces (Hahn et al., 2021, Sharma, 2018).

Furthermore, the intersection numbers are recovered as the symplectic invariants of the Airy spectral curve under Eynard–Orantin topological recursion, placing them at the crossroads of random matrix theory, integrable systems, and mirror symmetry. The Airy specialization of the general spectral curve formula produces precisely the Witten–Kontsevich invariants; the generating series solves the KdV hierarchy and encodes the physically significant tau-function (Eynard, 2011).

7. Applications and Broader Impact

Pure $\psi$-class intersection numbers permeate the study of moduli spaces, appearing in the enumeration of ribbon graphs, computation of Weil–Petersson volumes of moduli spaces, and as coefficients in the expansion of Hodge and tautological integrals. Their large-genus asymptotics directly inform the geometry of random Riemann surfaces and the behavior of geometric invariants in high complexity. These insights underpin connections between algebraic geometry, mathematical physics, and combinatorics, revealing rich universality and structure in moduli space intersection theory (Guo et al., 2021, Delecroix et al., 2020, Eynard, 2011).