symplectic field theory

symplectic field theory
Name	Symplectic field theory
Introduced	1990s
Main subjects	Symplectic topology, Contact geometry, Pseudoholomorphic curves

symplectic field theory Symplectic field theory is a program in mathematics introduced to unify techniques in symplectic topology, contact geometry, and low-dimensional topology by encoding counts of holomorphic curves in algebraic structures. It was formulated to connect methods from Morse theory, Floer homology, and the theory of Gromov–Witten invariants while drawing inspiration from ideas in quantum field theory and the Atiyah–Singer index theorem. The framework generates invariants for contact manifolds, symplectic cobordisms, and related objects, linking developments from many prominent figures and institutions.

History and Motivation

The motivation for the subject arose in the late 1990s amid developments by researchers at institutions such as Institute for Advanced Study, Princeton University, and ETH Zurich, building on prior breakthroughs including Gromov's theorem, Floer homology by Andreas Floer, and work on Seiberg–Witten invariants by Clifford Taubes. Early expositions placed emphasis on reconciling analytic foundations from Mikhail Gromov and algebraic formalisms connected to ideas from Edward Witten and the Donaldson–Thomas theory community. Subsequent workshops at venues like Mathematical Sciences Research Institute and conferences such as the International Congress of Mathematicians propagated techniques linking to the programs of John Milnor, Raoul Bott, and researchers associated with Princeton University and Stanford University.

Foundations and Definitions

Foundational aspects rest on analytical results pioneered in the work of Gromov (1985), the compactness frameworks of Helmut Hofer, and transversality techniques developed by groups at University of California, Berkeley and Cornell University. Central definitions use moduli spaces of pseudoholomorphic curves in symplectizations of contact manifolds and require virtual perturbation schemes influenced by work at Institute for Advanced Study and developments from Kuranishi structure proponents around Kenji Fukaya and collaborators. Fredholm theory, inspired by applications of the Atiyah–Singer index theorem and operators studied by S.-T. Yau-associated groups, provides index calculations that determine grading and expected dimensions for moduli spaces. The construction demands coherent orientations, gluing analyses akin to techniques from the Nash–Moser theorem literature, and compactness results echoing the contributions of Yasha Eliashberg and Maxim Kontsevich.

Algebraic Structures and Invariants

Algebraic outputs include rich structures such as differential graded algebras influenced by constructions in Morse theory and A∞-algebras familiar from research at Institut des Hautes Études Scientifiques, University of Oxford, and Harvard University. Invariants are organized into objects comparable to Gromov–Witten invariants and Floer homology groups, and relate to algebraic formalisms used by researchers in the Kontsevich–Soibelman circle. Operations mirror structures from quantum cohomology studied by groups at California Institute of Technology and University of Cambridge, and algebraic dualities reminiscent of work by Alexander Grothendieck and Pierre Deligne inform the behavior of pair-of-pants products and coproducts. Cohomological field theories appearing in computations connect to themes from Mirror symmetry programs at IHÉS and collaborations involving Maxim Kontsevich and Alexander Givental.

Computational Techniques and Examples

Computational techniques adapt tools from Floer homology computations at Imperial College London and explicit curve counts informed by classical examples such as the standard contact structure on S^3 and symplectic fillings related to Seifert fibered spaces and links explored by researchers at Columbia University and University of Michigan. Techniques include combinatorial approaches seen in work on Legendrian knot invariants influenced by research groups at Brown University and analytic gluing methods used in studies of symplectic cobordisms from teams at University of Chicago and Yale University. Examples computed in the literature involve contact homology for boundaries of Stein manifolds, computations influenced by contributions from Stein theory investigators and calculations analogous to those in Heegaard Floer homology research led by groups at Rutgers University and University of Texas at Austin.

Relations to Other Theories

The subject is deeply related to Gromov–Witten theory, Floer homology, and Contact homology traditions, and has conceptual ties to Seiberg–Witten theory as developed by researchers at Columbia University and McMaster University. Connections to Mirror symmetry link participants from Institute for Advanced Study and Princeton University communities, while formal similarities with algebraic structures from Topological quantum field theory research influenced by Michael Atiyah and Graeme Segal have been noted. The interplay with Legendrian knot theory and low-dimensional topology relates to work by scholars connected to University of Oxford and University of California, San Diego.

Applications and Open Problems

Applications span the classification of contact structures on manifolds studied in programs at University of California, Los Angeles and obstruction results for symplectic cobordisms that echo conjectures discussed at Mathematical Sciences Research Institute. Open problems include establishing fully rigorous virtual perturbation frameworks advocated by researchers at ETH Zurich and proving comprehensive functoriality properties reminiscent of goals set by communities around Princeton University and Stanford University. Other active directions involve computationally accessible invariants for higher-dimensional manifolds, formulating categorical extensions akin to ideas in Homological mirror symmetry and resolving conjectures inspired by symplectic geometers affiliated with Harvard University and University of Cambridge.

Category:Symplectic topology