KP hierarchy for Hodge integrals

  title={KP hierarchy for Hodge integrals},
  author={M. Kazarian},
  journal={Advances in Mathematics},
  • M. Kazarian
  • Published 2008
  • Mathematics
  • Advances in Mathematics
Abstract Starting from the ELSV formula, we derive a number of new equations on the generating functions for Hodge integrals over the moduli space of complex curves. This gives a new simple and uniform treatment of certain known results on Hodge integrals like Witten's conjecture, Virasoro constrains, Faber's λ g -conjecture, etc. Among other results we show that a properly arranged generating function for Hodge integrals satisfies the equations of the KP hierarchy. 
Dubrovin-Zhang hierarchy for the Hodge integrals
In this paper we prove that the generating series of the Hodge integrals over the moduli space of stable curves is a solution of a certain deformation of the KdV hierarchy. This hierarchy isExpand
From Kontsevich-Witten to linear Hodge integrals via Virasoro operators
We give a proof of Alexandrov's conjecture on a formula connecting the Kontsevich-Witten and Hodge tau-functions using only the Virasoro operators. This formula has been confirmed up to an unknownExpand
KP hierarchy for Hurwitz-type cohomological field theories
We generalise a result of Kazarian regarding Kadomtsev-Petviashvili integrability for single Hodge integrals to general cohomological field theories related to Hurwitz-type counting problems orExpand
KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions
In this paper, we continue our investigation of the triple Hodge integrals satisfying the Calabi–Yau condition. For the tau-functions, which generate these integrals, we derive the complete familiesExpand
Enumerative Geometry, Tau-Functions and Heisenberg–Virasoro Algebra
In this paper we establish relations between three enumerative geometry tau-functions, namely the Kontsevich–Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe theExpand
Hodge–GUE Correspondence and the Discrete KdV Equation
We prove the conjectural relationship recently proposed in [9] between certain special cubic Hodge integrals of the Gopakumar--Mari\~no--Vafa type [17, 28] and GUE correlators, and the conjectureExpand
Connecting Hodge Integrals to Gromov–Witten Invariants by Virasoro Operators
In this paper, we show that the generating function for linear Hodge integrals over moduli spaces of stable maps to a nonsingular projective variety X can be connected to the generating function forExpand
Polynomial recursion formula for linear Hodge integrals
We establish a polynomial recursion formula for linear Hodge integrals. It is obtained as the Laplace transform of the cut-and-join equation for the simple Hurwitz numbers. We show that the recursionExpand
We derive a quadratic recursion relation for the linear Hodge integrals of the form $\langle\tau_2^n\lambda_k\rangle$. These numbers are used in a formula for Masur-Veech volumes of moduli spaces ofExpand
Fe b 20 16 Connecting the Kontsevich-Witten and Hodge tau-functions by the ĜL ( ∞ ) operators
In this paper, we present an explicit formula that connects the KontsevichWitten tau-function and the Hodge tau-function by differential operators belonging to the ĜL(∞) group. Indeed, we show thatExpand


Hodge integrals and Gromov-Witten theory
Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory are studied. We find a universal system of differential equations which determines the generating function of theseExpand
Hurwitz numbers and intersections on moduli spaces of curves
This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with onlyExpand
Changes of variables in ELSV-type formulas
In [5] I. P. Goulden, D. M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjecturalExpand
Localization, Hurwitz Numbers and the Witten Conjecture
In this note, we use the method of [3] to give a simple proof of famous Witten conjecture. Combining the coefficients derived in our note and this method, we can derive more recursion formulas ofExpand
Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
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 isExpand
The Gromov–Witten Potential of A Point, Hurwitz Numbers, and Hodge Integrals
Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. TheExpand
New Trends in Algebraic Geometry: Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians
We describe algorithms for computing the intersection numbers of divisors and of Chern classes of the Hodge bundle on the moduli spaces of stable pointed curves. We also discuss the implementationsExpand
We describea formalism based on quantizationof quadratichamil- tonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures aboutExpand
An algebro-geometric proof of Witten's conjecture
We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumeratingExpand
Toda equations for Hurwitz numbers
We consider ramified coverings of P^1 with arbitrary ramification type over 0 and infinity and simple ramifications elsewhere and prove that the generating function for the numbers of such coveringsExpand