8.3 Surgery formula

If we want to compute Witten's invariant for a 3- manifold (without links) then we can proceed by using surgery. This means we consider cutting a tube S1 x D2 out of the manifold, twisting its boundary (the torus S1 x S1), and then inserting it back into the 3-manifold. Every 3-manifold can be obtained, starting from the 3-sphere, by a sequence of such surgeries.

The essential step in computing Witten's invariant by surgery is then to know:

the Hilbert space of a torus,

the action of the modular group SL(2, Z) (the group of components of Diff+ (Sl x S1)) on this Hilbert space.

The Hilbert space of a torus can be computed in various ways. Since the fundamental group is abelian the moduli space of representations is easily determined. Thus for SU(n) the moduli space is the complex projective space Pn_,(C) with its standard line-bundle. Hence the Hilbert space can be identified with the space of homogeneous polynomials in n variables of degree k. For n =2 this gives a (k+ l)-dimensional space.

From the point of view of loop groups the Hilbert space of a torus can be identified with the representations of LG of

level k. Moreover this identification is natural once we pick an interior (solid torus). This gives an explicit basis for the Hilbert space. The action of SL(2, Z) can be computed by using the Verlinde algebra. The essential point is to compute in the explicit basis) the matrix S representing the element

 

 

In principle it is also possible to compute the action of SL(2, Z) from the holomorphic quantization point of view. Since 7ri is abelian we only need to know the way ©-functions vary with the modulus of an elliptic curve as explained in Chapter 2.