Topics in symplectic Gromov–Witten theory

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Hirschi, Amanda 

The main focus of this thesis is on the Gromov--Witten theory of general symplectic manifolds. Mohan Swaminathan and I construct a framework to define a virtual fundamental class for the moduli space of stable maps to a general closed symplectic manifold. Our construction, inspired by [AMS21], works for all genera and leads to a more straightfoward definition of symplectic Gromov--Witten invariants as was previously available. We prove a formula for the Gromov--Witten invariants of a product of two symplectic manifolds, conjectured in [KM94].

I generalise the product formula to a formula for the Gromov--Witten invariants of a suitable fibre product of symplectic manifolds. Our invariants satisfy the Kontsevich-Manin axioms and are extended to descendent Gromov--Witten invariants. I show that our definition of Gromov--Witten invariants agrees with the classical Gromov--Witten invariants defined by [RT97] for semipositive symplectic manifolds.

Given a Hamiltonian group action on the target manifold, I construct equivariant Gromov--Witten invariants and prove a virtual Atiyah--Bott-type localisation formula, providing a tool for computations.

Together with Soham Chanda and Luya Wang, I construct infinitely many exotic Lagrangian tori in complex projective spaces of complex dimension higher than 2. We lift tori in P2, constructed by Vianna, and show that these lifts remain non-symplectomorphic, using an invariant derived from pseudoholomorphic disks.

Noah Porcelli and I use Ljusternik-Schnirelmann theory, applied to moduli spaces of pseudoholomorphic curves, and homotopy theory to prove lower bounds on the number of intersection points of two (possibly non-transverse) Lagrangians in terms of the cuplength of the Lagrangian in generalised cohomology theories, improving previous lower bounds by Hofer.

Keating, Ailsa
Gromov-Witten theory, symplectic
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge
Engineering and Physical Sciences Research Council (2434378)