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Some Symplectic Geometry
This section gives a small amount of background material on symplectic geometry and Lie group actions. The aim is mainly to give the conventions we use in this thesis regarding Hamiltonian vector fields, fundamental vector fields and moment maps. At the end we derive the formulae we will need for moment maps and symplectic structures for the natural actions of Lie groups on their cotangent bundles. Some general references for symplectic geometry are the works [6, 41, 66, 81]. This thesis deals exclusively with complex symplectic structures and for this the start of the paper [28] was useful. For symplectic fibrations, see [40, 81].
4.1. Fundamental Vector Fields. Firstly suppose M is a complex manifold and a complex Lie group G acts on M. That is, we have a map : G ×M−!M; (g,m) 7−! (g,m) = g · m such that (gh) ·m = g · (h ·m) and 1 ·m = m. Thus for any point m 2 M there is a map from G to M defined by m : G −! M; g 7−! g · m. If we differentiate m at the identity of G we obtain a linear map from the Lie algebra g of G to the tangent space TmM to M at m.
Definition 1.42. The fundamental vector field VF (X) on M associated to an element X 2 g and the action is the vector field on M such that, for any point m 2 M VF (X)m = −(m)(X) 2 Tm(M).
The Groups Gk and their Coadjoint Orbits
Let k be a positive integer and consider the ring C[]/(k) of polynomials in an indeterminate , modulo terms of order k.
• Gk is the complex Lie group consisting of invertible n × n matrices with entries in C[]/(k): Gk := GLn(C[]/(k)).
• Let gk denote the Lie algebra of Gk and gk the vector space dual of gk.
• Let Bk be the subgroup of Gk consisting of elements with constant term 1: Bk := { g 2 Gk g(0) = 1 }.
• Let bk be the Lie algebra of Bk and bk the vector space dual of bk.
For k = 1, the group Gk is just GLn(C) but for k > 1, Gk is not even reductive since Bk is then a nontrivial unipotent normal subgroup; there is an exact sequence of groups: (21) 1−!Bk−!Gk−!GLn(C)−!1 where the homomorphism onto GLn(C) is given by evaluation at = 0. This sequence splits because GLn(C) embeds in Gk as the subgroup of constant matrices. It follows that
Gk is the semi-direct product GLn(C) n Bk where GLn(C) acts on Bk by conjugation. Coadjoint orbits of the groups Gk will be the building blocks out of which the moduli spaces M(A) are formed, so they will be studied in some detail. An element g 2 Gk is of the form: g = g0 + g1 + · · · + gk−1k−1.
Extended Moduli Spaces
In this section we give explicit finite dimensional symplectic descriptions of spaces of compatibly framed meromorphic connections with fixed irregular types on trivial vector bundles over P1. The story begins similarly to the last section: things have now been set up so that we can literally just replace the coadjoint orbits of Section 1 by the extended orbits of Section 2.
Having done that we find that our study of the geometry of the extended orbits tells us a lot about these ‘extended’ moduli spaces. Firstly we see they decouple into a product of symplectic manifolds and we then deduce that the symplectic isomorphism class of any extended moduli space is not dependent on the choice of nice irregular types, but just on the pole orders k1, . . . , km and on the rank n (Corollary 2.44).
Secondly we see that for each pole there is a torus action which changes the choice of compatible framing at that pole. Moreover these actions are Hamiltonian (with respect to the symplectic structure we have defined) having moment maps given by the exponents of formal monodromy. It follows that the moduli spaces of the last section are obtained by taking symplectic quotients by these torus actions.
Symplectic Structure and Moment Map
Moduli spaces of flat (nonsingular) connections over compact surfaces (possibly with boundary) have been intensively studied recently. In particular they have natural symplectic or Poisson structures which give deep geometrical insight (see the lecture notes [11] of M. Audin for a very readable overview).
In this section we observe that the well known Atiyah-Bott symplectic structure on nonsingular connections naturally generalises to the singular case we have been studying. Moreover, as in the nonsingular case we find that the curvature is a moment map for the action of the gauge group. Thus the moduli spaces of flat connections, which were identified with M(A) and Mext(A) in the previous section, arise as infinite dimensional symplectic quotients. We will concentrate on the (better behaved) extended case here and find, as in the previous chapter, that M(A) is a finite dimensional (symplectic) quotient of Mext(A) by a torus.
The main technical difficulty here is that standard Sobolev/Banach space methods cannot be used since we want to fix infinite-jets of derivatives at the singular points ai 2 P1. Instead the infinite dimensional spaces here are naturally Fr´echet manifolds. We will not use any deep properties of Fr´echet spaces but do need a topology and differential structure (the explicitness of our situation means we can get by without using an implicit function theorem1). The reference used for Fr´echet spaces is Treves [101] and for Fr´echet manifolds or Lie groups see Hamilton [43] and Milnor [83]; we will give direct references to these works rather than full details here.
4.1. The Atiyah-Bott Symplectic Structure on Aext(A). Consider the complex vector space 1[D](P1, End(E)) of n × n matrices of global C1 singular one forms on P1 with poles on D (see p51). This is the space of sections of a C1 vector bundle and so can be given a Fr´echet topology in a standard way ([43] p68).
Monodromy Manifolds and Monodromy Maps
The basic idea is this: a compatibly framed meromorphic connection gives rise to canonical fundamental solutions on each sector at each singularity ai. All of these fundamental solutions extend to (multivalued) fundamental solutions on P1 \ {a1, . . . , am}. The (generalised) monodromy data is simply the collection of all the constant n×n matrices that occur as the ‘ratios’ of any of these fundamental solutions, together with the exponents of formal monodromy. To make all this more precise and to remove some of the redundancy here we will fix some auxiliary data (which will remain fixed throughout this chapter).
As in previous chapters, choose an effective divisor D = k1(a1) + · · · + km(am) on P1, a local coordinate zi vanishing at ai and nice formal normal forms A. For i = 1, . . . ,m choose disjoint closed discs ¯Di P1 centred at ai with interior Di. Label the anti-Stokes rays at ai as id1, . . . , idri and fix the radius of the open sector iSectj = Sect(idj , idj+1) to be equal to that of Di. Thus Di is a disjoint union of the point {ai}, the rays id1, . . . , idri and the sectors iSect1, . . . , iSectri . Choose a branch of log(zi) on id1 and extend it in a positive sense to all of Di \ ai as usual. Pick a base-point pi in the last sector iSectri at ai for each i and choose disjoint paths i : [0, 1] ! P1 \ {a1, . . . , am} joining p1 to pi for i = 2, . . . ,m (and not intersecting ¯Dj for j 6= 1, i). Write [ i] for the track i([0, 1]) of the ith path. Let li be a simple closed loop in Di based at pi and going once around ai in a positive sense (i = 1, . . . ,m). Without loss of generality we will assume the paths ihave been chosen such that the loop: ( −1 m · lm · m) · · · ( −1 3 · l3 · 3) · ( −1 (57) 2 · l2 · 2) · l1
Table of contents :
Chapter 0. Introduction
1. Motivation
2. Schlesinger’s Equations
3. Summary of Results
Chapter 1. Background Material
1. Meromorphic Connections and Linear Differential Systems
2. Stokes Factors, Torus Actions and Local Monodromy
3. Stokes Matrices
4. Some Symplectic Geometry
Chapter 2. Meromorphic Connections on Trivial Bundles
1. The Groups Gk and their Coadjoint Orbits
2. Extended Orbits
3. Moduli Spaces and Polar Parts Manifolds
4. Extended Moduli Spaces
5. Universal Family over Mext(A)
Chapter 3. C1 Approach to Meromorphic Connections
1. Singular Connections: C1 Connections with Poles
2. Smooth Local Picture
3. Globalisation
4. Symplectic Structure and Moment Map
Chapter 4. Monodromy
1. Monodromy Manifolds and Monodromy Maps
2. Monodromy of Flat Singular Connections
Chapter 5. The Monodromy Map is Symplectic
1. Factorising the Monodromy Map
2. Symplecticness of Lifted Monodromy Maps
Chapter 6. Isomonodromic Deformations
1. The Isomonodromy Connection
2. Full Flat Connections and the Deformation Equations
3. Isomonodromic Deformations are Symplectic
Chapter 7. One Plus Two Systems
1. General Set-Up
2. Poisson-Lie Groups
3. From Stokes Matrices to Poisson-Lie Groups
4. The Two by Two Case
Chapter 8. Frobenius Manifolds
1. Frobenius Manifolds and Poisson-Lie Groups
2. Explicit Local Frobenius Manifolds
Appendix A. Painlev´e Equations and Isomonodromy
Appendix B. Formal Isomorphisms
Appendix C. Asymptotic Expansions
Appendix D. Borel’s Theorem
Appendix E. Miscellaneous Proofs
Appendix F. Work in Progress
Appendix G. Notation
Bibliography
