The Algebraic Structure of Topological Field Theory

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9704320 Getzler In this project the investigator is to clarify the algebraic structures underlying certain recent discoveries in theoretical physics including the two-dimensional topological sigma model, topological gravity, and Gromov-Witten invariants. In particular, a structure theory for Gromov-Witten invariants is to be derived, and Gromov-Witten invariants of higher genus are to be computed. In mathematical terms, these invariants relate to various enumerative questions in algebraic geometry. For example, Gromov-Witten invariants of genus zero appear in the so called mirror symmetry phenomenon. Gromov-Witten invariants originally arose in theoretical physics - they describe a coupling of topological gravity with a model of conformal field theory. They play an important role in the popular 10-dimensional string model of the universe. The string theory model contains six hidden dimensions which form a tightly curved space known as Calabi-Yau manifold; Gromov-Witten invariants can be used to study various mathematical properties of Calabi-Yau manifolds.