Homotopy Quantum Field Theories and the Homotopy Cobordism Category in Dimension 1 + 1

Abstract

We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor from this category into a category of linear spaces. We then derive some very general properties of HCobord(n, X), including the fact that it only depends on the (n + 1)-homotopy type of X. We also prove that an HQFT with target space X and in dimension n + 1 implies the existence of geometrical structures in X; in particular, flat gerbes make their appearance. We give a complete characterization of HCobord(n, X) for n = 1 (or the 1 + 1 case) and X the Eilenberg-Maclane space K(G, 2). In the final section we derive state sum models for these HQFT's.

TQFT’s and gerbes

“…The manifolds involved in the TQFT now come equipped with maps to a target manifold on which the bundle or c Geometry & Topology Publications gerbe lives. This brings us into the realm of Segal's category C [23] and string connection [22], as well as Turaev's Homotopy Quantum Field Theory (HQFT) [24] and a related construction by Brightwell and Turner [8], as well as subsequent developments by Rodrigues [19], Bunke, Turner and Willerton [10] and Turner [25]. Here we will base our approach on a general framework for TQFT and related constructions by Semião and the author [18], which defines a TQFT to be a certain type of monoidal functor, without using the cobordism approach.…”

Abelian homotopy Dijkgraaf-Witten theory

“…This is, in fact, the motivating example and is a formulation of the "extended action" found in Freed and Quinn's work on Chern-Simons theory for finite gauge group [7]. HQFTs were defined by Turaev in [17] (and in a special case in [2] and further discussion of the connection between the two can be found in [14]). …”

Homotopy Quantum Field Theories and Related Ideas

“…By omitting this (and thus giving a role to higher homotopy) Turaev's results in [10] must be restated as applying to X an Eilenberg-Maclane space only. This is the position adopted by Rodrigues in [6], where a careful discussion of the axioms can be found.…”