Super Weil-Petersson Volumes

• Super Weil-Petersson volumes are geometric invariants defined by integrating spin-dependent cohomological classes over moduli spaces of super Riemann surfaces.
• They are encoded in generating functions tied to the KdV hierarchy and generalized BGW tau functions, revealing deep links between intersection theory and integrable systems.
• Recursion relations extend Mirzakhani’s classical approach by incorporating both Neveu–Schwarz and Ramond punctures, leading to deformations of polynomial volume formulas.

Super Weil-Petersson volumes are geometric invariants defined by integrating specific “super” cohomological classes—derived from spin structures—over the moduli space of super Riemann surfaces. They play a central role at the crossroads of intersection theory, algebraic geometry, the geometry of moduli spaces, integrable systems (in particular, the KdV hierarchy), and two-dimensional quantum gravity, most notably supersymmetric Jackiw–Teitelboim (JT) gravity. The modern theory synthesizes perspectives from matrix models, Virasoro constraints, and recursion relations, expanding on classical Weil–Petersson theory to the supergeometric/“spin curve” context. In recent developments, the extension to include Ramond as well as Neveu–Schwarz (NS) punctures on the worldsheet leads to a one-parameter deformation of the generating functions and associated integrable hierarchies.

1. Super Weil-Petersson Volumes: Definition and Algebraic Structures

Super Weil–Petersson volumes generalize classical Weil–Petersson volumes by passing from the (bosonic) moduli space of compact or bordered Riemann surfaces to the moduli space of super Riemann surfaces or, algebro-geometrically, the moduli space of stable spin curves $\overline{\mathcal{M}}_{g,n}^{\mathrm{spin}}$, endowed with extra spin structure data. At each marked point (boundary) the spin structure can be of Neveu–Schwarz (NS) or Ramond (R) type, specified by a labeling vector $\sigma \in \{0,1\}^n$.

For a given label $\sigma$, one defines classes $\Omega_{g,n}^{\sigma}$ in $H^*(\overline{\mathcal{M}}_{g,n}, \mathbb{Q})$ as pushforwards of the top Chern class of a vector bundle $E_{g,n}$ (arising from the spin structure) defined over $\overline{\mathcal{M}}_{g,n}^{\mathrm{spin}}$. The super Weil–Petersson volume with $n$ NS points and $m$ Ramond points is then

$$\mathrm{Vol}^{\mathrm{super}}_{g,n,m} = \int_{\overline{\mathcal{M}}_{g,n+m}} \Omega_{g,n+m}^{(1^n, 0^m)} \cdot \exp\left(2\pi^2 \kappa_1 + \frac12 \sum_{i=1}^{n+m} L_i^2 \psi_i \right)$$

where the $L_i$ are boundary lengths, $\kappa_1$ is the Miller–Morita–Mumford class, and $\psi_i$ are cotangent classes.

For NS-only punctures (i.e., $\sigma = (1,\ldots,1)$), these volumes coincide with polynomials satisfying a recursive structure. Allowing Ramond points leads to deformations (one-parameter families) of these polynomials and their recursions (Norbury, 2023, Alexandrov et al., 23 Dec 2024).

2. Generating Functions, KdV Hierarchy, and the BGW Tau Function

Super Weil–Petersson volumes are encoded in generating functions that can be identified with specialized tau functions of the KdV integrable hierarchy. The construction is as follows:

• The generating function for intersection numbers is

$$Z^{\Omega}(s, \hbar, \mathbf{t}) = \exp\left( \sum_{g, n} \frac{\hbar^{g-1}}{n!} \sum_{\vec{k}} \sum_{m \ge 0} \frac{s^m}{m!} \int_{\overline{\mathcal{M}}_{g,n+m}} \Omega_{g,n+m}^{(1^n, 0^m)} \prod_{i=1}^n \psi_i^{k_i} t_{k_i} \right),$$

where $s$ counts the number of Ramond points and $(t_0, t_1, \ldots)$ are coupling parameters or 'times'.

• For $s=0$ (no Ramond points), $Z^{\Omega}$ coincides with the Brézin–Gross–Witten (BGW) tau function of the KdV hierarchy, which solves the same string and Virasoro constraints as the generating function of NS-only super volumes (Alexandrov et al., 23 Dec 2024).
• Allowing $s \neq 0$ yields a one-parameter generalized BGW tau function, capturing both NS and R sector data.
• The function $$U = \hbar \frac{\partial^2}{\partial t_0^2} \log Z^{\Omega}$$ satisfies the KdV hierarchy of partial differential equations, with the tau function also subject to a set of (shifted) Virasoro constraints that enforce the geometric recursion (Alexandrov et al., 23 Dec 2024).

3. Recursion Relations: Stanford–Witten and Their Generalization

Super Weil–Petersson volumes satisfy a recursion relation closely akin to the classical Mirzakhani recursion, but now reflecting the supergeometric structure.

• For the NS-only sector, the recursion (Stanford–Witten) (Alexandrov et al., 23 Dec 2024, Norbury, 2023) is

$$L_1 \widehat{V}_{g,n}(s, L_1, \mathbf{L}_{K}) = \frac{1}{2} \iint x y D(L_1, x, y) P_{g,n+1}(x, y, \mathbf{L}_K) \,dx dy + \sum_{j=2}^n \int x R(L_1, L_j, x) \widehat{V}_{g,n-1}(s, x, \mathbf{L}_{K\setminus\{j\}})\,dx + \delta_{1,n}(\cdots)$$

where $D$ and $R$ are explicit hyperbolic geometry kernels, and additional data encodes the combinatorics of boundary decompositions.

• When Ramond points are included, the recursion is modified: the NS polynomial volumes are deformed, and the corresponding generating function is governed by the generalized BGW tau function. The variable $s$ tracks the number of Ramond punctures, and the recursion relation persists as a one-parameter deformation (Alexandrov et al., 23 Dec 2024).
• This recursion is equivalent to enforcing the Virasoro constraints on the generating function, thereby linking the geometric recursion with the integrable system structure.

4. Intersection Numbers and Their Polynomial Structure

Super Weil–Petersson volumes are intimately linked with intersection theory on the moduli space of stable curves, including the psi and kappa classes, and, in the super case, additional characteristic classes (Theta classes, or, more generally, spin classes). The coefficients in the polynomial expansion of the volumes are proportional to intersection numbers,

$$\int_{\overline{\mathcal{M}}_{g, n+m}} \Omega_{g, n+m}^{(1^n, 0^m)}\ \prod_{i=1}^n \psi_i^{d_i}$$

which also appear as coefficients in the series expansion of the tau functions. When only NS points are present, these coefficients are directly analogous to the classical intersection numbers appearing in the Kontsevich–Witten generating series, but for super moduli spaces they encode more refined data.

The polynomiality of these coefficients and their algorithmic computation have been established, including their full asymptotic expansion in the large genus limit (Huang, 14 Jan 2025). For example, with $n$ boundaries and multi-index $d=(d_1,\ldots,d_n)$,

$$[T_{d_1} \cdots T_{d_n}]_{g,n} = e_{0, d} + e_{1, d}/g + e_{2, d}/g^2 + \cdots$$

with each $e_{i, d}$ a polynomial in the number of boundaries $n$ and appropriate variables.

5. Ramond versus Neveu–Schwarz Punctures and Deformed Tau Functions

The behavior of the spin structure at marked points splits the moduli space and the resulting super volumes into sectors:

• Neveu–Schwarz (NS): At an NS point, the local monodromy is nontrivial. The resulting super volumes form a polynomial sequence satisfying the undeformed Stanford–Witten recursion. For these, the volumes coincide with those encoded by the standard BGW tau function.
• Ramond: At a Ramond point, the local monodromy is trivial. Including $m$ Ramond points is captured by a generating series with respect to $s$, leading to the generalized BGW tau function. Ramond boundary behavior produces a deformation of the NS polynomial volumes and modifies the recursion (Alexandrov et al., 23 Dec 2024, Norbury, 2023).

The generating function $Z^{\Omega}(s, \hbar, \mathbf{t})$ encodes all NS and Ramond sectors at once, and the additional parameter provides a deformation of the BGW tau function (homogeneity/dilaton and Virasoro constraints are parameter-dependent).

6. Virasoro Constraints, Integrable Systems, and Topological Recursion

Super Weil–Petersson generating functions satisfy a set of Virasoro constraints, which, in the integrable systems language, correspond to annihilation of the tau function by operators $L_m$ (generalized for super/deformed cases),

$$\left[(2m+1)!!\, t_m - L_m - \frac12 \hbar^{-1} s^2 \delta_{m,0}\right] Z^{\mathrm{BGW}} = 0$$

for all $m\geq 0$. These constraints ensure compatibility with the KdV hierarchy and algebraically generate the Stanford–Witten recursion.

The connection to (generalized) topological recursion is established via identification of the spectral curve (a genus-zero Riemann surface with additional data), whose correlators produce the super volumes by way of Laplace or inverse Laplace transformations. Allowing Ramond points changes the underlying spectral datum and transforms the recursion relations.

7. Mathematical Consequences and Open Directions

The super Weil–Petersson theory, especially as encoded by the generalized BGW tau function, achieves several significant advances:

• It establishes a deep link between intersection theory on spin moduli spaces and integrable systems, generalizing the classical relationship between the Kontsevich–Witten model and the KP/KdV hierarchy.
• By proving that super volumes (with both NS and Ramond sectors) satisfy a one-parameter family of recursions of Stanford–Witten type, the results provide new routes for determining explicit formulas and asymptotics for these invariants (Alexandrov et al., 23 Dec 2024, Huang, 14 Jan 2025).
• The generalized tau function approach yields new algebraic and analytic tools for the analysis of moduli spaces, and their role in supersymmetric quantum gravity and random geometry models.
• The formalism naturally extends to consideration of q-deformations, higher $\kappa$ classes, or asymptotic regimes, with recent work generalizing the framework to encompass these settings (see, e.g., (Do et al., 14 Oct 2025) for $q$-recursion, (Huang et al., 10 Jan 2025) for higher $\kappa$ data).

A plausible implication is that this unified algebraic–geometric–integrable structure underlies the interplay of moduli spaces and topological field theories across both classical and supersymmetric settings, with explicit computational and conceptual benefits for quantum geometry, enumerative invariants, and low-dimensional gravity models.

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Table: Sectors and Associated Generating Functions

Spin Structure at Points	Tau Function	Recursion Satisfied
All NS	BGW tau function	Stanford–Witten
Some Ramond	Generalized BGW tau func	Stanford–Witten (def.)

Here, "Stanford–Witten (def.)" denotes the deformed recursion incorporating Ramond points (Alexandrov et al., 23 Dec 2024).

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The super Weil–Petersson volume program thus provides a comprehensive, integrable, and computable extension of classical volume theory to supergeometry, with far-reaching relevance for both modern mathematical physics and algebraic geometry.