Homotopy Quantum Field Theory

Indiana University, Bloomington IN

Investigators

Abstract

Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory dealing with manifolds and cobordisms endowed with maps to a given target space. The aim of the project is to establish foundations of 3-dimensional HQFT in the case where the target space is the Eilenberg-MacLane space of type K(G,1) where G is a discrete group (finite or infinite). This should generalize the known results of the PI with N. Reshetikhin, O. Viro, and A. Virelizier. The project will specifically address the following problems:(1) produce a state sum construction of 3-dimensional HQFTs from spherical G-fusion G-categories; (2) introduce an appropriate notion of a G-modular G-category and give a surgery construction of a 3-dimensional HQFT from such a category. Solutions to these problems will include definitions of the required classes of G-categories and of the corresponding HQFTs. Finally, the PI will establish a fundamental relation between the constructions (1) and (2) via a G-version of the Drinfeld-Joyal-Street center of categories: the HQFT obtained by construction (1) from a spherical G-fusion G-category C is equivalent to the HQFT obtained by construction (2) from the appropriately defined G-center of C. The project develops new techniques in geometry arising from ideas of quantum physics. Specifically, the project will focus on the notion of Topological Field Theory introduced by the Fields medalist Edward Witten in the 1980's. Topological Field Theory allows topologists to analyze the shape of geometric objects from the microscopic viewpoint decomposing them into small elementary pieces. Mathematical formulation of Topological Field Theory given by the PI and co-authors has successfully led to creating bridges between previously unrelated areas of pure mathematics like the geometric theory of knotted strings in space on one hand, and the algebraic theory of representations and categories on the other hand. The project considerably enlarges the class of geometric objects which can be analyzed using Topological Field Theory. We include in consideration not only the objects themselves but also relations between them known in mathematics as maps or mappings. The principal aim of the project is to analyze the maps from the above mentioned microscopic viewpoint and to derive corresponding notions in algebra and theory of categories. The project will introduce and develop two different solutions to this problem and establish a subtle theorem relating these two solutions. This project will lead to a better understanding of geometric objects and their maps. It will also produce new powerful algebraic notions and techniques. Potential areas of applications outside of pure mathematics include theoretical physics and quantum computations.