Manifolds and cobordisms

Quantum field theory


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 XX.

Assumptions and general conventions

Fix an integer n0n \geq 0 and a field, KK. All vector spaces will be tacitly assumed to be finite dimensional. In general KK can be replaced by a commutative ring merely by replacing finite dimensional vector spaces by projective KK-modules of finite type, but we will not do this here.

A Bit of History

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 nn-dimensional manifolds and cobordisms between them, to manifolds with a simple bit of extra structure given by a continuous map to a background space, BB.

The category of BB-manifolds and BB-cobordisms: The basic objects on which an (n+1)(n+1)-homotopy quantum field theory is built are compact, oriented nn-manifolds together with maps to the ‘background’ space, BB. This space BB will be path connected with a fixed base point, *\ast. More precisely:


A BB-manifold is a pair (X,g)(X, g), where XX is a closed oriented nn-manifold (with a choice of base point m im_i in each connected component X iX_i of XX), and gg is a continuous map g:XBg : X \to B, called the characteristic map, such that g(m i)=*g(m_i) = \ast for each base point m im_i.

A BB-isomorphism between BB-manifolds, ϕ:(X,g)(Y,h)\phi : ( X, g) \to ( Y, h) is an isomorphism ϕ:XY\phi : X \to Y of the manifolds, preserving the orientation, taking base points into base points and such that hϕ=gh\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 Man(n,B)\mathbf{Man}(n,B) the category of nn-dimensional BB-manifolds and BB-isomorphisms. We define a ‘sum’ operation on this category using disjoint union. The disjoint union of BB-manifolds is defined by

(X,g)⨿(Y,h):=(X⨿Y,g⨿h),( X, g) \amalg ( Y, h) := ( X\amalg Y, g\amalg h),

with the obvious characteristic map, g⨿h:X⨿YBg\amalg h : X \amalg Y \to B. With this ‘sum’ operation, Man(n,B)\mathbf{Man}(n,B) becomes a symmetric monoidal category with the unit being given by the empty BB-manifold, \emptyset, with the empty characteristic map. Of course, this is an nn-manifold by default.

These BB-manifolds are the objects of interest, but they have to be related by the analogue of cobordisms for this setting.

A BB-cobordism, (W,F)(W,F), from (X 0,g)(X_0,g) to (X 1,h)(X_1,h) is a cobordism W:X 0X 1W : X_0 \to X_1 endowed with a homotopy class of maps F:WBF : W \to B relative to the boundary such that F X 0=gF|_{X_0} = g and F X 1=hF|_{X_1} = h.

Generally unless necessary in this entry, we will not make a notational distinction between the homotopy class FF and any of its representatives. Finally a BB-isomorphism of BB-cobordisms, ψ:(W,F)(W ,F )\psi : (W,F) \to (W^\prime, F^\prime), is an isomorphism ψ:WW \psi : W \to W^\prime such that

ψ( +W)= +W ,ψ( W)= W ,\psi (\partial_+W) = \partial_+W^\prime, \quad \psi (\partial_-W) = \partial_-W^\prime,

and F ψ=FF^\prime \psi = F, in the obvious sense of homotopy classes relative to the boundary.

We can glue BB-cobordisms along their boundaries, or more generally, along a BB-isomorphism between their boundaries, in the usual way.

The detailed structure of BB-cobordisms and the resulting category HCobord(n,B)\mathbf{HCobord}(n,B) is given in the Appendix to the paper by Rodriques, (see references), at least in the important case of differentiable BB-manifolds. This category is a monoidal category with strict duals.

Homotopy Quantum Field Theories

Definition (Categorical form)

A homotopy quantum field theory is a symmetric monoidal functor from HCobord(n,B)\mathbf{HCobord}(n,B) to the category, Vect , of finite dimensional vector spaces over the field KK.

However let us also give here a more basic definition of a homotopy quantum field theory.

Less categorical definition of HQFTs

A (n+1)(n + 1)-dimensional homotopy quantum field theory, τ\tau, with background BB assigns

  • to any nn-dimensional BB-manifold, (X,g)(X,g), a vector space, τ(X,g)\tau{(X,g)},

  • to any BB-isomorphism, ϕ:(X,g)(Y,h)\phi : (X, g) \to ( Y, h), of nn-dimensional BB-manifolds, a KK-linear isomorphism τ(ϕ):τ(X,g)τ(Y,h)\tau(\phi) : \tau{(X, g)} \to \tau{( Y, h)},


  • to any BB-cobordism, (W,F):(X 0,g 0)(X 1,g 1)(W,F) : (X_0,g_0) \to (X_1,g_1), a KK-linear transformation, τ(W):τ(X 0,g 0)τ(X 1,g 1)\tau(W) : \tau{(X_0,g_0)} \to \tau{(X_1,g_1)}.

These assignments are to satisfy the following axioms:

  1. τ\tau is functorial in Man(n,B)\mathbf{Man}(n,B), i.e., for two BB-isomorphisms, ψ:(X,g)(Y,h)\psi: (X, g) \to ( Y, h) and ϕ:(Y,h)(P,j)\phi : ( Y, h) \to (P,j), we have τ(ϕψ)=τ(ϕ)τ(ψ),\tau(\phi\psi) = \tau(\phi)\tau(\psi), and if 1 (X,g)1_{(X,g)} is the identity BB-isomorphism on (X,g)(X,g), then τ(1 (X,g))=1 τ(X,g)\tau(1_{(X,g)}) = 1_{\tau{(X,g)}}

  2. There are natural isomorphisms

    c (X,g),(Y,h):τ((X,g)⨿(Y,h))τ(X,g)τ(Y,h),c_{(X,g),(Y,h)} : \tau((X,g)\amalg (Y,h)) \cong \tau(X,g)\otimes \tau(Y,h),

    and an isomorphism, u:τ()Ku : \tau(\emptyset) \cong K, that satisfy the usual axioms for a symmetric monoidal functor.

  3. For BB-cobordisms, (W,F):(X,g)(Y,h)(W,F) : (X,g) \to (Y,h) and (V,G):(Y ,h )(P,j)(V,G): (Y^\prime, h^\prime) \to (P,j) glued along a BB-isomorphism ψ:(Y,h)(Y ,h )\psi :(Y,h) \to (Y^\prime,h^\prime), we have τ((W,F)⨿ ψ(V,G))=τ(V,G)τ(ψ)τ(W,F).\tau((W,F)\amalg_\psi (V,G))= \tau(V,G)\tau(\psi)\tau(W,F).

  4. For the identity BB-cobordism, 1 (X,g)=(I×X,1 g)1_{(X,g)} = (I\times X, 1_g), we have τ(1 (X,g))=1 τ(X,g).\tau( 1_{(X,g)}) = 1_{\tau(X,g)}.

  5. For BB-cobordisms (W,F):(X,g)(Y,h)(W,F) : (X,g) \to (Y,h) and (V,G):(X ,g )(Y ,h )(V,G) : (X^\prime,g^\prime) \to (Y^\prime,h^\prime) and (P,J):(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 BB-cobordism structure on I×XI \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)(n+1)-type of BB, cf. Rodrigues (2003).

  • With the revised version of the axioms, it becomes possible to attempt to classify HQFTs with a given nn and BB. Turaev did this in the original paper with n=2n = 2 and BB an Eilenberg-MacLane space, K(G,1)K(G,1). The results of Brightwell and Turner essentially gave the solution for BB a K(A,2)K(A,2).

1+1 dimensional HQFTs with background a K(G,1)K(G,1)

If we look at the case n=1n= 1 and with background an Eilenberg-Mac Lane space K(G,1)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, ZZ, corresponds to the Frobenius algebra, Z(S 1)Z(S^1). This is because the circle S 1S^1 is a Frobenius object? in the category Bord 2Bord_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 BB. Each one gives, combinatorially, a circle together with a labelling of the boundary by an element of GG. (It does not seem to be known how to get a GG-graded version of an abstract Frobenius object that will correspond to this situation, although this is probably not too hard to do.)

  • In the paper by Moore and Segal, (see below), they discuss GG-equivariant TFT?s and show how they naturally correspond to a simple case of Turaev’s HQFTs. They relate (1+1) equivariant TFTs to Turaev’s crossed G-algebras (which they call Turaev algebras).


  • 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 CC-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

Revised on June 26, 2012 01:25:28 by Byeongho Lee (