Afhandling

T he M oduli S pace of F lat C onnections on a S urface P oisson S tructures and Q uantization

A nders R eiter S kovborg

T he M oduli S pace of F lat C onnections on a S urface P oisson S tructures and Q uantization

AndersReiterSkovborg

PhD Dissertation July2006

Supervisor: JørgenEllegaardAndersen

Department ofMathematicalSciences Faculty ofScience, University ofAarhus

Contents

1 Introduction 1

2 The Moduli Space and the Algebra of Chord Diagrams 3

2.1 Lattice Gauge Field Theory . . . 3

2.2 Poisson Structures for Fat Graphs . . . 7

2.3 Poisson Structures on the Moduli Space . . . 13

2.4 The Poisson Algebra of Chord Diagrams . . . 17

2.5 Chord Diagrams as Functions on the Moduli Space . . . 19

3 Quantization of Poisson Algebras 24 3.1 Filtered and Graded Objects . . . 24

3.2 Deformation Quantization and-Products . . . 27

4 Quantization of the Algebra of Chord Diagrams 34 4.1 Chord Tangles . . . 34

4.2 Links and Non-Associative Tangles . . . 36

4.3 Universal Vassiliev Invariants . . . 38

4.4 -Products and Standard Situations . . . 41

5 Quantization of the Moduli Space 44 5.1 The General Linear Case . . . 44

5.2 The Special Linear Case . . . 62

5.3 Miscellaneous Results . . . 66

6 Quantization of the Loop Algebras 69 6.1 The Turaev-Vassiliev Quantization . . . 69

6.2 The AMR-Products and the Turaev-Vassiliev Quantization . . . 71

7 The CaseSL₂^pC^qRevisited 78 7.1 A Good Model for the Moduli Space . . . 78

7.2 The BFK-Product . . . 85

7.3 The AMR-Products and the BFK-Product . . . 87

7.4 Differentiability of the BFK-Product . . . 93

Bibliography 99

iii

Chapter 1

Introduction

Throughout this dissertation we work in the following set-up: We denote byΣan oriented, compact and connected surface, possibly with boundary. A basepointx0^PΣis fixed, and we let π1^pΣq π1^pΣ,x0^qdenote the fundamental group. Furthermore, G is a linearly reductive, affine algebraic group over the complex numbers (e.g. GLn^pC^q, SLn^pC^q, On^pC^q and Sp_2n^pC^qq. By a standard result (cf. [Hu]),Gis a closed subgroup of GLn^pC^qso that, in particular,Gis a Lie group; we writegfor its Lie algebra. The moduli space of flatG- connections onΣis denoted byM^pΣ;G^q. It is well-known that there is a canonical bijection Hol : M^pΣ;G^qÑHom^pπ₁^pΣq,G^qLG (1.1) where the G-action on Hom^pπ1^pΣq,G^qis by conjugation and Hol is given by taking the holonomy with respect to a flat connection along loops onΣbased atx0. LetΓ pΣqdenote the group of orientation preserving diffeomorphisms ofΣ. This group acts onM^pΣ;G^qvia pullback of connections:

g^rA^srpg^1qA^s, ^rA^sPM^pΣ;G^q, g^PΓ pΣq

so that the induced action on Fun^pM^pΣ;G^qqMap^pM^pΣ;G^q,C^qis given by

pgf^qprA^sq f^pg^1rA^sqf^prgA^sq, f ^PFun^pM^pΣ;G^qq, g^PΓ pΣq.

For a synopsis of the dissertation, the reader may consult the table of contents and the introductory paragraphs of the individual chapters.

It is presupposed that the reader is familiar with a few basic concepts and results from algebraic geometry and invariant theory (cf. [Fog]). For his convenience we recall the relevant material here. An affine algebraic setXV^pS^q„C^Nis the solution of a setSof polynomial equations inNvariables; associated to it is the ideal I^pX^{q „} C^rx1, . . . ,xN^sof polynomials vanishing onX. The Hilbert Nullstellensatz states that

IV^pa^q

?

a, aan ideal inC^rx1, . . . ,xN^s.

Occasionally the radical ideal^?a is denoted by Rad^pa^q. The ringO^pX^qof regular func- tions onXis isomorphic toC^rx1, . . . ,xN^s{I^pX^q. Algebraic morphisms between affine sets preserve regular functions; the action onO^pX^qinduced by an algebraicG-action onXis ra- tional. By the linear reductivity ofG, rational actions have well-behaved invariants. To be specific, ifGacts rationally on a complex vector spaceV, then the subspaceV^{G „}Vof fixed points has a uniqueG-invariant complementVG. The linear projection∇∇V:V^ÑV^G

1

2 Chapter 1Introduction

with kernelVG is called the Reynolds operator onV. The uniqueness ofVG implies that Reynolds operators are natural with respect toG-equivariant, linear mapsϕ:V ^ÑW, i.e., the diagram

V ^ϕ

∇

W

∇

V^{G ϕ|} W^G is commutative.

Remark 1.1. A simple, but important consequence of this is thatϕ|is surjective ifϕis.

IfVis an algebra (andGacts by algebra isomorphisms), then Reynolds’ identity

∇^pxy^q∇^px^qy, x^PV,y^PV^G (1.2) holds.

Chapter 2

The Moduli Space and the Algebra of Chord Diagrams

A Poisson structure on the algebra of functions onM^pΣ;G^qhas been studied by a number of people, e.g., Atiyah and Bott [AB], Goldman [G1, G2], Biswas and Guruprasad [BG], and Fock and Rosly [FR]. The authors approach the subject differently but common to all is that the Poisson bracket^t,^u_Bis defined in terms of anorthogonal structureonG, that is, a non-degenerate, symmetric, bilinear mapB:gg^ÑCinvariant under the adjoint action.

In this chapter we first construct the algebraO^pM^pΣ;G^qqof regular functions on the moduli space and then adapt the presentation in [FR] to define a Poisson bracket^t,^uton O^pM^pΣ;G^qqfor any symmetric Ad-invariant tensor t ^P g^bg; this generalizes the afore- mentioned Poisson structure since^t, ^uB ^t,^ut_B wheretB ^P g^bgis the symmetric Ad- invariant tensor corresponding toB^Ppg^bg^q under the isomorphismgginduced by Bitself. Afterwards we present the Poisson algebra of chord diagramsC^pΣ;G^qintroduced by Andersen, Mattes and Reshetikhin [AMR1]. One of the main results of this paper is that there exists a Poisson homomorphismΨ_B:C^pΣ;G^qÑpO^pM^pΣ;G^qq,^t,^uB^q; we generalize this to all Poisson brackets^t,^ut.

2.1 Lattice Gauge Field Theory

Agraph Kis a finite, 1-dimensional CW-complex with an orientation on each 1-cell. Its set of vertices is denoted byV^pK^qand its set of edges byE^pK^q. We also consider the setE^BpK^q of all endpoints of edges ofK. It is important to notice the distinction between vertices and endpoints; the two concepts are related by the obvious ‘incidence’ map

r s:E^BpK^qÑV^pK^q.

In the sequel we identify a vertex with its pre-image under this map. The endpoints of an edge are given by the maps

B ,^B:E^pK^qÑE^BpK^q.

An edgeα^PE^pK^qmay be traversed according to or counter to its orientation, yielding two curves,αandα¹, inK. ApathinKis a curve inKwhich is a composition of edge traversals, i.e., they have the formα^ǫ₁¹α^ǫnⁿ with the compatibility condition that^B α^ǫ_iⁱ :^Bǫ_iα_iand

B

α^ǫ_iⁱ ₁¹ :^Bǫ_{i 1}αi 1fori 1, . . . ,n1 are incident to the same vertex. Forloops(cyclic paths) we always work with indices modn.

3

4 Chapter 2The Moduli Space and the Algebra of Chord Diagrams

Acombinatorial complex K is a 2-dimensional CW-complex obtained from a graph by attaching a finite number of 2-cells along loops in the graph. The set ofgraph connections on K(more precisely, on its underlying graph) is simply the productG^EpKq; for a connection A^pAα^qPG^EpKqthe element Hol^Apα^q Aα ^PGis called theholonomywith respect toA alongα. We extend the concept of holonomy to paths in the obvious way:

Hol^Apα^ǫ₁¹α^ǫ_n^nqA^ǫ_α¹₁A^ǫ_αⁿ_n, A^PG^EpKq.

Notice that for a loop without a specified initial point the holonomy is still a well-defined conjugacy class. Therefore the equations

Hol^ApBF^q1, A^PG^EpKq (2.1)

withFrunning over all faces inKmake sense and define a subsetA^pK^qA^pK;G^q„G^EpKq called theG-connections on K. Thegauge groupofKis by definitionG^pK^qG^VpKq; it acts on the graph connections:

pgv^qpAα^qpg

rBα^sAαg¹

rB α^sq, ^pAα^qPG^EpKq, ^pgv^qPG^pK^q.

Since this action conjugates the holonomy along a loop, A^pK^qis an invariant subset; the orbit space

M^pK;G^qA^pK^qLG^pK^q

of the restricted action is called themoduli space of G-connections on K. The natural projection π:A^pK^qÑM^pK;G^qsets up a bijection

π: Fun^pM^pK;G^qqÑpFun^pA^pK^qqqGpKq.

AsA^pK^qis cut out ofG^EpKqby the algebraic equations (2.1), it is an affine algebraic set. The gauge group action on A^pK^qis clearly algebraic whence the induced action on functions preserves the property of being regular. Set

O^pM^pK;G^qqpπ^q1 O^pA^pK^qqGpKq„Fun^pM^pK;G^qq. (2.2) The notationO^pM^pK;G^qqis merely suggestive; we do not claim thatM^pK;G^qadmits the the structure of an algebraic variety.

Now supposeι:K^Ñ Σis an embedding such thatK ^„Σis a deformation retract; we callK^pK,ι^qamodelforΣ. There is a canonical bijection

Holι:M^pΣ;G^qÑM^pK;G^q

with the following description (cf. (1.1)): LetAbe a flat connection in a principalG-bundle P ^Ñ Σ. Choose for eachv ^PV^pK^qa basepoint in the fibre ofPoverv(a trivialization of P|v). This allows the holonomy with respect toAalong an edgeα^P E^pK^qto be expressed as an elementAα^PG, and we have

Holι^prA^sqrpAα^qsPM^pK;G^q, ^rA^sPM^pΣ;G^q. We define thealgebra of regular functions on the moduli spaceby

O^pM^pΣ;G^qqHol_ι^pO^pM^pK;G^qqq„Fun^pM^pΣ;G^qq.

2.1 Lattice Gauge Field Theory 5

Proposition 2.1. The subsetO^pM^pΣ;G^qq„Fun^pM^pΣ;G^qqis independent of the model used to define it.

Proof. Suppose we have two modelsιj: Kj ^ÑΣ, j 1, 2. Pick a mapρ:V^pK2^qÑV^pK1^q

and for eachv^PV^pK2^qa curveγvonΣfromρ^pv^qtov. Consider an edgeαofK2. SinceK1is a retract ofΣand by cellular approximation, the curveγ

rB

α^spι2^q|αγ¹

rB α^sonΣis homotopic rel endpoints to a pathPαinK1. Define a mapϕ:G^EpK1qÑG^EpK2qby

ϕ^pA^qαHol^ApPα^q, A^PG^EpK1q.

Assume thatA^PA^pK1^qand letP^ÑΣbe a principalG-bundle. Upon trivializingP^|V^pK₁^q, Adefines a flat connection ˜AonΣrepresenting Hol_ι1^1prA^sq. Now trivialize the fibre over v ^P V^pK2^qby parallel transporting the basepoint over ρ^pv^qwith respect to ˜A along γv; this choice implies that Holι₂^prA˜^sqis represented by the graph connectionϕ^pA^q. In conse- quence, not only doesϕmapA^pK1^qintoA^pK2^q, it also fits into the commutative diagram

G^{EpK1q ϕ} G^EpK2q

A^pK1^q ϕ|

π₁

A^pK2^q π2

M^pK1;G^{q T21} M^pK2;G^q

(2.3)

where the bottom map is the bijection Holι₂Hol_ι1¹:M^pK1;G^qÑM^pK2;G^q.

It is immediate from the construction thatϕis an algebraic morphism intertwining the gauge group actions:

ϕ^ppgv^qA^qρ^ppgv^qqϕ^pA^q, A^PG^EpK1q, ^pgv^qPG^pK1^q.

Hereρ: G^pK1^{q Ñ} G^pK2^qis the pullback via ρ. The induced map ϕ: Fun^pG^{EpK2qq Ñ} Fun^pG^EpK1qqtherefore preserves regular functions and also intertwines the actions:

pgv^qϕf ϕ^pρ^ppgv^qq f^q, f ^PFun^pG^EpK2qq, ^pgv^qPG^pK1^q.

The same statements hold true for the restrictionϕ|:A^pK1^qÑA^pK2^q. In particular,ϕand

pϕ| q

preserve fixed points (invariant functions). It thus follows that T₂₁ ^pπ₁^q1pϕ|

q

π₂: Fun^pM^pK2;G^qqÑFun^pM^pK1;G^qq

mapsO^pM^pK2;G^qqtoO^pM^pK1;G^qq; this proves the result sinceT₂₁ ^pHol_ι1^q1Hol_ι2.^l

6 Chapter 2The Moduli Space and the Algebra of Chord Diagrams

Remark 2.2. During the course of the above proof we established the diagram O^pM^pΣ;G^qq

O^pM^pK2;G^qq

Hol_ι2

T₂₁

π₂

O^pM^pK1;G^qq

Hol_ι

1

π₁

O^pA^pK2^qqG^pK₂^{q p}ϕ| q

O^pA^pK1^qqG^pK₁^q

O^pG^{EpK2qqGpK2q ϕ}

O^pG^EpK1qqGpK1q

(2.4)

We infer, in particular, that although the construction of ϕ: G^{EpK1q Ñ} G^EpK2qdepends on various choices, the induced map^pϕ|

q

is entirely canonical; namely it is equal to the com- position

τ₁₂ :π₁T₂₁ ^pπ₂^q1:O^pA^pK2^qqG^pK2^q

ÑO^pA^pK1^qqG^pK₁^q. (2.5) This will be useful later.

Corollary 2.3. The map Holι:M^pΣ;G^{q Ñ} M^pK;G^q depends only on the homotopy class of ι:K^ÑΣ.

Proof. Letιt:K^ÑΣbe a homotopy. Consider the construction of the transferϕ:G^EpKqÑ G^EpKq from the modelι0:K ^Ñ Σto the model ι1: K ^Ñ Σ. We may takeρ Id_VpK^qand γv^pt^qιt^pv^q. Restricting the homotopyιtto an edgeαofK, we conclude that

γ

rBα^spι1^q|αγ¹

rB α^spι0^q|αrel endpoints.

Therefore, by definition,ϕId_GE^pK^q, whence the result follows from diagram (2.3). ^l We finish this section with an important class of models forΣ. Let^xgλ,λ^PΛ|rµ,µ^P M^y be a finite presentationPofπ1^pΣq. There is an associated complexKP; its 1-skeletonK^p_P^1q consists of a single 0-cell v and an edge (loop) for each generator gλ. The relations rµ

determine glueing maps used to attach the 2-cells ofKP; it follows thatA^pKP^q„G^EpKPq G^Λis simply defined by the relations ofP. Hence the map EvP: Hom^pπ1^pΣq,G^qÑA^pKP^q

given by evaluating aG-representation ofπ1^pΣqon the generators fromPis a bijection. As G^pKP^qGacts by simultaneous conjugation onA^pKP^q, there is an induced bijection

EvP: Hom^pπ1^pΣq,G^qLG^ÑM^pKP;G^q.

Pre-composing this map with Hol : M^pΣ;G^qÑHom^pπ1^pΣq,G^q{G, we obtain a bijection EvPHol : M^pΣ;G^qÑM^pKP;G^q.

A choice of representatives for the generators gλ ^P π₁^pΣqgives rise to a mapι:K^p_P^{1q Ñ}Σ (sendingvtox0). The face boundaries ofKPare mapped to trivial loops onΣby construc- tion, soιextends to all ofKP. It is now a triviality that Holι EvPHol. Thus we have an induced bijection

HolEv_PHol_ι :O^pM^pKP;G^qqÑO^pM^pΣ;G^qq (2.6) depending solely onP.

2.2 Poisson Structures for Fat Graphs 7

2.2 Poisson Structures for Fat Graphs

In this sectionKdenotes afat graph, i.e., a graph equipped with a cyclic order on each of its vertices. In drawings of fat graphs the cyclic order will always agree with the coun- terclockwise order. Our goal is to define a Poisson bracket^t ,^ut onO^pG^EpKqqGpKqwhere t^Pg^bgis an Ad-invariant, symmetric element; we achieve this by giving a bivector field onG^EpKq. Writing down this tensor requires, however, a linearization^¤of the cyclic order at the vertices ofK; such a choice is termed aciliationsince the linear order at a vertex is indicated by a small cilium between the first and the last endpoint.

LetΓpG^EpKqqdenote the set of smooth vector fields onG^EpKq, and define linear operators X^κ:g^ÑΓpG^EpKqq, κ^PE^BpK^q

as follows: Forα ^P E^pK^qandb ^P g,X^{B αp}b^qis the left-invariant vector field correspond- ing tobassigned to the factorG^α ofG^EpKq, andX^Bαpb^qis the right-invariant vector field corresponding tobassigned to the factorG^αofG^EpKq. Define bivector fields onG^EpKqby

Bt^pv,^¤q

¸

κ,λ^Pv

ǫ^pκ,λ^qpX^κbX^λqpt^q, v^PV^pK^q where

ǫ^pκ,λ^q

$

'

&

'

%

1 ifκ λ 0 ifκλ

1 ifκ^¡λ We setBt^p¤q

°

v^PV^pK^qBt^pv,^¤q, and define

tf,g^ut^xBt^p¤q;d f^bdg^y, f,g^PO^pG^EpKqqGpKq. (2.7) Remark 2.4. Unlike Fock and Rosly, we employ no classicalr-matrix in the definition of Bt^p¤q; this approach is feasible because the corresponding bracket is defined for invariant functions only. In the case wheretcorresponds to an orthogonal structure onG, the next two results are covered in [FR].

Proposition 2.5. The formula (2.7) defines a map

t,^ut:O^pG^EpKqqGpKqO^pG^EpKqqGpKqÑO^pG^EpKqqGpKq which is independent of the ciliation on K.

Theorem 2.6. The bracket^t,^utdefines a poisson structure onO^pG^EpKqqGpKq.

We shall need various basic results concerningtand the mapsX^κ for the proofs of these statements. Often we work with abasisfor t; this is a set^te1, . . . ,en^{u „} gsuch thatt

°

iei^bei. Bases exist sincetis symmetric but are by no means unique. In fact, by the Ad- invariance oftthe set^tAdg^pe1^q, . . . , Adg^pen^quis another basis for anyg^PG; we will use this observation without further mention in the sequel. Applying the shorthandX_i^κ X^κpei^qP

ΓpG^EpKqq, we may write

tf,g^ut

¸

v^PV^pK^q

¸

κ,λ^Pv

ǫ^pκ,λ^q

¸

i

X_i^κf X_i^λg, f,g^PO^pG^EpKqqGpKq. (2.8)

8 Chapter 2The Moduli Space and the Algebra of Chord Diagrams

Consider the composite map

ϕ:g^ÝadÑEnd^pg^qÝbÝÝIdÑEnd^pg^bg^qÝEvÝÑt g^bg

and defineT^pϕ^bId^qpt^qPg^b3. It is significant thatTis invariantly defined; its expression in a basis is

T

¸

i,j

rej,ei^sbei^bej. (2.9) Lemma 2.7. T is an anti-invariant tensor.

Proof. Transposing the second and third factors ofg^b3obviously mapsTtoT. For any b^Pgwe differentiate the curve

s^ÞÝÑpAd_exppsb^qbAd_exppsb^qqpt^qt, s^PR ats0 to obtain

padb^bId Id^badb^qpt^q0.

Letting the 3-cycleσ^p1, 2, 3^qPS3act ong^b3and applying this fact tobei, we get σ^pT^q

¸

i,j

ej^brej,ei^sbei

¸

i

¸

j

ej^brei,ej^{s b}ei

¸

i

¸

j

rei,ej^sbej ^beiT

as desired. ^l

Remark 2.8. Iftcomes from an orthogonal structure, then any orthogonal basis ofgis a basis fort, andTis the structure tensor ofg.

Lemma 2.9. The linear maps X^κ: g ^Ñ ΓpG^EpKqqare independent Lie algebra homomorphisms, i.e.,

rX^κpb1^q,X^λpb2^qs

#

X^κprb1,b2^sq ifκλ

0 otherwise

for b1,b2^Pg.

Proof. Whenκ λthis is simply by definition of the Lie bracket of a Lie group. Ifκand λare endpoints of distinct edges, the claim is trivial. In caseκandλare the two endpoints of a single edge, the associativity ofG(left and right multiplication commute) implies the

result. ^l

The next two lemmas are easy consequences of the compatibility of the exponential map and the adjoint action:

gexp^psb^qexp^psAdg^pb^qqg; b^Pg,g^PG,s^PR.

Lemma 2.10. Letα^PE and b^Pg. Then

pX^{B αp}b^qqA^pX^BαpAd_Aα^pb^qqqA

for A^PG^EpKq.

2.2 Poisson Structures for Fat Graphs 9

Lemma 2.11. Letκ^PE^BpK^qand b^Pg. Then

pgv^qX^κpb^qX^κpAdgrκ^spb^qq where^pgv^qPG^pK^qand^pgv^qis the derivative of^pgv^q:G^EpKqÑG^EpKq. Finally, introduce the diagonal operators

X^∆pvq

¸

κ^Pv

X^κ:g^ÑΓpG^EpKqq, v^PV

whose importance is due to the next lemma.

Lemma 2.12. Let v^PV and b^Pg. Then

X^∆pvqpb^qf 0 for any f ^PO^pG^EpK^qqGpKq.

Proof. Letγ^v_bbe the curves^ÞÑexp^psb^qassigned to the factorG^vofG^pK^q. Then, trivially, d

ds^|s0 γ^v_b^ps^qAX^∆pvqpb^qA, A^PG^EpKq.

HenceX^∆pvqpb^qis tangential to theG^pK^q-orbits ofG^EpKqalong which f is constant. ^l Proof (Proposition 2.5). Let f,g ^P O^pG^EpKqqGpKq. From Lemma 2.11 it is immediate that Bt^pv,^¤q,v ^P V and hence alsoBt^p¤q are invariant under the gauge group action. Thus

tf,g^ut is an invariant function since both f and g enjoy this property. As O^pG^EpKqq is closed under left-invariant and right-invariant derivations (cf. [Hu]), the first statement of the proposition holds true.

For the second claim we must compare any two ciliations^¤and^¤1 ofK. It suffices to consider the case in which^¤1differs from^¤only at a single vertexvwhereκ₁  κn

andκ2 ^{1  1}κn ¹κ1. Then Bt^pv,^¤qBt^pv,^¤1_v^q2

¸

λ^Pv^tκ₁^u

pX^κ1bX^λqpt^qpX^λbX^κ1qpt^q

2

¸

λ^Pv

pX^κ1bX^λqpt^qpX^λbX^κ1qpt^q

2^rpX^κ1bX^∆pvqqpt^qpX^∆pvqbX^κ1qpt^qs. An application of Lemma 2.12 then yields

xBt^pv,^¤q;d f^bdg^yxBt^pv,^¤1q;d f^bdg^y

as desired. ^l

Of course, computations with the formula (2.8) still involves a ciliation, but we are free to choose a preferred one.