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Chris T. Woodward

Publications and source records attributed to Chris T. Woodward.

15 recordsLinked to original sources
arXiv2026

Augmentation varieties and disk potentials III

This is the third in a series of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we prove that for connected Legendrian covers of monotone Lagrangian tori, the augmentation variety is equal to the image of the zero level set of the disk potential, as suggested by Dimitroglou-Rizell-Golovko. In particular, we show that Legendrian lifts of Vianna's exotic tori are not Legendrian isotopic. Using related ideas, we show that the Legendrian lift of the Clifford torus admits no exact fillings, extending results of Dimitroglou-Rizell and Treumann-Zaslow in dimension two. We consider certain disconnected Legendrians, and show, similar to another suggestion of Aganagic-Ekholm-Ng-Vafa that the components of the augmentation variety correspond to certain partitions and each component is defined by a (not necessarily exact) Lagrangian filling.

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arXiv2026

Holomorphic disks and tropical Lagrangians

We develop a calculus for counting pseudoholomorphic disks with boundary in tropical Lagrangians contained in almost toric manifolds, using our previous work with Venugopalan. The results are mostly in dimension four under monotonicity assumptions although in principle the same technique works in any dimension and without monotonicity. The calculus is given as a sum over tropical graphs that interact with the tropical graph of the Lagrangian, generalizing results of Mikhalkin and Nishinou-Siebert for holomorphic spheres in toric varieties, and our previous result with Venugopalan which dealt with disks bounding almost toric moment fibers. The main contribution of this paper is the calculation of several multiplicities of vertices corresponding to disks, such as the holomorphic pant (half of the holomorphic pair of pants) and various univalent vertices occuring at trivalent vertices of the graph of the Lagrangian; a key tool is a Lagrangian isotopy from the Lagrangian pair of pants in the del Pezzo of degree seven to the inverse image of a diagonal, which is a special case of a results of Hind and Evans. We show that every integer eigenvalue of non-maximal modulus for quantum multiplication by the first Chern class is realized by such a sphere.

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arXiv2026

Augmentation varieties and disk potentials II

This is the second in a sequence of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we first define the Chekanov-Eliashberg algebra and its Legendrian contact homology. For a tame Lagrangian cobordism between Legendrians, we define a chain map between their Chekanov-Eliashberg algebras.

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arXiv2026

Augmentation varieties and disk potentials I

This is the first in a sequence of papers where we show that Lagrangian fillings such as the Harvey-Lawson filling in any dimension define augmentations of Chekanov-Eliashberg differential graded algebras by counting configurations of holomorphic disks connected by gradient trajectories, as in Aganagic-Ekholm-Ng-Vafa; we also prove that for Legendrian lifts of monotone tori, the augmentation variety is the zero level set of the Landau-Ginzburg potential of the Lagrangian projection, as suggested by Dimitroglou-Rizell-Golovko. In this part, we set up the foundations of moduli spaces of pseudoholomorphic buildings.

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arXiv2023

Fukaya categories of blowups

We compute the Fukaya category of the symplectic blowup of a compact rational symplectic manifold at a point in the following sense: Suppose a collection of Lagrangian branes satisfy Abouzaid's criterion for split-generation of a bulk-deformed Fukaya category of cleanly-intersecting Lagrangian branes. We show that for a small blow-up parameter, their inverse images in the blowup together with a collection of branes near the exceptional locus split-generate the Fukaya category of the blowup. This categorifies a result on quantum cohomology by Bayer and is an example of a more general conjectural description of the behavior of the Fukaya category under transitions occuring in the minimal model program, namely that mmp transitions generate additional summands.

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arXiv2023

A wall-crossing formula for Gromov-Witten invariants under variation of git quotient

We prove a quantum version of Kalkman's wall-crossing formula comparing Gromov-Witten invariants on geometric invariant theory (git) quotients related by a change in polarization. The wall-crossing terms are gauged Gromov-Witten invariants with smaller structure group. As an application, we show that the graph Gromov-Witten potentials of quotients related by wall-crossings of crepant type are equivalent up to a distribution in the quantum parameter that is almost everywhere zero. This is a version of the crepant transformation conjecture of Li-Ruan, Bryan-Graber, Coates-Ruan etc. in cases where the crepant transformation is obtained by variation of git.

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arXiv2022

Floer theory and flips

We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. As applications, we demonstrate the existence of Hamiltonian non-displaceable Lagrangian tori in, for example, small symplectic blow-ups of compact symplectic manifolds and moduli spaces of polygons. These results are part of a conjectural description of generators for the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program.

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arXiv2017

Quantum Kirwan morphism and Gromov-Witten invariants of quotients I

This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology $QH_G(X)$ of a smooth complex projective variety X with the action of a connected complex reductive group $G$ to the orbifold quantum cohomology $QH(X//G)$ of its geometric invariant theory quotient $X//G$, and prove that it intertwines the genus zero gauged Gromov-Witten potential of X with the genus zero Gromov-Witten graph potential of $X//G$.

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arXiv2017

Quantum Kirwan morphism and Gromov-Witten invariants of quotients II

This is the second in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. In this part we construct virtual fundamental classes on the moduli spaces used in the construction of the quantum Kirwan map and the gauged Gromov-Witten potential.

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arXiv2017

Quantum Kirwan morphism and Gromov-Witten invariants of quotients III

This is the third in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. We also give a formula for a solution to the quantum differential equation in terms of a localized gauged potential. These results overlap with those of Givental, Lian-Liu-Yau, Coates-Corti-Iritani-Tseng and Ciocan-Fontanine-Kim.

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arXiv2011

Quilted Floer trajectories with constant components

We fill a gap in the proof of the transversality result for quilted Floer trajectories in arXiv:0905.1370 by addressing trajectories for which some but not all components are constant. Namely we show that for generic sets of split Hamiltonian perturbations and split almost complex structures, the moduli spaces of parametrized quilted Floer trajectories of a given index are smooth of expected dimension. An additional benefit of the generic split Hamiltonian perturbations is that they perturb the given cyclic Lagrangian correspondence such that any geometric composition of its factors is transverse and hence immersed.

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arXiv2010

Functoriality for Lagrangian correspondences in Floer theory

Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric composition in the case that the composition is smooth and embedded. As a consequence we obtain 'categorification commutes with composition' for Lagrangian correspondences.

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arXiv2005

Localization for the norm-square of the moment map and the two-dimensional Yang-Mills integral

The first seven sections of the paper contain a version of localization for the norm-square of the moment map in equivariant de Rham theory, similar to that proved by P.-E. Paradan. The last section contains a definition and computation of the Yang-Mills path integral in two dimensions. The idea is to reverse the logic in Witten's paper and take the localization formula as the definition of the path integral. Using a symmetry argument we show that the path integral is given by the Migdal formula.

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