7.2 Stationary-phase approximations

To see if the above definitions make any sense, we first of all consider the stationary-phase approximation fe-> oo. One should think of the parameter k as something like l/fi where ti is Planck's constant. The classical limit comes from ft

For the rest of this section we shall only be concerned with Z( Y): the generalizations Z( Y, C\,..., Cr) are similar, and only slightly more complicated.

In the stationary-phase approximation, the dominant part comes from the stationary points of the exponent. That is, at points where

Then, the stationary-phase approximation to Z( Y) gives a sum of contributions, one from each of the representations a. Thus we need only look at the integral for Z( Y) locally. Suppose a is a flat connection, and

There are no linear terms, since di? = 0 at a. Here, da/3 is the covariant derivative of /3 with respect to the connection a.

 

where (,) is the inner product on Lie-algebra-valued 1-forms:

 

Define Q(/3) = (1/477-) Jy Tr (/3 a dj3). This is the quadratic term in the expansion of L(A) above. One can think of Q as a quadratic form in an infinite number of variables. Here:

Thus Q is given by a self-adjoint operator, —*da. This Q is related to the de Rham complex with respect to the coefficient system given by a. Let ga be the flat G-bundle on Y given by the connection a, with fibre the Lie algebra g. Then we have a de Rham complex:

 

 

Since a is flat, d^ = 0.

 

 

We shall assume for simplicity that a is a non-degenerate representation; i.e. that the above complex has no cohomology:

 

Here H°(Y, ga)=0 corresponds to a being an irreducible representation and Hl(Y,qa) = 0 corresponds to this rep­resentation being isolated (since dim//1 is essentially the number of deformation parameters of the representation).

In this case, Q is degenerate on d/2° , since *da vanishes on the image of

 

 

This corresponds to the fact that / is invariant under CS; dfi°a corresponds to infinitesimal gauge transformations. Factoring out d/2° , we find that Q is non-degenerate on O l/dfi°a.

Continuing analytically, and putting /1 = —iA, we obtain

 

V IT

Let us now digress to discuss classical Gaussian integrals. We start with the one-dimensional integral:

 

 

The n-dimensional form of this is as follows. Suppose Q is a non-degenerate quadratic form in xlf..., x„. Then

 

 

This holds for non-degenerate quadratic forms only (no zero eigenvalues).

Since H3, H2 are dual to H°, H\ these conditions essentially reduce to

 

Suppose we have the action of a compact group G (e.g. S1) on a Euclidean space, X, and Q{x) is a G-invariant quadratic form. Take a transversal slice of the space for the

G action. We must take into account some form of Jacobian: in fact, the appropriate quantity is the volume of an orbit.

 

 

Now G acts on X. At a point x e X we have a map:

 

 

This gives a map, B, from the tangent space of G at the identity to the tangent space of the orbit at x. The Jacobian of the map corresponds to the volume of the orbit. Thus

 

is the appropriate scaling factor. Hence we obtain the modulus

 

 

and

 

 

Application to our situation In our case,

 

B = (infinitesimal map from Lie algebra to tangent space of manifold)

 

the Laplace operator on fl°a.

 

Now Q is given by -*da on flx/âff. Consider the self- adjoint operator

 

 

acting on odd forms /2°dd where e = -1 on Ola and e = + 1 on fl3a. By duality one can replace I23a by I2°a, and thus P can be thought of as acting on Q°a®Qxa. P is closely related to Q.

We can think of I2la = V® W where

 

 

and W= in SI I. Then Q acts on W; and P acts on /2° eve W by

 

always has zero signature. Of course, in this case, we have not assigned meanings to det Q or sgn Q\ but in any 'sensible' definition one would hope that

 So, if we can make sense of det P, det B*B, then we can write down the local contribution to the stationary phase approxi­mation.

Here we have so far left out the level k In finite dimensions, such a factor changes the resultant integral by an appropriate power of k, which we will later see in our case is zero.

Regularization of determinants and signatures

 

Formally, one sees that

 

 

Let A be a Laplace operator, with positive eigenvalues A. Then we can define the zeta function:

 

 

The function £(s) is a meromorphic function, defined in the first instance for Re (s) sufficiently large. It can be analytically continued to the whole complex plane, having isolated poles. Here s = 0 is not a pole, and £(0), £'(0) are well defined.

Formally, £(0) is the dimension of the Hilbert space. In odd dimensions, £(0) = 0.

Following Ray and Singer [26] we define

The above definition makes sense as a real number, and is used by physicists to make sense of Gaussians occurring in QFT. We wish to do this for the Laplacian with twisted coefficients A°a. This makes det (A°a) well defined, and thus gives the B*B term.

 

and hence |det P| is well defined, giving |det <?|, from (iii). Thus one can evaluate (i), obtaining

 

Ray and Singer [26] proved that

 

Similarly, P2 = A°®A\ the direct sum of the Laplace operators on /2°, fl\ Thus

the square of the above expression is independent of Rieman- nian metric. The Riemannian metric is used to obtain a *-operator, which is necessary to make sense of the divergent quantities.

To prove independence of metric, one differentiates Ta with respect to the metric as parameter, and shows that this vanishes. Ray and Singer conjectured that Ta was the classical Reidemeister torsion. This conjecture was proved (indepen­dently) by Cheeger and Miiller. This is the first concrete encouragement for the Witten formula for Z( Y): the absolute value of the limit fc-»oo can be regularized, and the result is metric independent. This observation relating Ray-Singer tor­sion to the abelian Chern-Simons theory was made by Schwarz [28] in the late 1970s (and it extends fairly easily to the non-abelian case).

Phase factor

We now consider the phase factor as given by (ii). This involves sgn Q, which is related to sgn P, and was studied by Atiyah, Patodi and Singer [6].

 

Once again, rj can be analytically continued, and 17(0) is well defined. Formally,

 

Consider the situation where P is a self-adjoint operator with both positive and negative eigenvalues, and

 

 

Define

and it is thus natural to define sgn P= i?(0). Note however that this quantity is a real number, not an integer. Thus the resulting phase in (ii) will not be a root of unity in general. Then we have (cf. Proposition 4.20 of [61)

 

 

where r)a is the 17-function associated with Pa. We now have to investigate how sgn Qa depends on the metric.

Here a is a representation of irx(M), with no cohomology. Consider the trivial representation, and put

 

 

where 17d = dr}{ and -q{ corresponds to ordinary differential forms, without group fibres; d is the dimension of our Lie group. Then:

rja is independent of metric,

i?a(0) = (4/ v)8(G)L(a),

where 8(G) is a numerical invariant of G (it is n for SU(n): in general it depends on the value of the Casimir in the adjoint representation) and L is the Chern-Simons functional.Thus we obtain from the stationary-phase formula:

 

 

where C is a fixed multiplier, coming from -qd and C contains the only metric dependence in the formula; is metric independent. The phase factor is independent of G and the chosen representation, but depends on the choice of ground metric. The above formula for iL(0) leads to a shift

 

 

in the exponential multiplier arising from the value of the action at the critical point a. Such a shift is well known to physicists in various guises.

If we had succeeded in making an expression for Z(Y) independent of metric, we would have shown, for large k, how to make sense of the determinants and signatures by regularizing. This is very nearly true, but not quite - we have a phase ambiguity.

For this reason Witten has to choose a framing of the 3-manifold. For related reasons to define the invariants for links we have to choose (normal) framings for each com­ponent of the link. We shall say more about these framings in the next section.