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RESEARCH STATEMENT
PAUL JOHNSON
My research interests fall under the broad heading of enumerative geometry. The goal here is to count geometric objects satisfying certain constraints, and the methods lie at the intersection of many fields: algebraic geometry, combinatorics,representation theory, and recently symplectic geometry and physics. Typically in enumerative geometry there are a large number of invariants, and one of the main goals is to get a handle on this large amount of data by finding patterns in it. A famous example of this is the number N (d) of degree d rational curves through 3d − 1 points in P^2 ; Kontsevich found a recursive formula expressing N (d) in terms of N (x), with x < d.
I have two active research programs, each investigating patterns in different enumerative geometries of holomorphic maps f : C → X between curves. My main research program studies Gromov-Witten theory when X is an orbifold curve, and finds patterns in the form of nice families of differential equations. My main result here so far is my thesis [?], which solves the equivariant Gromov-Witten theory of one dimensional toric stacks. This work depends essentially on previous joint work with R. Pandharipande and H.-H. Tseng [7] extending the ELSV formula to orbifolds. There is at least one more paper in the works in this direction, building on the first two papers to prove the Virasoro conjecture for orbifold curves [?]. The first two sections give an overview of my results in this direction.
My second research program studies a more classical enumerative geometry of curves, Hurwtiz theory. Hurwitz theory counts covers of curves having given ramification conditions, and although it dates back to the 19th century it is in the middle of a resurgence, in large part due to connections with Gromov-Witten theory. My research attacks this classical problem with the modern technique of tropical geometry, which degenerates the curves in question to graphs. Joint work with R. Cavalieri and H. Markwig, shows that the tropical version of double Hurwitz numbers is equivalent to the classical problem [1]. In ongoing work we have applied this work to get strong results about double Hurwitz numbers [2], and we have had made preliminary investigations into using the same techniques to attack more complicated Hurwitz numbers. This work is summarized in the third section.
In addition to these programs investigating curves, I am interested in questions about the Gromov- Witten theory of higher dimensional orbifolds. The final section outlines preliminary work begun in this direction.
- Gromov-Witten theory of orbifold curves
In Gromov-Witten theory the geometric objects counted are holomorphic maps from f : C → X, where X is a fixed target space, and C is a complex curve, which varies. If we fix the genus g of C, and n points on it, and the homology class β = f∗[C] ∈ H 2 (X), the set of such maps form a moduli space Mg,n(X, β). Although potentially very singular, this moduli space has a well behaved intersection theory. We place conditions on our maps by intersecting cycles on this moduli space.
I am particularly interested in Gromov-Witten theory of orbifolds, or Deligne-Mumford stacks. Orb- ifolds can be thought of as manifolds with mild singularities allowed; they are spaces where every point x has a neighborhood isomorphic to Rn/Gx, where Gx is a finite group called the isotropy group of x. One reason for the importance of stacks is that moduli spaces of objects with automorphisms will
Date: September 12, 2008. 1
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form stacks; for instance, the moduli spaces Mg,n of holomorphic curves are orbifolds. The motivation for studying the Gromov-Witten theory of orbifolds comes from string theory; it was found that even though X is singular, its string theory - and hence Gromov-Witten theory - behaves as though X were smooth.
There are many patterns that occur Gromov-Witten theory independently of the target space X. A well understood example of this is the WDVV equation, which says that the genus 0 Gromov-Witten invariants can be used as the structure constants of an associative algebra, the quantum cohomology ring of X. Kontsevich’s recursion can be viewed as a special case of the WDVV equation.
With higher genus invariants things are more complicated and less understood, but for nice spaces X the relations among the Gromov-Witten invariants sometimes manifest themselves as integrable sys- tems. It is convient to package the Gromov-Witten invariants into a generating function, called the Gromov-Witten potential. Then, there is a tendency for the Gromov-Witten potential to satisfy an integrable hierarchy; that is, there is a large family of commuting differential operators which annihilate the generating function. A famous example of this phenomenon is the case when X is a point, in which case we are studying the intersection theory of Mg,n. The Witten-Kontsevich theorem says that in this case the Gromov-Witten potential satisfies the KdV hierarchy.
Another example is the Toda conjecture, first proven by Okounkov and Pandharipande ([9], [10]), which suggests that various flavors (equivariant, relative) of Gromov-Witten invariants of P^1 satisfy forms of the 2-Toda hierarchy.
For a general target space the Gromov-Witten potential might not satisfy an integrable hierarchy, the Virasoro conjecture suggests a different structure: there should be a family of differential operators generating half of a Virasoro algebra that annihilates the generating function. Using their work on the Toda conjecture, Okounkov and Pandharipande showed that the Virasoro conjecture holds for all smooth curves [11]; together, their three papers completely determine the Gromov-Witten theory of all smooth curves.
The goal of my main research program is to extend as much as possible of Okounkov and Pandhari- pande’s results to orbifold curves. The logical starting point of Okounkov and Pandharipande’s program is the equivariant Toda conjecture [10], and my main result extends this to one dimensional toric stacks. Interesting behavior arises if the orbifold is not effective - that is, if the action of an isotropy group Gx is non effective, and has some kernel K. Then, K is part of the isotropy group of every point of X. These copies of K can vary in a twisted way over X, similar to a principle bundle, and rather than producing singularities on X this isotropy can be viewed as an extra structure on X, called a gerbe. An example is M 1 , 1 , the moduli space of 1 pointed elliptic curves. Since every elliptic curve has an involution, M 1 , 1 is a Z/ 2 Z-gerbe. Physical reasoning has lead to a decomposition conjecture [4]: the Gromov-Witten theory of an ineffective orbifold decomposes as multiple copies of the Gromov-Witten theory of certain effective spaces produced from the orbifold.
In [5] I prove:
Theorem 1. Suppose that X is a 1-dimensional compact toric stack - that is, a non-effective orbifold P 1 with two singular points, such that all isotropy groups are abelian. Then, the Gromov-Witten theory of X satisfies multiple commuting copies of the 2-Toda hierarchy.
This result provides the first full verification of the decomposition conjecture. Through other methods, Milanov and Tseng had proven the effective version of Theorem 1 previously [8]. In addition to handling the interesting ineffective case, a major advantage of my approach is that it provides the preliminaries necessary to extend the degeneration method of [11] to prove the Virasoro conjecture for all effective orbifold curves [6]. Another future direction for this program is to consider more general non-effective orbifold structures, which is where the decomposition conjecture becomes
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terms of graphs with weights on the edges, and that this count of tropical Hurwitz numbers agrees with the count of usual Hurwitz numbers.
We are in the process of writing up results we have obtained as an application of our graph description of double Hurwitz numbers [2]. A key motivation for studying double Hurwitz numbers is the paper of Goulden, Jackson and Vakil [3]. They begin by showing that double Hurwitz numbers are piecewise polynomial; that is, if we fix integers m, n, the functions of m + n variables Hg(μ 1 ,... , μm; ν 1 ,... , νn) are piecewise polynomial, with the walls of the chambers of polynomial given by those values of μ and ν that could produce a possibly disconnected cover. When m = 1, the cover is necessarily connected, and hence the one part double Hurwitz numbers Hg(|ν|; ν 1 ,... , νn) are polynomial, with a particularly nice form. With this as a large part of the motivation, they conjecture that there is an ELSV type formula for the one part double Hurwitz numbers, where the moduli space of curves is replaced with a universal picard scheme; in genus 0 and 1 they identify this picard scheme by hand and verify their conjecture.
Our graph description recovers their piecewise polynomiality result in a more straightforward manner, and extends it in several directions. Perhaps most exciting, we see that all double Hurwitz numbers with fixed m + n can be viewed as one piecewise polynomial function as follows: if μm becomes negative, drop it from μ and add it to ν, changing from (m, n) Hurwtiz numbers to (m − 1 , n + 1) Hurwitz numbers. This itself could have been done before, but our methods reveal that this wall crossing is essentially no different in kind from the other wall crossings. In particular, while one part double Hurwitz numbers were special from the viewpoint of [3], in our point of view they are are no more special than other double Hurwitz numbers and are intimately related to them. An interesting question in light of this result is whether the conjectural ELSV-type formula can be extended to all double Hurwitz numbers, with the moduli spaces involved undergoing some sort of birational transformation when walls are crossed; a first step would be to investigate this in genus 0.
Another application of our result is to find wall crossing formulas for these polynomials; that is, to describe how the polynomials change when a wall is crossed. In [?], an explicit formula for these wall crossing formula in terms of simpler double Hurwitz numbers was found in case g = 0. Our graph description provides a much simpler proof, that with work extends to higher genus, which the methods of [13] cannot address.
Finally, we have made preliminary investigations into the application of tropical geometry to Hurwitz numbers with three or more arbitrary permutations, and hope that the technique will prove as fruitful there as it has for double Hurwitz numbers.
- Future Directions
Although my research so far has dealt with dimension 1 targets, future directions will look at higher dimensional orbifolds. An important first direction is formulas for higher dimensional Hurwitz-Hodge integrals. An important case is three dimensions, where results such as the Gopakumar-Marino-Vafa formula have culminated in the topological and equivariant vertex, leading to proofs in the toric case of the Gromov-Witten/Donaldson-Thomas equivalence.
Plans for future work involve extending some of these results to Hurwitz-Hodge integrals. Many of these identities are related to the physical notion of open-Gromov-Witten theory, and with R. Cavalieri we have investigated the very first steps of seeing this in the orbifold case. Furthermore, in part of an effort of Bryan, Pandharipande and collaborators to state an orbifold version of the GW/DT correspondence, I have helped calculate the GW invariants of some simple toric 3-fold stacks; to continue our effort, the next step would be to extend Okounkov and Pandharipande’s proof of the GMV formula to the orbifold case. Most of these results are in terms of symmetric functions, and the proofs tend to be rather indirect; a long term hope would be to have a more direct understanding of these in terms of Costello’s
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thesis, which shows that the genus g GW theory of X is expressible in terms of the genus 0 theory of an appropriate symmetric product of X.
References
[1] Cavalieri, R.; Johnson, P.; Markwig, H. Tropical Hurwitz Numbers, arXiv:0804. [2] Cavalieri, R.; Johnson, P.; Markwig, H. Piecewise Polynomiality of Double Hurwitz numbers, In preparation. [3] Goulden, I. P.; Jackson, D. M.; Vakil, R. Towards the geometry of double Hurwitz numbers, Adv. Math. 198 (2005), no. 1, 43–92. [4] Hellerman, S.; Henriques, A.: Pantev, T.; Sharpe, E.;Ando, M. Cluster decomposition, T-duality, and gerby CFT’s hep-th/ [5] Paul Johnson Equivariant Gromov-Witten theory of P^1 stacks, wreath products, and integrable hierarchies, in prepara- tion. [6] Paul Johnson The Virasoro Conjecture for Orbifold Curves, In preparation. [7] Johnson, P.; Pandharipande, R.; Tseng, H. Abelain Hurwitz Hodge Integrals, arXiv:0803. [8] Todor Milanov and Hsian-Hua Tseng, Equivariant orbifold sturctures on the projective line and integrable hierarchies, arXiv:0707.3172. [9] Andrei Okounkov and Rahul Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles Ann. of Math. 163 (2006), no. 2, 517-560. [10] Andrei Okounkov and Rahul Pandharipande, The equivariant Gromov-Witten theory of P^1 Ann. of Math. 163 (2006), no. 2, 561-605. [11] Andrei Okounkov and Rahul Pandharipande, Virasoro Constraints for target curves Invent. Math. 163 (2006) no. 1 47-81. [12] Paolo Rossi Gromov-Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of Seifert fibrations, arXiv:0808. [13] Shadrin, S.; Shapiro, M.; Vainshtein, A. Chamber behavior of double Hurwitz numbers in genus 0 Adv. Math. 217 (2008), no. 1, 79–96. E-mail address: [email protected]
