In [AR16] we weakened the stiffness criterion by adding the possibility of changing the bivector fieldG(V)σ by adding a term in the form ¯∂β(V)σ·ω˜ for any vector field β(V )σ∈C∞(Mσ, T0Mσ) . A Hitchin connection in a bundle Hˆk is a connection of the form (1) that preserves the subspaces Hσ(k) within each fiber Hˆk.
Splitting of the Tangent Bundle
Another construction, which we will need later, is the canonical line bundle of MJ, which is just defined as the outer upper power of the holomorphic tangent bundle. The Hermitian structure in the gain-induced T0MJ induces a Hermitian structure in the canonical line bundle, which we will also denote by h.
Complex Manifolds
We will use this fact extensively in later calculations, where we will see that ω vanishes when contracted with two vector fields of type (1,0).
Connections and Curvature
Divergence
We will also encounter the concept of divergenceδ(X) of a vector field X, which is initially only defined using the volume formωn and Lie derivatives of the formula. However, we will not see the divergence on this form, but it will arise when we calculate certain expressions involving the trace and the Levi-Civita connection, since the divergence on a Kähler manifold can be calculated by the expression.
First Chern Class
Hodge Theory and Ricci Potentials
Form in the kernel of one of the Laplace operators is called harmonic with respect to this Laplace, and on Kähler manifolds these are the same for each of the above. One of the main points for us later is the need to choose a polarization when doing geometric quantization.
Geometric Quantization
Prequantization
The pre-quantum space of the level k∈Ni is the infinite-dimensional vector space of sections of the kth tensor power from L. Then we define the pre-quantum map and send the function f ∈C∞(M) to the operator on Hk with the expression.
Polarization
Conversely, we obtain the polarization at any Kähler manifold if we let P = T00MJ, and all the polarizations we will study in this way arise from the Kähler structure at the manifold. It is a subspace of the prequantum space Hk and is finite-dimensional if M is compact.
Kähler Quantization
The proof of this relies on Tuynman's theorem [Tuy87], which states that quantum operators can be represented as Toeplitz operators. When combined with the theorem of Bordemann, Meinrenken and Schlichenmaier on the asymptotic behavior of the commutator of Toeplitz operators [BMS94], which we will return to after the introduction of Toeplitz operators in Section 2.4, the commutation relation follows quite easily.
Deformation Quantization
We can lose the subscriptJ when the connection to the complex structure is clear or irrelevant. With this construction we lose the commutation relation (Q3), but we do get the asymptotic relation (˜Q3), which is explicitly stated.
Berezin-Toeplitz Deformation Quantization
Besides constructing the Berezin-Toeplitz distortion quantization, the Toeplitz operators, as mentioned, are used to show that geometric quantization with Kähler polarization satisfies the asymptotic commutation relation. In the same paper, they show the following asymptotic result on the commutator of the Toeplitz operators.
Hitchin Connections
Before starting the construction of the Hithin compound in the next chapter, we examine the properties of families of Kähler structures on a symplectic manifold (M, ω). For each complex structure we obtain a splitting of the tangent bundle as described in section 1.2.
The Canonical Line Bundle of a Family
Then we consider the upper external power, which, as in the case of T0Mσ, gives a line bundle, and we will call this the canonical line bundle of the family of complex structures and denote it. Now we want to give an expression for the curvature ˆ∇K, but before we are ready to do that we will introduce some notation. We then let V, W be vector fields on T and calculate the variation using Leibniz's rule.
Holomorphic Families of Kähler Structures
It can be shown that the holomorphy of the family of complex structure as defined above corresponds to ˆJ being an integrable almost complex structure on T × M (see [AGL12]).
Rigid Families of Kähler Structures
For the last two terms we use the expression for the variation of the Levi-Civita connection (3.2), which we said before and using symmetries and type decompositions, we get Now we raise the index a and add the term G(V)au∇uG(W)bc to both sides of the equation, which gives us. Writing 6S(G(V)· ∇G(W)) and using symmetry gives exactly two of each of the first three terms on the right-hand side.
Weakly Restricted Families of Kähler Structures
Families of Ricci Potentials
We include this chapter in the special example of the modulo space of flat SU(n) connections, since this is first of all an interesting and well-studied example where our Hitchin connection constructions in the following chapters apply, and second because of its historical importance in the development of theory and its connection to physics.
Chern-Simons Theory and the Moduli Space
Here it turns out that the space of classical solutions is given by the space of modules of flat connections on Σ. For a flat connection, parallel transport depends only on the homotopy type of the loop, so the mapping restricts the homomorphism specification from π1(Σ) to SU(n). Moreover, choosing another point in the fiber would give a conjugate homomorphism so that the map is well defined from F → Hom(π1(Σ),SU(n)/SU(n) and since gauge equivalent links have conjugate holonomic representations the mapping is bounded to a mapping from M that is an isomorphism.
The Tangent Bundle and Symplectic Structure
Teichmüller Space and Kähler Structure
The Tangent Bundle and Symplectic Structure Since Z is central, this subspace is preserved by the conjugation operation, and since every If A is an irreducible flat connection, then by Hodge theory we have an identification of the tangent space with the harmonic forms. The subsetM∗ ⊂ Nσ is the smooth part, and if we denote the corresponding complex manifold by M∗σ, (M∗σ, ω) will in fact be a be Kähler manifold.
Quantization of Moduli Spaces
Through the work of Narasimhan and Seshadri [NS64], this is known to induce an almost complex integrable structure in M∗. In fact, the choice of the complex structure in Σ allows us to identify the entire module space of flat connectionsMme themodule space of the semi-unstable holomorphic vector. bundles of the trivial determinant. We denote this module space by Nσ.
The Mapping Class Group
In the following, we will make calculations moduloDn−2(M,Lk), and by this we simply mean that the equations hold for additions of differential operators of order −2 or lower. The proof is inductive starting at n= 2, but first we will start with the following small calculation, which we will use several times. Before saying this, we will introduce the following map on cohomology, which, together with the previous results, will play an important role in the existence theorems of the Hitchin connection in Chapter 7.
Sections vanishing up to order N − 1
Forn≥2, for every symmetric holomorphic tensor Gn∈C∞(Mσ,Sn(T0Mσ)), every holomorphic section s∈H0(Mσ,Lk), and every vector field Z ∈C∞(Mσ, T00Mσ), we get the identity . Combining Theorem 1.3 and Lemma 5.2 we immediately compute the following, the only additional ingredient being to apply the symmetry in the fifth equality. This is a result of the theory of jets or sections, and the proof follows the same idea, which is applied in the proof of Kodaira's embedding theorem.
Differential Operators Preserving the Subspace of Holomorphic sections
In this section, we go through two explicit constructions of the Hitchin connection in geometric quantization. We will assume that these quantum spaces form a smooth subsheaf of finite rank ˆH(k) of the trivial sheaf. The proof follows directly by inserting an expression for the Ricci form given in terms of the family of Ricci potentialsFσ, i.e. ρσ=nωσ+ 2id∂¯σFσ, in (6.2).
The Weakly Restricted Case
If none of the complex structures admit non-constant holomorphic functions on M, which is true for example if M is compact, we get that ψ(V) = 0. Note that if none of the complex structures admit non-constant holomorphic functions on M , we get that ψ(V) = 0, since the equationidψ(V) = ¯∂ψ(V) shows that ψ(V) is anti-holomorphic. If none of the complex structures admit non-constant holomorphic functions on M, we get that ψ(V) = 0.
Uniqueness of the Hitchin connection
Now we see that ˜ψ(V) is uniquely determined by this equation, since H0(M, T0Mσ) = 0 implies that M has only constant holomorphic functions. In this way, we came to the conclusion that all the symbols and thus the operator ˜u are uniquely determined, and so it must be the u constructed in Theorem 6.11.
Hitchin connection for smooth families of complex structures
In this section, we continue the study of the Hitchin connection in the weakly restricted case, which Andersen and I constructed in [AR16]. The proof relies heavily on G(V) being the (2,0) part of G(Ve ), which we cannot expect to repeat in the weakly restricted case. In the calculation we used that the Levi-Civita curvature is of type (1,1), and since Gβ is always of type (2,0), all the curvature terms contracted with it vanish.
Commutators of Differential Operators
Commutators in The Weakly Restricted Case
Clearly, the entries in −[∇TW, u(V)] are computed exactly equivalently, and we can easily extract the symbols from the above to obtain the expression required in the lemma. Things get much more complicated when we start calculating [u(V), u(W)] and this calculation will rely heavily on Lemmas 8.2 and 8.3. Due to the length of the expressions, we divided the calculation again into a series of smaller lemmas.
Curvature in the Weakly Restricted Case
We continue with the second order symbol, which gets contributions from C0, C1 and C2 and further collects the terms in powers of 2k+n, which we will expand on later. Curvature in the Weakly Restricted Case For the first order symbol we get contributions from C0, C1, C2 and C3. This time we get contribution from C0, C4 and C5, and we didn't do anything except add the terms together and we get the expression in the statement of.
Projective Flatness
Explicitly, in the case of the coadjoint orbit, we show that we get a Hitchin connection that cannot be projectively planar. This connection is invariant under the local action of the group of bundle automorphisms of the prequantum line bundle (L, h,∇) that includes the symplectomorphism group of (M, ω). Then each section Ψ will induce an isomorphism of the beams as above, so we get the following commutative diagram.
Pullback of the Hitchin connection on P n
We know that we get a projectively flat connection on the bundle of quantum states (H(k))|T restricted to the rigid partT of the compatible complex structures, since this space satisfies all the requirements of [AG14]. Retraction of the Hitchin connection onPn such that any set of lifts of these toCn+1 is a basis. In this chapter we give a number of examples where we can solve the weakly bounded criterion for open subsets of the entire family of complex structures on a given symplectic manifold, so that we can apply Theorem 6.11 to obtain a Hitchin connection on such subspace of all complex structures on the given symplectic manifold.
Symplectic Tori
We now withdraw the Hitchin connection in ˜H(k) to π∗H˜(k) and push it to H(k) by this isomorphism to obtain a projectively flat connection. We will now show that our construction of the Hitchin connection in the weakly restricted setting holds to give a construction of a natural partial connection in H(k) over Cω(M). The Moduli Space of Flat SU(n)-Connections We see that we can apply our Hitchin connection construction along the distribution.
The Moduli Space of Flat SU(n)-Connections
