THE MODULI SPACE OF FLAT CONNECTIONS ON SURFACES WITH BOUNDARY

AMBAR N. SENGUPTA

Abstract. A symplectic form is described for the space of connections on a surface with boundary, with boundary holonomies in given conjugacy classes. The induced symplectic structure on the moduli space of flat connections is studied.

1. Introduction

In this report we present certain results concerning a symplectic structure on the moduli space of flat connections on compact oriented surfaces with boundary. Some of the results are proven here; detailed proofs of the other results are in a longer article under preparation ([21]).

We work with a compact, connected, oriented 2-manifold Σ with boundary components ∂_{1}Σ, ..., ∂_{r}Σ, and a Lie group G. We assume that G is compact, connected, semisimple, and we choose an Ad-invariant metric ⟨·, ·⟩\underline{g} on its Lie algebra \underline{g} (several of the results are valid under weaker hypotheses on G). Let Θ_{i}, for i = 1, ..., r, be conjugacy classes in G. Thus Θ = Θ_{1} ×⋯×Θ_{r} is a conjugacy class in G^{r}. Let A(Θ) be the set of all connections on a principal G–bundle over Σ having holonomy around ∂_{i}Σ in Θ_{i}, for each i = 1, ..., r. Denote by G the group of gauge transformations, and by A^{0}(Θ) the subset of A(Θ) consisting of flat connections. We have then the moduli space M^{0}(Θ) = A^{0}(Θ)/G of flat connections with boundary holonomies in Θ. In this article we

• construct a 2–form Ω_{Θ} on A(Θ) and demonstrate that it is symplectic

• show that the curvature function is the moment map of the action of G on A(Θ)

• discuss the symplectic form induced by Ω_{Θ} on a (subspace of) the moduli space of flat connections

• describe the quantum Yang-Mills measure and give explicit formulas for loop expectation values • describe the relationship between the symplectic volume measure

Date: 10 August, 1999 .

on the moduli M^{0}(Θ) and the classical limit of the quantum Yang-Mills measure.

The moduli space of flat connections over surfaces with boundary/punctures has been studied from several points of view in the papers [2],[3], [8],[5],[6], [7],[9],[12, 13], [14, 15], [17, 18], [22, 23]. (This is only a sample of the literature, not an exhaustive list. I thank J. Huebschmann for bringing the papers [14, 15] to my attention). Interest in the moduli space of flat connections arises also from theoretical physics where it appears in Chern-Simons theory and (2+1)–dimensional gravity.

2. A 2-form for connections over a circle

Let A_{C} be the space of connections on a principal G–bundle π_{C} : P_{C} → C over an oriented circle C. Denote by G_{C} the set of all smooth bundle equivalences of π_{C}. Then G_{C} is a group under composition and acts on A_{C} by pullbacks.

We shall construct two 2–forms Ω^{1} and Ω^{2} on A_{C} which are G_{C}– invariant. We shall define Ω_{Θ}^{C} to be the sum Ω^{1} + Ω^{2}.

Let ω ∈ A_{C} and A, B ∈ T_{ω}A_{C}. Thus A, B are G–equivariant \underline{g}– valued 1–forms on P which vanish on vertical vectors.

Let Ω^{1} be the 2–form on A_{C} given by:

Ω_{ω}^{1}(A, B) =^{def} \frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} ϵ_{st}⟨A(c̃′_{ω}(s)), B(c̃′_{ω}(t))⟩\underline{g} dsdt (2.1)

where

ϵ_{st} = \{ \begin{matrix}1 & if s ≤ t \\ −1 & if s > t \end{matrix}

and c̃_{ω} : [0, 1] → P_{C} is any ω–horizontal path whose projection π_{C} ∘c̃_{ω} : [0, 1] → C traces out the circle C once positively. Since c̃_{ω} is unique up to global shifts by elements of G, the equivariance of A, B and the Adinvariance of ⟨·, ·⟩\underline{g} implies that Ω_{ω}^{1}(A, B) is independent of the choice of c̃_{ω}. Moreover, Ω_{ω}^{1}(A, B) is invariant under smooth reparametrizations of C. It is also readily verified that Ω^{1} is G_{C}–invariant.

We recall quickly some standard observations about bundles. If ϕ ∈ G_{C}, and p ∈ P_{C}, define ϕ̂(p) to be the element of G for which ϕ(p) = pϕ̂(p); then it is a readily verified standard fact that ϕ̂ is a smooth map P_{C} → G satisfying ϕ̂(pg) = g^{−1}ϕ̂(p)g for all (p, g) ∈ P_{C} × G, and the association ϕ ↦ ϕ̂ is an isomorphism which identifies G_{C} with the group (under pointwise multiplication) of all smooth maps ψ : P_{C} → G which satisfy ψ(pg) = g^{−1}ψ(p)g for all p ∈ P_{C} and g ∈ G. The action of G_{C} on A_{C}, in this version, takes the form: (ψ, ω) ↦ Ad (ψ^{−1})ω + ψ^{−1}dψ.

The Lie algebra LG_{C} can then be identified with the set of smooth Ad (G)–equivariant maps P_{C} → \underline{g}, which is indeed a Lie algebra under pointwise operations. If H ∈ LG_{C} then (see also (3.4) below) there is induced a “vector field” H̃ on A_{C} whose value at ω ∈ A_{C} is H̃_{ω} ∈ T_{ω}A_{C} defined by

H̃_{ω}(p) = \frac{d}{dt}|_{t=0} [Ad (exp(−tH(p))ω_{p} + exp(−tH(p))d exp(tH(p))]
= dH_{p} + [ω_{p}, H_{p}]

i.e.

H̃_{ω} = D^{ω}H =^{def} dH + [ω, H] (2.2)

Lemma 2.1. For any ω ∈ A_{C}, A ∈ T_{ω}A_{C} and H ∈ LG_{C},

Ω_{ω}^{1}(A, H̃(ω)) = − ∫_{0}^{1}⟨A(c̃′_{ω}(u), H(c̃_{ω}(u))⟩\underline{g} du − \frac{1}{2}⟨α(1), (Ad (c_{ω}^{−1}) + 1)H(c̃_{ω}(0))⟩\underline{g} (2.3)

where

α(1) = − ∫_{0}^{1} A(c̃′_{ω}(t)) dt (2.4)

and c̃_{ω} is a ω–horizontal curve projecting to c as before.

Proof. Let

α(t) = − ∫_{0}^{t} A(c̃′_{ω}(u)) du

and

β(t) = − ∫_{0}^{t} B(c̃′_{ω}(u)) du

where B = H̃(ω). Then

Ω_{ω}^{1}(A, B) = \frac{1}{2} [∫_{0}^{1}⟨α′(s), β(1) − β(s)⟩\underline{g} ds − ∫_{0}^{1}⟨α′(s), β(s)⟩\underline{g} ds]
= − ∫_{0}^{1}⟨α′(s), β(s)⟩\underline{g} ds + \frac{1}{2}⟨α(1), β(1)⟩\underline{g}

Now since c̃_{ω} is ω–horizontal, B = H̃ and equation (2.2) imply that

B(c̃′_{ω}(t)) = (H ∘ c̃_{ω})′(t)

and so

β(t) = H(c̃_{ω}(0)) − H(c̃_{ω}(t)) (2.5)

So

Ω_{ω}^{1}(A, B) = ∫_{0}^{1}⟨α′(s), H(c̃_{ω}(s))⟩\underline{g} ds − ⟨α(1), H(c̃_{ω}(0))⟩\underline{g} + \frac{1}{2}⟨α(1), H(c̃_{ω}(0)) − H(c̃_{ω}(1))⟩\underline{g}
= − ∫_{0}^{1}⟨A(c̃′_{ω}(u)), H(c̃′_{ω}(u))⟩\underline{g} du − \frac{1}{2}⟨α(1), H(c̃_{ω}(0)) + H(c̃_{ω}(1))⟩\underline{g}
= − ∫_{0}^{1}⟨A(c̃′_{ω}(u)), H(c̃′_{ω}(u))⟩\underline{g} du − \frac{1}{2}⟨α(1), (Ad (c_{ω}^{−1}) + 1)H(c̃_{ω}(0))⟩\underline{g}

where c_{ω} ∈ G is the holonomy of ω around C as given by c̃_{ω}(1) = c̃_{ω}(0)c_{ω}.

It is useful to recall a standard formula ([19, Equation (3.6g)]) concerning variation of parallel-transports. Let c : [0, 1] → Σ be a piecewise smooth path, and fix an initial point u on the fiber over c(0). If η is any connection on π : P → Σ denote by c̃_{η} the η–horizontal path, initiating at u, and projecting by π onto c. Let s ↦ ω_{s} ∈ A be such that (s, p) ↦ ω_{s}(p) is smooth, with ω_{0} = ω and let Y = \frac{d}{ds}|_{s=0}ω_{s}. Define h(s, t) ∈ G by requiring that c̃_{ω_{s}}(t) = c̃_{ω}(t)h(s, t). Then

\frac{∂}{∂s}|_{s=0}h(s, t) = − ∫_{c̃_{ω}|[0,t]} Y (2.6)

Let h(C; ω) denote the holonomy (with initial point u) of the connection ω around a loop C. Then, as a special case of the above formmula we have:

h(C; ω)^{−1} \frac{d}{ds}|_{s=0}h(C; ω_{s}) = − ∫C̃_{ω} Y (2.7)

Now consider another 2–form Ω_{C}^{2} on A_{C} : for A, B ∈ T_{ω}A_{C}, we set

Ω_{C}^{2}(A, B) =^{def} \frac{1}{2} ⟨(Ad (c_{ω}^{−1}) − 1)^{−1}α(1), (Ad (c_{ω}^{−1}) − Ad (c_{ω}))(Ad (c_{ω}^{−1}) − 1)^{−1}β(1)⟩\underline{g} (2.8)

where c̃_{ω} is a ω–horizontal curve projecting to c, c_{ω} ∈ G is the holonomy specified by

c̃_{ω}(1) = c̃_{ω}(0)c_{ω} (2.9)

and

α(1) = − ∫_{0}^{1} A(c̃′_{ω}(t)) dt = \frac{d}{ds}|_{s=0}c_{ω}^{−1}c_{ω_{s}} (2.10)

where s ↦ ω_{s} ∈ A is any map, with ω_{0} = ω, such that (s, p) ↦ ω_{s}(p) is smooth and \frac{d}{ds}|_{s=0}ω_{s}(p) = A(p), and β is defined similarly for B.

THE MODULI SPACE OF FLAT CONNECTIONS 5

We are using the convention that, for x ∈ G, (Ad (x^{−1})−1)^{−1} is 0 on Z_{x} (the Lie algebra of the centralizer Z_{x} of x) and so (Ad (x^{−1}) − 1)^{−1} maps \underline{g} onto LZ_{x}^{⊥} ⊂ \underline{g}. Lemma 2.2. If ω ∈ A_{C} and H ∈ LG_{C}, A ∈ T_{ω}A_{C},

Ω_{C}^{2}(A, H̃) = \frac{1}{2} ⟨α(1), (Ad c_{ω}^{−1} + 1)prH(c̃_{ω}(0))⟩\underline{g} (2.11)

where pr : \underline{g} → LZ_{c_{ω}}^{⊥} is the orthogonal projection. Proof. The expression for β(t) in (2.5) gives

β(1) = H(c̃_{ω}(0)) − H(c̃_{ω}(1))
= (1 − Ad c_{ω}^{−1})H(c̃_{ω}(0))

So, from the definition (2.8) of Ω_{C}^{2} we have:

Ω_{C}^{2}(A, H̃) = \frac{1}{2} ⟨(Ad c_{ω}^{−1} − 1)^{−1}α(1), (Ad c_{ω}^{−1} − Ad c_{ω})(Ad c_{ω}^{−1} − 1)^{−1}(1 − Ad c_{ω}^{−1})H(c̃_{ω}(0))⟩\underline{g}
= \frac{1}{2} ⟨(Ad c_{ω}^{−1} − 1)^{−1}α(1), (Ad c_{ω} − 1)(1 + Ad c_{ω}^{−1})prH(c̃_{ω}(0))⟩\underline{g}
= \frac{1}{2} ⟨α(1), (Ad c_{ω}^{−1} + 1)prH(c̃_{ω}(0))⟩\underline{g}

Let Θ be a conjugacy class in G and let A_{C}(Θ) be the set of all ω ∈ A whose holonomy around C is in Θ: (2.12) A_{C}(Θ) =^{def} {ω ∈ A_{C} : holonomy of ω around C is in Θ} Let T_{ω}A_{C}(Θ) be the set of all A ∈ T_{ω}A_{C} such that there is a map t ↦ ω_{t} ∈ A_{C}(Θ), with ω_{0} = ω, and (t, p) ↦ ω_{t}(p) smooth, satisfying \frac{d}{dt} |_{t=0}ω_{t} = A pointwise.

We introduce the 2–form Ω_{C}^{Θ} on A_{C}(Θ) given by

Ω_{C}^{Θ} =^{def} Ω_{C}^{1} + Ω_{C}^{2} (2.13)

Proposition 2.3. If ω ∈ A_{C}(Θ), A ∈ T_{ω}A_{C}(Θ), and H ∈ LG_{C}, then

Ω_{C}^{Θ}(A, H̃_{ω}) = − ∫_{0}^{1} ⟨A(c̃′_{ω}(t)), H(c̃_{ω}(t))⟩\underline{g} dt (2.14) Proof. In view of Lemmas 2.2 and 2.1, we need only check that α(1) ∈ LZ_{c_{ω}}^{⊥}, where α(t) = − ∫_{0}^{t} A(c̃′_{ω}(s)) ds. Now by equation (2.10),

α(1) = \frac{d}{ds}|_{s=0}c_{ω}^{−1}c_{ω_{s}}

where \frac{d}{ds}|_{s=0}ω_{s} = A. Since A ∈ T_{ω}A_{C}(Θ), we can take ω_{s} ∈ A_{C}(Θ). Then each c_{ω_{s}} belongs to Θ. Thus α(1) ∈ (Ad c_{ω}^{−1} − 1) (\underline{g}) = LZ_{c_{ω}}^{⊥}.

For a 2–form η on A_{C} define the exterior derivative dη to be the 3–form:

(2.15) dη_{ω}(A, B, C) = δ_{C}[η(A, B)] + cyclic permutation

where, on the right, δ_{C} is the “directional derivative” of functions on A_{C}. The following Lemma is essentially taken from [12, Remark 4.7]. We use the same letter C with two obviously different meanings.

Lemma 2.4. Let ω ∈ A_{C}, and A, B, C ∈ T_{ω}A_{C}. Then

dΩ_{C}^{1}(A, B, C) = \frac{1}{2} ⟨[α(1), β(1)], γ(1)⟩\underline{g} (2.16)

where α(1) = − ∫_{0}^{1} A(c̃′_{ω}(t)) dt and similarly for β, B and γ, C.

Proof. Let α(t) = − ∫_{0}^{t} A(c̃′_{ω}(s)) ds, and define β(t) and γ(t) similarly with respect to B and C, respectively. Then computation shows that

δ_{C}(α′(t)) = [α′(t), γ(t)]

(proof: Let s ↦ ω_{s} be such that (s, p) ↦ ω_{s}(p) is smooth and \frac{d}{ds}|_{s=0}ω_{s} = C, and let h(s, t) ∈ G be such that c̃_{ω_{s}}(t) = c̃_{ω}(t)h(s, t). Since A vanishes on vertical vectors, i.e. vectors tangent to fibers of the bundle π, and A(vg) = Ad (g^{−1})A(v) in obvious notation, it follows that A(c̃′_{ω_{s}}(t)) = Ad (h(s, t)^{−1})A(c̃′_{ω}(t)). Differentiating this with respect to s at s = 0 gives, upon using (2.6), [A(c̃′_{ω}(t)), γ(t)].)

So

δ_{C}[Ω_{ω}^{1}(A, B)] = \frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} ϵ_{st}⟨[α′(s), β′(t)], γ(t) − γ(s)⟩\underline{g} dsdt
= \frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} ∫_{s}^{t} ϵ_{st}⟨[α′(s), β′(t)], γ′(u)⟩\underline{g} dudsdt
= \frac{1}{2} ∫_{0≤s≤u≤t≤1}⟨[α′(s), β′(t)], γ′(u)⟩\underline{g} dudsdt + \frac{1}{2} ∫_{0≤t≤u≤s≤1}[⋯]

Adding up the cyclic permutations we obtain equation (2.16).

If Θ is a conjugacy class in G, let ν be the 2–form on Θ given by

ν(cA, cB) =^{def} \frac{1}{2}⟨(Ad c^{−1} − 1)^{−1}A, (Ad c^{−1} − Ad c)(Ad c^{−1} − 1)^{−1}B⟩\underline{g} (2.17)

Then a fairly straightforward calculation ([13, Proposition 3.9]) shows that

dν(cA, cB, cC) = −\frac{1}{2}⟨[A, B], C⟩\underline{g} (2.18)

Consider the map

(2.19) h : A_{C}(Θ) → G : ω ↦ h(ω) = holonomy of ω around C

with some chosen fixed initial point for determining holonomy. Then

h(ω)^{−1}dh_{ω}(A) = − ∫_{0}^{1} A(c̃′_{ω}(t)) dt = α(1) (2.20)

Combining this with the definition of ν and the expression for Ω_{C}^{2} in (2.8) we conclude that

h^{∗}ν = Ω_{C}^{2}|A(Θ) (2.21)

So it is reasonable to believe that d(Ω_{C}^{2}|A(Θ)) = h^{∗}dν, i.e.

dΩ_{C}^{2}(A, B, C) = −\frac{1}{2}⟨[α(1), β(1)], γ(1)⟩\underline{g} (2.22)

when A, B, C ∈ T_{ω}A_{C}(Θ).

Recall that

Ω_{C}^{Θ} = (Ω_{C}^{1} + Ω_{C}^{2})|A_{C}(Θ) (2.23)

So, by (2.16) and (2.22) we have

dΩ_{C}^{Θ} = 0 (2.24)

3. The symplectic structure for the space A(Θ)

We return now to the bundle π : P → Σ. We have the space A of all connections on π and the group G of all automorphisms of π, acting on A by pull backs.

There is a standard 2–form Ω on A, introduced by Atiyah and Bott in [1], defined as follows. If A, B ∈ T_{ω}A, we can form a well-defined 2–form ⟨A ∧ B⟩ on Σ: its value on vectors X, Y ∈ T_{x}Σ is

⟨A ∧ B⟩(X, Y) = ⟨A(X′), B(Y′)⟩\underline{g} − ⟨A(Y′), B(X′)⟩\underline{g}

where X′ and Y′ are vectors in T_{x′}P for some point x′ ∈ π^{−1}(x) such that π_{∗}X′ = X and π_{∗}Y′ = Y. The 2–form Ω is specified by the formula

Ω(A, B) = ∫_{Σ}⟨A ∧ B⟩

For any conjugacy class Θ = Θ_{1} × ⋯ Θ_{r} in G^{r}, there is a restriction map

ρ_{i} : A(Θ) → A_{∂_{i}Σ}(Θ_{i})

We can finally define the two-form Ω_{Θ} on A(Θ):

(3.1) Ω_{Θ} =^{def} Ω − ∑_{i=1}^{r} ρ_{i}^{∗}Ω_{∂_{i}Σ}^{Θ_{i}}

Since Ω is a constant 2–form on A, we have dΩ = 0. On the other hand, we know from (2.23) that dΩ_{∂_{i}Σ}^{Θ_{i}} is 0. So it is reasonable to conclude that

dΩ_{Θ} = 0, i.e. Ω_{Θ} is closed.

Next consider any non-zero A ∈ T_{ω}A(Θ). Define a \underline{g}–valued 1–form ∗A on P as follows. For any p ∈ P choose e_{1}, e_{2} ∈ T_{p}P which project to a positively oriented orthonormal basis (π_{∗}e_{1}, π_{∗}e_{2}) of T_{π(p)}Σ; define the 1–form ∗A at p by requiring that it vanish on the kernel of π_{∗} and satisfy ∗A(e_{2}) = A(e_{1}) and ∗A(e_{1}) = −A(e_{1}). Now let ϕ : Σ → [0, 1] be a smooth function, equal to zero near ∂Σ and equal to 1 in a non-empty open set over which A is not identically equal to zero. Let ψ = ϕ ∘ π. For any real number t, the connection ω + tψ(∗A) is equal to ω over ∂Σ, and so ψ(∗A) ∈ T_{ω}A(Θ). Moreover, Ω_{Θ}(A, ψ(∗A)) = Ω(A, ψ(∗A)) is strictly positive. We conclude that

Ω_{Θ} is non-degenerate.

So

Ω_{Θ} is symplectic on A(Θ)

The action of the group G on A(Θ) preserves the 2–form Ω_{Θ}. We shall determine the moment map of this action.

We can view LG^{∗} as the space of smooth \underline{g}–valued Ad-equivariant 2–forms on P which vanish on vertical vectors. The pairing of LG and LG^{∗} is by integration over Σ.

Consider the curvature map

J : A → LG^{∗} : ω ↦ F^{ω} =^{def} dω + \frac{1}{2}[ω, ω] (3.2)

where [ω, ω] is the \underline{g}–valued two-form whose value on (X, Y) is [ω(X), ω(Y)]− [ω(Y), ω(X)] = 2[ω(X), ω(Y)]. We take its derivative to be

dJ_{ω} : T_{ω}A → LG^{∗} : A ↦ D^{ω}A =^{def} dA + [ω, A] (3.3)

Let

γ_{ω} : G → A : ϕ ↦ ϕ^{∗}ω

be the orbit map through ω ∈ A. Then, as essentially already noted in (2.2),

γ′_{ω}H = dH + [ω, H] (3.4)

Proposition 3.1. Let ω ∈ A(Θ). Then for any X ∈ T_{ω}A(Θ) and H ∈ L(G)

Ω_{Θ}(A, γ′_{ω}H) = _{LG^{∗}}⟨dJ_{ω}(A), H⟩_{LG} (3.5)

In particular, if ω ∈ A^{0}(Θ) and A ∈ T_{ω}A^{0}(Θ) then Ω_{Θ}(A, γ′_{ω}(H)) is 0.

Proof. Since ∂Σ ≠ ∅, the bundle P is trivial. So, using a global section, we can take A to be the space of all smooth \underline{g}–valued 1–forms over Σ. Then

Ω(A, γ′_{ω}H) = ∫_{Σ}⟨A ∧ (dH + [ω, H])⟩
= ∫_{Σ} {⟨dA, H⟩ − d⟨A, H⟩ + ⟨[ω, A] ∧ H⟩}
=^{Stokes} ∫_{Σ}⟨D^{ω}A, H⟩ − ∫_{∂Σ}⟨A, H⟩

On the other hand, by Proposition 2.3,

∑_{i=1}^{r} ρ_{i}^{∗}Ω_{∂_{i}Σ}^{Θ_{i}}(A, γ′_{ω}H) = − ∫_{∂Σ}⟨A, H⟩

So

Ω_{Θ}(A, γ′_{ω}H) = ∫_{Σ}⟨D^{ω}A, H⟩ = _{LG^{∗}}⟨dJ_{ω}(A), H⟩_{LG}

Thus the quotient J^{−1}(0)/G is the moduli space of flat connections A^{0}(Θ)/G, where A^{0}(Θ) = {ω ∈ A(Θ) : ω is flat}.

4. The symplectic structure for the moduli space of flat connections

There are piecewise smooth loops

A_{1}, B_{1}, ..., A_{g}, B_{g}, C_{1}, ..., C_{r}

in Σ, all based at some point o ∈ Σ, such that they generate π_{1}(Σ, o) subject to the relation

C_{r} … C_{1}B̄_{g}Ā_{g}B_{g}A_{g} … B̄_{1}Ā_{1}B_{1}A_{1} = I, the identity in π_{1}(Σ, o)

Each C_{i} is of the form L̄_{i} · ∂_{i}Σ · L_{i} for some segment L_{i} running from o to ∂_{i}Σ. Consider the holonomy map

H : A → G^{2g+r} : ω ↦ (h(A_{1}; ω), h(B_{1}; ω), ..., h(A_{g}; ω), h(B_{g}; ω), h(C_{1}; ω), ..., h(C_{r}; ω)) (4.1)

It takes A(Θ) to G^{2g} × Θ, and A^{0}(Θ) to Π^{−1}(e) ⊂ G^{2g} × Θ, where Π : G^{2g} × Θ → G is given by:

Π(a_{1}, b_{1}, ..., a_{g}, b_{g}, c_{1}, ..., c_{r}) = c_{r} … c_{1}b_{g}^{−1}a_{g}^{−1}b_{g}a_{g} … b_{1}^{−1}a_{1}^{−1}b_{1}a_{1} (4.2)

The group G acts on A^{0}(Θ) and the group G acts on Π^{−1}(e). We have the homomorphism G → G : ϕ ↦ ϕ̃(u), where ϕ̃(u) ∈ G is given by ϕ(u) = uϕ̃(u).

Theorem 4.1. The map H has the following properties:

1. H : A^{0}(Θ) → Π^{−1}(e) is surjective and G − G–equivariant

2. H induces a bijection A^{0}(Θ)/G → Π^{−1}(e)/G

3. H′(ω)A is tangent to the G–orbit through H(ω) if and only A ∈ γ′_{ω}(G)

4. if Ã is tangent to a smooth path in Π^{−1}(e) at H(ω) then Ã = H′(ω)A for some A ∈ T_{ω}A(Θ).

The proof is along the lines of Corollary 4.2(a), (b) in [19] (where the case ∂Σ = ∅ is covered).

As a consequence we have:

Corollary 4.2. There is a unique 2–form Ω̃_{o,Θ} on Π^{−1}(e) ⊂ G^{2g} × Θ such that

H^{∗}Ω̃_{o,Θ} = Ω_{Θ} (4.3)

The space Π^{−1}(e) is, in general, not a submanifold of G^{2g} × Θ. In the corollary above, by 2-form we understand (for the moment) only a function on pairs of vectors tangent (at a common point) to smooth paths in G^{2g} × Θ lying entirely on Π^{−1}(e).

5. Explicit Formula for Ω̃_{o,Θ}

We shall write a typical element α = G^{2g+r} as α = ({α_{j}}_{j∈J}, α_{4g+1}, ..., α_{4g+r}), where J is the indexing set

J =^{def} {1, 2, 5, 6, ..., 4g − 3, 4g − 2}

Explicitly,

α = (α_{1}, α_{2}, α_{5}, α_{6}, ..., α_{4g−3}, α_{4g−2}, α_{4g+1}, ..., α_{4g+r})

Then we set

α_{3} = α_{1}^{−1}, α_{4} = α_{2}^{−1}, ..., α_{4g−1} = α_{4g−3}^{−1}, α_{4g} = α_{4g−2}^{−1}

i.e. we set

α_{j+2} =^{def} α_{j}^{−1} for all j ∈J

Next, we write the components of a tangent vector αH ∈ T_{α}(G^{2g} × G^{r}) as α_{j}H_{j}, with j running over J ∪ {4g + 1, ..., 4g + r}. Then we set

H_{j+2} = −Ad (α_{j})H_{j}

for j ∈ J.

Theorem 5.1. Let α ∈ Π^{−1}(e) ⊂ G^{2g}×Θ and αH^{(1)}, αH^{(2)} ∈ T_{α}(G^{2g}× Θ) two vectors tangent, at α, to smooth paths in G^{2g} × Θ lying entirely on Π^{−1}(e). Then

Ω̃_{o,Θ}(αH^{(1)}, αH^{(2)})

= \frac{1}{2} ∑_{i,j=1}^{4g+r} ϵ_{ij}⟨f_{i−1}^{−1}H_{i}^{(1)}, f_{j−1}^{−1}H_{j}^{(2)}⟩\underline{g}

+\frac{1}{2} ∑_{k=1}^{r}⟨(Ad c_{k}^{−1} − 1)^{−1}C_{k}^{(1)}, (Ad c_{k} − Ad c_{k}^{−1})(Ad c_{k}^{−1} − 1)C_{k}^{(2)}⟩\underline{g}

where we have taken α = ({α_{j}}_{j∈J}, c_{1}, ..., c_{r}) and set C_{k}^{(i)} = H_{4g+k}^{(i)}, and

ϵ_{ij} = \{ \begin{matrix}1 & if i < j \\ 0 & if i = j \\ −1 & if i > j \end{matrix} (5.1)

Sketch Proof. The proof uses the following strategy:

• consider two smooth paths [0, 1] → G^{2g} × Θ : t ↦ α^{(i)}(t), for i = 1, 2, lying entirely on Π^{−1}(e), which are tangent at α = α^{(i)}(0) to the vectors αH^{(1)} and αH^{(2)}, respectively;

• construct two paths t ↦ ω_{t}^{(i)}, with i = 1, 2, in A^{0}(Θ) such that H(ω_{t}^{(i)}) = α_{t}^{(i)};

• evaluate Ω_{Θ}(A^{1}, A^{2}), where A^{i} is the vector in T_{ω}A tangent to the path t ↦ ω_{t}^{(i)} at t = 0.

For the second step mentioned above, the construction of flat connections with specified holonomies is described in detail for the case of surfaces with no boundary in [19].

6. Symplectics and Volume of the Moduli Space

Recall from (4.2) the map Π : G^{2g} × Θ → G. Using the explicit formula for Ω̃_{o,Θ} mentioned above, we can prove:

Theorem 6.1. Let Θ be a conjugacy class in G^{r} such that: (i) e is a regular value of Π : G^{2g} × Θ → G and (ii) there is a non-empty open subset Π^{−1}(e)^{0} of Π^{−1}(e) such that the isotropy group of the conjugacy action of G on Π^{−1}(e) is Z(G) at every point of Π^{−1}(e)^{0}. Then on the manifold Π^{−1}(e)^{0}/G:

• Ω̃_{o,Θ} induces a 2–form \overline{Ω}_{Θ} on M^{0}(Θ)^{0} = Π^{−1}(e)^{0}/G

• the 2–form \overline{Ω}_{Θ} is symplectic

Sketch Proof. The fact that Ω̃_{o,Θ} induces a 2–form on Π^{−1}(e)^{0}/G is proven by checking directly that Ω̃_{o,Θ}(A, B) is 0 whenever A (or B) is tangent to the G–orbit.

The fact that \overline{Ω}_{Θ} is closed follows by straightforward calculation: we calculate dΩ̃_{o,Θ}(A, B, C) on vectors A, B, C tangent to (smooth paths lying entirely on) Π^{−1}(e) and this works out to 0 (very similar calculations have been done in [10, 12]).

The fact that \overline{Ω}_{Θ} is non-degenerate is proved by computing a determinant which is a non-zero multiple of the determinant of the matrix for \overline{Ω}_{Θ}. This determinant turns out to be non-zero (this strategy has been used earlier in [11, 17]).

Theorem 6.2. Assume that the genus of the surface Σ is greater than 1. Then there is a set D of full measure in G^{r} such that for any conjugacy class Θ containing a point of D the conditions of Theorem 6.1 hold. Moreover, the volume of M(Θ)^{0} relative to the symplectic structure \overline{Ω}_{Θ} is

(6.1) #Z(G)\frac{vol(G)^{2g+r−2}}{vol(T)^{r}} ∏_{i=1}^{r} \sqrt{F_{i}} ∑_{α} \frac{χ_{α}(c_{1})...χ_{α}(c_{r})}{(dim α)^{2g+r−2}}

where (c_{1}, ..., c_{r}) ∈ Θ, α runs over the irreducible representations of G, T is any maximal torus in G, and F_{i} = det (1 − Ad c_{i}^{−1}) |L(Z(c_{i}))^{⊥}, where L(Z(c_{i}))^{⊥} is the orthogonal complement of the Lie algebra of the centralizer Z(c_{i}) of c_{i}.

The volume formula for vol_{\overline{Ω}_{Θ}}(M^{0}(Θ)^{0}) agrees with that given by Witten in [22, equation (4.116)].

The proof of Theorem 6.1 uses the heat kernel method introduced by Witten.

7. The Quantum Theory and its Classical Limit

In this section we assume that Σ is a compact connected Riemannian 2–manifold with boundary ∂Σ having r ≥ 1 components ∂_{1}Σ, ..., ∂_{r}Σ. Note that we are not assuming that Σ is orientable or that G is semisimple. Fix basepoints on each boundary component ∂_{i}Σ. Let Θ_{1}, ..., Θ_{r} be conjugacy classes in G, Θ = Θ_{1} × ⋯ × Θ_{r}, and A(Θ) the set of all ω ∈ A for which (h(∂_{1}Σ; ω), ..., h(∂_{r}Σ; ω)) ∈ Θ, where h(∂_{i}Σ; ω) is the holonomy of ω around the loop ∂_{i}Σ. We are interested in a probability measure μ_{t}^{Θ} (here t > 0) on A(Θ) given by the formal expression

dμ_{t}^{Θ}(ω) = \frac{1}{Z_{t}(Θ)}e^{− \frac{S_{YM}(ω)}{t}} ∏_{i=1}^{r} δ_{Θ_{i}}(h(∂_{i}Σ; ω)) [Dω] (7.1)

where S_{YM} : A → R : ω ↦ \frac{1}{2} ∫_{Σ} |F^{ω}|2 ; \underline{g} dσ (F^{ω} is the curvature of ω and dσ is the Riemannian area-measure on Σ) is the Yang-Mills action, [Dω] is the informal “Lebesgue measure” on A pushed down to A/G , δ_{Θ_{i}} a “delta-function”, and Z_{t}(Θ) a “normalizing constant”.

There is a rigorously meaningful measure μ_{t}^{Θ} which corresponds, in a reasonable sense, to the expression (7.1).

Let q : D → Σ be the realization of Σ as a quotient of the unit disk D obtained by identifying certain pairs of arcs on ∂D. Let D_{L} and D_{U} be, respectively, the lower and upper (closed) halves of D, and let H_{L} be a Hilbert-Schmidt closure of L^{2}(D_{L}, dσ) and H_{R} that of L^{2}(D_{R}, dσ), where dσ is the area measure induced by q from that on Σ.

Let

S = H_{L} × H_{R} × G^{k} × Θ (7.2)

where k = 2g (genus of Σ) if Σ is oriented, and if Σ is unorientable then k is specified by the requirement that π_{1}(Σ) is generated by loops A_{1}, ..., A_{k}, C_{1}, ..., C_{r} subject to the relation

C_{r} ⋯ C_{1} · A_{k}^{2} ⋯ A_{1}^{2} = I

Then

• there is a probability measure μ_{t}^{Θ} defined on the Borel σ–algebra of S which can be constructed using the formal expression (7.1)

• for each “good” loop κ on the surface Σ, based at o, there is a μ_{t}^{Θ}–almost-everywhere defined G–valued random variable ω ↦ h(κ; ω), which is the extension of the usual notion of holonomy; the random variable h(κ; ω) is defined by interpreting the usual differential equation of parallel transport as a Stratonovich stochastic differential equation (this idea is due to L. Gross)

• let γ be the Gaussian measure on H_{L} × H_{R}, and λ the unit-mass measure on G^{k} × Θ invariant under the action of G (translation on G^{k} and conjugation on Θ); then μ_{t}^{Θ} is defined by conditioning the product measure γ × λ so that the holonomies around the generating loops of π_{1}(Σ, o) satisfy the geometrically imposed relationship between curvature and holonomies. Thus, in particular, the measure μ_{t}^{Θ} is concentrated on an infinite-dimensional “submanifold” of S.

This measure has been constructed in [17] for the case r = 1, i.e. when ∂Σ has one connected component. The construction in the case where ∂Σ has more than one component is exactly similar.

A curve on Σ will be called “good” if it is made up of the projection on Σ of radial and cross-radial segments on D. A “good triangulation” of Σ is a triangulation which is obtained by projecting by q onto Σ a triangulation of D consisting of radial and cross-radial segments. Let o = q(O), where O is the center of the disk D.

Let (t, x) ↦ Q_{t}(x) be the heat kernel function on G. This is a C^{∞}–function of (t, x) ∈ (0, ∞) × G satisfying

• the heat equation \frac{∂Q_{t}(x)}{∂t} = \frac{1}{2}Δ_{G}Q_{t}(x), where Δ_{G} is the Laplacian on G,

• and the initial condition lim_{t→0} ∫_{G} f(x)Q_{t}(x) dx = f(e), where e is the identity in G, for every continuous function f on G.

Then we have the following result on expectation values with respect to μ_{t}^{Θ}:

Theorem 7.1. Let κ_{1}, ..., κ_{n} be loops on Σ, all based at o, composed of oriented 1–simplices of a good triangulation T of Σ. Let f be a bounded measurable function on G^{n}. Then, for any conjugacy class Θ in G^{r},

∫ f(h(κ_{1}; ω), ..., h(κ_{n}; ω)) dμ_{t}^{Θ}(ω)

=  \frac{1}{N_{t}(Θ)} ∫_{G^{m}×Θ} f(x(κ_{1}), ..., x(κ_{n})) ∏_{i=1}^{r} δ(x(∂_{i}Σ)c_{i}^{−1}) ·

· ∏_{j=1}^{m} Q_{t|Δ_{j}|}(x(∂Δ_{j})) dx_{e_{1}}...dx_{e_{N}} d_{Θ}c (7.3)

where {Δ_{1}, ..., Δ_{m}} are the oriented 2–simplices of the triangulation, e_{1}, ē_{1}, ..., e_{M}, ē_{M} the edges, |Δ_{j}| is the area of Δ_{j}, and d_{Θ}c is the unitmass conjugation-invariant measure on Θ ⊂ G^{r}. The notation x(κ) means x(b_{n}) … x(b_{1}), where κ = b_{n} … b_{1}, with b_{1}, ..., b_{n} oriented 1– simplices. The presence of the delta-functions has to be interpreted as follows: for each ∂_{i}Σ, for any one particular edge e_{i}^{∗} on ∂_{i}Σ, x(e_{i}^{∗}) should be replaced by the value which makes x(∂_{i}Σ) = c_{i}, and the integration over dx(e_{i}^{∗}) dropped. The normalizing constant N_{t}(Θ) is given by

N_{t}(Θ) = \{ \begin{matrix}∫_{G^{2g}×Θ} Q_{t|Σ|}(c_{r} … c_{1}b_{g}^{−1}a_{g}^{−1}b_{g}a_{g} … b_{1}^{−1}a_{1}^{−1}b_{1}a_{1}) da_{1} … db_{g}d_{Θ_{1}}c_{1} … d_{Θ_{r}}c_{r} \\ if Σ is oriented of genus g; \\ ∫_{G^{k}×Θ} Q_{t|Σ|}(c_{r} … c_{1}a_{k}^{2} … a_{1}^{2}) da_{1} … da_{k} … d_{Θ_{1}}c_{1} … d_{Θ_{r}}c_{r} \\ if Σ is unorientable and π_{1}(Σ) is generated \\ by A_{1}, ..., A_{k}, C_{1}, ..., C_{r} subject to \\ C_{r} … C_{1}A_{k}A_{k} … A_{1}A_{1} = I, the identity in π_{1}(Σ). \end{matrix}

Next we have the following result on the limit of μ_{t}^{Θ}, as t → 0, for the case Σ is orientable.

Theorem 7.2. Assume that Σ is oriented of genus > 1. Then there is a dense subset D of full measure in G^{r} such that for any conjugacy class Θ in G^{r} containing a point of D, and for any continuous function f on G^{n} which is invariant under the conjugation action of G

lim_{t→0} ∫ f(h(κ_{1}; ω), ..., h(κ_{n}; ω)) dμ_{t}^{Θ}(ω) =

(7.4) \frac{1}{vol_{\overline{Ω}_{Θ}}(M(Θ)^{0})} ∫_{M(Θ)^{0}} f(h(κ_{1}; ω), ..., h(κ_{n}; ω)) dvol_{[\overline{Ω}_{Θ}]}(ω)

The proof is very similar to the case when ∂Σ has only one component [17, Theorem 8.3].

8. The discrete Yang-Mills measure and its classical limit

Let T be any triangulation of the surface Σ. Let E_{T} = {e_{1}, ē_{1}, ..., e_{M}, ē_{M}} be the set of all oriented edges of T and {Δ_{1}, ..., Δ_{m}} be the set of positively oriented 2–simplices of Σ. We pick positive numbers t, |Δ_{1}|, ..., |Δ_{n}|. Denote by V_{T} the set of all vertices of T .

Let A_{T} be the set of all maps

x : E_{T} → G

such that x(ē_{j})x(e_{j})^{−1} for each e_{j} ∈ E_{T}. Let A_{T} (Θ) be the set of those x ∈ A_{T} for which x(∂_{i}Σ) ∈ Θ_{i} for i = 1, ..., r, where Θ = Θ_{1} × ⋯ Θ_{r} is a conjugacy class in G^{r}.

Let ν_{t}^{T} be the \underline{discrete Yang-Mills measure} on A_{T} (Θ), specified

∫_{A_{T} (Θ)} f dν_{t}^{T} = ∫ f(x) ∏_{k=1}^{n} Q_{t|Δ_{k}|}(x(∂Δ_{k})) ∏_{i=1}^{r} δ(x(∂_{i}Σ)c_{i}^{−1}) dxd_{Θ}c (8.1)

Thus Theorem 7.1 can be restated as (with hypotheses as before)

∫ f(h(κ_{1}; ω), ..., h(κ_{n}; ω)) dμ_{t}^{Θ}(ω)

(8.2)  = \frac{1}{ν_{t}^{T} (A_{T})} ∫_{A_{T}} f(x(κ_{1}), ..., x(κ_{n})) dν_{t}^{T} (x)

Let

(8.3) A_{T}^{0} (Θ) =^{def} {x : x(∂Δ_{j}) = e, for i = 1, ..., n}

and

(8.4) G_{T} =^{def} {all maps ϕ : V_{T} → G}

This is a group acting on A_{T}^{0} (Θ) by

(ϕ, x) ↦ x^{ϕ} : x^{ϕ}(e) = ϕ(i)^{−1}x(e)ϕ(f)

where e is the 1–simplex running from i to f.

Then

A_{T}^{0} (Θ)/G_{T} ≃ M(Θ) ≃ Π^{−1}(e)/G

Replacing T by an isomorphic copy if necessary, there is a subdivision of T which contains loops A_{1}, ..., B_{g}, C_{1}, ..., C_{r}, all based at some vertex o, such that the map

A_{T} → G^{2g+r} : x ↦ (x(A_{1}), ..., x(B_{g}), x(C_{1}), ..., x(C_{r}))

induces a bijection

A_{T}^{0} (Θ)/G_{T} ≃ Π^{−1}(e)/G ≃ M^{0}(Θ) (8.5)

where Π : G^{2g} × Θ → G is the usual map

Π(a_{1}, b_{1}, ..., a_{g}, b_{g}, c_{1}, ..., c_{r}) = c_{r} … c_{1}b_{g}^{−1}a_{g}^{−1}b_{g}a_{g} … b_{1}^{−1}a_{1}^{−1}b_{1}a_{1}

Assume that Θ is such that Π is a submersion G^{2g} × Θ → G and 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).

Then (8.5) induces a symplectic structure, again denoted \overline{Ω}_{Θ} on the corresponding dense open subset A_{T}^{0} (Θ)^{0}/G_{T} of A_{T}^{0} (Θ)/G_{T}. The proof of Theorem 7.2 contains also the proof of

Theorem 8.1. Assume that Θ is such that Π is a submersion G^{2g} × Θ → G and 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). With notation and hypotheses as above, and f a G_{T} –invariant function A_{T} (Θ),

(8.6)  lim_{t→0} ∫_{A_{T} (Θ)} f dν_{t}^{T} = ∫_{A_{T}^{0} (Θ)^{0}/G_{T}} f̃dvol_{\overline{Ω}_{Θ}}

where f̃ is the function on A^{0}(Θ)/G_{T} induced by f.

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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA

  E-mail address: sengupta@math.lsu.edu