As described above, the groupsE generated by the lifts ρR act on sections in L⊗k, for eachσ ∈ T preserving the holomorphic sections. Thus, they define a fibrewise (i.e. inducing the identity on the base) action on the Verlinde bundle Zkover Teich ¨uller space.
Work is in progress by the author to show that this action is compatible with Hitchin’s connection, in the sense that for any sections : T →Ek, any vector fieldXonT and any liftρRof an elementR∈Hom(π1(S), µn), we have:
ρR(∇X(ρ−1R (s))) =∇X(s)
Remark7.4. Since the trivial connection∇tis clearly invariant under the action (the latter being the identity on the base), it remains only to show that the map uis invariant.
The final steps of the construction of a modular functor (and hence a topo- logical quantum field theory) go as follows. (For simplicity we fix a value ofk and remove it from the notation wherever possible.)
One would like to construct a line bundleLoverT with a connection∇op having the opposite curvature of∇H. This would allow a genuinely flat connec- tion to be defined in the tensor productZk⊗L.
There are obstructions towards doing this, however, in a way that preserves the action of the mapping class group. But allowing the passage to a central extension of the mapping class group, such obstructions disappear, and indeed, by transferring the entire situation to the realm of conformal field theory, An- dersen and Ueno have recently completed the construction and showed that it yields a modular functor.
In outline, conformal field theory produces a vector bundleV†, thebundle of conformal blocks, over Teichm ¨uller space, along with a projectively flat connec- tion∇†. (See [22].) It has been shown by Laszlo ([20]) that this construction is equivalent to the Verlinde vector bundle endowed with Hitchin’s connection.
Within the setting of conformal field theory, Andersen and Ueno have used a so-called opposite abelian theory to construct of a line bundleL†aboverT with a connection∇†ab, whose curvatureΩ∇absatisfies:
Ω∇†= c 2Ω∇†
ab
–Wherec is a rational number, called the central chargeof the conformal field theory. It is given by:c= (k·dim(SU(n)))
(k+n). The fibre ofL†abover a point σ∈ T is simplydet(H1(Σσ,O)).
Now, Teichm ¨uller space being contractible, one may construct the fractional tensor power
(L†ab)⊗−c/2 and endow the bundle:
V†⊗(L†ab)⊗−c/2 with the genuinely flat connection:
∇=∇H⊗1 + 1⊗(∇†ab)⊗−c/2
We may thus defineVk to be the space of covariantly constant sections in this bundle. Doing that, Andersen and Ueno ([5]) have shown, first of all, that the construction is compatible with the action of the mapping class group on Teichm ¨uller spaces, ensuring thatVk becomes a representation of a central ex- tension of the mapping class group.
Finally, Andersen and Ueno show that the association ofVkto the surfaceS satisfies all the axioms for a modular functor, and thus by the work of Walker and Grove ([15]) gives a topological quantum field theory in dimension2 + 1, withVas the TQFT vector spaces associated toS.
The action of the groupsE(n, d)goes through this final step easily, simply tensoring with the identity action on the fractional power ofL†ab. Supposing it preserves Hitchin’s connection, it then preserves covariantly constant sections, thus inducing a representation on the TQFT vector spaces.
Remark7.5. In a way, the representations on the TQFT vector spaces are not new.
From the topological viewpoint (using modular tensor categories to construct Reshetikhin-Turaev modular functors), Blanchet and others have constructed similar representations and decompositions of the Verlinde vector spaces. Since the two constructions are currently being shown to be equivalent ([6]), it would be interesting to understand the topological realisation of the results in the the- sis.
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