The studies were carried out under the supervision of Jørgen Ellegaard Andersen at the Department of Mathematics of Aarhus University from August 2000 to April 2005. This thesis represents the result of the work I did under the supervision of Jørgen Ellegaard Andersen at Aarhus University in the years 2000–.
Motivation and perspectives
In the paper [1], Andersen and Masbaum pursued the idea that the rotations from [14] should have an analogy in the theoretical setting of the gauge. This thesis generalizes Andersen and Masbaum's idea to orders higher than 2, aiming to "rediscover" group representations in Verlinde spaces in the gauge-theoretic setting.
Notational conventions
It has long been assumed, and is now becoming known ([6]), that the TQFT vector spaces resulting from sln(C) are the same as the gauge theoretical ones. Both are given by the amazing Verlinde formula:. for the topological version and [13] for a version in algebraic geometry.).
Outline and main results
Here|M(n,O)|ζ,ζa,b,c′,ζ′′ denotes the intersection of the fixed point components of α, β and γ, corresponding to the pairs (ζ, a), (ζ′, b) and (ζ′′, c), respectively, under the identification mentioned in theorem 1. This chapter covers some of the background material needed in the rest of this thesis.
The Weil pairing
The assumption ensures that this is independent of the path chosen for the integration. The following properties of the Weil pairing are all consequences of the definition and calculation rules mentioned above.
Holomorphic coverings and the norm map
The following result can be obtained by using the fact that M(Σ) is "quasi-algebraically closed." See [28] for a sketchy proof. In this case, Nm(f)forf ∈ M(Σ′) is simply the function onΣ′/G= Σinduced by the G-invariant function Q.
Stable and semistable bundles
Due to the uniqueness of the graded bundle, every direct sum of stable even-slope bundles is isomorphic to its graded bundle. Therefore, every equivalence class S has a representative that is the direct sum of stable bundles of equal slope, and is unique up to isomorphism.
Moduli spaces, general theory
Hom(−, M) satisfying the following two conditions:. M,Φ) is universal in the sense that for every other varietyN and every natural transformationΨ :J. To see that the induced morphism matches T({x}) on sets, we simply need to trace 1M1 around the following diagram for each elementx∈M1:.
Moduli spaces of vector bundles
This compactification is usually called the "module space of semistable sheaves", Mss(n, d). For a morphism : S → S′and a family F parameterized by S′, the retreat F byf is simply the retreat under f×1Σ.
Primitive torsion points, associated coverings
It moves with the action of α for the same reason as in the previous lemma. Furthermore, the operation of μn by multiplication on the eLα threads is limited to deck transformations on Σα.
Equivariant bundles, direct images
For any holomorphic bundleEonΣα, we define its push down or direct image πα∗(E) as the descent of the equivariant bundle⊕ζ′∈µnζ′∗(E). As mentioned before, all of the above is valid for any finite, cyclic Galois cover. There are only n ways to give OΣα the structure of a µn-equivariant bundle, because the action of the generator ζn ∈ µn on OΣα is given by multiplication by a holomorphic (therefore constant) function f on Σα with 1 = Nmα(f) = fn.
Hence, the following diagram commutes (where each of the vertical maps is the canonical action of ζ∈µn).
Fixed points as direct images
Since Lχ0 ∼= Lα for the canonical generatorχ0 of µbn (i.e. the inclusionµn ֒→C∗) by mapping [x, λ]χ0 7→λx, this action is exactly the same as that of hLαi ⊆J(n). Since πα∗ naturally extends to family functors, the previous theorem combined with the methods in 2.5 and lemma 3.13 shows that πα∗ induces a morphism:Picd(Σα)→ |M(n, d)|α. L⊗q−1α ⊗E combined with the fact that Ei ⊗L⊗jα is stable and therefore indecomposable for everyi, j shows by the theorem of Krull-Remak-Schmidt that for everyj∈ {1,.
Therefore, the action on the left-hand side of (3.9) is multiplied by the determinant of the correct representation.
The kernel of Nm α
It is easy to see that the equivariant action of ζ ∈ µn on OΣα that makes φ an equivariant isomorphism is (x, λ) 7→ (ζ(x), ζ−1λ). Identifying Pick(Σα) = Pic0(Σα) = J(Σα) as complex varieties, the results in 2.6 guarantee that Φkα are morphisms. J(Σα) being complete and connected, it follows that ImΦkα are connected and closed . The inclusion on the left is irrelevant, it remains only to show that the union is split.
Fixing D with a prime divisor as in Lemma 2.14, we can assume that D and(gα) have split support.
Description of the fixed point varieties
First, we need a small calculation, which is surprisingly difficult to show directly from the definitions, due to the assumption that divisors must be disjoint to calculate the Weil pairing. By definition 3.37, the image under πα∗ of the first is equal to|M(n,∆d)|ζa1 and the image of the latter is equal to. If q is even (for example, when d and thus r is odd), the component |M(n,∆d)|ζa is independent of the choice of.
Finally, we consider how the operation of the remaining elements in J(n) behaves at the fix point for α.
Non-primitive torsion points
Remarks 3.51. Note that many times qdividesn2,L⊗n/2α is trivial, and therefore some of the precautions taken below are in fact completely empty. The only remaining obstacle to the identification of components of the fixed-point variety lies in the fact that µm no longer acts transiently on the strands of πα∗. Indeed, in the general case, k= (n,k)k (n, k) and (n,k)k being a simple tone, Corollary 3.43 reduces the statement of the proposition to that with αreplaced by (n, k) k αhset by(n, k).).
Finally, as an excuse for the notation, I provide an overview of the values of the various variables in some specific cases.
Pairwise intersections
In the cases where the fixed-point manifolds are not connected, how do the individual components intersect. Since the right-hand sides of these equations are non-isomorphic (according to Lemma 3.15 and assuming that hαi ∩ hβi= 0), ζ must be of order and Theorem 3.19 shows that πα∗(L) is stable. Since the action of β is transitive on the fibers of πα∗,J(n)/hα, β acts freely and transitively on each layer.
In this case, the varieties with fixed points each have = 2 components, and the number of layers is equal to 2.
The covering associated to a pair
In both cases, each of the component intersections has n2g−2r2 elements in each of the a(α, β, d) layers. The projection images from Lα⊕LβtoLα,LβandLα⊗Lβ ∼=Lα+βinduce images πα, πβandπγso that the following diagram commutes:. With suitable choices of the points pα,pβenpγ in definition 3.35, there exists a divisor,F of degreedn(q−1)2 onΣ, such that:˜.
Therefore, one can easily check that the following matrix describes a divisor F with the desired properties.
Triple intersections
For the second statement, continue with the proof of lemma 4.13, only this time with pα=πα(˜p),pβ=πβ(˜p) and pγ=πγ(˜p). Where we used the assumption datdis even, and hence q | n2, again in the penultimate equality. Finally, we try to reinforce the result of theorem 4.18 in the special case where d= 0ennis twice an odd number.
As discussed in the proof of Proposition 3.20, the action ζ ∈ µn on Σα covers the action ζ2 on Σα1 as well as the action ζn2 on Σα2, µ2 ⊆µn and µn˜⊆µn, which are Galois groups for the coverings of πα1 and πα2, respectively.
The Hecke correspondence
The groups constitute the central extensions of J(n), and they turn out to be independently determined by the complex structure in Σ. In Chapter 7 we will see further how elevators induce actions on Verlinde bundles on the Teichm ¨uller space of the lower surface of Σ, and are thus represented in Verlinde vector spaces in the construction of topological quantum field theories from gauge theory and conformal field. theory. During the investigation of groups we gradually strengthen the assumptions, finally giving a complete representation of elevator groups in the special case where is an odd prime.
Retraction by α induces an endomorphism α∗ of Pic(M(n,∆d))∼=Z with (α∗)n=1. Of course, the "−1" option is only relevant if niche is even.) But since Ln,dis ampleet, α∗(Ln,d)and by Kodaira's vanishing theorem,L−1n,dis not large.
Investigation of the groups of lifts
This riser acts by multiplying by ζ within the fiber over the point in P(Ep∗) corresponding tojpα(ζ∗(L)). Under the identification P(Ep∗)∼=q−11 ([E]) from Remark 5.9, liftRα, defined in the above proof, acts by multiplying by ζ on the fibers of OP(E∗p)(1) above the point that corresponds tojpα( ζ∗(L))∈q1−1([E]). The endomorphismsTα,TβandTγ defined onEp∗ at the beginning of the proof are related as follows:.
Hence, on this basis, Tα corresponds to multiplication by the matrixΛ introduced in the proof of lemma 6.25 above.
The projectively flat connection
Narasimhan and Seshadri ([10], see also section 2 in [16]) have shown that this in fact gives an integrable complex structure to M, making M a K¨ahler manifold,(Mσ, Iσ, ω). Narasimhan and Seshadri ([9]) have shown that (M, Iσ, ω) can be identified with the underlying K¨ahler manifold of Msσ(n,∆d). Since Mssσ(n,∆d) has rational singularities, (a version of) Hartog's theorem ensures that any algebraic section in L⊗kn,d can be extended over Msσ(n,∆d) over the singularities to Mssσ( n,∆d).
As mentioned earlier, Hitchin constructs the connection in a rather different way, using hypercohomology and Kodaira-Spencer deformation theory.
The group actions from the topological viewpoint
Moreover, the Narasimhan-Seshadri map associates to a class[φ]∈M the vector bundle E= ˜Σ×π1(S)Cn, whereπ1(S)acts on Cnviaφ. Therefore, when Sis gets a complex structure, the Narasimhan-Seshadri map takes the action from Hom(π1(S), µn)onM to the action of J(n)onM(n,O). For each pointσ∈T, the identification of M with Msσ(n,∆d) takes the elevators defined above to those defined in Definition 6.5.
Of course, the presentation of the groups of elevators given in Chapter 6 is far from clear with this topological definition.
The action on the TQFT spaces
This will allow a truly flat connection to be defined in the tensor product Zk⊗L. However, there are obstacles to doing this in a way that preserves the behavior of the mapping class group. The action of the groups E(n, d) goes easily through this last step, simply with the identity action on the fractional power of L†ab. From the topological point of view (using modular tensor categories to construct Reshetikhin-Turaev modular functors), Blanchet and others constructed similar representations and decompositions of the Verlinde vector spaces.
Andersen, Asymptotic Faithfulness of the QuantumSU(n) Representations of the Mapping Class Groups, for at blive vist i Ann.
