A homotopy quantum field theory is a TQFT on cobordisms with extra topological structure: a representation of a cobordism category of cobordisms that are equipped with the extra structure of a continuous function into a fixed topological space $X$.
Fix an integer $n \geq 0$ and a field, $K$. All vector spaces will be tacitly assumed to be finite dimensional. In general $K$ can be replaced by a commutative ring merely by replacing finite dimensional vector spaces by projective $K$-modules of finite type, but we will not do this here.
HQFTs were introduced in 1999 by Vladimir Turaev for 2-dimensional manifolds. He extended them to 3-dimensional ones the following year. At about the same time, Brightwell and Turner (1999) looked at what they called the homotopy surface category and its representations. There are two viewpoints which interact and complement each other. Turaev’s seems to be to see HQFTs as an extension of the tool kit for studying manifolds given by TQFTs, whilst in Brightwell and Turner’s, it is the ‘background space’, which is probed by the surfaces in the sense of sigma-models.
The idea of Turaev was to extend the basic ideas of TQFTs from $n$-dimensional manifolds and cobordisms between them, to manifolds with a simple bit of extra structure given by a continuous map to a background space, $B$.
The category of $B$-manifolds and $B$-cobordisms: The basic objects on which an $(n+1)$-homotopy quantum field theory is built are compact, oriented $n$-manifolds together with maps to the ‘background’ space, $B$. This space $B$ will be path connected with a fixed base point, $\ast$. More precisely:
Definition
A $B$-manifold is a pair $(X, g)$, where $X$ is a closed oriented $n$-manifold (with a choice of base point $m_i$ in each connected component $X_i$ of $X$), and $g$ is a continuous map $g : X \to B$, called the characteristic map, such that $g(m_i) = \ast$ for each base point $m_i$.
A $B$-isomorphism between $B$-manifolds, $\phi : ( X, g) \to ( Y, h)$ is an isomorphism $\phi : X \to Y$ of the manifolds, preserving the orientation, taking base points into base points and such that $h\phi = g$.
If as is often the case, the manifolds under consideration will be differentiable and then ‘isomorphism’ is interpreted as ‘diffeomorphism’, but equally well we can position the theory in the category of PL-manifolds or triangulable topological manifolds with the obvious changes. In fact for some of the time it is convenient to develop constructions for simplicial complexes rather than manifolds, as it is triangulations that provide the basis for the combinatorial descriptions of the structures that we will be using.
Denote by $\mathbf{Man}(n,B)$ the category of $n$-dimensional $B$-manifolds and $B$-isomorphisms. We define a ‘sum’ operation on this category using disjoint union. The disjoint union of $B$-manifolds is defined by
with the obvious characteristic map, $g\amalg h : X \amalg Y \to B$. With this ‘sum’ operation, $\mathbf{Man}(n,B)$ becomes a symmetric monoidal category with the unit being given by the empty $B$-manifold, $\emptyset$, with the empty characteristic map. Of course, this is an $n$-manifold by default.
These $B$-manifolds are the objects of interest, but they have to be related by the analogue of cobordisms for this setting.
A $B$-cobordism, $(W,F)$, from $(X_0,g)$ to $(X_1,h)$ is a cobordism $W : X_0 \to X_1$ endowed with a homotopy class of maps $F : W \to B$ relative to the boundary such that $F|_{X_0} = g$ and $F|_{X_1} = h$.
Generally unless necessary in this entry, we will not make a notational distinction between the homotopy class $F$ and any of its representatives. Finally a $B$-isomorphism of $B$-cobordisms, $\psi : (W,F) \to (W^\prime, F^\prime)$, is an isomorphism $\psi : W \to W^\prime$ such that
and $F^\prime \psi = F$, in the obvious sense of homotopy classes relative to the boundary.
We can glue $B$-cobordisms along their boundaries, or more generally, along a $B$-isomorphism between their boundaries, in the usual way.
The detailed structure of $B$-cobordisms and the resulting category $\mathbf{HCobord}(n,B)$ is given in the Appendix to the paper by Rodriques, (see references), at least in the important case of differentiable $B$-manifolds. This category is a monoidal category with strict duals.
A homotopy quantum field theory is a symmetric monoidal functor from $\mathbf{HCobord}(n,B)$ to the category, Vect , of finite dimensional vector spaces over the field $K$.
However let us also give here a more basic definition of a homotopy quantum field theory.
A $(n + 1)$-dimensional homotopy quantum field theory, $\tau$, with background $B$ assigns
to any $n$-dimensional $B$-manifold, $(X,g)$, a vector space, $\tau{(X,g)}$,
to any $B$-isomorphism, $\phi : (X, g) \to ( Y, h)$, of $n$-dimensional $B$-manifolds, a $K$-linear isomorphism $\tau(\phi) : \tau{(X, g)} \to \tau{( Y, h)}$,
and
These assignments are to satisfy the following axioms:
$\tau$ is functorial in $\mathbf{Man}(n,B)$, i.e., for two $B$-isomorphisms, $\psi: (X, g) \to ( Y, h)$ and $\phi : ( Y, h) \to (P,j)$, we have $\tau(\phi\psi) = \tau(\phi)\tau(\psi),$ and if $1_{(X,g)}$ is the identity $B$-isomorphism on $(X,g)$, then $\tau(1_{(X,g)}) = 1_{\tau{(X,g)}}$
There are natural isomorphisms
and an isomorphism, $u : \tau(\emptyset) \cong K$, that satisfy the usual axioms for a symmetric monoidal functor.
For $B$-cobordisms, $(W,F) : (X,g) \to (Y,h)$ and $(V,G): (Y^\prime, h^\prime) \to (P,j)$ glued along a $B$-isomorphism $\psi :(Y,h) \to (Y^\prime,h^\prime)$, we have $\tau((W,F)\amalg_\psi (V,G))= \tau(V,G)\tau(\psi)\tau(W,F).$
For the identity $B$-cobordism, $1_{(X,g)} = (I\times X, 1_g)$, we have $\tau( 1_{(X,g)}) = 1_{\tau(X,g)}.$
For $B$-cobordisms $(W,F) : (X,g) \to (Y,h)$ and $(V,G) : (X^\prime,g^\prime) \to (Y^\prime,h^\prime)$ and $(P,J): \emptyset \to \emptyset$, some fairly obvious diagrams are commutative.
These axioms are slightly different from those given in the original paper of Turaev in 1999. The really significant difference is in axiom 4, which is weaker than as originally formulated, where any $B$-cobordism structure on $I \times X$ was considered as trivial. The effect of this change is important as it is now the case that the HQFT is determined by the $(n+1)$-type of $B$, cf. Rodrigues (2003).
With the revised version of the axioms, it becomes possible to attempt to classify HQFTs with a given $n$ and $B$. Turaev did this in the original paper with $n = 2$ and $B$ an Eilenberg-MacLane space, $K(G,1)$. The results of Brightwell and Turner essentially gave the solution for $B$ a $K(A,2)$.
If we look at the case $n= 1$ and with background an Eilenberg-Mac Lane space $K(G,1)$, then HQFTs correspond to crossed G-algebras, in much the same way that commutative Frobenius algebras correspond to 2d TQFTs. There the correspondence is given by a 2d TQFT, $Z$, corresponds to the Frobenius algebra, $Z(S^1)$. This is because the circle $S^1$ is a Frobenius object? in the category $Bord_2$ of 2d-cobordisms between 1-manifolds.
In the case of HQFTs, the role of the circle is replaced by the family of circles with characteristic maps to $B$. Each one gives, combinatorially, a circle together with a labelling of the boundary by an element of $G$. (It does not seem to be known how to get a $G$-graded version of an abstract Frobenius object that will correspond to this situation, although this is probably not too hard to do.)
V. Turaev, Homotopy field theory in dimension 2 and group-algebras, preprint arXiv: arXiv:math.QA/9910010
V. Turaev, Homotopy field theory in dimension 3 and crossed group-categories, preprint arXiv:math.GT/0005291v1.
V. Turaev, Homotopy Quantum Field Theory, Tracts in Mathematics 10, (with Appendices by Michael Muger and Alexis Vurelizier), European Mathematical Society, June 2010.
M. Brightwell and P. Turner, Representations of the homotopy surface category of a simply connected space, J. Knot Theory and its Ramifications, 9 (2000), 855–864.
G. Rodrigues. Homotopy Quantum Field Theories and the Homotopy Cobordism Category in Dimension 1 + 1, J. Knot Theory and its Ramifications, 12 (2003) 287–317 (previously available as arXiv:math.QA/0105018).
T. Porter and V. Turaev, Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed $C$-algebras, Journal of Homotopy and Related Structures 3(1), 2008, 113–159. (arXiv:math.QA/0512032).
T. Porter, Formal Homotopy Quantum Field Theories II: Simplicial Formal Maps, in Cont. Math. 431, p. 375 - 404 (Streetfest volume: Categories in Algebra, Geometry and Mathematical Physics - edited by A. Davydov, M. Batanin, and M. Johnson, S. Lack, and A. Neeman) (arXiv:math.QA/0512034)
A treatment of HQFTs that includes some details of the links with TQFTs is given in HQFTs meet the Menagerie, which is a set of notes prepared by Tim Porter for a school and workshop in Lisbon, Feb. 2011.
Related ideas are discussed in