2.2 Functions

We now introduce the integer lattice and correspondingly the quotient torus

 

 

where we identify U(l) with R/Z in the standard way by 0 -> exp (2vi0). Roughly speaking quantizing the torus should be the same as taking the /1-invariant part of the quantization 3€(I) of the vector space Hl(2, R). The different models of then lead to different models for the quantization of the torus.

 

The complex quantization is in many ways the easiest to understand and is the most relevant for our purposes. For this we pick a complex structure tr on Hl(Z, R), which might come from a complex structure roni itself. This makes the torus H'(I, U{ 1)) into an abelian variety A„, which is the Jacobian JT if a- = r. The complex line-bundle L on Hl(h, R) with curvature 2iriw descends to become a holomorphic line- bundle on A„ with first Chern class represented by u>. This has 'degree' 1, in the sense that the Liouville volume

(2.3.1)

The line-bundle L obtained in this way is not uniquely defined by its curvature since the torus is not simply connected. We can alter L by tensoring with any flat line-bundle. These different choices correspond to different actions of A on L, which we did not specify.

There are various equivalent ways to get rid of this ambiguity in L. The classical algebro-geometric way, when a- = r, is to consider first the degree (g-1) Jacobian J?"1, i.e. the moduli space of holomorphic line-bundles of degree g -1 over ST. This has a natural divisor D given by the image of the (g - l)st symmetric product. This divisor, called the 'theta- divisor', represents line-bundles of degree g -1 on Ir which have a non-zero holomorphic section. The line-bundle [D]on J?-1 defined by this divisor is then unambiguously defined. Moreover it has the correct first Chern class. To shift back to JT (i.e. the degree 0 Jacobian) we have to pick a base point on JgT~l. This can be done by choosing a spin structure on 2T or equivalently a square root of the canonical line-bundle. Having chosen such a spin structure we shift [D] back to Jr and this becomes our choice of L.

To quantize JT we then take the space of holomorphic sections of L. This is just one-dimensional, corresponding on J? 1 to the section of [D] vanishing on D. This also follows from the Riemann-Roch theorem using (2.3.1).

Quantizing at level k means replacing L by Lk and, again by Riemann-Roch, we get a space of dimension k8.

The basic section of L is given transcendentally by the classical ©-function. This is obtained by considering directly the action of A on the holomorphic sections of L and finding the unique fixed vector. Note that this is not strictly a vector in the Hilbert space       it is holomorphic but not squareintegrable.

More generally the sections of Lk are given by the 0- functions of level k. If the complex structure <r = r is repre­sented in if by the g x g complex symmetric matrix Z (with positive definite imaginary part), and u e Cg, then

 

 

is the explicit formula for the basic ©-functions of level k. Here me(Z/k)g runs over the ks basic elements and the torus A„ is the quotient of Cg by the g basis vectors and the g columns of Z.

Although we concentrated on the case of the Jacobian, i.e. for complex structures <t = t, formula (2.3.2) defines ©-func­tions for general a- (i.e. for general Z).

Thus the quantization of         U{ 1)) at level k produces

a vector space V„ of dimension kg, for each a- e Sf. These form

a holomorphic vector bundle V over Sf. As with the quantiz­ation of a linear space we expect all the projective spaces P(V,0.) to be naturally isomorphic. If was actually a sub- space of the Hilbert space H„{2) (at level k) this would be automatic. Unfortunately, as was pointed out earlier, V„ lies in some completion of H„. However, the explicit formulae for ©-functions can be used to derive the necessary iden­tifications. The result is that the vector bundle V over y has a natural connection which is projectively flat. Moreover the cen­tral (scalar) curvature can be computed explicitly.

The connection arises from the fact that the 0m of (2.3.2) obviously satisfy a differential equation

 

 

This shows that the 0m are covariant constant sections of a connection over if. This connection is not however totally 'natural'; it is not invariant under the action of Sp(2g, Z). The natural connection differs by a central factor.

The projective flatness of the spaces P( V„) can also be interpreted as a cohomological rigidity. In fact we can form the finite Heisenberg group Tk from the Z/fc-module H\2, Zk) and P{ V„) is essentially the Heisenberg representa­tion of rk.

We now have at least the beginnings of the data needed for a topological quantum theory as described in § 2.1. We have associated a projective space to each oriented surface 2. The next set of data would be to show how a 3-manifold Y with dY = 2 picks out a point in this projective space. Now it is not hard to see that the image of

 

 

is a Lagrangian sub-module W (i.e. a maximal sub-module on which the symplectic form vanishes). We could now use this Lagrangian subspace to construct the Heisenberg rep-

resentation of rk and W would then define a natural 'vacuum vector' in this space.

This would be the outline of the abelian theory where G= (7(1). We shall not pursue this (rather uninteresting) case in further detail. Instead we move on to study the non-abelian case, beginning in Chapter 3 with the classical tlieory gen­eralizing that of the Jacobian.