Open string topology and holomorphic curves

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

The Principal Investigator plans to study algebraic structures that arise naturally from invariants of Legendrian submanifolds in contact manifolds as well as arbitrary submanifolds in smooth manifolds. More specifically, the invariants for a Legendrian submanifold come from holomorphic curves in certain symplectic manifolds, while the invariants for a smooth submanifold arise from open string topology which is based on the intersection theory of the manifold's path space. Both theories have algebraic structures guided by topological field theory. Part of this project is to refine the language of differential graded operad algebras to express the two theories in the similar algebraic language. There is a growing body of evidence that suggests a connection between the string topology of a manifold and the holomorphic invariants for its lift in the symplectic cotangent bundle or contact unit cotangent bundle. The Principal Investigator plans to define and compute some of these invariants, hopefully leading to a connection between the two theories when the smooth submanifold is a knot in Euclidean 3-space. The Principal Investigator also proposes several more computational projects related to knots in 3-space and smooth surfaces in 4-space, also defined using holomorphic curves. Contact geometry makes many appearances in physics, from optics to thermodynamics to classical mechanics. For example, particles obeying the Least Action Principal from mechanics translate into objects in contact geometry (or its closely related field, symplectic geometry) known as holomorphic curves. Studying these holomorphic curves have led to some powerful and sometimes surprising discoveries about contact rigidity and contact dynamics. Knot theory has applications in understanding large and small aspects of the universe, as well as long DNA strands confined to small space. A central question in knot theory is determining the complexity of knots which in turn requires developing computable non-trivial knot invariants. Again holomorphic curves has recently provided a plethora of such useful knot invariants. The Principal Investigator plans to develop other knot invariants, motivated by string theory, that should be connected to these invariants based on holomorphic curves.