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Yang-Mills on Surfaces with Boundary :

Quantum Theory and Symplectic Limit

Ambar SenguptaDepartment of Mathematics

Louisiana State University

Baton Rouge, Louisiana 70803-4918, USA

e-mail : [email protected]

Abstract. The quantum field measure for gauge fields over a com-pact surface with boundary, with holonomy around the boundarycomponents specified, is constructed. Loop expectation values forgeneral loop configurations are computed. For a compact orientedsurface with one boundary component, let M(Θ) be the modulispace of flat connections with boundary holonomy lying in a conju-gacy class Θ in the gauge group G. We prove that a certain naturalclosed 2−form on M(Θ), introduced in an earlier work by C. Kingand the author, is a symplectic structure on the generic stratum ofM(Θ) for generic Θ. We then prove that the quantum Yang-Millsmeasure, with the boundary holonomy constrained to lie in Θ, con-verges in a natural sense to the corresponding symplectic volumemeasure in the classical limit. We conclude with a detailed treat-ment of the case G = SU(2), and determine the symplectic volumeof this moduli space.

1. Introduction and Overview of Results

This paper presents the construction a quantum gauge field measure over compactsurfaces, with specified boundary holonomies, and a determination of the classical limit ofthis measure when the surface is oriented and has one boundary component.

Results concerning the quantum field measure

The construction of the measure and determination and study of the loop expectationvalues are carried out in sections 1-5. In these sections :(i) we construct the Euclidean quantum field measure for gauge theory over a compact

surface with boundary, with boundary holonomy (or its conjugacy class) specified (thegauge group is a compact connected Lie group)

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(ii) loop expectation values are computed explicitly, and it is shown that they are invari-ant under appropriate area-preserving surface homeomorphisms

The loop expectation value formulas we obtain are quite natural in view of the free-boundary expectation value formulas available in [Fi, Se2,3, Wi1]. Thus one could takethem simply as a starting point, rather than a conclusion, from the point of view of latticegauge theory. However, our objective in this paper has been to derive these expectationvalue formulas from a continuum theory.

Results concerning the classical limit and sympectic volume

In sections 6-11, we focus attention on a compact oriented surface Σ of positive genuswith one boundary component ∂Σ. Let M(Θ) denote the moduli space of flat connectionsover Σ with holonomy around ∂Σ lying in a given conjugacy class Θ in the gauge group G.The gauge group is now taken to be compact connected semisimple. In [KS2] a 2−formΩΘ was defined on the space A(Θ) of all connections whose holonomy around ∂Σ lie inΘ. It was shown that ΩΘ descends naturally to a closed 2−form ΩΘ on the moduli spaceM(Θ) = A0(Θ)/G, where A0(Θ) is the set of flat connections in A(Θ) and G is the groupof gauge transformations. In [KS2] it was proven that ΩΘ is non-degenerate on the ‘smoothpart’ of M(Θ) when the conjugacy class Θ passes through a certain neighborhood of theidentity in G; the proof of non-degeneracy in [KS2] was obtained by ‘perturbation’ of thecase Θ = e, the latter case being dealt with by means of an earlier result in [KS1]. Inthe present paper we prove the following results:(iii) ΩΘ is non-degenerate on (the ‘smooth part’ of) M(Θ) for generic conjugacy classes

Θ, i.e. for all Θ passing through a dense open subset of G - the proof rests on adeterminant identity (7.8.1) proven in Lemma 7.8

(iv) the Yang-Mills quantum-field measure

dµΘT (ω) = ZT (Θ)−1e−SYM(ω)/T δΘ (h(C;ω)) [Dω]

converges, as T ↓ 0, to the normalized symplectic volume measure on M(Θ). A pre-cise statement and the notation will be explained later; the determinant identity(7.8.1) is again the key

(v) the symplectic volume of M(Θ), in the case G = SU(2), is computed in Theorem 9.1:

volΩΘ(M(Θ)) =

2π(π − θ)[

vol(SU(2))2π2

]23

if g = 1

4π sin θ[

vol(SU(2))2π2

]23

vol (SU(2))2g−2∑∞

n=1χn(c)n2g−1 if g ≥ 2

wherein c is any element of Θ, χn(c) = sinnθ/ sin θ, with θ specified by Tr c = 2 cos θ.(The formula for g ≥ 2 is also valid for g = 1)

Related recent works

Recent interest in 2−dimensional quantum gauge theory, attested to by, for instance,the works [AIK, Be, Di, Fi, Fo, Je, KS1-3, RR, Se1-5, Wi1-3], stems in part from questionsassociated to a ‘classical limit’ of the quantum theory; in particular, in determining the

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relationship of the classical limit of the quantum field measure, over oriented surfaces,to a symplectic volume measure on the moduli space of flat connections. The results ofsections 6-11 of the present paper address the natural extension of this question to the casewhere we consider connections over surfaces with boundary with the holonomies aroundthe boundary components known up to conjugation. The investigation of the limitingquantum Yang-Mills measure arose in the case of closed surfaces in [Se1] and in Witten’spapers [Wi1,2]; a description of some of the questions in this area is given in [Wi3]. Thework [Fo] of Forman, and [Se4,5] are also devoted to the case of closed oriented surfaces.The most influential early works on the symplectic structure on the moduli space of flatconnections on closed oriented surfaces are by Atiyah and Bott [AB] and Goldman [Go]. Itseems likely that the symplectic structure that is the subject of the present work is the sameas the one obtained through group-cohomological techniques by Biswas and Guruprasad[BG].

In [Wi1], Witten gives formulas for the symplectic volumes of moduli spaces of flatconnections over a surface with punctures, with holonomies around these punctures lyingin specified conjugacy classes. For a genus g Riemann surface with one puncture, with theholonomy lying in a conjugacy class Θ, the volume given by Witten ((3.18) or (4.116) in[Wi1]) is

21

2g−1π2g−1

∞∑

n=1

sinnθ

n2g−1

where θ is such that Tr a = 2 cos θ for a ∈ Θ. To compare this with our volume formulagiven in (v) above (or Theorem 9.1) we need to note that : (i) the metric on SU(2) used in[Wi1] is given by 〈a, b〉 = −Tr (ab), and (ii) the symplectic form used in [Wi1] is 1

4π2 timesours (equation (2.29) in [Wi1]). This inner-product on SU(2) corresponds to taking SU(2)to be a 3-sphere of radius 21/2; its volume then is 2π2(21/2)3. Moreover, since Witten’ssymplectic form is (4π2)−1 times ours, and since M(Θ) has dimension 6g − 4, we mustmultiply our volume formula by (4π2)−(6g−4)/2. Putting all these pieces together in thevolume formula given in (v) above, we get

1

(4π2)(6g−4)/24π sin θ · 2 · (2π223/2)2g−2

∞∑

n=1

sinnθ/ sin θ

n2g−1

= 21

2g−1π2g−1

∞∑

n=1

sinnθ

n2g−1

in pleasant agreement with the expression given by Witten.

2. Notation and Background for Construction of the Yang-Mills Measure

2.1. The surface Σ as a quotient of the disk D. We shall work with a compact2−dimensional Riemannian manifold Σ.

In section 3 and 4, Σ is a torus with one hole, i.e. with one boundary component. Itwill be convenient to view this Σ as a quotient of the diskD = (x, y) ∈ R2 : x2+y2 ≤ 1 in

the following way. Consider the path t 7→ xtdef= (cos 2πt, sin 2πt) tracing out ∂D. If r < s

then xrxs will denote the path t 7→ xt with t ∈ [r, s]. Divide ∂D into arcs K1, K2, ..., K7,

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where Kj is given by xt with t ∈ [ j−12π ,

j2π ], j ∈ 1, ..., 7. Identify K1 with K3 (the reverse

of K3), K2 with K4, and K5 with K7, linearly. This yields the quotient space Σ and thequotient map q : D → Σ. We shall equip Σ with the orientation which makes q orientationpreserving. The point o = q(O), where O = (0, 0) is the center of D, will serve as abasepoint on Σ. The loops

A = q(x0O)q(K1)q(Ox0)

B = q(x0O)q(K2)q(Ox0)

C = q(x0O)q(K7K6K5)q(Ox0)

generate π1(Σ, o) subject to the relation

CB ABA = I (2.1.1)

wherein I is the identity in π1(Σ, o). The presence of K5 and K7 is not essential; however,when Σ is a general compact surface with more than one boundary component (as willbe the case in section 5) then arcs like K5 and K7 must be included, and this is why wechoose to keep them in our framework.

2.2. The G-bundle π : P → Σ, space of connections A, holonomies h(κ;ω),and cur-

vature Ωω. We shall work with a principal G−bundle π : P → Σ, where G is a compactconnected Lie group with Lie algebra g having an Ad-invariant inner-product 〈·, ·〉g onit. The set of all connections on P will be denoted A. The metrics on Σ and on g in-

duce a metric on A in a standard way. Fix once and for all a basepoint u ∈ π−1(o). Ifκ : [r, s] → Σ is a path on Σ with κ(r) = o, and if ω ∈ A , then we denote by τω(κ)u theparallel translate of u along κ with respect to ω. Thus if κ is a loop based at o then theholonomy of ω around κ, with u as initial point, is h(κ;ω) ∈ G given by τω(κ)u = uh(κ;ω).The curvature of ω ∈ A will be denoted Ωω; thus Ωω = dω + 1

2 [ω, ω].

2.3. The gauge transformation groups G, G0, acting on A. We shall use the setG of automorphisms of P , i.e. diffeomorphisms φ : P → P for which π φ = π andφ(pg) = φ(p)g for all p ∈ P , g ∈ G. This is a group under composition, and the subgroup

Godef= φ ∈ G : φ(u) = u will be of use. These groups act on A by (φ, ω) 7→ φ∗ω.

2.4. The Yang-Mills action. If ω ∈ A then the Yang-Mills action SYM(ω) is given by

SYM(ω) =1

2

∫

Σ

||Ωω||2g dσ (2.4.1)

where dσ is the Riemannian surface area measure on Σ, and ||Ωω||2g is the function on Σ

given by

||Ωω||2g(x) = ||Ωω(e1, e2)||2g

where x runs over Σ, and e1, e2 ∈ TpP , for any p ∈ π−1(x), are such that (π∗e1, π∗e2) is anorthonormal basis of TxΣ. Since Ωω vanishes when applied vertical vectors and since it isa 2−form, it follows from the Ad-invariance of 〈·, ·〉g that ||Ωω||2g is a well-defined function

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Σ. Furthermore, SYM is invariant under the G-action and therefore defines a function onA/G and on A/Go.

2.5. The curvature function Fω, and the parallel-transport equation. Let ω ∈ A. We

shall use the map sω : D → P given by sω(x) = τω

(

q(Ox))

u, where Ox is the radial path

from O to x. A convenient way to express the curvature is by means of the map

Fω : D → g : x 7→ Fω(x)def= Ωω(e1, e2) (2.5.1)

where on the right e1, e2 ∈ TpP , with any p ∈ π−1q(x), are such that (π1e1, π∗e2) is apositively oriented orthonormal basis of Tq(x)Σ. (If Σ were an unorientable surface theorientation on Tq(x)Σ here would be the one which would make q orientation-preserving ina neighborhood of x). Then

SYM(ω) =1

2

∫

D

||Fω||2g dσ (2.5.2)

with dσ here being the surface area measure for Σ pulled up to D by q.We shall almost always work with admissible curves on Σ; by an admissible curve we

mean a curve of the form q κ, where κ : [0, 1] → D : t 7→ κt is a path which can beexpressed (i.e. reparametrized) in polar coordinates ‘r = r(θ)’ (thus κ cuts every radius,excluding O, at most once). If κ is such a path and if q κ is piecewise smooth then foreach t ∈ [0, 1] we have a loop q(κtO) · q κ|[0, t] · q(Oκ0); these loops will be very useful.The holonomy

ht(ω)def= h (q(κtO) · q κ|[0, t] · q(Oκ0);ω) (2.5.3)

satisfies the differential equation of parallel-transport :

dht(ω)ht(ω)−1 = −d

(∫

Dκt

Fω dσ

)

(2.5.4)

where Dκt is the subset of D whose positive boundary is q(κtO) · q κ|[0, t] · q(Oκ0), i.e.

Dκt = rxs : r ∈ [0, κ(s)], s ∈ [0, t].

2.6. The map ω 7→(

Fω, h(A;ω), h(B;ω), h(C;ω))

. The map

ω 7→(

Fω, h(A;ω), h(B;ω), h(C;ω))

(2.6.1)

induces a one-to-one map from the quotient space A/Go. However, the image is constrainedby the condition (cf. (2.1.1))

h(C;ω)h(B;ω)−1h(A;ω)−1h(B;ω)h(A;ω) = h(L0 · ∂D · L0;ω) (2.6.2)

where L0 is the radial path from O to x0 = (1, 0), and h(L0 · ∂D · L0;ω) is computable,by means of (2.5.4), in terms of only Fω.

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2.7. Yang-Mills measures µT , µcT , µ

ΘT for the spaces A, Ac, AΘ. Let T > 0. The

Yang-Mills measure for A is, informally, a probability measure µT on A/G given by theheuristic formula

dµT ([ω]) =1

ZTe−SYM(ω)/T [Dω] (2.7.1)

where [Dω] is the formal Riemannian volume measure on A pushed down to A/G, and ZTis a ‘normalizing constant’. A rigorous construction of µT is given in [Se2].

Let c ∈ G, and consider

Ac = ω ∈ A : h(C;ω) = c (2.7.2)

If Θ is a conjugacy class in G, we consider also

AΘ = ω ∈ A : h(C;ω) ∈ Θ (2.7.3)

Our goal is to construct a probability measure µc on (an appropriate completion of) Ac/Go,and a probability measure µΘ on AΘ/Go which are given heuristically by

dµcT (ω) =1

ZT (c)δ(

h(C;ω)c−1)

e−SYM(ω)/T [Dω] on Ac/Go (2.7.4)

dµΘT (ω) =

1ZT (Θ)δΘ

(

h(C;ω))

e−SYM(ω)[Dω] on A/G (2.7.5)

wherein ZT (c) and ZT (Θ) are ‘normalizing constants’, and δΘ is the δ−function specified

by “∫

Gf(x)δΘ(x) dx” =

∫

Θf(x) dΘ(x)

def=∫

Gf(kθk−1) dk for any θ ∈ Θ (and dk is the

unit-mass Haar measure on G).Actually we will realize these measures on certain larger spaces Ωc and ΩΘ.

3. Construction of the measures µcT and µΘT

3.1. The Strategy. Since SYM(ω) =12||Fω||2L2(D;g), the expression (2.7.5) for µΘ

T sug-

gests that it is reasonable to construct µΘT as

a Gaussian measure on L2(D; g) times Haar measure on G3 (corresponding to theholonomies h(A;ω), h(B;ω) and h(C;ω)) conditioned to satisfy the constraint (2.6.2)as well as the constraint h(C;ω) ∈ Θ (recall that Θ is a conjugacy class in G).

Similarly, for µcT we must use the constraint h(C;ω) = c in addition to (2.6.2).3.2. Stochastic parallel transport and holonomy over the disk D. As is well known,

the standard Gaussian measure ‘on’ L2(D; g) actually lives on a Hilbert-Schmidt closure

L2(D; g). Henceforth we shall write Ωdisk for the space L2(D; g), and µT,disk for the Gaus-sian measure, with covariance scaled by T > 0, on it. The probability space for quantumgauge theory over the disk D is:

(Ωdisk, µT,disk) =(

L2(D; g), µT,disk

)

For ω ∈ Ωdisk, Fω now corresponds to a g−valued white-noise; i.e. to each Borel set

E ⊂ R2 there is a Gaussian g−valued random variable ω 7→∫

EFω dσ

def= (Fω, 1E); a

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more detailed account is given in section A3 of the Appendix. In order to impose theconstraint (2.6.2) the meaning of the holonomy h(L0 · ∂D · L0;ω) needs to be understoodfor ω ∈ Ωdisk. More generally we would like to (and will need) give appropriate meaningto the differential equation for parallel-transport (2.5.4).

Thus let κ : [0, 1] → D be an admissible path, and define parallel-transport along κby reinterpreting (2.5.4) as a Stratonovich stochastic differential equation (this idea is dueto L. Gross). Thus in place of (2.5.4) we consider the Stratonovich stochastic differentialequation

dht(ω) ht(ω)−1 = −d

(

∫

Dκt

Fω dσ

)

(3.2.1)

with initial condition h0(ω) = e

where Dκt is, as before, the region rxs : 0 ≤ r ≤ κ(s), 0 ≤ s ≤ t.

The solution ht(ω) of (3.2.1) can be obtained as a (probabilistic) limit of products of

the form exp(

∫

Dκtj

\Dκtj−1

Fω dσ)

, where t0 = 0 < t1 < · · · < tN = t and max |tj−tj−1| → 0.

Now t 7→∫

Dκt

Fω dσ is, in law, a g−valued Brownian motion with time clocked by the

quadratic variation which, given that Fω is Gaussian as described before, is simply T |Dκt |,

where |Dκt | is the area of Dκ

t . Thus the solution of (3.2.1) is, in law, simply a Brownianmotion on G with time clocked by T |Dκ

t | instead of by t.In particular, the density of ht(·), with respect to unit-mass Haar measure on G, is

QT |Dκt |(·), where Qt(x) is the heat kernel on G normalized to

∫

GQt(x) dx = 1 (here dx is

unit-mass Haar measure on G).Furthermore, it is proven in [Se2,3] that if κ1 and κ2 are admissible loops in D whose

interiors do not overlap then h(κ1; ·) and h(κ2; ·) are independent G-valued random vari-ables on the Gaussian probability space (Ωdisk, µT,disk).

Thus, if κ1, ..., κn are non-overlapping admissible loops, base at O, in D then the joint

distribution measure, under µT,disk, of(

h(κ1;ω), ..., h(κn;ω))

, as a random variable in ω

running over Ωdisk, is

QTA1(y1) · · ·QTAn

(yn) dy1 · · ·dyn (3.2.2)

where Ai is the area enclosed by κi and dyi is unit-mass Haar measure on G.

3.3. Construction of the measures µTc and µΘT . The construction of the conditional

probability, satisfying the constraint (2.6.2), requires the technical artifice of dividing Dinto a ‘lower-half’ DL and an ‘upper-half’ DU and working with h(L0 · ∂D · L0;ω) =h(∂DL;ω)h(∂DU ;ω). The full technical details of this construction in a general settingare presented in [Se2,3] and so we shall give here only a conceptual account. In particular,we shall not make explicit the technical role played by DL and DU . In the Appendixwe shall quote the relevant tools from [Se2,3] and explain how they apply in the presentcontext.

As explained in the Appendix (section A3), for any c ∈ G there is a probabilitymeasure µcT on the space

Ωcdef= Ωdisk ×G2 (3.3.1)

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which satisifies (cf. the constraint (2.6.2))

h(L0 · ∂D · L0;ω) = cb−1a−1ba for µcT -a.e. ω = (ω1, a, b) ∈ Ωc (3.3.2)

and is, in a natural and precise sense, the conditioning of the probability measure

µT,disk × (Haar on G2)

to satisfy the constraint (3.3.2).Analogously, for any conjugacy class Θ in G, there is a probability measure µΘ

T on

ΩΘdef= Ωdisk ×G2 ×Θ (3.3.3)

such that

h(L0 · ∂D · L0;ω) = cb−1a−1ba for µΘT -a.e. ω = (ω1, a, b, c) ∈ ΩΘ (3.3.4)

and µΘT is the conditioning of the probability measure

µT,disk × (Haar on G2)× (G−invariant measure on Θ)

to satisfy the constraint (3.3.4).

3.4. Expectation values. Consider a measurable function φ = (φ1, ..., φn) : Ωdisk → Gn

which has a bounded density ρφ on Gn, and consider a bounded measurable function f onGn ×G2. Suppose φk...φ1 = h(L0 · ∂D · L0; · ) for some k ∈ 1, ..., n. Then under simpleconditions (detalied in Proposition A6 and the discussion following it) on φ and ψ :∫

Ωc

f (φ(ω1), a, b) dµcT (ω1, a, b)

=1

NT (c)

∫

G2

da db

∫

Gn

dx f(x, a, b)ρφ(x)δ(

xk · · ·x1(cb−1a−1ba)−1

)

(3.4.1)where da and db are the unit-mass Haar measure on G, dx = dx1 · · ·dxn, and

NT (c) =

∫

G2

QT |Σ|(cb−1a−1ba) da db (3.4.2)

with Qt being the heat kernel on G, as in section 3.2, and |Σ| the area of Σ. The significanceof the ‘delta function’ δ(xk · · ·x1 · · ·) in (3.4.1) is simply that one of the variables xj shouldbe replaced by the value which makes the product xk · · ·x1 · · · equal to the identity (andthe integration dxj dropped).

The corresponding result for ΩΘ is :∫

ΩΘ

f (φ(ω1), a, b) dµΘT (ω1, a, b, c)

=1

NT (Θ)

∫

Θ

dΘc

∫

G2

da db

∫

Gn

dx f(x, a, b)ρφ(x)δ(

xk · · ·x1(cb−1a−1ba)−1

)

(3.4.3)

8

where dΘc is the unit-mass G−invariant measure on the conjugacy class Θ specified by theintegration formula

∫

ΘF (c) dΘc =

∫

GF (kck−1) dk for any c ∈ Θ (3.4.4)

and the normalizer NT (Θ) is given by:

NT (Θ) =

∫

G2

QT |Σ|(cb−1a−1ba) da db dΘc (3.4.5)

Thus NT (Θ) = NT (c) if c ∈ Θ.These results are taken from Propostion 4.5 of [Se2]; for ease of reference, we quote

the exact result in the Appendix (section A6).The expectation value formulas (3.4.1) and (3.4.4) follow from Proposition A6 (quoted

from [Se2,3]) in the manner explained in the discussion following A6 in the Appendix.

4. Loop Expectation Values

4.1. Triangulation of Σ. We shall always work with a certain type of triangulation(‘admissible triangulation’) S of Σ obtained by means of the quotient map q : D → Σfrom a triangulation T of D. The triangulation T of D will consist of radii and cross-radial segments (i.e. those which can be parametrized in polar coordinates in the formr = r(θ)), and will include the origin as a vertex. We will also assume that the arcs on ∂Dcorresponding to the loops A, B and C are made up of 1−simplices of the triangulation(this can always be ensured by subdividing the original triangulation). These technicalrestrictions are to ensure that the parallel-transport equation (3.2.1) is applicable and sothat then the relevant holonomies are meaningful.

4.2. G−fields and lassos. An assignment e 7→ ye ∈ G, with e running over theoriented 1−simplices of a triangulation, satisfying ye = y−1

e will be called a G−field over

the triangulation. If κ is a path consisting of oriented 1−simplices of the triangulation,κ = bj · · · b1, then we define

y(κ)def= ybj · · · yb1

Let T be a triangulation of D as described in section 4.1, and ∆ a positively oriented2−simplex (triangle) in T . We can join an appropriate vertex of ∆ to the origin O by aradial path consisting of 1−simplexes of T and thus form a loop l∆ based at O. In this waywe form loops l∆1

, ..., l∆m, one for each positively oriented 2−simplex ∆j in T . The loops

∆i will be called the lassos of the triangulation. This can be done, with the ∆i orderedappropriately, in such a way that the following hold :(∗) For any G−field y over T :

y(l∆m) · · · y(l∆1

) = y(x0O) · y(∂D) · y(Ox0) (4.2.1)

9

where ∂D is taken to start from x0 = (1, 0) ∈ ∂D. A complete description of theconstruction of the loops l∆i

is given in Theorem 3.1, and the discussion preceding it,of [Se2].

(∗∗) To each loop κ in T , based at o, we can associate a sequence of the lassos l∆i1, l∆i2

...

such that y(κ) = y(l∆i1)±1y(l∆i2

)±1 · · · holds for any G−field y over T .

These are simple graphical facts and are actually independent of considerations of G−fields(Lemma A3 in [Se2]).

4.3. Stochastic holonomy over Σ, for µcT and µΘT . We shall work with loops on Σ

based at O, which consist of oriented 1−simplices of some triangulation S of Σ of the typedescribed in section 4.1. If κ is such a loop then there exist a sequence of loops κ1, ..., κrsuch that:(a) each κj is either the projection by q of a loop κj in D or is one of the basic loops

A,B,C and their reverses(b) for any G−field y over S, y(κr) · · ·y(κ1) = y(κ).The existence of such a decomposition is seen by breaking κ into suitable segments andlifting these up to D (a detailed account is given in section 5 of [Se2]). Recall that for theloops κj on D the stochastic holonomy h(κj ;ω) ∈ G is meaningful as a random variablein ω ∈ ΩDisk.

For ω = (ω1, a, b) ∈ Ωc, or for ω = (ω1, a, b, c) ∈ ΩΘ, we define

h(κ;ω)def= h(κr;ω) · · ·h(κ1;ω) (4.3.1)

where

h(κj ;ω) =

h(κj ;ω1) if κj is the loop in D projecting to κja±1 if κj is A or Ab±1 if κj is B or Bc±1 if κj is C or C

(4.3.2)

Then h(κ;ω) is a well-defined G−valued random variable on Ωc and ΩΘ (that h(κ;ω) isindependent of the choice of the κj is proven in Lemma A2 of [Se2]).

4.4. Theorem. Let κ1, ..., κn be loops on Σ, all based at o, composed of oriented1−simplices of a triangulation S of Σ (of the type described in section 4.1). Let f be abounded measurable function on Gn. Consider any c ∈ G. Then∫

f (h(κ1;ω), ..., h(κn;ω)) dµcT (ω)

=1

NT (c)

∫

f (x(κ1), ..., x(κn)) δ(

x(C)c−1)

m∏

j=1

QT |∆j|

(

x(∂∆j))

dxe1 · · ·dxeM

(4.4.1)where ∆1, ...,∆m are the positively oriented 2−simplexes of S, |∆j | is the area of ∆j ,e1, ..., eM and their reverses are the distinct oriented 1−simplices of S, Qt(·) is the heatkernel on G normalized to

∫

GQt(x) dx = 1, dx being unit-mass Haar measure on G, and

NT (c) =

∫

G2

QT |Σ|(cb−1a−1ba) da db (4.4.2)

10

The meaning of the δ−function in (4.4.1) is that for some arbitrary bond ej lying on ∂Σand appearing in the loop C, the variable xej should be replaced by the value which makesx(C) = c and dxej should be dropped from the integration.

Proof. In view of the observation (∗∗) concerning the decomposition of a loop in D interms of the lassos l∆i

, and in view of the definition of h(κ;ω) in (4.3.1), we see that eachh(κi;ω) is a product of certain of the h(l∆i

;ω)±1’s, h(A;ω)±1, h(B;ω)±1, and h(C;ω)±1

(the latter being c±1).Thus to f we may associate a bounded measurable function F such that

f (h(κ1;ω), ..., h(κn;ω)) = F (h(l∆1;ω), ..., h(l∆m

;ω), h(A;ω), h(B;ω), h(C;ω)) (4.4.3)

and, more generally,

f (x(κ1), ..., x(κn)) = F (x(l∆1), ..., x(l∆m

), a, b, c) (4.4.4)

for any G−field x over the triangulation S (of course, the c−term, being constant, couldbe dropped from F but that would make the form of F dependent on c).

Recalling the µc−expectation-value formula (3.4.1) and the µ−distribution of therandom variables h(l∆i

; ·) given in (3.2.2), we have :

∫

f (h(κ1;ω), ..., h(κn;ω)) dµcT (ω)

=1

NT (c)

∫

Gn×G2

F (y∆1, ..., y∆m

, a, b, c) ·

· δ(

y∆m· · · y∆1

(cb−1a−1ba)−1)

dadb

m∏

j=1

QT |∆j |(y∆j)dy∆j

(4.4.5)

where NT (c) is as in (4.4.2) (which is the same as stated earlier in (3.4.2)).A combinatorial argument (Lemma A1 in [Se2]) shows that if we take the y∆i

’s anda, b, c such that y∆m

· · · y∆1= cb−1a−1ba (as is required by the δ(·) term in (4.4.5)) then

it is possible to associate to this data a G−field x over S in such a way that

x(l∆i) = y∆i

for i ∈ 1, ..., m (4.4.6a)

x(A) = a, x(B) = b, x(C) = c (4.4.6b)

In (4.4.6a) we have tacitly raised x to a G−field, also denoted x, on the triangulation ofD which projects by q : D → Σ to the triangulation S of Σ.

The goal now is to change variables (y∆i, a, b) 7→ xei. In order to do this it is

necessary to understand how∏mi=1 dy∆i

dadb is transformed. It is shown in the proof ofLemma A1 in [Se2] that the G−field x can be chosen in the following way :(i) for a certain collection of bonds e, the variables xe are chosen arbitrarily; for the

remaining bonds :(ii) for one bond e∗A lying on q(∂D) and appearing in the loop A (i.e. e∗A lies on the part

of A on q(∂D)), x(e∗A) is the product of a with some of the ‘previously’ assigned valuesof x;

11

(iii) for one bond e∗B lying on q(∂D) and appearing in the loop B, x(e∗B) is product of bwith some of the ‘previously’ assigned values of x;

(iv) for one bond e∗C lying on ∂Σ and appearing in the loop C, x(e∗C) is product of c withsome of the ‘previously’ assigned values of x;

(v) there is one bond bi corresponding to each ∆i with xbi chosen to be y∆itimes some

of the ‘previously’ assigned values of x.(A more precise formulation, along with a specification of what ‘previously assigned’ means,is given in the proof of Lemma A1 of [Se2].)

From (i)-(v), and the translation-invariance of Haar measure, it follows that in the inte-gral on the right side of (4.4.5), we can introduce the variables xej instead of (y∆i

, a, b),and then the right side of (4.4.5) equals :

1

NT (c)

∫

F (x(∆1), ..., x(∆m), x(A), x(B), c) ·

· δ(

x(C)c−1)

m∏

j=1

QT |∆j |

(

x(∂∆j))

dxe1 · · ·dxeM

(4.4.7)This calls for some explanation : recall from (i) that certain xei can be chosen arbitrarily,these are integrated over G in (4.4.7) without changing the value of the integral since∫

Gdx = 1; the term δ

(

y∆m· · · y∆1

(cb−1a−1ba)−1)

disappeared since (4.4.6a, b) and (4.2.1)

imply that y∆n· · · y∆1

(cb−1a−1ba)−1 = e is satisfied automatically by our choice of the

xei . Finally the new δ(·)−term δ(

x(C)c−1)

appears because, by condition (iii) above, one

of the bond variables xe, with e = e∗C on ∂Σ and appearing in C, must be chosen in such away that x(C) = c; this says that this xe is not really a ‘variable’ but is determined by theother xei (and c). That the choice of e∗C does not affect the value of the integral (4.4.7)follows by a change-of-variables argument (if e′C is another bond on ∂Σ appearing in Cthen, given all the other variables xe, a new variable x(e′C) may be introduced in place ofx(e∗C) by requiring that x(e′C) be such that x(C) = c, a relationship that expresses x(e′C)as a product of x(e∗C) with certain of the other bond-variables xe).

Now (4.4.4) and (4.4.6b) imply that (4.4.7) is equal to

1

NT (c)

∫

f (x(κ1), ..., x(κn)) δ(

x(C)c−1)

m∏

i=j

QT |∆j |

(

x(∂∆j))

dxe1 · · ·dxeM (4.4.8)

Thus, recalling that we started our algebraic manipulations with the loop expectation-value formula (4.4.5), we see that

∫

f (h(κ1;ω), ..., h(κn;ω)) dµcT (ω) is equal to (4.4.8),

thereby proving (4.4.1).

4.5. Invariance of loop expectation values for µcT . Unlike in the case of the “free-boundary” theory developed in [Se2], we cannot expect the loop expectation values

∫

f (h(κ1;ω), ..., h(κn;ω)) dµcT (ω)

12

to be invariant under area-preserving transformations of Σ. This is because the loop Chas been selected out as a special loop and so we must consider only area-preservinghomeomorphisms of Σ which fix C.

Thus we wish to show that if φ is an area-preserving homeomorphism of Σ and φC =C then

∫

f (x(κ1), ..., x(κn)) δ(

x(C)c−1)

m∏

j=1

QT |∆j |

(

x(∂∆j))

dxe1 · · ·dxeM

=

∫

f (x(φ κ1), ..., x(φ κn)) δ(

x(C)c−1)

m′

∏

j=1

QT |∆′

j|

(

x(∂∆′j))

dxe′1· · ·dxe′

M′

(4.5.1)where on the left we are using a triangulation S while on the right we are using a triangu-lation S′. The loops κi and C are composed of simplices in S, and the loops φ κi and Care composed of simplices in S′.

First we observe that either side of (4.5.1) is invariant under subdivisions of thetriangulation. This is a consequence of the convolution property of the heat kernel Qt(·) :

∫

G

Qt(xy−1)Qs(yz) dy = Qt+s(xz),

which we can use to ‘collapse’ adjacent 2−simplices ∆∗1 and ∆∗

2 of a subdivision of S:

∫

QT |∆∗

1|

(

x(∂∆∗1))

QT |∆∗

2|

(

x(∂∆∗2))

dxb = QT |∆∗|

(

x(∂∆∗))

(4.5.2)

wherein ∆∗ is the region formed by collapsing ∆∗1 and ∆∗

2 along their common edge b.Extra bond-variables which remain after collapsing such common edges can be integratedaway (or combined into bond variables for the undivided triangulation) using

∫

dxb = 1.Since the only way area appears in the right of (4.5.2) is through |∆∗| = |∆∗

1|+ |∆∗2|, it

follows that the left side of (4.5.1) depends on areas only through the areas of the connectedcomponents R1, ...,RN of Σ \ ∪n+1

i=1 Im(κi) (here Im(κi) is the image of the path κi as a

subset of Σ, and we have set κn+1def= C, for convenience). A more detailed argument is

available in Fact 1 of the Appendix in [Se2].By the ‘Hauptvermutung of topology’ (Theorem 4.6 in [Br]), S and S′ have subdivi-

sions S and S′ and there is a simplicial isomorphism φ : S → S′ obtained by an isotopy ofφ, preserving the sets Im(κi) and Im(C), such that φC = C, and φκi and φκi are thesame when taken as sequences of bonds in S′. In view of the invariance of the integrals inconsideration under subdivisions we may and will assume that S = S and S′ = S′. Thusφ is a simplicial isomorphism between S and S′.

By simple relabelling of xe as xφ(e), the left side of (4.5.1) equals :

∫

f(

x(φ κ1), ..., x(φ κn))

δ(

x(φ C)c−1)

m∏

j=1

QT |∆j |

(

x(

∂φ(∆j))

)

dxe′1· · ·dxe′

M

(4.5.3)

13

Since φ κi and φ κi are the same as sequences of bonds in S′, and since φ C = C,relabelling the simplices makes (4.5.3) equal to :

∫

f (x(φ κ1), ..., x(φ κn)) δ(

x(C)c−1)

m∏

j=1

QT |φ−1(∆′

j)|

(

x(∂∆′j))

dxe′1· · ·dxe′

M′

(4.5.4)

(the number M ′ of edges in S′ equals the number M because in the present situation Sand S′ are isomorphic). Now, as observed earlier (after (4.5.2)), (4.5.4) depends on theareas |φ−1(∆j)| only through certain of their sums; specifically, through the areas of the

regions φ−1φ(Ri), where, as before, the R1, ...,RN are the connected components of thecomplement of Im(k1)∪· · ·∪ Im(kn)∪ Im(C) in Σ. Since φ is an isotopy of φ through mapstaking the subset Im(k1) ∪ · · · ∪ Im(kn) ∪ Im(C) always into the same subset, it followsthat φ−1φ(Ri) = Ri. Therefore, (4.5.4) equals

∫

f (x(φ κ1), ..., x(φ κn)) δ(

x(C)c−1)

m′

∏

j=1

QT |∆′

j|

(

x(∂∆′j))

dxe′1· · ·dxe′

M′

(4.5.5)

and this being the right side of (4.5.1), we have proven (4.5.1).

Observations:(1) The above arguments show that the restriction that φ is the identity on C may be

removed if C were to be replaced by φ(C) on the right side of (4.5.1).(2) Since the left side of (4.5.1) involves a triangulation S and the right side involves S′,

we may set φ to be the identity and conclude that either side is independent of the

specific triangulation used.(3) The observation (2) is not surprising in view of the fact that the integrals in (4.5.1)

are, up to the constant NT (c), the loop expectation-value as given in Theorem 4.4.Nevertheless, the arguments above give an independent and direct proof (one whichalso does not depend on the triangulations being of the type described in section 4.1).

We turn now to the corresponding results for µΘT . For this we use the notation dΘc to

denote the G−invariant unit-mass measure on a conjugacy class Θ in G; thus∫

ΘF (c) dΘc =

∫

GF (kck−1) dk for any c ∈ Θ.

4.6. Theorem. Let κ1, ..., κn be loops on Σ, all based at o, composed of oriented1−simplices of a good triangulation S of Σ (of the type described in section 4.1). Let fbe a bounded measurable function on Gn. Consider any conjugacy class Θ in G. Then∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω)

=1

NT (Θ)

∫

f (x(κ1), ..., x(κn)) δ(

x(C)c−1)

m∏

j=1

QT |∆j |

(

x(∂∆j))

dxe1 · · ·dxeMdΘc

(4.6.1)

14

where ∆1, ...,∆m are the positively oriented 2−simplexes of S, e1, ..., eM and their reversesare the distinct oriented 1−simplices of S, Qt(·) is the heat kernel on G normalized to∫

GQt(x) dx = 1, dx being unit-mass Haar measure on G, and

NT (Θ) =

∫

G2

QT |Σ|(cb−1a−1ba) da db dΘc (4.6.2)

The meaning of the δ−function in (4.6.1) is that for some arbitrary bond ej lying on ∂Σand appearing in the loop C, the variable xej should be replaced by the value which makesx(C) = c and dxej should be dropped from the integration.

Proof. The proof is virtually identical to that of Theorem 4.4, except that we use theµΘT -expectation-value formula (3.4.3) as starting point instead of the µcT−formula (3.4.1)

used for proving Theorem 4.4.

4.7. Theorem. The loop expectation values∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω) re-

main invariant if each κi is replaced by φ κi, where φ is any area-preserving homeomor-phism of Σ which preserves the orientation of C.

Proof. Recall, from Observation (1) in section 4.5, that the general form of (4.5.3) is :

∫

f (x(κ1), ..., x(κn)) δ(

x(C)c−1)

m∏

i=1

QT |∆j |

(

x(∂∆j))

dxe1 · · ·dxeM

=

∫

f (x(φ κ1), ..., x(φ κn)) δ(

x(φ C)c−1)

m′

∏

j=1

QT |∆′

j|

(

x(∂∆′j))

dxe′1· · ·dxe′

M′

(4.7.1)the notation being as for (4.5.1), and c any element of G. We will show that either integraldepends not on the full curve C but only on the part of C which lies on ∂Σ.

In (4.7.1) we can take c ∈ Θ and integrate by the G−conjugation-invariant measuredΘc; this yields a formula of the form :

∫

[· · ·]δ(

x(C)c−1)

[· · ·] dΘc =

∫

[· · ·]′δ(

x(φ C)c−1)

[· · ·]′ dΘc (4.7.2)

Now the loop C can be expressed as

C = L · C∂ · L (4.7.3)

where C∂ lies entirely on ∂Σ, and L is a path from o to a point on ∂Σ where C ‘first hits’∂Σ. Then

x(C) = x(L)−1x(C∂)x(L) (4.7.4)

We substitute this into (4.7.2) and observe that the δ(· · ·) term on the left can be reorderedinto to the form

δ(

x(C∂)x(L)c−1x(L)−1

)

(4.7.5a)

15

Now since dΘc is aG−conjugation-invariant measure on Θ it follows that in the integrationsin (4.7.2), the δ−term (4.7.5a) can be replaced by

δ(

x(C∂)c−1)

(4.7.5b)

Returning to (4.7.2), we then have

∫

[· · ·]δ(

x(C∂)c−1)

[· · ·] dΘc =

∫

[· · ·]′δ(

x(φ C∂)c−1)

[· · ·]′ dΘc (4.7.6)

Now C∂ and φC∂ can differ only in their starting points and orientation. We have assumedthat φC and C have the same orientation; thus the loops C∂ and φC∂ can only differ intheir starting points. But then x(φ C∂), as a product of the bond-variables xe, is a cyclicpermutations of x(C∂), i.e. a conjugate of x(C∂). Since dΘc is G−conjugation-invariant itfollows that in the right side of (4.7.6) φC∂ can be replaced by C∂ . Tracing our argumentsback to (4.7.1) we conclude that :

∫

f (x(κ1), ..., x(κn)) δ(

x(C)c−1)

m∏

j=1

QT |∆j |

(

x(∂∆j))

dxe1 · · ·dxeM

=

∫

f (x(φ κ1), ..., x(φ κn)) δ(

x(C)c−1)

m′

∏

j=1

QT |∆′

j|

(

x(∂∆′j))

dxe′1· · ·dxe′

M′

(4.7.7)and this what we wished to prove. Note that, as in section 4.5, taking φ to be the identitymap in (4.7.7) shows that the loop-expectation-value formulas are actually independentthe particular triangulation used.

5. Other Surfaces

In this section we will sketch the construction of the boundary-holonomy-restrictedquantum gauge field measures for a general compact 2−dimensional Riemannian manifoldwith boundary, and write down the loop expectation-value formulas.

5.1. Hypotheses on Σ, and generators of π1(Σ, o). In this section, Σ is a compact2−dimensional Riemannian manifold with boundary.

Instead of the loops A,B,C of the earlier sections, we now have the following gener-ators of π1(Σ, o):

(Σ1) if Σ is orientable, has genus g ≥ 0, and has p boundary-components, then we haveloops

A1, B1, ..., Ag, Bg, C1, ..., Cp

generating π1(Σ, o) subject to the relation that

Cp · · ·C1BgAgBgAg · · ·B1A1B1A1 gives the identity in π1(Σ, o)

The loops Ci are of the form LiC′iLi, where C

′i traces a loop around boundary com-

ponent #i and Li is a simple path from o to a point of C′i (the point where Ci ‘first

hits’ ∂Σ)

16

(Σ2) if Σ is unorientable and has p boundary-components, then we have loops

A1, ..., Ag, C1, ..., Cp

generating π1(Σ, o) subject to the relation that

Cp · · ·C1AgAg · · ·A1A1 gives the identity in π1(Σ, o)

The loops Ci are of the form LiC′iLi, as described for (Σ1).

5.2. The connection spaces A(Θ) and A(c). We are interested in connections ω withspecified values, or restrictions, on the values of the holonomies h(Ci;ω). For simplicitywe shall work with simultaneous restrictions on all of the Ci; restrictions on only some ofthe Ci can be handled similarly. Thus we consider the spaces of connections:

A(c1, ..., cp)def= ω ∈ A : h(C1;ω) = c1, ..., h(Cp;ω) = cp, for any (c1, ..., cp) ∈ Gp

(5.2.1)

A(Θ)def=

ω ∈ A :(

h(C1;ω), ..., h(Cp;ω))

∈ Θ

, for any G-conugation orbit Θ in Gp

(5.2.2)Here, by G-conjugation orbit we mean an orbit of the conjugation action of G on Gp, i.e.a subset of the form (xc1x

−1, ..., xcpx−1) : x ∈ G.

We shall usec

def= (c1, ..., cp) (5.2.3)

5.3. The measures µcT and µΘ

T . Let T be a fixed positive real number. On A(c)/Go weare interested in the measure

dµcT ([ω]) =

1

ZT (c)e−SYM(ω)/T δ

(

(

h(C1;ω), ..., h(Cp;ω))

c−1)

[Dω] (5.3.1)

while on A(Θ)/G we are interested in the measure

dµΘT ([ω]) =

1

ZT (Θ)e−SYM(ω)/T δΘ

(

(

h(C1;ω), ..., h(Cp;ω))

)

[Dω] (5.3.2)

wherein δΘ is specified in the manner explained after (2.7.5).As in the case of the torus, we start with the Gaussian probability space

(Ωdisk, µT,disk) for the disk D (5.3.3)

Then we define

Ωcdef=

Ωdisk ×G2g if Σ is orientable, i.e. satisfies (Σ1)Ωdisk ×Gg if Σ satisifies (Σ2)

(5.3.4)

Similarly,

ΩΘdef=

Ωdisk ×G2g ×Θ if Σ satisfies (Σ1)Ωdisk ×Gg ×Θ if Σ satisifies (Σ2)

(5.3.5)

17

We define the measure µc to be µT,disk × (Haar on Gσg ) (the σ in the exponent being 2if (Σ1) applies and 1 if (Σ2) applies) conditioned to satisfy the following constraint (recallthat L0 is the radial path from O to x0 = (1, 0) ∈ ∂D) :

h(L0 · ∂D · L0;ω1) =

cp · · · c1b−1g a−1

g bgag · · · b−11 a−1

1 ba if (Σ1) appliescp · · · c1a

2g · · ·a

21 if (Σ2) applies

(5.3.6)

where

ωdef=

(ω1, a1, b1, ..., ag, bg) if (Σ1) applies(ω1, a1, ..., ag) if (Σ2) applies

(5.3.7)

Thus the constraint will hold µcT -almost-everywhere.

To define µΘT we use the same constraint condition on the measure

µT,disk × (Haar on Gσg)× (G-invariant unit-mass on Θ) (5.3.8)

wherein σ is as before. The space on which µΘT will sometimes be denoted A(Θ)/G.

Expectation values with respect to µcT and µΘ

T may be obtained in a way exactlyanalogous to that explained in section 3.4 for the one-holed torus.

Stochastic holonomies with respect to µcT and µΘ

T are also defined exactly analogouslyto (4.3.1) in section 4.3.

Carrying out essentially the same arguments as in section 4 we obtain the followinggeneralized version of Theorems 4.4. and 4.6.

5.4. Theorem. Let κ1, ..., κn be loops on Σ, all based at the point o, composed oforiented 1−simplices of a good triangulation S of Σ (of the type described in section 4.1).Let f be a bounded measurable function on Gn. Consider any c = (c1, ..., cp) ∈ Gp. Then

∫

f (h(κ1;ω), ..., h(κn;ω)) dµcT (ω)

=1

NT (c)

∫

f (x(κ1), ..., x(κm))

p∏

i=1

δ(

x(Ci)c−1i

)

m∏

j=1

QT |∆j |

(

x(∂∆j))

dxe1 · · ·dxeM

(5.4a)

where ∆1, ...,∆m are the positively oriented 2−simplices of S, e1, ..., eM and their reversesare the distinct oriented 1−simplices of S, Qt(·) is the heat kernel on G normalized to∫

GQt(x) dx = 1, dx being unit-mass Haar measure on G, and

NT (c)def=

∫

G2g QT |Σ|(cp · · · c1b−1g a−1

g bgag · · · b−11 a−1

1 b1a1) da1 · · ·dbg if (Σ1) applies∫

Gg QT |Σ|(cp · · · c1a2g · · ·a

21) da1 · · ·dag if (Σ2) applies

The meaning of the δ functions in (5.4a) is that, for each Ci, for some arbitrary bond ejlying on ∂Σ and appearing in the loop Ci, the variable xej should be replaced by the valuewhich makes x(Ci) = ci and dxej should be dropped from the integration.

18

For µΘT , we have :

∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω)

=1

NT (Θ)

∫

f (x(κ1), ..., x(κn))

p∏

i=1

δΘ

(

x(Ci)c−1i

)

dΘci

m∏

j=1

QT |∆j |

(

x(∂∆j))

dxe1 · · ·dxeM

(5.4b)The meaning of the delta-function δΘ and the integrator dΘc are explained in section 2.7.The normalizer NT (Θ) is

∫

Gp NT (c) dΘc.

The normalizer NT (c) has appeared in the appendix of [Se2] as a topologically-invariant function associated to the system of loops C1, ..., Cp on Σ. The functions NT (Θ)have appeared in [Wi1].

As in section 4, the loop expectation value formulas (5.4a, b) are invariant under area-preserving homeomorphisms of Σ which fix the loops Ci pointwise. However, for p > 1,the analog of Theorem 4.7 need not hold.

6. Spaces of connections and 2−forms on them

In this section we describe the surface, spaces of connections, and 2−forms whichwe shall be working with in sections 6-9. Some of the notation and definitions are lifteddirectly from [KS2,3].

6.1. The surface Σ, group G, the connection spaces A, A(Θ), M(Θ). Henceforth, Σwill denote a compact connected oriented 2−dimensional manifold of genus g ≥ 1, witha connected non-empty boundary ∂Σ. In sections 8 and 9, Σ will be taken to be alsoa Riemannian manifold. We shall use a fixed basepoint o in Σ, and piecewise smoothloops A1, B1, ..., Ag, Bg, C : [0, 1] → Σ, all based at a point o ∈ Σ, which generate thefundamental group π1(Σ, o), subject to the relation that CBgAgBgAg · · · ...B1A1B1A1 ishomotopic to the constant loop at o, wherein we denote by X the reverse of any path X .The loop C is of the form LC∗L, where C∗ is a simple loop around the boundary ∂Σ andL is a path from o to the initial point of C∗; the loops Ai and Bi are also of the formAi = LA∗

iL and Bi = LB∗i L. A detailed description of these loops is given in [KS3 :Section

6.1] (and in a wider setting in [Se2]), but we shall not need such details here.We work with a principal G-bundle π : P → Σ, where the gauge group G is now

assumed to be compact, connected, and semisimple. The semisimplicity hypothesis is notof essential significance but we impose it to focus on the more significant issues.

Since Σ has boundary, the bundle P is trivial and so there is a smooth section s : Σ →P . Connections and other forms on P can be pulled down to Σ by using s. In this waywe can and will identify the space A of all connetions on P with the space of all smoothg-valued 1-forms over Σ; thus we take

A = space of all smooth g-valued 1-forms onr Σ (6.1.1)

The section s is used only for convenienve and the constructions (such as symplectic forms)and results we discuss are independent of the choice of s.

19

A connection ω ∈ A is flat if its curvature is zero, i.e. if dω + 12 [ω, ω] = 0. We will be

interested in the set of all flat connections :

A0 def= ω : ω ∈ A and ω is flat (6.1.2)

We work with a fixed basepoint u ∈ π−1(o), where o is the basepoint on Σ. If κ is apiecewise smooth loop on Σ based at o, then, as explained in section 2.2, h(κ;ω) denotesthe holonomy of ω around κ, with u as initial point.

We shall denote by Θ a conjugacy class in the group G; we shall work with the spaces

A(Θ) = ω ∈ A : h(C;ω) ∈ Θ and A0(Θ) = ω ∈ A0 : h(C;ω) ∈ Θ (6.1.3)

The group G (section 2.3) of automorphisms of P can be identified, via the section s,with the set of all smooth maps Σ → G; the group structure is now simply pointwisemultiplication. The subgroup Go now consists of those φ ∈ Σ for which φ(o) = e. Theaction of G on A is given by:

A× G → A : (A, φ) 7→ A · φdef= Ad(φ−1)A+ φ−1dφ (6.1.4)

This action carries the sets A(Θ) and A0(Θ) into themselves, and we have the modulispace

M(Θ) = A0(Θ)/G (6.1.4)

6.2. The 2−forms Ω and ΩΘ. We are interested in a certain 2−form ΩΘ on A(Θ)which was introduced in [KS3]. The definition is:

ΩΘdef= Ω+ Ωex + Ων (6.2.1)

We shall explain now the definitions of the 2−forms Ω, Ωex and Ων . The standard 2−formΩ on A (as described in [AB]) is given by :

Ω(A,B)def=

∫

Σ

〈A ∧B〉 (6.2.2)

where A and B are g−valued 1−forms on Σ and the product 〈A ∧B〉 is the 2−form on Σgiven by 〈A ∧B〉(X, Y ) = 〈A(X), B(Y )〉g − 〈A(Y ), B(X)〉g.

Recall, from section 6.1, the loop C, part of which goes around ∂Σ. Let A be a tangentvector to A (i.e. A is a g−valued 1−form on Σ), and define

α : [0, 1] → g : t 7→ α(t) = −

∫ t

0

Ad(h−1s )A

(

C′(s))

ds (6.2.3)

where s 7→ hs describes parallel transport along C : h′(s)h(s)−1 = −A (C′(s)) withh(0) = e. It is known (and readily verifiable) that α(t) is the variation in ht corresponding

20

to the variation A in the connection ω. Define β : [0, 1] → G similarly with respect to atangent vector B to A. The 2−form Ωex is defined by:

Ωex(A,B)def= −

1

2

∫ 1

0

∫ 1

0

ǫst〈α′(s), β′(t)〉 ds dt (6.2.4)

where

ǫst =

1 if s ≤ t;−1 if s > t.

Next, the 2−form Ων on A(Θ) is defined by :

Ων∣

∣

ω(A,B) = −

1

2〈α(1),

(

Ad (c)− 1)−1

β(1)〉+1

2〈β(1),

(

Ad (c)− 1)−1α(1)〉 (6.2.5)

where c = h(C;ω), C being the loop for ∂Σ as described in Section 6.1, and α, β are as in

(6.2.3). The terms involving(

Ad (c) − 1)−1

are understood by setting (Ad (c)− 1)−1

to

be 0 on ker [Ad (c)− 1] =[

(Ad (c)− 1) (g)]⊥

. Thus all terms on the right of the definition(6.2.1) of ΩΘ have been specified.

6.3. The results of [KS3]. For ease of reference, we shall record here certain factsrelating to M(Θ) and ΩΘ, including most of the results of [KS3].

(i) The 2−form ΩΘ is equivariant under the action of G. Let A,B ∈ TωA(Θ) (an elementof TωA(Θ) is, by definition, of the form ∂ωt/∂t|t=0 where t 7→ ωt ∈ A(Θ) is suchthat (t, p) 7→ ωt(p) is smooth and ω0 = ω) and suppose that A or B is tangent tothe G−orbit through ω, in the sense that it is equal to ∂(ω · φt)/∂t|t=0 for some patht 7→ φt ∈ G with (t, p) 7→ φt(p) smooth and φ0 = id. Then ΩΘ(A,B) = 0. (Proposition5.1 in [KS3])

(ii) The holonomy representation

H : ω 7→ (h(A1;ω), h(B1;ω), ..., h(Ag;ω), h(Bg;ω), h(C;ω)) ∈ G2g ×Θ (6.3.1)

induces a bijection A(Θ)/Go → Π−1(e), where Go is the subgroup of G consisting ofall φ with φ(o) = e, and

Π : G2g ×Θ → G : (a1, b1, ..., ag, bg, c) 7→ cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1 (6.3.2)

The group G acts on the right of Π−1(e) by conjugation, and the holonomy mappingH is G − G equivariant with respect to the homomorphism G → G : φ 7→ φ(o), andinduces a bijection of the full moduli spaces

M(Θ) → Π−1(e)/G (6.3.3)

(iii) There is a dense open subset D of G such that for every conjugacy class Θ passingthrough any point in D, the set Π−1(e) is a smooth submanifold of G2g ×Θ.

21

(iv) There is a 2−form Ωo,Θ on G2g ×Θ whose restriction to Π−1(e) pulls back to ΩΘ bythe holonomy map H.

(v) If α ∈ Π−1(e) and A,B ∈ TαΠ−1(e) (this being taken, by defintion, to be ker dΠα)

then Ωo,Θ(A,B) is 0 if A or B is tangent to the G−orbit through α; moreover, Ωo,Θis G-equivariant. In this sense Ωo,Θ descends to a 2−form, also denoted ΩΘ, onΠ−1(e)/G. (Theorem 3.6 in [KS3])

(vi) The 2−form ΩΘ is closed (i.e. Ωo,Θ on Π−1(e) is closed); it is non-degenerate onthe smooth part of Π−1(e)/G for all Θ passing through a dense open subset of acertain neighborhood of the identity in G. (Theorem 4.1 in [KS3]). By the ‘generic(or smooth) part of Π−1(e)/G’ we mean the subset of points corresponding to thepoints on Π−1(e) where the derivative dΠ is surjective and where the isotropy of theG−action is minimal within any component. (Section 7.4 below gives a more detailedexplanation of ‘generic part’ of M(Θ))

(vii) Formula specifying ΩΘ: Let (αH(1), cC(1)), (αH(2), cC(2)) ∈ T(α,c)

(

G2g ×Θ)

; then

Ωo,Θ

(

(αH(1), cC(1)), (αH(2), cC(2)))

=1

2

∑

1≤i,k≤4g

ǫik〈Ad (αi−1 · · ·α1)−1H

(1)i ,Ad (αk−1 · · ·α1)

−1H(2)k 〉g

−1

2〈(

Ad c−1 − 1)−1

C(1),(

Ad c−1 − Ad c) (

Ad c−1 − 1)−1

C(2)〉g

(6.3.4)where

ǫik =

1 if i < k0 if i = k−1 if i > k

and we have used the convenient if unusual notation of setting

α = (α1, α2, α5, α6, ..., α4g−3, α4g−2) (6.3.5)

andαj+2 = α−1

j for all j ∈ Jdef= 1, 2, 5, 6, ..., 4g− 3, 4g− 2 (6.3.6)

and, analogously, H(r) = (H(r)1 , H

(r)2 , H

(r)5 , H

(r)6 , ..., H

(r)4g−3, H

(r)4g−2), and

Hj+2 = −Ad (αj)Hj for all j ∈ J (6.3.7)

In (6.3.4), the terms (Ad c−1 − 1)−1 are meaningful because cC(1), cC(2) ∈ TcΘ and,as is readily seen, c−1TcΘ = (Ad c−1 − 1)(g).

Note that (6.3.4) specifies the 2−form Ωo,Θ on all of G2g × Θ; the 2−form ΩΘ isobtained from its restriction to Π−1(e).

6.4. The sense in which H induces ΩΘ from ΩΘ. We wish to make a technicalremark here concerning section 6.3(iv). A precise statement of 6.3(iv) is as follows. Let

22

(α, c) ∈ Π−1(e) ⊂ G2g × Θ, and consider vectors v1, v2 ∈ T(α,c)(

G2g ×Θ)

which aretangent to smooth paths lying entirely on Π−1(e). It has been shown in sections 4.1-4.4 of[KS2] that:(a) there are paths [0, 1] → A0(Θ) : t 7→ ωit, with i = 1, 2, such that (t, p) 7→ ωit(p) is

smooth and H(

ωit)

is initially tangent to vi, for i = 1, 2

(b) ΩΘ (V1, V2) = Ωo,Θ(v1, v2), where Vi = ∂ωti/∂t|t=0, for i = 1, 2.In view of section 6.3(i), it is then reasonable to say that Ωo,Θ gives the 2−form onΠ−1(e) ≃ A(Θ)/Go (as in section 6.3(ii)) induced by ΩΘ, and hence that the 2−form ΩΘ

for M(Θ) is induced by ΩΘ. However, to make this a strictly logical conclusion one shouldverify that if A,B ∈ TωA

0(Θ) are such that H′(ω)A or H′(ω)B (the derivatives beingpointwise partial derivatives, for instance H′(ω)A = ∂H(ωt)/∂t|t=0 if t 7→ ωt ∈ A0(Θ) issuch that (t, p) 7→ ωt(p) is smooth and ∂ωt(p)/∂t|t=0 = A) is 0 then ΩΘ(A,B) = 0. Forthe case of closed surfaces this has been proven in Theorem 5.9.1(i) of [Se6] by showingthat H′(ω)A is 0 if and only if A is tangent to the Go−orbit through ω. We expect thatthis result (and, with minor modifications, its proof) holds for surfaces with boundary. Afull treatment of this issue in the setting of compact surfaces with any number of boundarycomponents is postponed to a future investigation. For the purposes of this paper it willsuffice to take the relationship of ΩΘ and ΩΘ as specified above by (a) and (b).

7. Properties of ΩΘ, a determinant identity, and non-degeneracy for ΩΘ

The goal of this section is to prove that ΩΘ is non-degenerate. For this we shall firstprove a determinant identity ((7.8.1)) which will be useful again in sections 8 and 9.

7.1. Notation. We record here some notation some of which has already been used insection 6. We work with a conjugacy class Θ in G, and we use the map

Π : G2g ×Θ → G : (a1, b1, ..., ag, bg, c) 7→ cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1 (7.1.1)

The indexing set

Jdef= 1, 2, 5, 6, ..., 4g− 3, 4g− 2 (7.1.2)

is convenient to label a typical point in G2g ×Θ as

α = (αjj∈J , c)

Thus, comparing with the usual notation (a1, b1, ..., ag, bg, c) we have set α1 = a1, α2 =b1,..., α4g−3 = ag, α4g−2 = bg. As seen in the expression for Ωo,Θ in (6.3.4), it is alsoconvenient to introduce

α3 = α−11 , α4 = α−1

2 , ..., α4g−1 = α−14g−3, α4g = α−1

4g−2 (7.1.3)

Correspondingly, a vector in T(α,c)(G2g × Θ) has the form (αjHjj∈J , cC), with C ∈

c−1TcΘ ⊂ g (not to be confused with the loop C itself), and we set

Hj+2 = −(Adαj)Hj for j ∈ J (7.1.4)

23

It may be convenient to view the above notation as an expression of the imbedding

G2g ×Θ → G4g ×Θ : (a1, b1, ..., ag, bg, c) 7→ (a1, b1, a−11 , b−1

1 , ..., ag, bg, a−1g , b−1

g , c) (7.1.5)

The derivative of Π at a point (α, c) ∈ G2g ×Θ will be taken, by means of left translation,as a map T(α,c)

(

G2g ×Θ)

→ g; it is given by

dΠ(α,c)(αH, cC) =

4g∑

i=1

f−1i−1Hi + f−1

4g C (7.1.6)

whereinfi = Ad (αi · · ·α1) (7.1.7)

Alternatively,

dΠ(α,c)(αH, cC) =∑

j∈J

(f−1j−1 − f−1

j+2)Hj + f−14g C (7.1.8)

Here we have used the fact that for any j ∈ J ,

Ad(f−1j+1)Hj+2 = −Ad(f−1

j+2)Hj (7.1.9)

This form is useful for determining the adjoint dΠ∗(α,c), which, again by left transla-

tions, we will take as a map g → (g2g)⊕(

c−1TcΘ)

; it is given by:

dΠ∗(α,c)X =

(

(fj−1 − fj+2)Xj∈J , pr2g+1f4gX)

(7.1.10)

where pr2g+1 is the orthogonal projection g → c−1TcΘ. For the orbit map

γ(α,c) : G→ G2g ×Θ : x 7→ γ(α,c)(x) =(

xαjx−1j∈J , xcx

−1)

(7.1.11)

the derivative is the map g → g2g ⊕(

c−1TcΘ)

given by

γ′(α,c)(αH, cC) =(

(

Adα−1j − 1

)

Hjj∈J ,(

Ad c−1 − 1)

C)

(7.1.12)

Here C is of course an element of c−1TcΘ ⊂ g, and is not the loop C of section 6.1.

7.2. Interpretation of maps of the form (Ad c− 1)−1

. We record some simple facts,conventions, and observations which will be useful later in computations. For c ∈ G, let

Zcdef= X ∈ g : (Ad c)X = X (7.2.1)

Since(Ad c− 1)

∗= Ad c−1 − 1 (7.2.2)

it follows that(

Ad c±1 − 1)

(g) = Z⊥c = c−1TcΘ (7.2.3)

24

and (Ad c±1 − 1) map Z⊥c into itself isomorphically; by

(

Ad c±1 − 1)−1

we shall meanthe inverse of this map. We have encountered and shall encounter again the composite(Ad c−1 −Ad c)(Ad c−1 − 1)−1. Observe that

Ad c−1 − Ad c = −(Ad c− 1)(1 + Ad c−1) = (1 + Ad c)(Ad c−1 − 1) (7.2.4)

Thus (Ad c−1 − Ad c) maps g into Z⊥c , and so (Ad c−1 − Ad c)(Ad c−1 − 1)−1 maps Z⊥

c

into itself. Splitting any X ∈ g into a component in Zc and one in Z⊥c shows that

(Ad c−1 −Ad c)(Ad c−1 − 1)−1(Ad c−1 − 1)X = (Ad c−1 − Ad c)X (7.2.5)

From (7.2.4) we have, for any C(1) ∈ Z⊥c and X ∈ g,

⟨

(

Ad c−1 − 1)−1

C(1),(

Ad c−1 − Ad c)

X⟩

g= −

⟨

C(1),(

1 + Ad c−1)

X⟩

g

=⟨

C(1), (Ad c− 1)−1 (

Ad c−1 − Ad c)

X⟩

(7.2.6)where in the second equality we used the hypothesis that C(1) ∈ Z⊥

c and the fact (followingfrom (7.2.4)) that

(Ad c− 1)−1 (

Ad c−1 −Ad c)

X = −(

1 + Ad c−1)

X + an element of Zc

The following result (similar to a result discussed in [KS1] for closed surfaces) says thatthe map Π, introduced in (6.3.2), plays a role somewhat analogous to that of a momentmap.

7.3. Lemma. Let (α, c) ∈ Π−1(e) ⊂ G2g ×Θ, (αH(1), cC) ∈ T(α,c)(G2g ×Θ), and X ∈ g.

Then

Ωo,Θ

(

(

αH(1), cC(1))

, γ′(α,c)(X))

=

⟨

dΠ(α,c)(αH(1), cC(1)) ,

(

1 + Ad c

2

)

X

⟩

g

(7.3.1)

Proof. Recall that α = αjj∈J , and αj+2 = α−1j , for every j ∈ J , and

fidef= Ad (αi · · ·α1) for every i ∈ 1, ..., 4g. Recall also the expression (7.1.12) for γ′(α,c).

25

Then the expression for Ωo,Θ in (6.3.4) gives :

Ωo,Θ

(

(αH(1), cC(1)), γ′(α,c)(X))

(7.2.5)=

1

2

4g∑

i,k=1

ǫik

⟨

f−1i−1H

(1)i , f−1

k−1

(

Adα−1k − 1

)

X⟩

−1

2

⟨

(

Ad c−1 − 1)−1

C(1),(

Ad c−1 − Ad c)

X⟩

(7.2.6)=

1

2

4g∑

i=1

⟨

f−1i−1H

(1)i , (1− f−1

i−1) + (f−14g − f−1

i )X⟩

+1

2

⟨

C(1),(

1 + Ad c−1)

X⟩

(7.1.9)=

1

2

∑

j∈J

⟨

H(1)j ,

[

fj−1

(

1 + f−14g − f−1

j−1 − f−1j

)

− fj+2

(

1 + f−14g − f−1

j+1 − f−1j+2

)]

X⟩

+1

2

⟨

(Ad c)C(1), (1 + Ad c)X⟩

=∑

j∈J

⟨

H(1)j , (fj−1 − fj+2)

(

1 + f−14g

2

)

X

⟩

+

⟨

(Ad c)C(1),

(

1 + Ad c

2

)

X

⟩

(7.1.8)=

⟨

dΠ(α,c)(αH(1), cC(1)),

(

1 + Ad c

2

)

X

⟩

where the last line is obtained from the hypothesis that (α, c) ∈ Π−1(e) which implies that(Ad c)f4g = 1.

7.4. The generic startum of M(Θ). The moduli space M(Θ) is, in general, not amanifold. For the purposes of this paper we shall define the tangent space T(α,c)Π

−1(e) tobe ker dΠ(α,c); this could contain vectors which are not tangent to any paths in Π−1(e). Bya k−form on Π−1(e) we mean the restriction of a k−form on G2g×Θ to (the tangent spacesof) Π−1(e). A G−equivariant k−form η on Π−1(e) for which η(v1, ..., vk) = 0 wheneverany vi is in the image of the derivative of the orbit map of the conjugation action of G willbe taken to be a k−form η over M(Θ). Such a form η is closed if it arises from a form η′

on G2g ×Θ for which the restriction of dη′ to the tangent spaces of Π−1(e) is 0.A simple argument based on the expressions for dΠ∗

(α,c) and γ′(α,c) given in (7.1.10)

and (7.1.12) shows that if (α, c) ∈ Π−1(e) then

ker dΠ∗(α,c) = ker γ′(α,c) (7.4.1)

It has been shown by means of Sard’s theorem in section 3.2 of [KS4] that there is adense open subset of G such that if Θ includes a point in this set then Π is a submersionat every point on Π−1(e) and thus Π−1(e) is a smooth submanifold of G2g ×Θ. We shallfocus on such Θ, to be called generic Θ.

For generic Θ, we see by (7.4.1) that the conjugation action of G on Π−1(e) is locallyfree. Standard results of transformation group theory ([Bre]) imply that in each connected

26

component of Π−1(e) there is a dense open subset such that the quotient of this under theaction of G is a smooth manifold and the quotient map is a submersion. It follows thatthere is a dense open subset of M(Θ) which is a manifold, to be called the generic part (orstratum) of M(Θ), of dimension (2g − 2) dimG + dim(Θ). The quotient map Π−1(e) →M(Θ) over this generic part is a submersion.

It will be convenient to define the following ‘tangent space’:

TpM(Θ)def=(

dΠ∗p(g) + γ′p(g)

)⊥⊂ TpΠ

−1(e) (7.4.2)

The subspace in (7.4.2) is equivariant under the conjugation action of G.If p projects to a point p in the generic part of M(Θ) then the derivative at p of the

quotient map sets up

a G-invariant isomorphism q∗ of TpM(Θ) onto TpM(Θ). (7.4.3)

Smooth G-equivariant k-forms on Π−1(e) which vanish on the directions γ′p(g) correspondone-to-one by q∗ to smooth k−forms on the generic part of M(Θ).

We equip TpM(Θ) with the inner-product which makes

the isomorphism q∗ in (7.4.2) an isometry(7.4.4)

Some of the discussion and results below carry over to more general Θ, and we expectthat suitable sharper versions of the results below exist for all conjugacy classes Θ. Towardsthis, note that for general Θ we may define q∗ΩΘ at p ∈ Π−1(e) to mean the restriction ofΩo,Θ to TpM(Θ), and define det Ωo,Θ to be the determinant of the restriction of Ωo,Θ onTpM(Θ) with respect to any orthonormal basis in TpM(Θ) (section 7.6 contains more onsuch determinants).

The case where Θ consists of one point (this being any point lying necessarily in thecenter of G) becomes essentially identical to the theory for closed surfaces, i.e. the theorycovered by [KS1].

7.5. Lemma. Split T(α,c)(G2g ×Θ) as a direct sum of orthogonal pieces

T(α,c)(G2g ×Θ) = dΠ∗

(α,c)(g)⊕ γ′(α,c)(g)⊕ T(α,c)M(Θ) (7.5.1)

Then, with respect to this decomposition, Ωo,Θ has the ‘matrix form’

∗ Q ∗−Qt 0 0∗ 0 q∗ΩΘ

(7.5.2)

where Q is the bilinear map dΠ∗(α,c)(g)× γ′(α,c)(g) → R given by

Q(

Y, γ′(α,c)X)

=

⟨

dΠ(α,c)Y ,1 + Ad c

2X

⟩

(7.5.3)

27

and Qt is the bilnear map γ′(α,c)(g)× dΠ∗(α,c)(g) → R given on

(

γ′(α,c)X, Y)

by the right

side of (7.5.3).Proof. First we observe that (α, c) ∈ Π−1(e) implies that Π γ(α,c)(x) = e for every

x ∈ G, and so dΠ∗(α,c)(g) and γ

′(α,c)(g) are indeed orthogonal. From (7.3.1) it follows that

Ωo,Θ

(

Y, γ′(α,c)X)

is 0 when Y ∈ ker dΠ(α,c); since dΠ∗(α,c)(g)

⊥ = ker dΠ(α,c), this explains

the two zeros in the second column. The top block Q in the second column is, by definitionof the ‘matrix form’,

Q(A,B)def= ΩΘ(A,B)

and so the expression (7.5.3) for Q is simply a restatement of (7.3.1). The second block inthe first column now follows by skew-symmetry of Ωo,Θ. The bottom corner block followsfrom the fact that ΩΘ is, by defintion, the image of Ωo,Θ under the isomorphism q∗ of(7.4.4).

7.6. Determinants. Let A : V →W be a linear map between finite-dimensional inner-product spaces. If kerA 6= 0, or if V = 0, then we define det(A) = 0. If A ( 6= 0) is anisomorphism onto its image A(V ), then by detA we shall mean the determinant of a matrixof A relative to orthonormal bases in V and A(V ). Thus detA is determined up to sign,but is otherwise independent of the choice of bases. Consideration of matrices shows thatdet(

A|(kerA)⊥)

= det (A∗|A(V )). If A is an isomorphism onto W , and if B : W → Z is alinear map into a finite dimensional inner-product space Z, then det(BA) = det(B) det(A).

If P : V ×W → R is a bilinear form, where V and W are finite-dimensional inner-product spaces of equal dimension, then by detP (determined up to sign) we shall meanthe determinant of the matrix [P (vi, wj)], where vi and wj are orthonormal basesin V and W , respectively; detP is taken to be 0 if V and W are 0−dimensional. Thedeterminant det ΩΘ, which we shall use below, is to be understood in this sense.

7.7. Lemma. Let (α, c) ∈ Π−1(e). Denote by 〈dΠ⊗dΠ〉 the bilinear form on T(α,c)(G2g×

Θ) defined by

〈dΠ⊗ dΠ〉(X, Y ) = 〈dΠ(α,c)X, dΠ(α,c)Y 〉g (7.7.1)

and let 〈dΠ⊗ (Ad c)pr2g+1〉 be the bilinear form on T(α,c)(G2g ×Θ) given by

〈dΠ⊗ (Ad c)pr2g+1〉(X, Y ) = 〈dΠ(α,c)X, (Ad c)pr2g+1Y 〉 (7.7.2)

where pr2g+1 is the projection of T(α,c)(

G2g ×Θ)

on the last factor c−1TcΘ. Then, as-suming that c is not in the center of G, ( i.e. Θ consists of more than one point)

det

(

Ωo,Θ −1

2〈dΠ⊗ dΠ〉+

1

2〈dΠ⊗ (Ad c)pr2g+1〉

)

= det

(

−1

2(Ad c− 1)−1(Ad c−1 − Ad c)(Ad c−1 − 1)−1

) (7.7.3)

where (Ad c−1 − 1)−1 and (Ad c− 1)−1 are taken as maps from (Ad c−1 − 1)(g) into itself.

28

Proof. Let (α, c) ∈ Π−1(e) , and consider vectors (αH(1), cC(1)), (αH(2), cC(2)) ∈T(α,c)(G

2g ×Θ). Recall the expression for Ωo,Θ in (6.3.4) :

Ωo,Θ

(

(αH(1), cC(1)), (αH(2), cC(2)))

=1

2

4g∑

i,k=1

ǫik

⟨

f−1i−1H

(1)i , f−1

k−1H(2)k

⟩

−1

2

⟨

(

Ad c−1 − 1)−1

C(1),(

Ad c−1 − Ad c) (

Ad c−1 − 1)−1

C(2)⟩

(7.7.4a)where

fi = Ad (αi · · ·α1),

(α1, ..., α4g) = (a1, b1, a−11 , b−1

1 , ..., ag, bg, a−1g , b−1

g )

α = (a1, b1, ..., ag, bg) = αjj∈J

J = 1, 2, 5, 6, ..., 4g− 3, 4g− 2

H(r)j+2 = − (Adαj)H

(r)j for every j ∈ J

all as explained in section 7.1. From these we have

Ωo,Θ

(

(αH(1), cC(1)), (αH(2), cC(2)))

=∑

j,k∈J

〈H(1)j ,ΩjkH

(2)k 〉+ 〈C(1),Ω2g+12g+1C

(2)〉

(7.7.4b)wherein (upon using the relationship (7.1.9) between Hj+2 and Hj in the expression forΩo,Θ in (7.7.4a))

Ωjk =1

2

[

fj−1

(

ǫjkf−1k−1 − ǫj k+2f

−1k+2

)

− fj+2

(

ǫj+2 kf−1k−1 − ǫjkf

−1k+2

)]

(7.7.5a)

and, using the second equality in (7.2.6),

Ω2g+1 2g+1 = −1

2(Ad c− 1)−1(Ad c−1 − Ad c)(Ad c−1 − 1)−1 (7.7.5b)

Recall that

Jdef= 1, 2, 5, 6, ..., 4g− 3, 4g− 2

so that J ∪ (J + 2) = 1, 2, ..., 4g.So, from (7.7.5a), for j, k ∈ J with j + 2 < k,

Ωjk =1

2(fj−1 − fj+2)(f

−1k−1 − f−1

k+2) (7.7.6)

29

On the other hand, from the expression for dΠ(α,c) in (7.1.8) we have :

〈dΠ⊗ dΠ〉(

(αH(1), cC(1)), (αH(2), cC(2)))

(7.1.8)=

∑

j,k∈J

⟨

(f−1j−1 − f−1

j+2)H(1)j , (f−1

k−1 − f−1k+2)H

(2)k

⟩

+∑

j∈J

⟨

(f−1j−1 − f−1

j+2)H(1)j , (Ad c)C(2)

⟩

+⟨

(Ad c)C(1),(

f−1k−1 − f−1

k+2

)

H(2)k

⟩

+ 〈C(1), C(2)〉

(7.7.7)

and

〈dΠ⊗ (Ad c)pr2g+1〉(

(αH(1), cC(1)), (αH(2), cC(2))

(7.1.8)=

⟨

∑

j∈J

(f−1j−1 − f−1

j+2)H(1)j , (Ad c)C(2)

⟩

+ 〈C(1), C(2)〉(7.7.8)

Thus, setting

Ω′ def= Ωo,Θ −

1

2〈dΠ⊗ dΠ〉+

1

2〈dΠ⊗ (Ad c)pr2g+1〉 (7.7.9)

we can write

Ω′(

(αH(1), cC(1)), (αH(2), cC(2)))

=∑

j,k∈J

⟨

H(1)j ,Ω′

jkH(2)k

⟩

+∑

j∈J

⟨

H(1)j ,Ω′

j 2g+1C(2)⟩

+∑

k∈J

⟨

C(2),Ω′2g+1kH

(2)k

⟩

+⟨

C(1),Ω′2g+12g+1C

(2)⟩

(7.7.10)where

Ω′2g+1 2g+1 = −

1

2(Ad c− 1)−1(Ad c−1 −Ad c)(Ad c−1 − 1)−1 (7.7.11)

and, from (7.7.6), (7.7.7) and (7.7.8),

Ω′jk = 1

2(fj−1 − fj+2)(f−1k−1 − f−1

k+2)−12 (fj−1 − fj+2)(f

−1k−1 − f−1

k+2) = 0

for j, k ∈ J with j + 2 < k(7.7.12)

and from (7.7.4b), (7.7.7) and (7.7.8),

Ω′j 2g+1 = 0− 1

2(fj−1 − fj+2)Ad c+ 1

2(fj−1 − fj+2)Ad c = 0 for j ∈ J (7.7.13)

and, similarly,

Ω′2g+1k = −1

2Ad c−1(

f−1k−1 − f−1

k+2

)

for k ∈ J (7.7.14)

30

Thus the ‘matrix’ [Ω′jk]j,k∈J for Ω′ has an upper triangular form :

Ω′ =

D1 0 0 · · · · · · 0∗ D5 0 · · · · · · 0∗ ∗ · · · · · · · · · 0· · · · · · · · · Dj · · · 0...

......

......

...∗ ∗ ∗ ∗ D4g−3 0∗ ∗ ∗ ∗ ∗ Ω′

2g+1 2g+1

(7.7.15)

where the diagonal entry Dj , for j, j+1 ∈ J , is given from (7.7.5a), (7.7.7) and (7.7.9), by

Djdef=

[

Ω′jj Ω′

j j+1

Ω′j+1 j Ω′

j+1 j+1

]

=

[

fj+2f−1j−1 − 1 fj+2f

−1j

−fjf−1j−1 + fj+3f

−1j−1 − fj+3f

−1j+2 fj+3f

−1j − 1

]

=

[

Ad (α−1j αj+1αj)− 1 Ad (α−1

j αj+1)

−Adαj +Ad (α−1j+1α

−1j αj+1αj)− Adα−1

j+1 Ad (α−1j+1α

−1j αj+1)− 1

]

(7.7.16)

This factorizes (as observed in [KS1]) as

Dj =

(

a−1 00 b−1

) (

0 1−1 a−1 − 1

) (

1 0b− 1 1

) (

a 00 b

)

(7.7.17)

where a = Adαj and b = Adαj+1. This implies that det(Dj) = 1. Combining thisobservation with the form for Ω′ given in (7.7.15), we have:

det Ω′ = detΩ′2g+1 2g+1

∏

j∈J ′

detDj = detΩ′2g+1 2g+1 (7.7.18)

where J ′ = j ∈ J : j + 1 ∈ J = 1, 5, ..., 4g− 3. Recalling the expression for Ω′2g+1 2g+1

given in (7.7.11), the proof is complete.

7.8. Lemma. Let (α, c) ∈ Π−1(e) ⊂ G2g × Θ, and suppose that Π : G2g × Θ → G is asubmersion at (α, c). Then

detΩΘ =

(

det γ′(α,c)det dΠ∗

(α,c)

)2

det[

(1−Ad c)−1]

(7.8.1)

where det[

(1− Ad c)−1]

is the determinant of the map

(1− Ad c)−1 : Z⊥c → Z⊥

c

31

with Z⊥c being the subspace (ker(Ad c− 1))

⊥= (1−Ad c) (g).

In particular, ΩΘ is non-degenerate on the smooth part of Π−1(e)/G.Proof. Recall from Lemma 7.5 the splitting T(α,c)(G

2g × Θ) as a direct sum oforthogonal pieces

T(α,c)(G2g ×Θ) = dΠ∗

(α,c)(g)⊕ γ′(α,c)(g)⊕ T(α,c)M(Θ) (7.8.2)

and the corresponding ‘matrix form’ of Ωo,Θ given by

∗ Q ∗−Qt 0 0∗ 0 q∗ΩΘ

(7.8.3)

where

Q(

Y, γ′(α,c)X)

=

⟨

dΠ(α,c)Y ,1 + Ad c

2X

⟩

(7.8.4)

for every Y ∈ dΠ∗(α,c)(g), and q∗ : T(α,c)M(Θ) → T

(α,c)M(Θ) is as explained in section

7.4.Note that in the decomposition (7.8.2), the sum of the second and third summands is

ker dΠ(α,c). Using this we see that the ‘matrix’ for 12〈dΠ⊗ dΠ〉 at (α, c) has the form

∗ 0 00 0 00 0 0

(7.8.5)

and the matrix for 12〈dΠ⊗ (Ad c)pr2g+1〉 at (α, c) has the form

∗ Q1 ∗0 0 00 0 0

(7.8.6)

where Q1 is the bilinear map dΠ∗(α,c)(g)× γ′(α,c)(g) → R given by

Q1

(

Y, γ′(α,c)X)

=1

2〈dΠ(α,c)Y, (Ad c)(Ad c−1 − 1)X〉 (7.8.7)

Recall the bilinear form Ω′ on T(α,c)(G2g ×Θ) given in (7.7.9):

Ω′ def= Ωo,Θ −

1

2〈dΠ⊗ dΠ〉+

1

2〈dΠ⊗ (Ad c)pr2g+1〉

From the preceding observations we see that, relative to the splitting of T(α,c)(G2g ×Θ) as

in (7.8.1), Ω′ has the matrix form:

∗ Q2 ∗−Qt 0 00 ∗ q∗ΩΘ

(7.8.8)

32

where Q2 = Q+Q1 is the bilinear map dΠ∗(α,c)(g)× γ′(α,c)(g) → R given by

Q2

(

Y, γ′(α,c)X)

=⟨

dΠ(α,c)Y,X⟩

=⟨

Y , dΠ∗(α,c)X

⟩

(7.8.9)

and −Qt is the bilinear map γ′(α,c)(g)× dΠ∗(α,c)(g) → R given by

−Qt(

γ′(α,c)X, Y)

= −

⟨

dΠ(α,c)Y,1 + Ad c

2X

⟩

From (7.8.8) we have, as usual not worrying about signs,

det Ω′ = detQ2 det(−Qt) detΩΘ

and the expressions for Q2 and −Qt then imply that

det Ω′ =

det dΠ∗(α,c)

det γ′(α,c)·det dΠ∗

(α,c)

det γ′(α,c)det

(

−1 + Ad c

2

)

detΩΘ (7.8.10)

Now decomposing g as the orthogonal sum of Zc = ker (Ad c− 1) and Z⊥c , we have

det

(

1 + Ad c

2

)

= det

(

1 + Ad c

2

∣

∣

∣Z⊥c → Z⊥

c

)

(7.8.11)

Combining the expression for det Ω′ given in (7.8.10) with that obtained earlier in (7.7.3),and using (7.8.11), we obtain

det ΩΘ =

(

det γ′(α,c)

det dΠ∗(α,c)

)2det[

(Ad c− 1)−1(Ad c− Ad c−1)(Ad c−1 − 1)−1]

det(

(1 + Ad c)∣

∣

∣Z⊥c

)

(7.2.4)=

(

det γ′(α,c)det dΠ∗

(α,c)

)2 det (1−Ad c)−1

det(

(1 + Ad c)∣

∣

∣Z⊥c

)

det(

(1 + Ad c)∣

∣

∣Z⊥c

)

(7.8.12)

thus proving the determinant identity (7.8.1).

Recall from section 6.3(iii) that there is a dense open subset D ⊂ G such that forevery conjugacy class Θ passing through any point of D, Π is a submersion at every pointof the level set Π−1(e) and thus Π−1(e) is a smooth submanifold of G2g × Θ. In section7.4 we saw that ΩΘ is a smooth 2−form on a subset (which we call the generic part) ofM(Θ) which is a smooth manifold; as noted in section 6.3(vi), it has been proven in [KS2]that ΩΘ is a closed 2−form on this generic part of M(Θ). Combining this with Lemma7.8 we obtain the following result.

7.9. Theorem. The 2−form ΩΘ is a symplectic structure on the generic part of M(Θ)for every generic conjugacy class Θ in G.

33

8. Limiting quantum Yang-Mills and symplectic volume

In this section Σ is equipped with a Riemannian metric; i.e. Σ is a compact connectedsmooth Riemannian manifold with one connected boundary component ∂Σ. We will provethat as T ↓ 0 the measures µΘ

T (given heuristically by (2.7.5) and rigorously in section 5.3)converge, in a sense specified below, to a volume measure on M(Θ) = A0(Θ)/G whichcorresponds locally to the symplectic structure ΩΘ on M(Θ) (if λ is a symplectic 2−formon a 2d−dimensional space, then the corresponding volume form volλ is the exterior power∧dλ/d!).

Recall that the measure µΘT , as constructed in section 5.3, is a probability measure

on a space A(Θ)/Go. For each well-behaved loop κ on Σ based at o there is a randomvariable ω 7→ h(κ;ω) ∈ G which corresponds to the the holonomy of ω around κ. Forour present purposes we shall not need any details of the definition of A(Θ)/Go, nor ofthe stochastic random holonomies, nor of what ‘well-behaved’ loops mean. What we shallneed is summarized in the following special case of Theorem 5.4, which may be taken as aspecification of µΘ

T , thus providing a choice of a starting point for our present discussions.The triangulations of Σ that we use are assumed to be ‘admissible’ in the sense of

section 4.1. Alternatively, for the purposes of the present section, we may work with anyarbitrary triangulation of Σ and consider µΘ

T as being a measure defined by means of (8.1.1)below.

8.1. Loop expectation values for µΘT . Let S be a (well-behaved) triangulation of

Σ, with the basepoint o as a 0-simplex. Then for any loops κ1, ..., κn based at o andconsisting of oriented 1-simplices of S, and any bounded measurable function f on Gn,

∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω)

= NT (Θ)−1

∫

f (x(κ1), ..., x(κn)) δ(

x(C)c−1)

m∏

j=1

QT |∆j | (x(∂∆j)) dxe1 · · ·dxeMdΘc

(8.1.1)where the notation is as explained in Theorem 5.4. The normalizer NT (Θ) is given by:

NT (Θ) =

∫

G2g

QT |Σ|(cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1) da db dΘc (8.1.2)

By appropriate subdivision of S and by joining o to a vertex of each ∆j by an ap-propriate curve lj , each loop κ in S based at o can be expressed as a ‘composite’ of asequence of the loops of the form lj · ∆j · lj and their reverses, and the loops Ai, Bi, Cand their reverses (by ‘composite’ here we include the operation of successively droppingedges which are traversed in opposite directions consecutively in the usual composite ofcurves). This has been described in more detail in section 4.2. Each x(κi) in (8.1.1) isthen a product of the x(lj)

−1x(∂∆j)±1x(lj) and x(Ai)

±1, x(Bi)±1, and x(C)±1. Thus

we can express f (x(κ1), ..., x(κn)) as F(

y∆1, ..., y

∆m, ai, bi, c

)

(8.1.3)

for some function F , wherey∆j

= x(lj)−1x(∂∆j)x(lj) (8.1.4)

34

ai = x(Ai), bi = x(Bi), c = x(C) (8.1.5)

Conversely, given y∆1

,..., y∆m

, ai, bi, c in G, satisfying

y∆m

· · ·y∆1

= cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1

there is an assignment e 7→ xe, with xe = x−1e for every oriented 1−simplex e of S, such

that (8.1.4) and (8.1.5) hold.With this change of variables it follows, as for (4.4.5),

∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω) =

1

NT (Θ)

∫

Gm×G2g×Θ

F(

y∆1, ..., y

∆m, ai, bi, c

)

·

· δ(

y∆m

· · · y∆1

(cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1)−1)

dΘc

g∏

i=1

daidbi

m∏

j=1

QT |∆j |(y∆j)dy

∆j

(8.1.6)where the δ(·) term means that any one of the y

∆jmay be replaced in the integrand by

the value which makes the argument in δ(·) equal to e, and the corresponding integrationdy

∆j, along with the δ(·)−term, dropped from the integration.

Taking T ↓ 0 we obtain the following Lemma which is very close to Lemma 2.14 in[Se4]; since the sketch proof presented in [Se4] is sloppy enough to be viewed as incorrectwe present a detailed argument here.

8.2. Lemma. Let κ1, ..., κn be loops based at o and composed of oriented 1−simplices ofthe triangulation S, let f be a continuous function on Gn, and let F be associated to f asdescribed above in (8.1.3). Then

limT↓0

∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω) = lim

T→0NT (Θ)−1

∫

F (e, ..., e, ai, bi, c)·

·QT |Σ|

(

cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1)

da1db1 · · ·dagdbgdΘc(8.2.1)

provided that the limit on the right side exists and provided that the limit

N0(Θ)def= lim

T→0

∫

G2g

QT |Σ|(cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1) da1db1 · · ·dagdbgdΘc (8.2.2)

exists and is positive.Proof. Let λT be the Borel measure on Gm ×G2g ×Θ specified by requiring that for

every continuous function h on Gm ×G2g ×Θ,

∫

h dλT =

∫

Gm−1×G2g×Θ

h (y1, ...ym−1, y′m, a1, ..., bg, c) ·

·QT |∆m|(y′m)

m−1∏

j=1

QT |∆j |(yj)dyj

(

g∏

i=1

daidbi

)

dΘc

(8.2.3)

35

whereiny′m = Π(a1, ..., bg, c) (ym−1 · · · y1)

−1 (8.2.4)

and, as usual,Π(a1, ..., bg, c) = cb−1

g a−1g bgag · · · b

−11 a−1

1 b1a1 (8.2.5)

By a change-of-variables argument, the role of y′m (and of ∆m) can be replaced by that ofy′j (and of ∆j) in the integration (8.2.3).

The loop expectation value formula (8.1.6) says that

∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω)

=1

NT (Θ)

∫

F (y1, ..., ym, ai, bi, c) dλT (yj, ai, bi, c)(8.2.6)

So we estimate

∣

∣

∣

∫

F (y1, ..., ym, ai, bi, c)dλT (yj, ai, bi, c)

−

∫

F (e, ..., e, ai, bi, c) dλT (yj, ai, bi, c)∣

∣

∣

≤m∑

j=1

∫

∣

∣

∣F (e, ..., e, yj, ..., ym, ai, bi, c)

− F (e, ..., e, yj+1, ..., ym, ai, bi, c)∣

∣

∣dλT (yj, ai, bi, c)

(8.2.7)wherein the j = m term is interpreted using F (e, ..., e, ai, bi, c) as the second term in theintegrand.

Let ǫ > 0. By (uniform) continuity of F , there is a neighborhood Uǫ of e such that,for any j ∈ 1, ..., m, if yj ∈ Uǫ then the integrand on the right side in (8.2.7) is less thanǫ. Thus, since the total mass of λT is NT (Θ),

left side of (8.2.7) ≤ mǫ ·NT (Θ) + 2||F ||sup

m∑

j=1

λT (Sj,ǫ) (8.2.8)

whereSj,ǫ

def= (y1, ..., ym, ai, bi, c) ∈ Gm+2g ×Θ : yj /∈ Uǫ (8.2.9)

Using the convolution property∫

Qt(ab)Qs(b−1c) dadb = Qt+s(ac), the conjugation invari-

ance of Qt, and∫

GQt(x) dx = 1, we have

λT (Sj,ǫ) =

∫

y/∈Uǫ

QT |∆j |(y)QT |Σ|−T |∆j|

(

Π(a1, ..., bg, c)y−1)

dyda1...dbgdΘc

≤ supy/∈Uǫ

QT |∆j |(y)(8.2.10)

36

Since limt↓0 supy/∈U Qt(y) = 0 for any neighborhood U of e, we can divide (8.2.8) by NT (Θ),let T ↓ 0, and use the hypothesis that N0(Θ) exists and is positive to conclude that (8.2.1)holds.

For the following, recall from section 8.3 that for any Θ which passes through a certaindense open subset of G, the map Π : G2g×Θ → G is a submersion at every point of Π−1(e).

8.3. Theorem. Assume that(i) Θ is such that Π is a submersion at every point of Π−1(e)(ii) Π−1(e) has a dense open subset Π−1(e)0 on which the isotropy of the conjugation

action of G is Z(G)(iii) volΩΘ

(

M(Θ)0)

<∞, where M(Θ)0 is the projection of Π−1(e)0 onto M(Θ) by the

projection map Π−1(e) → Π−1(e)/G ≃ M(Θ) (as in (6.3.3)), and volΩΘis the volume

with respect to the symplectic structure ΩΘ on M(Θ)0.(iv) κ1,...,κn are loops on Σ based at o, as in section 8.1.

Then for any continuous function f on Gn which is invariant under the conjugationaction of G,

limT↓0

∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω)

=1

volΩΘ(M(Θ)0)

∫

M(Θ)0f (h(κ1;ω), ..., h(κn;ω)) dvolΩΘ

([ω])(8.3.1)

Proof. In view of the limiting formulas given in Lemma 8.2, we shall focus on comput-ing limt→0

∫

G2g×ΘH(a1, ..., bg, c)Qt (Π(a1, ..., bg, c)) da1...dbgdΘc, for continuous functions

H.By the submersivity hypothesis, Π−1(e) is a submanifold of G2g × Θ. Picking any

point x∗ ∈ Π−1(e) we have, again by submersivity of Π at x∗, a coordinate system χ on aneighborhood W of x∗ in G2g ×Θ and a coordinate system on a neighborhood U of Π(x∗)in G such that Π(W ) = U and Π|W corresponds, in the coordinates, to projection on thefirst dim(G) coordinates. Let

V = (Π|W )−1(e) = Π−1(e) ∩W

Thus, taking the coordinate system χ such that χ(W ) is a cube, there is a diffeomorphism

Φ : U × V →W

such that Π Φ : U × V → U is the projection on the first factor. Therefore, writing

w = Φ(u, v)

the derivative dΦ(u,v) : g ⊕ TvΠ−1(e) → TΦ(u,v)W = dΠ∗

w(g) + ker dΠw can be expressedas a matrix of the form

dΠ∗w(g)

ker dΠw

g TvΠ−1(e)

[

B 0∗ D2Φ(u,v)

] (8.3.2)

37

where D2Φ(u,v) : TvΠ−1(e) → ker dΠΦ(u,v) is the partial deriative of Φ in the second

variable, and B−1 is the restriction of dΠw to dΠ∗w(g) = (ker dΠw)

⊥. It should be noted

that the diagonal blocks listed in (8.3.2) are indeed ‘square blocks’. Moreover, detB−1 =det dΠ∗

w. Consequently,

| det dΦ(u,v)| =| detD2Φ(u,v)|

| det dΠ∗Φ(u,v)|

(8.3.3)

Let H be a continuous function on G2g × Θ with compact support contained in W . Forthe following computations it is necessary to bear in mind that we use the unit-massinvariant measures on G and Θ, and these differ by constant volume factors from therespective Riemannian volume measures. Then, upon using (8.3.3), we have the followingchange-of-variables formula

vol(G)2gvol(Θ)

∫

G2g×Θ

H (a1, ..., bg, c)Qt (Π(a1, ..., bg, c)) da1...dbgdΘc

= vol(G)

∫

U×V

H (Φ(u, v))Qt(u)| detD2Φ(u,v)|

| detdΠ∗Φ(u,v)|

dudσ(v)

(8.3.4)

where du is the (restriction to U of the) usual unit-mass Haar measure on G, dσ(v) theRiemannian volume measure on V ⊂ Π−1(e), and vol denotes volume measured withrespect to the Riemannian metrics on G and Θ induced by 〈·, ·〉g. Since H is contin-uous and has compact support contained in W , we can apply the heat-kernel propertylimt→0

∫

Gh(x)Qt(x) dx = h(e) valid for all continuous functions h on G, to conclude that

limt↓0

∫

G2g×Θ

H (a1, ..., bg, c)Qt (Π(a1, ..., bg, c)) da1...dbgdΘc

= vol(G)1−2gvol(Θ)−1

∫

V

H (Φ(e, v))| detD2Φ(e,v)|

| det dΠ∗Φ(e,v)|

dσ(v)

(8.3.5)Since | detD2Φ(e,v)| is the Jacobian, at v, of the map V → V : v 7→ Φ(e, v), we have

limt↓0

∫

G2g×Θ

H (a1, ..., bg, c)Qt (Π(a1, ..., bg, c)) da1...dbgdΘc

= vol(G)1−2gvol(Θ)−1

∫

V

H(v)1

| detdΠ∗v|dσ(v)

= vol(G)1−2gvol(Θ)−1

∫

Π−1(e)

H(v)1

| detdΠ∗v|dσ(v)

(8.3.6a)

where in the last line we used again the fact that H is supported in W . By a standardpartition-of-unity argument, we conclude that

limt↓0

∫

G2g×Θ

H (a1, ..., bg, c)Qt (Π(a1, ..., bg, c)) da1...dbgdΘc

= vol(G)1−2gvol(Θ)−1

∫

Π−1(e)

H(v)1

| det dΠ∗v|dσ(v)

(8.3.6b)

38

holds for every continuous function H on G2g ×Θ.Applying this to the limiting formulas (8.2.1) and (8.2.2) we have

limT↓0

∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω)

= N0(Θ)−1vol(G)1−2gvol(Θ)−1

∫

Π−1(e)

F (e, ..., e, α, c) | det(dΠ(α,c))∗|−1 dσ(α, c)

(8.3.7)with

N0(Θ) = vol(G)1−2gvol(Θ)−1

∫

Π−1(e)

| det(dΠ(α,c))∗|−1 dσ(α, c) (8.3.8)

and dσ being the Riemannian volume-measure on Π−1(e); the limiting formula (8.3.7),and the existence of the right side of (8.3.7), is contingent upon N0(Θ) being positive andfinite. Positivity of N0(Θ) is clear from (8.3.8) since Π−1(e) 6= ∅; finiteness will be shownbelow in (8.3.13).

Let Π−1(e)0 be the dense open subset of Π−1(e) on which the isotropy of the G-action on the manifold is Z(G). Then by standard facts from the theory of transformationgroups (section 16.4.1(i), problem 16.10.1 and problem 12.10.1(a) in [Die]), the projectionΠ−1(e)0 → M(Θ)0 is a principal G/Z(G)−bundle over the manifold M(Θ)0. Since fis G−invariant, it follows that so is F and hence the function F (e, ..., e, (·))|Π−1(e)0 isconstant on the fibers; we will sometimes view F (e, ..., e, (·)) as a function on the quotientΠ−1(e)0/G. Writing F (e, ..., e, (·))|Π−1(e)0 as a sum of functions supported on subsetsof Π−1(e)0 on which there are local-trivializations, it follows from (8.3.7) (see [KS1] orLemma 5.8 of [Se5] for details of this argument) that

limT↓0

∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω)

= N0(Θ)−1 vol(G)1−2g

vol(Θ)vol (G/Z(G))

∫

Π−1(e)0/G

F (e, ..., e, α, c)| det γ′(α,c)|

| det(dΠ(α,c))∗|dσ(α, c)

(8.3.9)and

N0(Θ) =vol(G)2−2g

vol(Θ) ·#Z(G)

∫

Π−1(e)0/G

| det γ′(α,c)|

| det(dΠ(α,c))∗|dσ(α, c) (8.3.10)

with dσ being the volume-measure on Π−1(e)0/G corresponding to the Riemannian stru-cure on Π−1(e)0/G (this being as in (7.4.4)). Recall from Lemma 7.8 that

| det γ′(α,c)|

| det dΠ∗(α,c)|

= |Pf(ΩΘ)|∣

∣det(1− Ad c)−1∣

∣

−1/2(8.3.11)

where the Pfaffian |Pf(ΩΘ)| is the square-root of | detΩΘ|. Furthermore, since Pf(ΩΘ)dσ =dvolΩΘ

, we see that the right side of (8.3.9) is given by

N0(Θ)−1∣

∣det(1−Ad c)−1∣

∣

−1/2 vol(G)1−2g

vol(Θ)vol (G/Z(G)) ·

·

∫

Π−1(e)0/G

F (e, ..., e, α, c) dvolΩΘ(α, c)

(8.3.12)

39

and, recalling the identifications Π−1(e)/G ≃ M(Θ) and Π−1(e)0/G ≃ M(Θ)0,

N0(Θ) =∣

∣det(1−Ad c)−1∣

∣

−1/2 vol(G)2−2g

vol(Θ) ·#Z(G)volΩΘ

(

M(Θ)0)

(8.3.13)

where c is any point in Θ. The hypothesis that volΩΘ

(

M(Θ)0)

is finite now shows thatN0(Θ) is finite and this justifies (8.3.7) and hence also (8.3.9) and (8.3.12).

Now returning to the relationship between f and F explained in (8.1.3), (8.1.5) andin the remarks following (8.1.5), we have

f (h(κ1;ω), ..., h(κn;ω)) = F (e, ..., e, a1, ..., bg, c) (8.3.14)

for any flat connection ω with h(Ai;ω) = ai, h(Bi;ω) = bi and h(C;ω) = c. Combining(8.3.9), (8.3.12) and (8.3.14), the proof is complete.

9. The case of SU(2) bundles

In this section we specialize the considerations of the preceding sections to the caseG = SU(2) and show that a sharp form of the semiclassical limit formula holds. We alsodetermine the symplectic volume of M(Θ) explicitly.

We shall exclude the case where Θ consists of one point (which must be one of thematrices ±I). The excluded cases are essentially contained in the theory for SU(2) andSO(3) flat connections over closed surfaces, and this theory is treated fully in [Se5].

9.1. Theorem. Let the group G be SU(2), and let Θ be a conjugacy class containingmore than one point (i.e. Θ is not the conjugacy class of I or of −I). Then :(i) M(Θ) is a smooth (6g− 4)−dimensional manifold which is connected.(ii) the 2−form ΩΘ on M(Θ) is symplectic and the corresponding volume of M(Θ) is

volΩΘ(M(Θ)) =

2π(π − θc)[

vol(SU(2))2π2

]23

if g = 1

4π sin θc

[

vol(SU(2))2π2

]23

vol (SU(2))2g−2∑∞

n=1χn(c)n2g−1 if g ≥ 2

(9.1.1)where χn is the character function of SU(2) specified below in (9.1.7), c is any pointin the conjugacy class Θ, θc is the number in (0, π) for which cos θc = Tr(c)/2, andvol (SU(2))is the volume of SU(2) with respect to the fixed metric 〈·, ·〉g on its Liealgebra.

(iii) if S be a triangulation of Σ, and κ1, ..., κn loops based at o made up of oriented1−simplices of S as in section 4, then for any continuous conjugation-invariant func-tion f on Gn,

limT↓0

∫

f (h(κ1;ω), ..., h(κn;ω)) dµΘT (ω)

=1

volΩΘ(M(Θ))

∫

M(Θ)

f (h(κ1;ω), ..., h(κn;ω)) dvolΩΘ([ω])

(9.1.2)

40

Proof. (i) In SU(2) any element not in the center lies in a unique maximal torus, andtwo elements commute if and only if both lie in the same maximal torus. This implies thatthe isotropy of the G conjugation action at any (a1, ..., bg, c) ∈ Π−1(I) ⊂ G2g × Θ (withI being the identity matrix, and Θ being other than the one-point conjugacy class I) isZ(G) = ±I. Consequently, from (7.4.1), Π is a submersion at every point of Π−1(I),and so Π−1(I) is a smooth submanifold of G2g ×Θ, of dimension 3(2g) + 2− 3 = 6g− 1.Since the isotropy group of the G action is ±I everywhere it follows, by standard factsfrom transformation group theory alluded to earlier ([Die]), that Π−1(I)/G ≃ M(Θ) isa smooth manifold of dimension 6g − 4, and Π−1(I) → Π−1(I)/G ≃ M(Θ) is a smoothprincipal SU(2)/±I−bundle. Let (α1, c1), (α2, c2) ∈ Π−1(I); in particular, c1, c2 ∈ Θ.Since G = SU(2) is connected, there is a continuous path [0, 1] → G : t 7→ xt such thatx0 = I and x1c1x

−11 = c2. Therefore t 7→ (xtα1x

−1t , xtc1x

−1t ) is a path in Π−1(I) from

(α1, c1) to a point (α′2, c2) where α

′2 lies in K−1(c−1

2 ), where K is the product commutatormap

K : G2g → G : (a1, b1, ..., ag, bg) 7→ b−1g a−1

g bgag · · · b−11 a−1

1 b1a1.

In Proposition 3.10 of [Se5], it was proven that the set K−1(c−1) is connected for everyc ∈ SU(2). Consequently, α′

2 can be connected to α2 by a continuous path lying inK−1(c−1

2 ). Combining all these observations we conclude that (α1, c1) can be connectedto (α2, c2) by a continuous path in Π−1(I).

(ii) Recall from (8.3.13) that, since now M(Θ)0 = M(Θ),

limt→0

∫

G2g×Θ

Qt(

cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1)

da1 · · ·dbgdΘc

=∣

∣det(1− Ad c)−1∣

∣

−1/2 vol(G)2−2g

vol(Θ) ·#Z(G)volΩΘ

(M(Θ))

(9.1.3)By definition of dΘc,

∫

Θh(c)dΘc =

∫

Gh(xc1x

−1) dx for any c1 ∈ Θ. It follows then byconjugation invariance of Qt that for any c ∈ Θ,

∫

G2g×Θ

Qt(

cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1)

da1 · · ·dbgdΘc

=

∫

G2g

Qt(

cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1)

da1 · · ·dbg

(9.1.4)

The heat kernel Qt has a standard expansion in terms of characters of the group G. Usingthis, it is proven in Lemma 5.5 of [Se5] (see also [Wi1]) that

limt→0

∫

G2g

Qt(

cb−1g a−1

g bgag · · · b−11 a−1

1 b1a1)

da1 · · ·dbg =

π−θc2 sin θc

if g = 1∑∞n=1

χn(c)n2g−1 if g ≥ 2

(9.1.5)

where

θc ∈ (0, π) is defined by cos θc = Tr(c)/2, (9.1.6)

41

i.e. c is conjugate to

(

eiθc 00 e−iθc

)

, and χn is the character of the n−dimensional irre-

ducible representation of SU(2) :

χn(c) =sin[n cos−1 1

2Tr(c)]

sin[cos−1 12Tr(c)]

for every c ∈ SU(2) \ ±I (9.1.7)

Combining this with (9.1.3) we have, with θc as in (9.1.6),

volΩΘ(M(Θ)) =

2∣

∣det(1− Ad c)−1∣

∣

12 vol(Θ) π−θc

2 sin θcif g = 1

2∣

∣det(1− Ad c)−1∣

∣

12 vol(Θ)vol(G)2g−2

∑∞n=1

χn(c)n2g−1 if g ≥ 2

(9.1.8)

Next we compute vol(Θ). Fix c ∈ Θ. If we view SU(2) as a 3-sphere in R4 in the usualway, the angle between c and I is θc. So the surface area of the 2-sphere Θ is

vol(Θ) = 4π sin2 θc

[

vol (SU(2))

2π2

]23

(9.1.9)

where of course the volume of SU(2) is with respect to the metric 〈·, ·〉g on the Lie algebra

of SU(2).

The mapping eiθ 7→ Ad

(

eiθ 00 e−iθ

)

is a homomorphism of the circle group onto the

group of rotations in a plane in g and has kernel ±1; therefore Ad c is rotation by angle2θc in a plane orthogonal to the maximal torus of c, and so

| det(1− Ad c)−1|12 = det

[

1− cos 2θc sin 2θc− sin 2θc 1− cos 2θc

]− 12

=1

2 sin θc(9.1.10)

Substituting this into (9.1.8) yields the volume formula (9.1.1).

(iii) is a special case of Theorem 8.3, bearing in mind that now M(Θ)0 = M(Θ), andthat, by (ii), this has finite volume.

9.2. Remarks.(i) As already noted, the cases Θ = I and Θ = −I are essentially contained in the

study of the moduli space of flat connections over the trivial SU(2)−bundle and thenon-trivial SO(3)−bundle over the closed surface of genus g dealt with in [Se4,5].The following observations are based on [Se4,5]. The moduli space M(I) consists of anumber of strata whose structure and volumes have been determined in [Se4,5]. Themoduli space M(−I) consists of one point if g = 1; if g ≥ 2 then M(−I) is a smoothconnected 3(2g − 2)−dimensional manifold. The 2−form ΩΘ is symplectic and thecorresponding volume is obtained by considering

limt→0+

∫

G2g

Qt(

−I · b−1g a−1

g bgag · · · b−11 a−1

1 b1a1)

da1 · · ·dbg;

42

this leads to the value 2 ·vol (SU(2))2g−2∑∞

n=1(−1)n−1/n2g−2 (which is the same as isobtained if we set vol(Θ) and the determinant factor equal to 1 in the formula (9.1.8)).

(ii) If in the first formula in (9.1.1) (the case g = 1) we set θc = 0, we obtain the value

2π2[

vol(SU(2))2π2

]23

which is exactly the volume computed in the closed-surface theory

of [Se5] (and in Lemma 3.11 in [Se4]). Such a comparison cannot be made for g ≥ 2since in this case the dimension of M(Θ) collapses at Θ = I.

(iii) If we set g = 1 in the second formula (i.e. the one for g ≥ 2) in (9.1.1) and usethe trigonometric sum formula

∑∞n=1(sinnθ)/n = (π − θ)/2 for θ ∈ (0, π), then what

results is the first formula in (9.1.1) (i.e. the one for g = 1); thus the second formulaactually covers all cases.

Appendix

We will quote some results from [Se2,3] and indicate briefly how they are applied insections 3 and 5 to the construction of µcT and µΘ

T .The basic conditional proability result we need is (from [Se2]):

A1. Theorem. Let (Ωi,Fi, Pi), for i = 1, 2 be probability spaces, where Ω1 and Ω2

are complete separable metric spaces and the corresponding Fi are the Borel σ−algebras.Let G be a compact Lie group. Consider random variables hi : Ωi → G, for i = 1, 2,each having positive density with respect to Haar measure on G. Then there is a uniqueassignment

(F1 ⊗ F2)×G→ [0, 1] : (E, z) 7→ P (E|h1h2 = z)

such that the following hold:(i) E 7→ P (E|h1h2 = z) is a probability measure for every z ∈ G(ii) z 7→ P (E|h1h2 = z) is measurable for every E ∈ F1 ⊗F2, and

∫

V

P (E|h1h2 = z) dp12(z) = (P1 ⊗ P2)(E ∩ h1h2 ∈ V )

holds for every Borel V ⊂ G, where p12 is the probability measure on G describingthe distribution of h1h2

(iii) z 7→ P (A×B|h1h2 = z) is continuous for every A ∈ F1 and B ∈ F2.Moreover,

P (h1h2 = z|h1h2 = z) = 1

Using the conditional measure described above, we can construct another conditionalmeasure, which is the one we use for the measures µcT and µΘ

T .

A2. Definition. Let (Ωi,Fi, Pi), for i = 1, 2, 3, be probability spaces, with (Ω1,F1, P1)and (Ω2,F2, P2) being as in Theorem A1. Let G be a compact Lie group and considerG−valued random variables hi on Ωi, with h1 and h2 having positive densities with respectto Haar measure on G. Consider the product space (Ω,F) = (Ω1,F1)×(Ω2,F2)×(Ω3,F3);for E ∈ F we define

P (E|h1h2 = h3)def=

1

Z

∫

Ω3

dP3(ω3)P(

Eω3 |h1h2 = h3(ω3))

ρ12(

h3(ω3))

(A2.1)

43

where ρ12 is the density of h1h2 with respect to Haar measure on G, and Eω3 = (ω1, ω2) :(ω1, ω2, ω3) ∈ E, and

Z =

∫

Ω3

ρ12(

h3(ω3))

dP3(ω3) (A2.2)

Then P (·|h1h2 = h3) makes sense, is a probability measure on (Ω,F), and satisfies

P (h1h2 = h3|h1h2 = h3) = 1 (A2.3)

Thus P (·|h1h2 = h3) really lives on the subset of Ω where h1h2 = h3.

A3. Construction of µcT . Fix a positive real number T . We shall describe the detailsof the construction of µcT , for any c ∈ G, in terms of the conditional probability measuredescribed in Definition A2. For the sake of notational simplicity we shall only describe thecase considered in section 3, i.e. of the torus with one hole. The general case is exactlyanalogous and is discussed briefly in section A5 below.

For the construction of µcT , divide the disk D into an upper half DU and a lower halfDL. The boundaries ∂DL and ∂DU , when considered as loops, will be taken to start fromthe center O of the diskD. Thus, with L0 being the radial path from O to x0 = (1, 0) ∈ ∂D,we have

h(∂DL;ω)h(∂DU ;ω) = h(L0 · ∂D · L0;ω) (A3.1)

for any connection ω over D.Equip D with the measure which corresponds, via the quotient map q : D → Σ, to

the Riemannian area measure on Σ. For any subset A ⊂ D (such as A = DU or A = DL)let L2(A; g) be the corresponding Hilbert space of g-valued square-integrable functions onD vanishing outside A.

Let Ω1 be a Hilbert-Schmidt closure of L2(DU ; g) and PT1 , or simply P1, the standard

Gaussian measure, with variance scaled by T , on the Borel σ−algebra F1 of Ω1 :thus for each f ∈ L2(DU ; g) there is a Gaussian random variable f on Ω1, such that

for any f1, f2 ∈ L2(DU ; g), 〈f1, f2〉L2(P1) = T 〈f1, f2〉L2(DU ;g).

The g−valued white-noise referred to in section 3.2 (or at least the part of it overDU ) is obtained, for instance, by choosing any orthonormal basis e1, ..., ed of g and setting

Fω(E) =∑di=1 (1Eei)

˜ei for every Borel E ⊂ DU .Let (Ω2,F2, P2) be the corresponding space of DL (we are suppressing the superscript

T in PT2 ).We modify the definition of Ωdisk given in section 3.2 to :

Ωdisk = ΩU × ΩL (A3.2)

Finally, let Ω3 = G2, and let P3 be the unit-mass Haar measure on G2. The functions hiare as follows :

h1(ω1) = h(∂DU ;ω1) (A3.4)

h2(ω2) = h(∂DL;ω2) (A3.5)

h3(a, b) = cb−1a−1ba (A3.6)

44

In (A3.4) and (A3.5), the left sides are defined in terms of solutions of the stochastic dif-ferential equation (3.2.1) and, as pointed out after (3.2.1), are G−valued random variableswith densities QT |DU |(·) and QT |DL|(·), respectively.

As in section 3.3, we defineΩc = Ωdisk ×G2 (A3.7)

and we define the measure µcT by

µcT = P (·|h1h2 = h3) (A3.8)

where the right side is the probability measure specified above in Definition A2.Although Ωc does not depend on c, it follows from (A2.3) that the measure µc really

lives on the subspace of ω for which the constraint

h(L0 · ∂D · L0;ω) = cb−1a−1ba

holds (the left side is defined to be the product h(∂DL;ω2)h(∂DU ;ω1), following (A3.1)).From the expression in (A2.2) for the ‘normalizer’ Z = ZT (c), and from the observa-

tions made above concering the densities of h1 and h2, we have :

ZT (c) =

∫

G2

QT |Σ|(cb−1a−1ba) da db

which is the same as our earlier value (3.4.2).

A4. Construction of µΘT . The constrction of µΘ

T , where Θ is any conjugacy class in G,is similar to that for µcT . The only difference is that now we set

Ω3 = G2 ×Θ (A4.1)

take the measure P3 to be (unit-mass Haar on G2)×(the G−invariant unit-mass measureon Θ), take F3 to be given by h3(a, b, c) = cb−1a−1ba, and define

ΩΘ = Ωdisk ×G2 ×Θ (A4.2)

The spaces (Ωi,Fi, Pi), for i = 1, 2, and the corresponding variables h1 and h2 are as forµcT . With this, we define

µΘT

def= P (·|h1h2 = h3) (A4.5)

The corresponding normalizer Z = NT (Θ) is then :

NT (Θ) =

∫

G2×Θ

QT |Σ|(cb−1a−1ba) da db dΘc (A4.4)

which agrees with the value in (3.4.5).

A5. Other surfaces. For general compact surfaces (as described in section 5), Ω3 istaken to be Gσg×Gp (with notation as in section 5; σ is 1 if Σ is unorientable, 2 otherwise),

45

and h3 is taken to be the appropriate function of the ai, bi, ci given by the right side of(5.3.6).

Finally, we quote from (Proposition 4.5 of) [Se2] the exact expectation-value formulawhich was alluded to in section 3.4.

A6. Proposition. With notation and hypotheses as in sections A2 and A3, let φ =(φ1, ..., φm) : Ω1 → Gm and ψ = (ψ1, ..., ψn) : Ω2 → Gn be measurable functions. Supposethat φk · · ·φ1 = h1 and ψl · · ·ψ1 = h2, where the hi are as in section A1. Suppose alsothat φ has a bounded density ρφ on Gm, and ψ has a bounded density ρψ on Gn. Then forany bounded measurable function f on Gm ×Gn × Ω3, we have (with ω = (ω1, ω2, ω3)):

∫

f(

φ(ω1), ψ(ω2), ω3

)

dP (ω|h1h2 = h3)

=1

Z

∫

f(x, y, ω3)ρφ(x)ρψ(y)δ(

xk · · ·x1yl · · · ylh3(ω3)−1)

dP3(ω3)

where δ(·) means that we can drop any xj (with 1 ≤ j ≤ k) or yj (with 1 ≤ j ≤ l) from theintegration and replace it in the integrand with the value which makes xk · · ·x1yl · · · y1 =h3(ω3), and Z is the normalizing constant given in (A2.2).

In applications, we work typically with random variables of the form h(κ;ω) where κis an admissible loop in D based at O; breaking κ into pieces in DL and pieces in DU (andadjoining appropriate additional radial segments) we can express h(κ;ω) as a product ofvariables h(κ′;ω) where κ′ is a loop in DL or in DU . Thus Proposition A6 is applicable insituations where φi and ψi are holonomy variables h(κ′; ·).

Acknowledgement. At various stages of this work the author has received supportfrom NSF grant DMS 9400961, U.S. Army Research Office #DAAH04-94-G-0249, and fromthe Alexander von Humboldt Foundation. This paper was written in large part during mystay at the Institut fur Mathematik, Ruhr-Universitat Bochum, and it is a pleasure tothank Professor S. Albeverio and colleagues for their hospitality.

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