Notes on prequantization of moduli of G-bundles with connection on Riemann surfaces

Let X → S be a smooth proper family of complex curves (i.e. family of Riemann surfaces), and F a G-bundle over X with connection along the fibres X → S. We construct a line bundle with connection (LF ,∇F ) on S (also in cases when the connection on F has regular singularities). We discuss the resulting (LF ,∇F ) mainly in the case G = C∗. For instance when S is the moduli space of line bundles with connection over a Riemann surface X, X = X × S, and F is the Poincaré bundle over X , we show that (LF ,∇F ) provides a prequantization of S.


Introduction
Of special interest in physics are line bundles with connection over various moduli spaces of G-bundles with connection over Riemann surfaces.Such line bundles are used to construct conformal field theories, which for example produce interesting 3-manifold (topological) invariants.We consider the problem of constructing such line bundles as a problem of constructing Deligne cohomology classes.
O * [1], and hence classes in H 2 (S, Z(1)) correspond to isomorphism classes of line bundles on S. Further, classes in H 2 (S, Z(2)) correspond to isomorphism classes of line bundles with connection on S. So our objective is to construct classes in H 2 (S, Z(2)) where S is one of the moduli spaces being considered.
In topology, the theory of characteristic classes constructs cohomology classes on spaces which are equiped with say G-bundles over them.Let us review how this is done.Recall that a G-bundle on a space Y is described by a classifying map Y → BG (defined up to homotopy).Also H (BG, C) ∼ = (Sym g * ) G as graded algebras, with elements of g * being of degree 2. So a G-bundle on Y yields for instance classes in H 4 (Y, C) corresponding to G-invariant bilinear forms on g.
In the setting of objects with algebraic structure, one may carry out an analogous procedure to construct classes in Deligne cohomology.
Suppose Y is an algebraic variety, G a reductive algebraic group, and F a G-bundle over Y .Instead of a classifying map we have where ∆G is the standard simplicial G-scheme model of EG and BG := ∆G/G .But since ∆G ∼ pt. as a simplicial scheme, And there is a natural map H 2m (BG, Z(m)) → H 2m top (BG, Z) which is actually an isomorphism [1].So again, we have classes in H 4 (Y, Z(2)) corresponding to certain bilinear G-invariant forms on g.
Consider the particular situation of a proper smooth family of complex curves (i.e.family of compact Riemann surfaces) X → S, and a G-bundle F on X .We may construct classes in H 4 (X , Z(2)) as above.And further, classes in H 4 (X , Z(2)) may be integrated down to a obtain classes in H 2 (S, Z(1)).
We consider arbitrary X → S as above, and F with (relative) connection; and fix a class in H 4 (BG, Z(2)).By following the same construction with slightly different complexes we produce a class in H 2 (S, Z(2)).i.e. a line bundle with connection on S. In the cases relevant to physics the curvature is the natural 2-form which the considered moduli spaces carry; which makes our objects prime candidates for physical applications.
Here we discuss the complexes involved in the construction, and compute the curvature of the resulting objects in the case when G = C * .
The ideas discussed here grew out of conversations with A. Beilinson, who in particular suggested a version of our main construction.

Regular case.
Consider X π → S a family of proper smooth curves, F a G-bundle on X with connection along the fibres.Put ∆F := F × G ∆G; and denote q : ∆F → X , p : ∆F → BG.
We will construct complexes Z ∆ (n), Z π (n) on ∆F, X respectively, for which a diagram of the form First of all recall that for any complex C on S, there is a natural We shall now construct Z ∆ on ∆F such that p : ∆F → BG, q : ∆F → X induce Suppose U is a neighbourhood in X over which the relative connection F can be extended to a total flat connection and further there is a (flat) trivialization showing that that is the case locally.
Consider U as in the construction of Z ∆ (n), and t U : ∆F U → S × ∆G as before.Since ∆G pt., Finally, the map induced by p * on cohomology is with the first map being an isomorphism again because ∆G pt.
now yields an isomorphism class (L F , ∇ F ).

Case of regular singularities.
We shall discuss the case of G compact, and then comment about the case of general G.
X → S, F, ∆F as before, and non-intersecting sections S σ i → X , along which the connection on F has regular singularities.Assume that the isomorphism type of the F s at the marked points is constant, and fix trivializations of the underlying bundle of F at the σ i (s)'s.
Remark: Let Y have a G-torsor H, and suppose G acts freely on E. Then there is a canonical map Consider S being the moduli space of line bundles with connection over X and F the Picard bundle over S. For any s ∈ S, T s S can be identified with H 1 (X, C).So H 1 (X, C) ⊗ H 1 (X, C) → H 2 (X, C) → C defines a nondegenerate bilinear form on T s S, which actually endows S with a symplectic structure.Denote the symplectic form by ω S .
Proposition.ω S = ω F .i.e. (L F , ∇ F ) is a prequantization of S. This is a direct consequence of the previous two claims.