The study of intersection numbers of ψ-classes on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves represents one of the most beautiful confluences of algebraic geometry, mathematical physics, and integrable systems. Witten's 1991 conjecture—that the generating function of these intersection numbers is a τ-function for the KdV hierarchy—launched a research program that continues to yield fundamental results. This survey covers foundational work from the 1990s through cutting-edge developments in resurgent asymptotics from 2024-2025, with complete bibliographic information for over 50 key papers.
The central objects are the ψ-classes $\psi_i = c_1(\mathbb{L}i)$, where $\mathbb{L}i$ is the cotangent line bundle at the $i$-th marked point. The intersection numbers $\langle\tau{d_1}\cdots\tau{d_n}\rangle_g = \int_{\overline{\mathcal{M}}{g,n}}\psi_1^{d_1}\cdots\psi_n^{d_n}$ vanish unless $\sum d_i = 3g-3+n$ (the dimension of $\overline{\mathcal{M}}{g,n}$), and their complete determination was achieved through the remarkable connection to integrable hierarchies.
The foundational paper establishing the connection between 2D quantum gravity and intersection theory appeared in Witten's 1991 survey article. Witten conjectured that two different approaches to 2D quantum gravity—one via matrix models and one via topological field theory—yield identical partition functions, implying that the generating function $F(t_0,t_1,\ldots) = \sum \langle\tau_{d_1}\cdots\tau_{d_n}\rangle\prod(t_{d_i}/d_i!)$ satisfies the KdV hierarchy of integrable differential equations.
@incollection{Witten1991,
author = {Witten, Edward},
title = {Two-dimensional gravity and intersection theory on moduli space},
booktitle = {Surveys in Differential Geometry},
volume = {1},
pages = {243--310},
publisher = {Lehigh University},
address = {Bethlehem, PA},
year = {1991},
mrnumber = {1144529}
}Kontsevich's proof, published in 1992, established the conjecture using a combinatorial description of moduli spaces via ribbon graphs (fatgraphs arising from Strebel differentials). The key insight was identifying intersection numbers with asymptotic coefficients of a Hermitian matrix integral—the matrix Airy function. By demonstrating that this matrix integral is a τ-function for KdV, Kontsevich completed the proof.
@article{Kontsevich1992,
author = {Kontsevich, Maxim},
title = {Intersection theory on the moduli space of curves and the matrix {A}iry function},
journal = {Communications in Mathematical Physics},
volume = {147},
number = {1},
pages = {1--23},
year = {1992},
doi = {10.1007/BF02099526},
mrnumber = {1171758}
}The Witten-Kontsevich theorem has been reproven through remarkably diverse approaches, each illuminating different structural aspects. Kazarian and Lando (2007) gave a purely algebro-geometric proof exploiting the ELSV formula connecting intersection numbers to Hurwitz numbers:
@article{KazarianLando2007,
author = {Kazarian, M. E. and Lando, Sergei K.},
title = {An algebro-geometric proof of {W}itten's conjecture},
journal = {Journal of the American Mathematical Society},
volume = {20},
number = {4},
pages = {1079--1089},
year = {2007},
doi = {10.1090/S0894-0347-07-00566-8},
eprint = {math/0601760},
archiveprefix = {arXiv}
}Mirzakhani's celebrated proof (2007) approached the problem through hyperbolic geometry, establishing that Weil-Petersson volumes $V_{g,n}(b_1,\ldots,b_n)$ of moduli spaces of hyperbolic surfaces with geodesic boundaries are polynomials whose coefficients encode ψ-class intersections. Her recursion formula, derived from McShane-type identities, implies the Virasoro constraints:
@article{Mirzakhani2007JAMS,
author = {Mirzakhani, Maryam},
title = {Weil-{P}etersson volumes and intersection theory on the moduli space of curves},
journal = {Journal of the American Mathematical Society},
volume = {20},
number = {1},
pages = {1--23},
year = {2007},
doi = {10.1090/S0894-0347-06-00526-1},
mrnumber = {2233719}
}@article{Mirzakhani2007Invent,
author = {Mirzakhani, Maryam},
title = {Simple geodesics and {W}eil-{P}etersson volumes of moduli spaces of bordered {R}iemann surfaces},
journal = {Inventiones Mathematicae},
volume = {167},
number = {1},
pages = {179--222},
year = {2007},
doi = {10.1007/s00222-006-0013-2},
mrnumber = {2264808}
}Okounkov and Pandharipande developed an approach through Gromov-Witten theory and Hurwitz numbers, establishing the GW/H correspondence:
@incollection{OkounkovPandharipande2009,
author = {Okounkov, Andrei and Pandharipande, Rahul},
title = {Gromov-{W}itten theory, {H}urwitz numbers, and Matrix models, {I}},
booktitle = {Algebraic Geometry---{S}eattle 2005, {P}art 1},
series = {Proceedings of Symposia in Pure Mathematics},
volume = {80},
pages = {325--414},
publisher = {American Mathematical Society},
year = {2009},
eprint = {math/0101147},
archiveprefix = {arXiv}
}The r-spin generalization was proven by Faber, Shadrin, and Zvonkine, showing that intersection numbers on moduli spaces of r-spin structures form τ-functions for the r-KdV (Gelfand-Dickey) hierarchy:
@article{FaberShadrinZvonkine2010,
author = {Faber, Carel and Shadrin, Sergey and Zvonkine, Dimitri},
title = {Tautological relations and the $r$-spin {W}itten conjecture},
journal = {Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure},
series = {4},
volume = {43},
number = {4},
pages = {621--658},
year = {2010},
doi = {10.24033/asens.2130},
eprint = {math/0612510},
archiveprefix = {arXiv}
}The simplest recursive relations are the string equation and dilaton equation, which follow directly from the geometry of forgetful maps:
These equations, combined with the initial condition $\langle\tau_0^3\rangle_0 = 1$, determine all genus-0 intersection numbers completely.
Dijkgraaf, Verlinde, and Verlinde (1991) derived loop equations for 2D gravity equivalent to an infinite set of Virasoro constraints $L_n\cdot Z = 0$ for $n \geq -1$, where $Z = \exp(F)$. These constraints completely determine all intersection numbers recursively:
@article{DVV1991,
author = {Dijkgraaf, Robbert and Verlinde, Herman and Verlinde, Erik},
title = {Loop equations and {V}irasoro constraints in non-perturbative two-dimensional quantum gravity},
journal = {Nuclear Physics B},
volume = {348},
number = {3},
pages = {435--456},
year = {1991},
doi = {10.1016/0550-3213(91)90199-8}
}Dijkgraaf's comprehensive review connects intersection theory, KdV hierarchy, and topological field theory:
@incollection{Dijkgraaf1992,
author = {Dijkgraaf, Robbert},
title = {Intersection Theory, Integrable Hierarchies and Topological Field Theory},
booktitle = {New Symmetry Principles in Quantum Field Theory},
series = {NATO ASI Series},
pages = {95--158},
publisher = {Springer},
year = {1992},
eprint = {hep-th/9201003},
archiveprefix = {arXiv}
}The Eynard-Orantin topological recursion (2007) provides a universal framework producing symplectic invariants $\omega_{g,n}$ from any spectral curve. For the Airy curve $y^2 = x$, the recursion recovers ψ-class intersection numbers:
@article{EynardOrantin2007,
author = {Eynard, Bertrand and Orantin, Nicolas},
title = {Invariants of algebraic curves and topological expansion},
journal = {Communications in Number Theory and Physics},
volume = {1},
number = {2},
pages = {347--452},
year = {2007},
eprint = {math-ph/0702045},
archiveprefix = {arXiv}
}Zhou proved the equivalence between Virasoro constraints for Weil-Petersson volumes and Eynard-Orantin topological recursion:
@article{Zhou2013,
author = {Zhou, Jian},
title = {Topological Recursions of {E}ynard-{O}rantin Type for Intersection Numbers on Moduli Spaces of Curves},
journal = {Letters in Mathematical Physics},
volume = {103},
pages = {587--615},
year = {2013},
doi = {10.1007/s11005-013-0632-7}
}Mulase and Safnuk established that Mirzakhani's recursion is equivalent to the Virasoro constraints and showed the generating function is a 1-parameter KdV solution:
@article{MulaseSafnuk2008,
author = {Mulase, Motohico and Safnuk, Brad},
title = {Mirzakhani's recursion relations, {V}irasoro constraints and the {KdV} hierarchy},
journal = {Indiana University Mathematics Journal},
volume = {50},
pages = {189--218},
year = {2008},
eprint = {math/0601194},
archiveprefix = {arXiv}
}Borot's lecture notes provide an excellent modern exposition of topological recursion and its applications:
@article{Borot2017,
author = {Borot, Ga{\"e}tan},
title = {Lecture notes on topological recursion and geometry},
journal = {arXiv preprint},
eprint = {1705.09986},
archiveprefix = {arXiv},
primaryclass = {math-ph},
year = {2017}
}For genus 0 with $n$ marked points, the string equation yields a closed formula: $$\langle\tau_{d_1}\cdots\tau_{d_n}\rangle_0 = \frac{(n-3)!}{d_1!\cdots d_n!}$$ when $\sum d_i = n-3$, and zero otherwise. This appears in Kaufmann-Manin-Zagier:
@article{KaufmannManinZagier1996,
author = {Kaufmann, R. and Manin, Yu. and Zagier, D.},
title = {Higher {Weil-Petersson} volumes of moduli spaces of stable $n$-pointed curves},
journal = {Communications in Mathematical Physics},
volume = {181},
number = {3},
pages = {763--787},
year = {1996},
doi = {10.1007/BF02101297}
}Liu and Xu developed generating function techniques providing explicit formulas for n-point functions, which encode all intersection numbers:
@article{LiuXu2011,
author = {Liu, Kefeng and Xu, Hao},
title = {The $n$-point functions for intersection numbers on moduli spaces of curves},
journal = {Advances in Theoretical and Mathematical Physics},
volume = {15},
number = {5},
pages = {1201--1236},
year = {2011},
eprint = {math/0701319},
archiveprefix = {arXiv}
}Key formulas from Liu-Xu:
Okounkov's generating function approach connects to random matrix theory edge scaling:
@article{Okounkov2002,
author = {Okounkov, Andrei},
title = {Generating functions for intersection numbers on moduli spaces of curves},
journal = {International Mathematics Research Notices},
year = {2002},
number = {18},
pages = {933--957},
eprint = {math/0101201},
archiveprefix = {arXiv}
}Faber and Pandharipande proved the λ_g conjecture, providing closed formulas for integrals involving the top Chern class of the Hodge bundle:
@article{FaberPandharipande2003,
author = {Faber, Carel and Pandharipande, Rahul},
title = {Hodge integrals, partition matrices, and the $\lambda_g$ conjecture},
journal = {Annals of Mathematics},
volume = {157},
number = {1},
pages = {97--124},
year = {2003},
doi = {10.4007/annals.2003.157.97},
eprint = {math/9908052},
archiveprefix = {arXiv}
}The λ_g theorem states: $$\int_{\overline{\mathcal{M}}{g,n}}\psi_1^{a_1}\cdots\psi_n^{a_n}\lambda_g = \binom{2g+n-3}{a_1,\ldots,a_n}\int{\overline{\mathcal{M}}_{g,1}}\psi_1^{2g-2}\lambda_g$$
where the one-point integral equals $\frac{2^{2g-1}-1}{2^{2g-1}}\cdot\frac{|B_{2g}|}{(2g)!}$ with $B_{2g}$ the Bernoulli numbers.
Their earlier paper on Hodge integrals established the remarkable formula for pure ψ-class one-point integrals:
@article{FaberPandharipande2000,
author = {Faber, Carel and Pandharipande, Rahul},
title = {Hodge integrals and {Gromov-Witten} theory},
journal = {Inventiones Mathematicae},
volume = {139},
number = {1},
pages = {173--199},
year = {2000},
doi = {10.1007/s002229900028},
eprint = {math/9810173},
archiveprefix = {arXiv}
}The one-point generating function: $1 + \sum_{g>0}t^g\int_{\overline{\mathcal{M}}_{g,1}}\psi^{3g-2} = \exp(t/24)$.
Eynard and Lewański developed natural bases for intersection numbers providing new closed formulas for small genus:
@article{EynardLewanski2023,
author = {Eynard, Bertrand and Lewa{\'n}ski, Danilo},
title = {A natural basis for intersection numbers},
journal = {Rendiconti dell'Istituto di Matematica dell'Universit{\`a} di Trieste},
volume = {55},
year = {2023},
eprint = {2108.00226},
archiveprefix = {arXiv}
}The study of large genus asymptotics was initiated by Mirzakhani and Zograf (2015), who computed the diverging factor in the asymptotic expansion:
@article{MirzakhaniZograf2015,
author = {Mirzakhani, Maryam and Zograf, Peter},
title = {Towards large genus asymptotics of intersection numbers on moduli spaces of curves},
journal = {Geometric and Functional Analysis},
volume = {25},
number = {4},
pages = {1258--1289},
year = {2015},
doi = {10.1007/s00039-015-0336-5},
eprint = {1112.1151},
archiveprefix = {arXiv}
}Zograf's earlier conjectural work provided numerical evidence and explicit formulas:
@article{Zograf2008,
author = {Zograf, Peter},
title = {On the large genus asymptotics of {W}eil-{P}etersson volumes},
year = {2008},
eprint = {0812.0544},
archiveprefix = {arXiv},
primaryclass = {math.AG}
}Amol Aggarwal's work (2020-2021) established uniform large genus asymptotics through combinatorial analysis of Virasoro recursions:
@article{Aggarwal2021,
author = {Aggarwal, Amol},
title = {Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials},
journal = {Inventiones mathematicae},
volume = {226},
number = {3},
pages = {897--1010},
year = {2021},
doi = {10.1007/s00222-021-01059-9},
eprint = {1912.04482},
archiveprefix = {arXiv}
}@article{Aggarwal2020JAMS,
author = {Aggarwal, Amol},
title = {Large genus asymptotics for volumes of strata of abelian differentials},
journal = {Journal of the American Mathematical Society},
volume = {33},
number = {4},
pages = {941--989},
year = {2020},
doi = {10.1090/jams/947}
}Delecroix, Goujard, Zograf, and Zorich established deep connections between Masur-Veech volumes and intersection numbers:
@article{DGZZ2021Duke,
author = {Delecroix, Vincent and Goujard, {\'E}lise and Zograf, Peter and Zorich, Anton},
title = {Masur-{V}eech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves},
journal = {Duke Mathematical Journal},
volume = {170},
number = {12},
pages = {2633--2718},
year = {2021},
doi = {10.1215/00127094-2021-0054},
eprint = {1908.08611},
archiveprefix = {arXiv}
}@article{DGZZ2022Invent,
author = {Delecroix, Vincent and Goujard, {\'E}lise and Zograf, Peter and Zorich, Anton},
title = {Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves},
journal = {Inventiones mathematicae},
volume = {230},
pages = {123--224},
year = {2022},
doi = {10.1007/s00222-022-01123-y},
eprint = {2007.04740},
archiveprefix = {arXiv}
}Guo and Yang proved the polynomiality phenomenon—that asymptotic coefficients are polynomials in $n$ and the multiplicities:
@article{GuoYang2024,
author = {Guo, Jindong and Yang, Di},
title = {On the large genus asymptotics of psi-class intersection numbers},
journal = {Mathematische Annalen},
volume = {388},
pages = {61--97},
year = {2024},
doi = {10.1007/s00208-022-02519-6},
eprint = {2110.06774},
archiveprefix = {arXiv}
}The most sophisticated recent work uses resurgent analysis to obtain all subleading corrections:
@article{EGFGGL2023,
author = {Eynard, Bertrand and Garcia-Failde, Elba and Giacchetto, Alessandro and Gregori, Paolo and Lewa{\'n}ski, Danilo},
title = {Resurgent large genus asymptotics of intersection numbers},
year = {2023},
eprint = {2309.03143},
archiveprefix = {arXiv},
primaryclass = {math.AG}
}This paper proves the Guo-Yang polynomiality conjecture and provides new results for r-spin and Θ-class intersection numbers, with Stokes constant $S_A = 1$ and action $A = 2/3$ from the Airy ODE.
The most recent developments include:
@article{GNYZ2024,
author = {Guo, Jindong and Norbury, Paul and Yang, Di and Zagier, Don},
title = {Combinatorics and large genus asymptotics of the {B}r{\'e}zin-{G}ross-{W}itten numbers},
year = {2024},
eprint = {2412.20388},
archiveprefix = {arXiv},
primaryclass = {math.AG}
}@article{HideThomas2025,
author = {Hide, Will and Thomas, Joe},
title = {Large-$n$ asymptotics for {W}eil-{P}etersson volumes of moduli spaces of bordered hyperbolic surfaces},
journal = {Communications in Mathematical Physics},
volume = {406},
number = {9},
pages = {203},
year = {2025},
doi = {10.1007/s00220-025-05369-4},
eprint = {2312.11412},
archiveprefix = {arXiv}
}The ELSV formula (Ekedahl-Lando-Shapiro-Vainshtein, 2001) provides a fundamental bridge between Hurwitz numbers and intersection theory: $$H_{g,n}(\mu) = \prod_i \frac{\mu_i^{\mu_i}}{\mu_i!} \int_{\overline{\mathcal{M}}_{g,n}} \frac{c(\Lambda^\vee)}{\prod_i(1-\mu_i\psi_i)}$$
@article{ELSV2001,
author = {Ekedahl, Torsten and Lando, Sergei and Shapiro, Michael and Vainshtein, Alek},
title = {Hurwitz numbers and intersections on moduli spaces of curves},
journal = {Inventiones Mathematicae},
volume = {146},
number = {2},
pages = {297--327},
year = {2001},
doi = {10.1007/s002220100164}
}Okounkov and Pandharipande established the GW/H correspondence relating Gromov-Witten theory to Hurwitz enumeration:
@article{OkounkovPandharipande2006,
author = {Okounkov, Andrei and Pandharipande, Rahul},
title = {Gromov-{W}itten theory, {H}urwitz theory, and completed cycles},
journal = {Annals of Mathematics},
volume = {163},
number = {2},
pages = {517--560},
year = {2006},
doi = {10.4007/annals.2006.163.517},
eprint = {math/0204305},
archiveprefix = {arXiv}
}Recent work extends these connections to double Hurwitz numbers:
@article{BorotEtAl2023,
author = {Borot, Ga{\"e}tan and Do, Norman and Karev, Maksim and Lewa{\'n}ski, Danilo and Moskovsky, Ellena},
title = {Double {H}urwitz numbers: polynomiality, topological recursion and intersection theory},
journal = {Mathematische Annalen},
volume = {386},
pages = {179--243},
year = {2023},
doi = {10.1007/s00208-022-02457-x},
eprint = {2002.00900},
archiveprefix = {arXiv}
}Dubrovin, Yang, and Zagier developed new formulas for KdV τ-functions with direct applications to intersection numbers:
@article{DubrovinYangZagier2021,
author = {Dubrovin, Boris and Yang, Di and Zagier, Don},
title = {On tau-functions for the {KdV} hierarchy},
journal = {Selecta Mathematica (New Series)},
volume = {27},
number = {1},
pages = {Article 12},
year = {2021},
doi = {10.1007/s00029-021-00620-x},
eprint = {1812.08488},
archiveprefix = {arXiv}
}Alexandrov identified connections to the BKP hierarchy:
@article{Alexandrov2021BKP,
author = {Alexandrov, Alexander},
title = {Intersection numbers on $\overline{\mathcal{M}}_{g,n}$ and {BKP} hierarchy},
journal = {Journal of High Energy Physics},
year = {2021},
pages = {Article 13},
eprint = {2012.07573},
archiveprefix = {arXiv},
primaryclass = {math-ph}
}Dijkgraaf and Witten's review provides an accessible introduction to the physics connections:
@article{DijkgraafWitten2018,
author = {Dijkgraaf, Robbert and Witten, Edward},
title = {Developments in topological gravity},
journal = {International Journal of Modern Physics A},
volume = {33},
number = {30},
pages = {1830029},
year = {2018},
doi = {10.1142/S0217751X18300296},
eprint = {1804.03275},
archiveprefix = {arXiv}
}The primary tool for modern computations is admcycles, providing comprehensive functionality for the tautological ring:
@article{DelecroixSchmittvanZelm2021,
author = {Delecroix, Vincent and Schmitt, Johannes and van Zelm, Jason},
title = {admcycles -- a {S}age package for calculations in the tautological ring of the moduli space of stable curves},
journal = {Journal of Software for Algebra and Geometry},
volume = {11},
number = {1},
pages = {89--112},
year = {2021},
doi = {10.2140/jsag.2021.11.89},
eprint = {2002.01709},
archiveprefix = {arXiv}
}Repository: https://gitlab.com/modulispaces/admcycles
Key features include: generators and products of tautological classes; intersection number computation; verification of tautological relations via Pixton's relations; pushforwards/pullbacks under gluing and forgetful morphisms; double ramification cycles; strata of k-differentials.
HodgeIntegrals for Macaulay2:
@article{Yang2010,
author = {Yang, Stephanie},
title = {Intersection numbers on $\bar{M}_{g,n}$},
journal = {The Journal of Software for Algebra and Geometry: Macaulay2},
volume = {2},
year = {2010},
doi = {10.2140/jsag.2010.2.1}
}Faber's foundational KaLaPs Maple program:
@incollection{Faber1999Algorithms,
author = {Faber, Carel},
title = {Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of {J}acobians},
booktitle = {New Trends in Algebraic Geometry ({W}arwick, 1996)},
series = {London Math. Soc. Lecture Note Ser.},
volume = {264},
pages = {93--109},
publisher = {Cambridge Univ. Press},
year = {1999},
eprint = {alg-geom/9706006},
archiveprefix = {arXiv}
}The diffstrata submodule handles strata of differentials:
@article{CostantiniMollerZachhuber2023,
author = {Costantini, Matteo and M{\"o}ller, Martin and Zachhuber, Jonathan},
title = {diffstrata---{A} {S}age Package for Calculations in the Tautological Ring of the Moduli Space of {A}belian Differentials},
journal = {Experimental Mathematics},
year = {2023},
eprint = {2006.12815},
archiveprefix = {arXiv},
primaryclass = {math.AG}
}Computation times increase substantially with genus and marked points. For top-degree ψ-class integrals, string and dilaton equations provide efficient O(n) recursions. Mixed intersections with λ and κ classes require more complex procedures. Most packages work efficiently for g ≤ 10 and n ≤ 15, with automated lookup of pre-calculated relations helping for known cases.
The theory of ψ-class intersection numbers stands as a remarkable achievement connecting algebraic geometry, mathematical physics, combinatorics, and integrable systems. Three decades after Witten's conjecture, the field continues to produce fundamental results—from the complete understanding of large genus asymptotics via resurgent analysis to powerful computational tools enabling exploration of new phenomena.
Key recent developments include the proof of polynomiality in asymptotic expansions (Guo-Yang, confirmed by Eynard et al.), explicit computation of all subleading corrections through Borel resummation, and extension to super Weil-Petersson volumes relevant to JT supergravity. The admcycles package has democratized computation, enabling verification of tautological relations and exploration of new conjectures.
Open directions include understanding the nonperturbative structure beyond resurgent asymptotics, extending results to moduli of higher differentials and super Riemann surfaces, and developing more efficient algorithms for high-genus/many-point computations. The profound connections established between seemingly disparate areas—hyperbolic geometry (Mirzakhani), enumerative geometry (Hurwitz theory), random matrices (Kontsevich), and integrable systems (KdV)—suggest that ψ-class intersection numbers will continue to reveal deep mathematical structures.