An Interpretation of Some Hitchin Hamiltonians In Terms of Isomonodromic Deformation

0.27 MB·English

by  Michael Lennox Wong

An Interpretation of Some Hitchin Hamiltonians
In Terms of Isomonodromic Deformation
2 Michael Lennox Wong
1
0
2 January 26, 2013

n
a
J
3 Abstract
2
This paper deals with moduli spaces of framed principal bundles with connec-

] tions with irregular singularities over a compact Riemann surface. These spaces
G
have been constructed by Boalch by means of an infinite-dimensional symplectic
A
reduction. It is proved that the symplectic structure induced from the Atiyah–Bott
.
h form agrees with the one given in terms of hypercohomology. The main results of
t
a this paper adapt work of Krichever and of Hurtubise to give an interpretation of
m
someHitchinHamiltoniansasyieldingHamiltonianvectorfieldsonmodulispaces
[ ofirregularconnectionsthatarisefromdifferencesofisomonodromicflowsdefined

2 in two different ways. This relies on a realization of open sets in the moduli space
v ofbundlesasarisingviaHeckemodificationofafixedbundle.
0
5
9
5 Introduction
.
5
0 Thestudyoftheisomonodromicdeformationsofconnectionsonholomorphicbun-
1
dles over Riemann surfaces has its roots in Hilbert’s twenty-first problem, or the
1
: Riemann–Hilbert problem, which askswhetheronecan realize agiven representa-
v
i tion of the fundamental group of a punctured surface as the monodromy of some
X
meromorphic connection whose poles lie at the punctures. Since the fundamental
r
a group is unchanged as we vary the locations of the punctures on the surface, one
may seek to determine the precise constraints on thesemovements ensure that the
resulting monodromy remains the same. This is the problem of isomonodromic
deformation. For simple poles on CP1, the answers lie in Schlesinger’s equations
[Sch12].
For higher order poles, simply defining the monodromy data is a delicate task
(see [JMU81, Boa01, Boa02]). Over CP1, the isomonodromy equations were ob-
served by N. Hitchin to define a (complex) Poisson manifold, which is symplec-
tic over a dense open set [Hit97]. The symplectic point of view has been further
pursuedbyP.Boalch[Boa01,Boa07],whohasdescribedspacesofirregularconnec-
tions as infinite-dimensional symplectic reductions in the style of Atiyah and Bott
[AB82]. Furthermore, using the theory of quasi-Hamiltonian reduction developed
1
by A. Alekseev, A. Malkin and E. Meinrenken [AMM98], he has shown that the
spaceofmonodromydataisendowedwithanaturalsymplecticstructure,andthat
the map taking a generic compatibly framed connection to its monodromy data is
symplectic.
I.Kricheverconsideredisomonodromicdeformationforvectorbundlesandcon-
nectionsoverRiemann surfacesofarbitrary genusg [Kri02]; in thiscase, thedefor-
mation parameters include the moduli of the punctured Riemann surface and the
irregularpolarpartoftheconnection. UsingtheTjurinparametrizationofthemod-
uli space of vector bundles of rank n and degree n(g −1)+n, he also represented
theflowsasnon-autonomousHamiltonian vectorfieldsonthemodulispaceinthe
regular singular case. Following on this, J. Hurtubise extended this to bundles of
arbitrarydegree[Hur08].
We begin with a review of the definition of the monodromy data for a mero-
morphic connection with a higher order pole at the origin of the unit disc, that is,
the Stokes data associated to the connection. While this is done in [Boa02], cer-
tain aspects of the construction are used later so we give a short exposition for the
sakeofclarity. Section2describesvariousmodulispacesofbundleswithmeromor-
phic connections over a fixed compact Riemann surface, with poles bounded by a
fixed divisor, and their symplectic reductions. As various authors have observed
for Higgs bundles [BR94, Mar94, Bot95b], and following Hurtubise’s description
for vector bundles [Hur08], the relevant deformation spaces are given by the first
hypercohomologyof an appropriate one-stepcomplex; the associated Poisson and
symplectic structures are also given in these terms. The spaces constructed here
will be the fibres of a bundle on which the isomonodromy connection will later be
described.
The isomonodromy connection is constructed by showing that a connection is
determined by its monodromy data, which lie in a space independent of the holo-
morphic data of the modulus of the Riemann surface or the isomorphism class of
the bundle. Section 3 begins with a review of how the space of monodromy data
is constructed. Wecite Boalch’s resultsontheconstructionofasymplecticformon
this space and the fact that the monodromy map, which associates to a triple con-
sisting of a bundle, connection and a compatible framing its monodromy data, is
symplectic. To make the link between Boalch’s construction of the moduli spaces
asinfinite-dimensionalsymplecticreductionsinheritingtheAtiyah–Bottsymplectic
form[Boa07,§4]andthehypercohomologyrealizationofthesymplecticformgiven
intheprevioussection2,wejustifywhytheseformsagree.
The main results of the paper are given in the final section, but the story told
there relies upon being able to realize large open sets in the moduli space of prin-
cipal bundlesasHeckemodificationsofafixedbundle. Variouscaseswherethisis
possiblewereworkedoutin[Won10];Section4givesabriefreviewofthis,provid-
ingwhatisnecessaryforthesubsequentdiscussion.
Section 5 begins with a description of isomonodromic deformation as a local
splitting of, or Ehresmann connection on, a bundle over the space of deformation
parametersconsistingofthemoduliofcomplexstructuresona genusg surface to-
gether with a divisor of poles, as well as the irregular part of a connection at the
2
divisor. The fibres are the spaces of generic compatibly framed connections with
fixed irregular part, constructed in Section 2. The rest of the section describes and
proves the primary results of this paper. The main idea is as follows. Given a tan-
gent vector to the base of the just described bundle (i.e., a deformation of either
the modulus of the punctured surface or the irregular part of the connection), the
isomonodromyconnectionproducesauniquelift. Sincewearethinkingofthemod-
uli of bundles as arising from modifications of a fixed bundle, the isomonodromic
deformationofthefixedbundlegivesusasecondlift. Thedifferencebetweenthese
liftsisthereforetangenttothefibre,thusproducingavectorfieldonamodulispace
ofconnectionsoverafixedRiemannsurfaceanddivisor. Afunctiononthismoduli
space is then constructed using invariant polynomials, i.e. a Hitchin Hamiltonian,
whichturnsouttobeaHamiltonianforthevectorfielddescribedabove.
Afirstdraftofthematerialappearingherewaswrittenaspartofadoctoralthe-
sisunderthesupervisionofJacquesHurtubise. Ithankhimwarmly forexplaining
many of the ideas that appear here. I am also grateful to Marco Gualtieri for his
interest in and several discussions on the subject and to Indranil Biswas for clar-
ification on several points. I would also like to thank Ronnie Sebastian for some
troubleshootinghelp. Debtisalsoowedtotherefereewhocaughtseveralinaccura-
cies,indicated thesubstanceofRemark5.11and askedforsomeclarification in the
finalsection.
1 Local Monodromy
Let G be a semisimple complex algebraic group with Lie algebra g, let T ⊆ G be
a maximal torus with Lie algebra t, and let Φ be the associated root system with
#Φ =: 2r. Let ∆ ⊆ C be the unit disc with coordinate z and let P → ∆ be a
principalG-bundle,necessarilytrivial,andlet∇beameromorphicconnectioninP
withapoleonlyattheorigin. Inthissection,wewillbrieflyreviewwhatoneneeds
toobtainthemonodromydata,alsooftenreferredtoasStokesdata,associatedto∇.
Asmentionedin theintroduction,thedefinitionrequiressomecare andis doneby
P. Boalch in [Boa02, §2]. Since we are unlikely to improve upon his exposition, we
will describe only what is necessary for our discussion of moduli spaces and refer
thereadertherefordetailsoftheconstruction.
We will assume that ∇ has a pole of order k ≥ 2 at 0. A framing of P at 0 is
a choice of element s ∈ P in the fibre of P above 0, which we may think of as
0 0
a section of P over the single point 0. A triple (P,s ,∇) will be referred to as a
0
framed connection. Let s : ∆ → P be a section for which s(0) = s . With respect to
0
this section,∇ becomes a g-valued meromorphic1-form, and wemay considerthe
lowestordertermintheLaurentseriesexpansion:
A−k
.
zk
ThetermA−k ∈ gdependsonlyons0 andnotons.
A root α ∈ Φ may be thought of α as an element of t∗, so that ker α ⊆ t will
be a hyperplane; recall that the set t of regular elements of t is defined to be the
reg
3
complementofallsuchhyperplanes:
t := t\ ker α.
reg
α[∈Φ
A framed connection (P,s0,∇) is called compatibly framed if A−k ∈ t; it is called
genericornon-resonantifA−k ∈ treg.
Suppose now that (P,s ,∇) is a generic compatibly framed connection with
0
leading coefficient A−k ∈ treg. Then there is a unique formal transformation (i.e.
transformation in G(C[[z]]), so given by a power series which may not converge)
whose leading term is the identity with respect to which the connection form is of
theform
A0 := A−k + A−(k−1) +···+ A−2 + Λ dz,
(cid:18) zk zk−1 z2 z(cid:19)
where A ,Λ ∈ t,−k ≤ j ≤ −2. A0 is called the formal type of (P,s ,∇); the sum of
j 0
thenon-logarithmicterms,i.e.A0−Λ/zdz,iscalledtheirregulartype;andΛiscalled
theexponentofformalmonodromy.
′ ′ ′
Two compatibly framed connections(P,s ,∇),(P ,s ,∇)are said tobe isomor-
0 0
′ ′
phic if there exists an isomorphism of G-bundles ϕ : P → P such that ϕ(s ) = s
0 0
∗ ′
andϕ ∇ = ∇. Inthiscase,oneisgenericifand onlyiftheotheris,and iftheyare
generic,thentheyhavethesameformaltype.
Consider the set H (A0) of isomorphism classes of generic compatibly framed
connections with a fixed formal type A0. Let B+,B− ⊆ G be opposite Borel sub-
groups containing T and let U+,U− be their unipotent radicals. Given a generic
compatibly framed connection (P,s ,∇) ∈ H (A0), there are overlapping sectors
0
in the unit disc on each of which fundamental solutions for the connection (i.e. G-
valuedfunctionsgforwhichtheconnectionformisgivenbydgg−1)exist. Todefine
the Stokes data, one chooses an initial sector, as well as a branch of the logarithm
function to specify an initial solution. On the overlaps of the sectors, the solutions
willdifferbyaconstantelementofG;theseelements,theStokesmultipliers,willlie
inU+ andU− foralternatesectorsaswegoaroundthedisc. Weobtainamapping
H (A0) → (U+ ×U−)k−1,
calledtheirregular Riemann–Hilbertmap.
Theorem 1.1. [Boa02, Theorem 2.8] The irregular Riemann–Hilbert map is a bijec-
tion. Inparticular,H (A0)isisomorphictoanaffinespaceofdimension#Φ(k−1)=
2r(k−1).
2 Connections
In this section we will be workingover a fixed compact Riemann surface X with a
fixedeffectivedivisorD ofdegreed. Wewillwrite
m m
D = k x , D := x ,
j j red j
Xj=1 Xj=1
4
m
withthex distinctsothatd = k ,degD = m.
j j=1 j red
We will let G,g,T,t,Φ be asPin Section 1. By GD we will mean the group of G-
valued functionson D (in theschematic sense),sothat G = G(O ), that is, G is
D D D
thegroupofD-valued pointsofG. Similarly, thenotationofg willoftenbeused.
D
Wewilltypically thinkofelementsofg as polynomials in local coordinatesat the
D
supportofD withcoefficientsing.
2.1 Symplectic and Poisson Structures and Reductions
Weconsiderpairs(P,∇),whereP isaholomorphicprincipalG-bundleonX and∇
isameromorphicconnectioninP whosepolesareboundedbyD. Inthecasewhere
D is reduced, i.e. we are considering logarithmic connections, the relevant moduli
spacecanbeconstructedasin[Nit93](seealso[Sim94]);forarbitraryD,i.e.allowing
for irregular poles, the only known construction of the moduli space appears to
be an analytic one by an infinite-dimensional symplectic reduction [Boa01, Boa07].
We will denote by L (ε,D) the moduli space of such pairs whose bundle is of
X,G
topologicaltypeε∈ π (G),abbreviatingtoL(ε,D)ifX andGareunderstood.For
1
apair(P,∇) ∈ L(ε,D),thedeformationcomplexis
−∇·
adP −−→ adP ⊗K(D). (2.1)
Thatis,thespaceofinfinitesimaldeformationsof(P,∇)isgivenbythefirsthyper-
cohomology group of this complex (cf. [BR94, Theorem 2.3], [Bot95b, Propositions
3.1.2,3.1.3],[Mar94,Proposition7.1]):
H1(−∇·).
Since the Killing form on g is Ad-invariant, it gives a well-defined pairing be-
t
tweensectionsofadP andsoitfollowsthat (−∇·) = ∇·. Hencethedualcomplex
to(2.1)is
∇·
adP(−D) −→ adP ⊗K,
thenthediagram
adP(−D) ∇· // adP ⊗K
− (2.2)
1(cid:15)(cid:15) (cid:15)(cid:15) 1
adP //adP ⊗K(D),
−∇·
the top row being the cotangentcomplex and the bottomthe tangent complex, de-
fines a Poisson structure on L [Mar94, §6,7]. The vanishing of the Schouten–
ε,D
Nijenhuisbracket,i.e.theJacobiidentity,can beprovedasin[Bot95a,§5]or[Pol98,
§6].
Wewillwanttorealize thespacesL(ε,D)slightlydifferently. Wewillconsider
triples(P,s,∇),where(P,∇) ∈L(ε,D)andsisalevelstructureofP overD,i.e.a
5
section of P over D or, equivalently, a trivialization of P over D; the space ofsuch
triples will be denoted P(ε,D). Since the space of infinitesimal deformations of
a level structure (P,s) is given by H1(X,adP(−D)), the deformation complex for
(P,s,∇)is
−∇·
adP(−D)−−→ adP ⊗K(D)
anditsdualcomplex
∇·
adP(−D)−→ adP ⊗K(D).
Constructing a diagram as in (2.2), since this time we get an isomorphism of com-
plexes,theresultingPoissonstructureisnon-degenerateandweobtainasymplectic
formonP(ε,D). Observethat
dimP(ε,D) = 2dimG(g−1+d). (2.3)
The space P(ε,D) admits a free action of G with g ∈ G acting on the level
D D
structureby
g·(P,s,∇) = (P,s·g−1,∇),
anditisclearthatwemaymaketheidentification
L(ε,D) = P(ε,D)/G .
D
Itfollowsthat
dimL(ε,D) = dimP(ε,D)−dimG = dimG 2(g−1)+d . (2.4)
D
(cid:0) (cid:1)
The reason for introducing the level structures is that the symplectic leaves are
theneasilyidentifiedusingsymplecticreduction(cf.[Mar94,§6.2]).
Proposition 2.5. The G -action on P(ε,D) is Hamiltonian with moment map µ :
D
P(ε,D) → g∗ givenby
D
(P,s,∇) 7→ (s∇) .
pol
Here, (s∇) is the Laurent polynomial of g-valued 1-forms we obtain by triv-
pol
ializing ∇ with respect to the section s. It can be paired with an element of g via
D
theinvariantbilinearformandtakingresidues,andhenceyieldsanelementofg∗ .
D
Thus, the symplectic leaves of L(ε,D), which are the symplectic reductions of
P(ε,D), consist of those pairs (P,∇) for which the polar part of ∇ lies in a fixed
coadjointorbiting∗ . Again,withoutatrivialization,thepolarpartof∇isnotwell-
D
defined,but its coadjoint orbit is. If γ ⊆ g∗ denotesa coadjoint orbit, then we will
D
denotethecorrespondingsymplecticleafinL(ε,D)byL(ε,D)γ.
6
2.2 Irregular Parts
We now consider the subgroup H of G consisting of elements whose leading
D D
term is the identity, i.e., the kernel of the map G = G(O ) → G(O ). (This
D D D
red
group is referred to as B in [Boa01, §2] and as B in [Hur08, §4], but we use H
k D D
soasnottogive theimpressionthatweare referringtoaBorelsubgroup.) TheLie
algebrah ofH willthenbethekernelofg = g(O ) → g(O ),soifwethinkof
D D D D D
red
g as polynomials in the local coordinate with coefficients in g, then h is thesub-
D D
algebraofpolynomialswithzeroconstantterm. Dually,ifg∗ isrealized asLaurent
D
polynomialswithcoefficientsing,thenh∗ consistsofthosewhoselogarithmicterm
D
vanishes.
InparallelwithProposition2.5,wehavethefollowing.
Proposition 2.6. The H -action on P(ε,D) is Hamiltonian with moment map µ :
D
P(ε,D) → h∗ givenby
D
(P,s,∇) 7→ (s∇)
irr
where(s∇) istheirregularcomponentofthepolarpartofs∇,i.e.itis(s∇) with
irr pol
thelogarithmictermomitted.
Observe that the quotientL(ε,D)/H is theset oftriples (P,s,∇) where s is a
D
trivialization of P| , so that in a neighbourhoodof each x ∈ suppD, we obtain
D j
red
a framed connection; we will denote this quotient by L(ε,D) . The symplectic
cf
reductions L(ε,D)γ arising from this action therefore consist of triples (P,s,∇),
cf
wheresisalevelstructureoverD andforwhichtheirregularpolarpartof∇lies
red
inafixedcoadjointH -orbitγ ⊆ h∗ .
D D
2.3 Further Reductions
Let W be the Weyl group associated to the root system Φ; it may be realized as
W = N (T)/T,whereN (T)isthenormalizerofT inG. AsinSection1,r = 1#Φ
G G 2
will be the number of positive roots and l := rkG = dimT will be the rank of G
so that dimG = 2r +l. Let T := T(D) = T(O ) be the group of T-valued maps
D D
on D and t := t(O ) its Lie algebra. Let P(ε,D,T) ⊆ P(ε,D) be the subspace
D D
consisting of triples (P,s,∇) for which (s∇) takes values in t and hence may be
pol
consideredas an element of t∗ and for which s := s| is a genericcompatible
D red Dred
framingasinSection1.
ForafixedP and∇,itisnothardtoseethatanytwolevelstructuresthatgiveel-
ementsinP(ε,D,T)mustdifferbyanelementofN (T) ,thegroupofmapsfrom
G D
D intoN (T). SowegetanN (T) -torsor;indeed,wemaythinkofP(ε,D,T)as
G G D
a(left)N (T) -bundleoveranopensetinL(ε,D). Therefore,using(2.4),
G D
dimP(ε,D,T) = 2dimG(g−1)+(dimG+l)d =2 dimG(g−1)+(r+l)d .
(cid:0) (cid:1)(2.7)
SinceT ⊆ N (T) ,itactsonP(ε,D,T),andasbefore,wehavethefollowing.
D G D
7
Proposition 2.8. The T -action on P(ε,D,T) is Hamiltonian with moment map
D
µ :P(ε,D,T) → t∗ definedas
D
(P,s,∇) 7→ (s∇) .
pol
Let us consider the quotient, which we will denote as L(ε,D,T), its symplec-
tic leaves L(ε,D,T)η, and how they compare to those of P(ε,D). Elements of
L(ε,D,T)aretriples(P,w,∇), wherew isaclassoflevelstructurewith(w∇) ∈
pol
t∗ . Thereisaninducedmap
D
L(ε,D,T) = P(ε,D,T)/T → P(ε,D)/G = L(ε,D)
D D
taking
(P,w,∇) 7→ (P,∇).
Fromthisexpression,itisclearthatthefibresareW -torsors.
D
Sincethecoadjointorbitsint∗ aresingletons,agivensymplecticleafL(ε,D,T)η
D
of the quotient L(ε,D,T) consists of those triples for which (w∇) = η ∈ t∗ is
pol D
fixed (note that this is independent of the representative of w). The preimage of
L(ε,D)γ ⊆ L(ε,D) consists of those (P,w,∇) with (w∇) ∈ γ ∩t∗ . But this is
pol D
theunionofP(ε,D,T)η withη ∈ γ ∩t∗ ;thisintersectionispreciselytheW -orbit
D D
ofanyoneofitselements. Thus,themap
P(ε,D,T)η → L(ε,D)γ
η∈[γ∩t∗
D
isacoveringandsoanisomorphismoneachP(ε,D,T)η.
Similarly,thereisan(T ∩H )-actionandwerecordthefollowing.
D D
Proposition2.9. The(T ∩H )-actiononP(ε,D,T)isHamiltonianwithmoment
D D
mapµ : P(ε,D,T) → (t ∩h )∗ givenby
D D
(P,s,∇) 7→ (s∇) .
irr
Wemaythinkofanelementg ∈ N (T) ∩H asanN (T)-valuedfunctionon
G D D G
D that is the identity on D . But this means that the image of g must lie in the
red
identitycomponentofN (T),whichispreciselyT. Thisjustifiesthefollowing.
G
Lemma2.10. IfN (T) = N (T)(O ),then
G D G D
N (T) ∩H = T ∩H .
G D D D D
Corollary 2.11. Ifγ ⊆ h∗ is acoadjoint H -orbit, thenγ ∩(t ∩h )∗ consistsofat
D D D D
mostasinglepoint.
Fromthelemmaitfollowsthattheinducedmap
L(ε,D,T) := P(ε,D,T)/(T ∩H )= P(ε,D,T)/(N (T) ∩H ) → P(ε,D)/H
cf D D G D D D
8
isanisomorphismontoitsimage,andthatifη ∈ (t ∩h )∗ andγ isitsH -orbitin
D D D
h∗ ,thenthesymplecticreductionscanbeidentified:
D
L(ε,D,T)η = L(ε,D)γ.
cf cf
Mixingnotation,wewillwriteL(ε,D)η for thesespaces,forη ∈ (t ∩h )∗. Since
cf D D
dim(T ∩H ) = l(d−m),wehave
D D
dimL(ε,D)η = 2 dimG(g−1)+rd+lm . (2.12)
cf
(cid:0) (cid:1)
Remark 2.13. One will observenow that for η ∈ (t ∩h )∗, elementsof L(ε,D)η
D D cf
aretriples(P,s,∇),wheresisagenericcompatibleframingfor∇,and∇isoffixed
irregulartypeateachpointofsuppD.
3 Global Monodromy
Foruseinthissectionandthelast,wewilldefineaschemeLby
m
L := SpecC[z]/(zkj), (3.1)
ja=1
sothatListhedisjointunionofthe(k −1)thformalneighbourhoodsoftheorigin
j
inCfor1 ≤ j ≤ m. Fornow,wewillonlyuseLasawayofdenotingmpointswith
fixedmultiplicitiesk ,...,k ,butitwillplaymoreofaroleinSection5.
1 m
3.1 The Space of Monodromy Data
WedefinethemanifoldofmonodromydatafollowingBoalch[Boa01,§3]asfollows.
For1≤ j ≤ m,weset
Cj := G×(U+ ×U−)kj−1×t,
where, in the case k = 1, wee replace t by the dense open set (though not Zariski
j
open)
t′ := t\ α−1(Z).
α[∈Φ
Weseethat
dim C = dimG+(dimG−l)(k −1)+l = k dimG−l(k −2).
j j j j
Wenowconsideertheproduct
G2g ×C ×···×C
1 m
andobservethatitadmitsaG-action: ieneachfacteorC ,g ∈ Gactsby
j
g·(g ,Kj,Λ )= (g g−1,Kj,eΛ ),
j j j j
9
andineachfactorofG2g,theactionisbyconjugation. IfX0 referstothesubman-
g,L
ifoldoftheproductsatisfying
f
[A ,B ]···[A ,B ]g exp(2πiΛ )g−1···g exp(2πiΛ )g−1 = e, (3.2)
1 1 g g 1 1 1 m m m
thenthespaceofmonodromydataisthendefinedtobe
X := G\X0 .
g,L g,L
Itsdimensionisgivenby f
m
d dimG−l(k −2) +2gdimG−2dimG = 2[dimG(g−1)+rd+lm].
j j
Xj=1(cid:0) (cid:1)
Comparingwith(2.12), weobservethatthisispreciselydimL(ε,D)η.
cf
Without delving into the theory of quasi-Hamiltonian G-spaces and reduction
developed in [AMM98], which provides a variation of the well-known theory of
symplectic reduction where the moment maps are G-valued, we give a brief and
rough explanation of how it gives a more geometric construction of X together
g,L
with a holomorphic symplectic form. The spaces G2 = G × G can be thought of
as spaces of representations of the fundamental group of a punctured torus; they
are quasi-Hamiltonian G-spaces [AMM98, Proposition 3.2]. Glueing tori together
corresponds to what is known as the fusion product [AMM98, §6], so the data for
representations of a genus g surface will come from a g-fold fusion product G2 ⊛
··· ⊛ G2. Boalch shows that the spaces C are quasi-Hamiltonian (G × T)-spaces
i
[Boa07,Theorem5],sothatthefusionproduct
e
G2⊛···⊛G2⊛C ⊛···⊛C
1 m
isaquasi-Hamiltonian(G×Tm)-space,whoeseG-reduectionispreciselyX asde-
g,L
scribedabove. Thishasthefollowingconsequence.
Theorem 3.3. [Boa07, Theorems 3, 4, 5] The manifold X carries a holomorphic
g,L
symplecticform.
3.2 The Monodromy Map
We return to the spaces L(ε,D)η described at the end of Section 2.3. As pointed
cf
outin Remark 2.13, an elementis atriple (P,s,∇), whereP is aG-bundle, satriv-
ialization over D , and ∇ a connection with poles bounded by D and such that
red
(s∇) = η ∈ (t ∩ h )∗ is fixed. These spaces are precisely those constructed
pol D D
by Boalch via an infinite-dimensional symplectic reduction [Boa07, Definition 15,
Theorem 9] (cf. [Boa01, §4,5]), as such they are endowed with a complex analytic
symplectic form, which we will call the Atiyah–Bott form, as it is induced from a
∞
symplecticformonaspaceofC connections(cf.[AB82,§9]).
Section 1indicated howto definemonodromydataat each pole. There wesaw
that the data needed to define the monodromy data of a meromorphic connection
intheneighbourhoodofasinglepolewas:
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