arXiv subjects

Jim Bryan

Publications and source records attributed to Jim Bryan.

At least 37 records · Page 2Linked to original sources

arXiv2010

The Orbifold Topological Vertex

We define Donaldson-Thomas invariants of Calabi-Yau orbifolds and we develop a topological vertex formalism for computing them. The basic combinatorial object is the orbifold vertex, a generating function for the number of 3D partitions asymptotic to three given 2D partitions and colored by representations of a finite Abelian group G acting on C^3. In the case where G=Z_n acting on C^3 with transverse A_{n-1} quotient singularities, we give an explicit formula for the vertex in terms of Schur functions. We discuss applications of our formalism to the Donaldson-Thomas Crepant Resolution Conjecture and to the orbifold Donaldson-Thomas/Gromov-Witten correspondence. We also explicitly compute the Donaldson-Thomas partition function for some simple orbifold geometries: the local football and the local BZ_2 gerbe.

math.AG↗

arXiv2009

The Quantum McKay Correspondence for polyhedral singularities

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C^3/G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry of Y in terms of R, an ADE root system associated to G. Namely, we give an explicit formula for the Gromov-Witten partition function of Y as a product over the positive roots of R. In terms of counts of BPS states (Gopakumar-Vafa invariants), our result can be stated as a correspondence: each positive root of R corresponds to one half of a genus zero BPS state. As an application, we use the crepant resolution conjecture to provide a full prediction for the orbifold Gromov-Witten invariants of [C^3/G].

math.AG↗

arXiv2009

BPS invariants for resolutions of polyhedral singularities

We study the BPS invariants of the preferred Calabi-Yau resolution of ADE polyhedral singularities C^3/G given by Nakamura's G-Hilbert schemes. Genus 0 BPS invariants are defined by means of the moduli space of torsion sheaves as proposed by Sheldon Katz. We show that these invariants are equal to half the number of certain positive roots of an ADE root system associated to G. This is in agreement with the prediction via Gromov-Witten theory.

math.AG↗

arXiv2009

On the rigidity of stable maps to Calabi-Yau threefolds

If X is a nonsingular curve in a Calabi--Yau threefold Y whose normal bundle N_{X/Y} is a generic semistable bundle, are the local Gromov-Witten invariants of X well defined? For X of genus two or higher, the issues are subtle. We will formulate a precise line of inquiry and present some results, some positive and some negative.

math.AG↗

arXiv2008

Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov--Reshetikhin--Vafa for the easy case G = Z/n; to handle the considerably more difficult case G=Z/2 x Z/2, we will also use a refinement of the author's recent q--enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold C^3/G. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.

math.CO↗

arXiv2007

Hurwitz-Hodge Integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture

Let G be the group A_4 or Z_2xZ_2. We compute the integral of λ_g on the Hurwitz locus H_G\subset M_g of curves admitting a degree 4 cover of P^1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E_6 and D_4 root systems respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds [C^3/A_4] and [C^3/(Z_2xZ_2)].

math.AG↗

arXiv2007

Root Systems and the Quantum Cohomology of ADE resolutions

We compute the C*-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C^2/G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to non-simply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov-Witten potential of [C^2/G].

math.AG↗

arXiv2007

The Crepant Resolution Conjecture

For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the Gromov-Witten theories of the orbifold and the resolution. We prove the conjecture for the equivariant Gromov-Witten theories of the nth symmetric product of the complex plane and the Hilbert scheme of n points in the plane.

math.AG↗

arXiv2006

The local Gromov-Witten theory of curves

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of P^1. A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a compete and effective solution. The local Gromov-Witten theory of curves is equivalent to the local Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of C^2, and the orbifold quantum cohomology the symmetric product of C^2. The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.

math.AG↗

arXiv2006

Super-rigid Donaldson-Thomas invariants

We solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gromov-Witten/Donaldson-Thomas correspondence for contributions from super-rigid rational curves. In particular, we prove the full GW/DT correspondence for the quintic threefold in degrees one and two.

math.AG↗

arXiv2005

The orbifold quantum cohomology of C^2/Z_3 and Hurwitz-Hodge integrals

Let Z_3 act on C^2 by non-trivial opposite characters. Let X =[C^2/Z_3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials of X and Y are equal after a change of variables -- verifying the Crepant Resolution Conjecture for the pair (X,Y). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of independent interest. In a self contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.

math.AG↗

arXiv2004

Curves in Calabi-Yau 3-folds and Topological Quantum Field Theory

We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau 3-folds. We define relative invariants for the local theory which give rise to a 1+1-dimensional TQFT taking values in the ring Q[[t]]. The associated Frobenius algebra over Q[[t]] is semisimple. Consequently, we obtain a structure result for the local invariants. As an easy consequence of our structure formula, we recover the closed formulas for the local invariants in case either the target genus or the degree equals 1.

math.AG↗

arXiv2003

The closed topological vertex via the Cremona transform

We compute the local Gromov-Witten invariants of the "closed vertex", that is, a configuration of three rational curves meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of CP^3 and then to compute those invariants via the geometry of the Cremona transformation.

math.AG↗

arXiv2002

BPS states of curves in Calabi--Yau 3--folds

The Gopakumar-Vafa conjecture is defined and studied for the local geometry of a curve in a Calabi-Yau 3-fold. The integrality predicted in Gromov-Witten theory by the Gopakumar-Vafa BPS count is verified in a natural series of cases in this local geometry. The method involves Gromov-Witten computations, Mobius inversion, and a combinatorial analysis of the numbers of etale covers of a curve.

math.AG↗

arXiv2002

Surface bundles over surfaces of small genus

We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby's list of problems in low-dimensional topology. Namely, we show that 2 is the smallest possible base genus that can occur in a 4-manifold of non-zero signature which is an oriented fiber bundle over a Riemann surface. A refined version of the problem asks for the minimal base genus for fixed signature and fiber genus. Our constructions also provide new (asymptotic) upper bounds for these numbers.

math.AG↗

arXiv2000

Evidence for a conjecture of Pandharipande

In his paper "Hodge integrals and degenerate contributions", Pandharipande studied the relationship between the enumerative geometry of certain 3-folds and the Gromov-Witten invariants. In some good cases, enumerative invariants (which are manifestly integers) can be expressed as a rational combination of Gromov-Witten invariants. Pandharipande speculated that the same combination of invariants should yield integers even when they do not have any enumerative significance on the 3-fold. In the case when the 3-fold is the product of a complex surface and an elliptic curve, Pandharipande has computed this combination of invariants on the 3-fold in terms of the Gromov-Witten invariants of the surface. This computation yields surprising conjectural predictions about the genus 0 and genus 1 Gromov-Witten invariants of complex surfaces. The conjecture states that certain rational combinations of the genus 0 and genus 1 Gromov-Witten invariants are always integers. Since the Gromov-Witten invariants for surfaces are often enumerative (as oppose to 3-folds), this conjecture can often also be interpreted as giving certain congruence relations among the various enumerative invariants of a surface. In this note, we state Pandharipande's conjecture and we prove it for an infinite series of classes in the case of the projective plane blown-up at 9 points. In this case, we find generating functions for the numbers appearing in the conjecture in terms of quasi-modular forms. We then prove the integrality of the numbers by proving a certain a congruence property of modular forms that is reminiscent of Ramanujan's mod 5 congruences of the partition function.

math.AG↗

arXiv2000

G-bundles on Abelian surfaces, hyperkahler manifolds, and stringy Hodge numbers

We study the moduli space M(G,A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, M(G,A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable G_C-bundles with trivial Chern classes where G_C is the complexified group. M(G,A) has the structure of a hyperkahler orbifold. We show that when G is Sp(n) or SU(n), M(G,A) has a natural hyperkahler desingularization which we exhibit as a moduli space of G_C-bundles with an altered stability condition. In this way, we obtain the two known families of hyperkahler manifolds, the Hilbert scheme of points on a K3 surface and the generalized Kummer varieties. We show that for G not Sp(n) or SU(n), the moduli space M(G,A) does not admit a hyperkahler resolution, in fact, it does not have a crepant resolution. Inspired by the physicists Vafa and Zaslow, Batyrev and Dais define ``stringy Hodge numbers'' for certain orbifolds. These numbers are conjectured to agree with the Hodge numbers of a crepant resolution (when it exists). We compute the stringy Hodge numbers of M(SU(n),A) and M(Sp(n),A) and verify the conjecture in these cases.

math.AG↗

arXiv1999

Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds

We study the contribution of multiple covers of an irreducible rational curve C in a Calabi-Yau threefold Y to the genus 0 Gromov-Witten invariants in the following cases. (1) If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multiple covers of degree d is given by the sum of all 1/n^3 where n divides d. (2) For a smoothly embedded contractable curve C in Y we define schemes C_i for i=1,...,l where C_i is supported on C and has multiplicity i, and the integer l (0 l). In the latter case we also get a formula for arbitrary genus. These results show that the curve C contributes an integer amount to the so-called instanton numbers that are defined recursively in terms of the Gromov-Witten invariants and are conjectured to be integers.

math.AG↗