Gromov-Witten theory, Hurwitz numbers, and Matrix models, I

By A. Okounkov, R. Pandharipande. 2001

291 citations recorded by the source corpus.

The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is to present an exposition of the circle of ideas involved: Hurwitz numbers, Gromov-Witten theory of the projective line, matrix integrals, and the theory of random trees. Further topics will be treated in a sequel.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2fadad02d66fa40ed97aafb8046be03347bb5f2e

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Citations in the corpus

The Equivariant Gromov-Witten theory of P**1 (2002, methodology): “However, while the known derivations of the KdV equations for the point require the analysis of elaborate auxiliary constructions (see [1] [12] [14] [20] [21]), the Toda equations for P 1 follow directly, almost in textbook fashion, from the operator description of the theory.”
LOCALIZATION IN GROMOV-WITTEN THEORY AND ORBIFOLD GROMOV-WITTEN THEORY (2011, unknown): “By now, Witten’s conjecture has been reproved many times (Okounkov-Pandharipande [48], Mirzakhani [45], Kim-Liu [35], Kazarian-Lando [34], Chen-Li-Liu [10], Kazarian [33], Mulase-Zhang [46], etc.”
Algebra of differential operators associated with Young diagrams (2012, background): “−1,∆k−1,∆k >, (see, e.g., [2]). The Hurwitz numbers appear in diﬀerent framewroks: strings and QCD [8], mirror symmetry [7], theory of singularities [1], matrix models [16, 20], integrable systems [26, 14, 25], Yang-Mills theory [6, 14] and the theory of moduli of curves [13, 12, 20] and other br”
Polynomial recursion formula for linear Hodge integrals (2009, methodology): “The celebrated Ekedahl-Lando-Shapiro-Vainshtein formula [6, 16, 29 ] relates Hurwitz numbers and linear Hodge integrals on the Deligne-Mumford moduli stackMg;‘ consisting of stable algebraic curves of genus g with ‘ distinct nonsingular marked points subject to the stability condition 2g 2 +‘ > 0. ”
THE MODULI SPACE OF CURVES AND GROMOV-WITTEN THEORY (2006, unknown): “en (2n+1) ∂3 ∂t n∂t2 0 F= ∂2 ∂t n−1∂t 0 F ∂3 ∂t3 0 F +2 ∂3 ∂t n−1∂t 2 0 F ∂2 ∂t2 0 F + 1 4 ∂5 ∂t n−1∂t 4 0 F. Witten’sconjecturenowhasmanyproofs,byKontsevich[Ko1],Okounkov-Pandharipande [OP], Mirzakhani [Mi], and Kim-Liu [KiL]. It is a sign of the richness of this conjecture that these proofs ”
Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials (2019, result): “First, we point out an interesting comparison with the proofs by Mirzakhani [38], by Kontsevich [31], and by Okounkov–Pandharipande [41] of Witten’s conjecture: the generating function of ψ-class intersections on moduli spaces of curves is a solution of the KdV hierarchy of partial differential equa”