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45
The KP hierarchy, branched covers, and triangulations
, 2008
"... The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plücker relations from geometry. We give a soluti ..."
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Cited by 28 (1 self)
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The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plücker relations from geometry. We give a solution to the Plücker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generating series for a general class of transitive ordered factorizations in the symmetric group satisfies the KP hierarchy. These factorizations appear in geometry as encodings of branched covers, and thus by specializing our transitive factorization result, we are able to prove that the generating series for two classes of branched covers satisfies the KP hierarchy. For the first of these, the double Hurwitz series, this result has been previously given by Okounkov. The second of these, that we call the mhypermap series, contains the double Hurwitz series polynomially, as the leading coefficient in m. The mhypermap series also specializes further, first to the series for hypermaps and then to the series for maps, both in an orientable surface. For the latter series, we apply one of the KP equations to obtain a new and remarkably simple recurrence for triangulations in a surface of given genus, with a given number of faces. This recurrence leads to explicit asymptotics for the number of triangulations with given genus and number of faces, in recent work by Bender, Gao and Richmond.
The moduli space of curves and GromovWitten theory
, 2006
"... The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology r ..."
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Cited by 26 (4 self)
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The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Faber’s intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of mathematicians.
From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group
, 2010
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Tropical Hurwitz numbers
"... Abstract. Hurwitz numbers count genus g, degree d covers of P 1 with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piecewise linear objects called tropical curves. This paper develops a ..."
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Abstract. Hurwitz numbers count genus g, degree d covers of P 1 with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piecewise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers. 1.
Generating functions for HurwitzHodge integrals
, 2008
"... In this paper we describe explicit generating functions for a large class of HurwitzHodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are i ..."
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Cited by 10 (5 self)
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In this paper we describe explicit generating functions for a large class of HurwitzHodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are interesting mathematical objects on their own, but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves, and the orbifold GromovWitten theory of DM stacks. Our main tool is AtiyahBott localization: its underlying philosophy is to translate an interesting geometric problem into a purely combinatorial one.
MIRROR SYMMETRY FOR ORBIFOLD HURWITZ NUMBERS
"... Abstract. We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the rLambert curve. We ..."
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Cited by 9 (5 self)
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Abstract. We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the rLambert curve. We argue that the rLambert curve also arises in the infinite framing limit of orbifold GromovWitten theory of [C 3 /(Z/rZ)]. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.
Changes of variables in ELSVtype formulas
, 2006
"... In [3] 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 conjectural Picard variety. We are going to use their formula to study the intersection theory on this variety (if ..."
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Cited by 9 (1 self)
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In [3] 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 conjectural Picard variety. We are going to use their formula to study the intersection theory on this variety (if it is ever to be constructed) by methods close to those of M. Kazarian and S. Lando in [5]. In particular, we prove a WittenKontsevichtype theorem relating the intersection theory and integrable hierarchies. We also extend the results of [5] to include the Hodge integrals over the moduli spaces, involving one λclass. 1
On double Hurwitz numbers with completed cycles
 J. Lond. Math. Soc
, 2012
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Hodgetype integrals on moduli spaces of Admissible Covers
, 2009
"... Hodge integrals are a class of intersection numbers on moduli spaces of curves involving the tautological classes λi, which are the Chern classes of the Hodge ..."
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Cited by 7 (6 self)
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Hodge integrals are a class of intersection numbers on moduli spaces of curves involving the tautological classes λi, which are the Chern classes of the Hodge