2.1 Axioms for a topological QFT

The notion of a topological quantum field theory (QFT), i.e. not depending on any background geometry, is one which has emerged recently in the work of Witten [35] [36]. The Jones polynomial fits into such a theory so we shall begin by reviewing briefly what is meant by a topological QFT. It is convenient to give an axiomatic approach since this emphasizes the mathematical structures involved. The physics can be viewed as motivating background.

A more extensive treatment of topological QFTs can be found in [2]. The reader may also wish to consult [3] and, for closely related ideas on conformal field theories, the treat­ment in [29] may be helpful.

A topological QFT in dimension d is a functor Z which assigns

a finite-dimensional complex vector space Z(Z) to each compact oriented smooth d-dimensional manifold I,

a vector Z( Y) € Z(2) for each compact oriented (d +1)- dimensional manifold Y with boundary X

This functor satisfies the following axioms.

A1 (Involutory) Z(I*) = Z(I*), where 1* denotes 2 with the opposite orientation and Z{2)* is the dual space.

A2 (Multiplicativity) Z(SluI2) = Z(I1)®Z(I2), where u is the disjoint union.

A3 (Associativity) For a composite cobordism

y=y1uaiy2 .



[Note: In this associative axiom we have used the previous two axioms to view Z(Y,) and Z(Y2) as homomorphisms Zihi)^ Z(h2) and Z(22)-> Z(23) respectively.] In addition we impose the non-triviality axioms.

A4 Z(0)= C for the empty d-manifold.

A5 Z(Z x I) is the identity endomorphism of Z(I).

The functoriality of Z together with A5 imply homotopy invariance. This means that the group Diff+ (.£) of orientation- preserving diffeomorphisms of h acts on Z{1) via its group of components r(Z).

For a closed (d + l)-dimensional manifold Y the boundary is empty and so, by A4, the vector Z( Y) is just a complex number. Thus such a topological QFT assigns numerical invariants to closed (d + l)-dimensional manifolds. Moreover cutting Y along a d-manifold 1 and applying A3 (with = X, = 0) we see that




the pairing (,) being between the dual spaces Z(I) and Z{h*). Thus the numerical invariants of a closed (d + 1)- manifold can be computed from any decomposition Y = y, ux Y2.

The character of this representation of r(Z) on Z(Z) is determined by the axioms. If /e Diff+ (.£) we can form the

manifold If from the product 2*1 by using / to identify 2 xO and 2x1. Formula (2.1.1) then implies that

Trace Z(f) = Z(Zf)   (2.1.2)

where Z(f) is the induced transformation on Z{h). In par­ticular, taking / to be the identity

dim Z(I) = Z(IxSl).   (2.1.3)

These formulae should be compared with the Feynman integral formula (1.1.1) in the physical interpretation which we come to next.

At this stage it may be helpful to make some remarks on the physical interpretation of our axioms. The idea is that, for a closed (d + l)-manifold Y, the invariant Z(Y) is the partition function given by some Feynman integral as dis­cussed in Chapter 1. Of course only very special Lagrangians will give rise to topologically invariant partition functions. The vector space Z(2) is then the 'Hilbert space' of the theory on the 'space' 1. The endomorphism of Z{1) given by Z(2 x I) should be the 'imaginary time' evolution operator e"™ (where T is the length of the interval /), but axiom A5 implies that the Hamiltonian H = 0. Thus, in a topological QFT there is no dynamics. All states are ground states and this is related to the finite dimensionality of the 'Hilbert space' Z(1). Although there is no interesting propagation along a cylinder there is interesting propagation across a non-trivial cobordism, i.e. across singular surfaces which change the topology of h. This 'topological propagation' is the essential content of the theory from the Hamiltonian point of view. Relativistic invariance asserts that the final numerical invariants, such as Z( V), are independent of the time variable which one may pick to slice Y.

We are now going to concentrate on the situation that is relevant to the Jones-Witten theory. In particular we will put d =2 so that I, is a surface. In fact we need to refine andsupplement the basic axioms above in a number of ways. In the first place our theory will be a unitary one. This means that the vector spaces Z(£) all have natural Hermitian metrics (i.e. they are finite-dimensional Hilbert spaces) and if dY = I,2 u the linear maps are adjoints of each other. In particular, for a closed 3- manifold Y, when Z( Y) is just a complex number,

Z(Y*) = Z(Y).

It is this property which eventually explains the ability of the Jones polynomial to distinguish mirror images.

So far we have axiomatized an 'absolute' theory and this will lead to Witten's invariants for closed 3-manifolds. However, to get invariants for links in 3-manifolds, and hence the Jones polynomials, we have to relativize our axioms. We therefore consider a pair (Y, L) where Y is an oriented 3- manifold as before and L<= Y is an oriented 1-manifold. If Y has boundary 2 then L is assumed transversal to 2, and so dLd is an oriented 0-manifold, i.e. a collection of signed points. A typical picture is depicted below.



If Y is closed then L is just an oriented link in Y. Our link L, and hence its boundary, is also assumed to carry some further information. In an abstract form this could

be just an index from some given indexing set I. This is the formulation given in [29] for conformal field theories. However, for the concrete case of the Witten-Jones theory, I is just the set of irreducible representations of the compact Lie group G. Thus for each component of L (or to each signed point in 1) we assign an irreducible representation A of G. Reversing the orientation of the component and simul­taneously replacing A by its dual A* we regard as giving equivalent data.

In this framework our topological QFT is a functor Z which assigns a vector space Z(h, P, A) to each surface X with points P = (P1,...,Pr) marked with representations A = (Aj,..., Ar). Note that we can give each point P, a + sign by picking A, or A? as necessary. If Y is a 3-manifold with an oriented link L marked with representations fi, then Z assigns a vector


where A =dfi is the induced marking on the signed set of points P = dL.

The axioms for Z have to be modified in a relatively obvious manner. Note that, for a closed 3-manifold Y with a marked link L we get a numerical invariant

Z(Y, L, fi)e C.

Taking Y = S\ G = SU(2) and the standard two- dimensional representation this invariant will eventually be identified with a certain value of the Jones polynomial. Note that the group of components of Diff+ (S2, P) acts on the vector space Z(S2,P,/i), provided all are equal. This is closely related to the braid group representations of the Jones theory. In fact Bn is the group of components of orientation- preserving diffeomorphisms of S2 with n +1 marked points Pu..., Pn+1 where Pn+l = oo is distinguished and kept fixed, while the others are permuted.





means that the bundle of Hilbert spaces has a natural connec­tion which is projectively flat. Alternatively, tensoring the bundle by a suitable line-bundle over £p, we get a flat connec­tion. The required line-bundle L is actually K~1/2 where K is the canonical line-bundle of if.

For a full treatment of the quantization process we refer the reader, for example, to [14] or [38].

Notice that a natural way to get a complex structure <r on R) is to fix a complex structure r on X This enables us to identify the dual of the space of holomorphic differentials on X The complex structures on h modulo the identity component of Diff+ (1) are parametrized by the Teichmiiller space ST and r->o- defines an embedding ST^if, essentially the period mapping. The quantizations we are now discussing can be carried out over the whole of if, but in the non-abelian situation to be discussed later this will no longer be true and we shall be restricted to 9~.

If, in all this story, we rescale the symplectic form w by a factor k then nothing essentially alters except that the Heisen- berg commutation rules now pick up a similar factor. Physi­cally k plays the role of the inverse of Planck's constant h. Geometrically, however, our compact surface I provides a natural normalization for the 2-form u>. This normalization becomes more significant when, in the next section, we intro­duce the integer lattice Hl(X Z) and the associated ©-func­tions. The factor k can then only take integer values and is called the level of the theory.