Gromov–Witten theory

Gromov–Witten theory
Name	Gromov–Witten theory
Field	Gromov-related symplectic and algebraic geometry
Introduced	1990s
Contributors	Mikhail Gromov, Edward Witten, Maxim Kontsevich, Yuri Manin, Dusa McDuff, Dietmar Salamon, Aleksandr Givental, Richard Thomas, Paul Seidel, Katherine Wehrheim

Gromov–Witten theory is a mathematical framework that assigns enumerative invariants to symplectic and algebraic targets via counts of holomorphic curves, arising from interactions among ideas of Mikhail Gromov, Edward Witten, and Maxim Kontsevich. The subject connects techniques from representation theory, enumerative geometry, and gauge theory to produce invariants that are deformation‑invariant under families studied by Dusa McDuff and Dietmar Salamon. It has driven advances linked to the Mirror symmetry conjecture proposed by Philip Candelas, Brian Greene, and Andrew Strominger, and influenced developments in string theory research led by Edward Witten and Cumrun Vafa.

Introduction

Gromov–Witten theory originated from work of Mikhail Gromov on pseudoholomorphic curves, formalized in contexts influenced by Edward Witten and rigorized by researchers such as Maxim Kontsevich and Yuri Manin. The theory constructs invariants using moduli of maps into targets such as Calabi–Yau manifolds, Fano varieties, and K3 surfaces and interfaces with conjectures connecting Mirror symmetry and Hodge theory. Major contributors include Aleksandr Givental, Aleksey Zinger, Jun Li, Vakil, and Richard Thomas, while foundational analytic underpinnings were strengthened by Dusa McDuff and Paul Seidel.

Mathematical Foundations

The analytic roots lie in the study of pseudoholomorphic curves introduced by Mikhail Gromov and developed by Dusa McDuff and Dietmar Salamon within symplectic topology. Algebraic foundations were given by constructions of stable maps due to Maxim Kontsevich and moduli compactifications advanced by Jun Li and Kontsevich collaborators; intersection theory inputs derive from work of Pierre Deligne and Alexander Grothendieck in algebraic geometry. Virtual fundamental class techniques involve contributions from Kai Behrend, Barbara Fantechi, Tom Graber, and Ryan Kiem and are informed by deformation–obstruction theories used by Richard Thomas and Jacob Lurie. Transversality and perturbation frameworks relate to constructions by Fukaya, Kenji Fukaya, Kazushi Ueda, and Katrin Wehrheim, while Gromov compactness builds on analytic estimates reminiscent of work by Sergei Novikov in other contexts.

Moduli Spaces of Stable Maps

Moduli spaces of stable maps, central to the theory, generalize classical moduli such as Deligne–Mumford stacks studied by Pierre Deligne, David Mumford, and John Milnor. Compactifications use stability notions introduced by Maxim Kontsevich and compactifications for targets parallel constructions by Jun Li. Virtual cycles on these spaces depend on obstruction theories formalized by Kai Behrend and Barbara Fantechi and analytic gluing methods developed by Dusa McDuff and Dietmar Salamon. The interplay with tautological classes references work by Carel Faber, Ravi Vakil, and Alexander Zvonkin, while degeneration techniques connect to studies by Mark Gross and Bernd Siebert.

Gromov–Witten Invariants

Gromov–Witten invariants are intersection numbers on moduli spaces defined via evaluation maps and psi classes, building on enumerative frameworks from Yuri Manin and cohomological field theory formalisms by Alexander Givental. The invariants satisfy axioms analogous to those in TQFT articulated by Graeme Segal and obey relations predicted by Mirror symmetry work of Philip Candelas and computed explicitly in examples by Aleksey Zinger and Jun Li. Structures such as quantum cohomology rings were formalized by Alexander Givental and Richard Pandharipande; reconstruction theorems and integrable systems perspectives were developed by E. Witten and Maxim Kontsevich.

Computation and Techniques

Computational techniques exploit localization methods from equivariant cohomology due to Michael Atiyah and Raoul Bott and virtual localization introduced by Tom Graber and Pandharipande. Degeneration formulas owe to Jun Li and relative theories advanced by Eleny Ionel and Thomas Parker. Tropical geometry approaches from Grigory Mikhalkin and combinatorial counts by Bertrand Toën intersect with vertex formalism by Aleksey Okounkov and Nekrasov techniques, while mirror computations use periods studied by Richard Thomas and Paul Seidel. Computational packages and algorithms influenced by Maxim Kontsevich and Andrei Okounkov enable explicit enumerative predictions.

Relations to Other Theories

The theory interfaces with Mirror symmetry, Donaldson–Thomas theory developed by Simon Donaldson and Richard Thomas, and Floer homology pioneered by Andreas Floer and Paul Seidel. Connections to Hodge theory relate to work by Phillip Griffiths and Wilfried Schmid, while links to Integrable systems reflect insights of Boris Dubrovin and Edward Witten. Wall‑crossing phenomena echo results from Maxim Kontsevich and Yuri Manin and relate to stability conditions studied by Tom Bridgeland. Interactions with Quantum field theory trace back to Edward Witten and Cumrun Vafa.

Applications and Examples

Applications include enumerative counts on Calabi–Yau threefolds central to predictions by Philip Candelas, computations on projective spaces driven by Aleksey Givental, and cases on K3 surfaces investigated by Shing-Tung Yau and Richard Thomas. Examples in toric geometry exploit combinatorics from Victor Guillemin and Reid, while relative and orbifold theories connect to work by Ian Morrison and Martin Olsson. The framework underlies advances in Mirror symmetry tests performed by Brian Greene and computational confirmations by Andrei Okounkov, influencing research programs at institutions such as IAS, MSRI, and Clay.

Category:Symplectic topology