arXiv subjects

Dustin Ross

Publications and source records attributed to Dustin Ross.

26 records · Page 2Linked to original sources

arXiv2015

Sigma Models and Phase Transitions for Complete Intersections

We study a one-parameter family of gauged linear sigma models (GLSMs) naturally associated to a complete intersection in weighted projective space. In the positive phase of the family we recover Gromov-Witten theory of the complete intersection, while in the negative phase we obtain a Landau--Ginzburg-type theory. Focusing on the negative phase, we develop foundational properties which allow us to state and prove a genus-zero comparison theorem that generalizes the multiple log-canonical correspondence and should be viewed as analogous to quantum Serre duality in the positive phase. Using this comparison result, along with the crepant transformation conjecture and quantum Serre duality, we prove a genus-zero correspondence between the GLSMs which arise at the two phases, thereby generalizing the Landau-Ginzburg/Calabi-Yau correspondence to complete intersections.

math.AG↗

arXiv2015

The Loop Murnaghan-Nakayama Rule

We give a combinatorial proof of a natural generalization of the Murnaghan-Nakayama rule to loop Schur functions. We also define shifted loop Schur functions and prove that they satisfy a similar relation.

math.CO↗

arXiv2015

Wall-Crossing in Genus Zero Landau-Ginzburg Theory

We study genus zero wall-crossing for a family of moduli spaces introduced recently by Fan-Farvis-Ruan. The family has a wall and chamber structure relative to a positive rational parameter. For a Fermat quasi-homogeneous polynomial W (not necessarily Calabi-Yau type), we study natural generating functions of invariants associated to these moduli spaces. Our wall-crossing formula relates the generating functions by showing that they all lie on the same Lagrangian cone associated to the Fan-Jarvis-Ruan-Witten theory of W. For arbitrarily small parameter, a specialization of our generating function is a hypergeometric series called the big I-function which determines the entire Lagrangian cone. As a special case of our wall-crossing, we obtain a new geometric interpretation of the Landau-Ginzburg mirror theorem.

math.AG↗

arXiv2015

The Gerby Gopakumar-Mariño-Vafa Formula

We prove a formula for certain cubic $\Z_n$-Hodge integrals in terms of loop Schur functions. We use this identity to prove the Gromov-Witten/Donaldson-Thomas correspondence for local $\Z_n$-gerbes over $\proj^1$.

math.AG↗

arXiv2014

The open string McKay correspondence for type A singularities

We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Calabi-Yau orbifolds by viewing the open theories as sections of Givental's symplectic vector space and the correspondence as a linear map of Givental spaces which identifies them. We deduce a Bryan-Graber-type statement for disk invariants and extend it to arbitrary genus zero topologies in the Hard Lefschetz case. Upon leveraging Iritani's theory of integral structures to equivariant quantum cohomology, we conjecture a general form of the symplectomorphism entering the OCRC which arises from a geometric correspondence at the equivariant K-theory level. We give a complete proof of this in the case of minimal resolutions of threefold A_n singularities. Our methods rely on a new description of the equivariant quantum D-modules underlying the Gromov-Witten theory of this class of targets.

math.AG↗

arXiv2014

Cyclic Hodge Integrals and Loop Schur Functions

We conjecture an evaluation of three-partition cyclic Hodge integrals in terms of loop Schur functions. Our formula implies the orbifold Gromov-Witten/Donaldson-Thomas correspondence for toric Calabi-Yau threefolds with transverse type A singularities. We prove the formula in the case where one of the partitions is empty, and thus establish the orbifold Gromov- Witten/Donaldson-Thomas correspondence for local toric surfaces with transverse type A singularities.

math.AG↗

arXiv2012

Localization and Gluing of Orbifold Amplitudes: The Gromov-Witten Orbifold Vertex

We define a formalism for computing open orbifold GW invariants of [C^3/G] where G is any finite abelian group. We prove that this formalism and a suitable gluing algorithm can be used to compute GW invariants in all genera of any toric CY orbifold of dimension 3. We conjecture a correspondence with the DT orbifold vertex of Bryan-Cadman-Young.

math.AG↗

arXiv2011

Open Gromov-Witten Theory and the Crepant Resolution Conjecture

We compute open GW invariants for $\mathcal{K}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}$, open orbifold GW invariants for $[\C^3/\Z_2]$, formulate an open crepant resolution conjecture and verify it for this pair. We show that open invariants can be glued together to deduce the Bryan-Graber closed crepant resolution conjecture for the orbifold $[\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)/\Z_2]$ and its crepant resolution $\mathcal{K}_{\mathbb{P}^1\times\mathbb{P}^1}$.

math.AG↗