On Algebraic Structures Implicit in Topological Quantum Field Theories

In the course of the development of our understanding of topological quantum field theory (TQFT) [1,2], it has emerged that the structures of generators and relations for the construction of low dimensional TQFTs by various combinatorial methods are equivalent to the structures of various fundamental objects in abstract algebra. Thus, 2D-TQFTs can be constructed from commutative Frobenius algebras [3] or from semisimple associative algebras [4]; while 3D theories can be constructed either from nicely behaved braided monoidal categories [5,6 7,8,9] or from Hopf algebras [10,11]. In [12], a possible method was proposed to extend this picture to D=4. Namely, it was shown how to construct a 4D TQFT from a new type of algebraic structure called a Hopf category. The purpose of this paper is to show that under physically reasonable hypotheses, the seemingly exotic algebraic structures used in the constructions above arise naturally from 3D and 4D TQFT’s. We shall show that any 3D-TQFT with a property which we call factorizability, which any TQFT which came from a path integral with a topological lagrangian would be expected to satisfy, contains a braided monoidal category Hopf in its structure, and that this category arises from a generalized Hopf algebra by a construction first proposed by Yetter. We shall show moreover that any factorizable 4D-TQFT gives rise to a Hopf category object in a certain reasonably concrete bicategory. This theorem lends weight to the conjecture in [12] that the 4D-TQFT which is believed to be constructable from Donaldson-Floer theory is related to the Hopf category constructed in [12] from the canonical basis of a quantum group [13]. The importance of the procedure we outline in this paper is greatly increased by the recent breakthrough in the understanding of Donaldson-Floer theory made by Witten [14]. The pair of differential equations whose solutions are related to DF theory in this new approach is much more tractible than the self duality equation [15]. In particular, there is a well behaved version of them on manifolds with boundary. Thus, we can take the geometric constructions in this paper as a prescription: consider the space of solutions of Witten’s equations on the manifolds with boundaries or corners we are examining, there must then appear certain algebraic operations on them, from which the TQFT can be reconstructed. The contents of this paper are as follows: Section 2 gives the definition of a 3D-TQFT with factorizability and physical motivation for the definition. In Section 3, we prove that every 3DTQFT with factorizability contains a Hopf algebra object, and show the relation between this object and the category used to define factorization. Section 4 recapitulates the definition of a Hopf category. In Section 5, we explain the extension of the definition of factorizable TQFT to D=4, and prove of the main theorem in 4D. Finally, in Section 6, we outline some extensions of our argument and suggest directions for further work. Throughout the paper all manifolds (with or without boundary or corners) are compact and oriented.