Seiberg-Witten and Gromov invariants of symplectic 4-orbifolds and some applications

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DMS-0304956Weimin ChenThe Principal Investigator plans to continue his work in gaugetheory and symplectic, low-dimensional topology. More concretely,he plans to study the following problems: (1) general properties offinite groups of symplectomorphisms of a symplectic 4-manifold,(2) whether a symplectic circle action on a 6-dimensional manifoldis Hamiltonian if its fixed-point set is non-empty and finite, and(3) general, global restrictions on the occurrence of quotient singularitiesin an algebraic surface. A unifying theme in these studies is a theory,yet to be developed in this project, on the Seiberg-Witten and Gromovinvariants of a symplectic 4-orbifold, which will be built on andsubstantially extend the related fundamental work of Taubes onsymplectic 4-manifolds.A 4-dimensional orbifold is a space which locally looks like the space-time we all live in, except that there is a certain degree of ambiguity caused bya finite symmetry occured locally. Many examples of this kind of space can befound easily and naturally in Mathematics (e.g. topology and geometry) as wellas models in Theoretical Physics (e.g. orbifold string theory). The proposedresearch seeks to understand a fundamental duality in a 4-dimensional orbifoldbetween a certain type of fields that are distributed all over the space and a certaintype of worldsheet that is wiped out by a collection of strings in the space whichis condensed in a 2-dimensional subspace. When there is no such ambiguity of finite symmetries, such a duality is well-understood, and is a fundamental piece ofmathematics.