1 Introduction

Let G be a simply connected compact Lie group, and let \(\Sigma \) be a closed oriented 2-manifold of genus \(g > 0\). In [7] Ramadas, Singer and Weitsman construct a line bundle \(\mathcal {L}\) over the moduli space of gauge equivalence classes of flat connections \(\mathcal {A}_F(\Sigma )/{\mathcal G}\) on a trivial G-bundle on \(\Sigma .\) This bundle possesses a natural connection, whose curvature is a scalar multiple of Goldman’s symplectic form.

The purpose of this paper is to compute the degree (that is, the first Chern class) of the line bundle described in [7] in the case \(G=SU(2)\). Our main theorem is

Theorem 1.1

The degree of the line bundle is 1.

As we will explain, the second integral cohomology group of the moduli space is infinite cyclic, and the theorem implies that the first Chern class of \(\mathcal {L}\) is a generator. There is in fact a preferred generator (depending on the orientation of \(\Sigma \)), which agrees with \(c_1 (\mathcal {L})\).

In view of Question 5.6 of [5], Theorem 1.1 has the following corollary:

Corollary 1.2

Let \(\Sigma \) be a closed oriented 2-manifold of genus \(g > 0\). Let \({\mathcal G}_n\) be the gauge group of the trivial SU(n)-bundle on \(\Sigma \), and let \(\mathcal {A}_F^{SU(n)}(\Sigma )\) denote the space of flat connections on this bundle. The classifying maps for the line bundles

$$\begin{aligned} \mathcal {L}\rightarrow \mathcal {A}_F^{SU(n)}(\Sigma )/{\mathcal G}_n \end{aligned}$$

induce a homotopy equivalence \({{\mathrm{colim}}}_{n\rightarrow \infty } \mathcal {A}_F^{SU(n)}(\Sigma )/{\mathcal G}_n \simeq {\mathbb C}P^\infty \).

It was previously shown in [5] that the stable moduli space

$$\begin{aligned} {{\mathrm{colim}}}_{n\rightarrow \infty } \mathcal {A}_F^{SU(n)}(\Sigma )/{\mathcal G}_n \cong {{\mathrm{colim}}}_{n\rightarrow \infty } \,\mathrm{Hom}\,(\pi _1 \Sigma , SU(n))/SU(n) \end{aligned}$$

is a \(K(\mathbb {Z}, 2)\) space and hence is homotopy equivalent to \({\mathbb C}P^\infty \). This corollary gives a geometric viewpoint on this homotopy equivalence. In Sect. 4, we also obtain a geometric viewpoint on the homotopy equivalence \({{\mathrm{colim}}}_{n\rightarrow \infty } \,\mathrm{Hom}\,(\pi _1 \Sigma , U(n))/U(n) \simeq (S^1)^{2g} \times {\mathbb C}P^\infty \) from [8].

Our computation of \(c_1 (\mathcal {L})\) in genus 1 (Sect. 3) is similar to Kirk–Klassen [4, Theorem 2.1].Footnote 1 For related work in the algebraic category, see Drezet–Narasimhan [3].

We remark that it would be interesting to extend this degree calculation to other simply connected compact Lie groups.

2 The Chern–Simons line bundle

Let G be a simply connected, compact Lie group, equipped with a chosen faithful representation into \(\text {GL}(n, {\mathbb C})\), and let \(\mathfrak {g}\) be the Lie algebra of G (viewed as a subalgebra of \(\mathfrak {gl}(n, {\mathbb C})\)). The space of connections on the trivial G-bundle over \(\Sigma \) will be denoted by \(\mathcal {A}= \Omega ^1 (\Sigma , \mathfrak {g})\), and the gauge group of this bundle will be denoted by \(\mathcal {G} = C^\infty (\Sigma ,G)\).

The line bundle from [7] is defined using the Chern–Simons cocycle ( [7], p. 411) \(\Theta : \mathcal {A}\times {\mathcal G}\rightarrow {\mathbb C}\) defined by

$$\begin{aligned} \Theta (A, g) = \exp i (CS (\mathbf{A}^\mathbf{g}) - CS(\mathbf{A}) ). \end{aligned}$$

The Chern–Simons functional \(CS(\mathbf{A})\) is defined by

$$\begin{aligned} CS(\mathbf{A}) = \frac{1}{4 \pi } \int _N \mathrm{Trace} (\mathbf{A} d\mathbf{A} + \frac{2}{3} \mathbf{A}^3) \end{aligned}$$

where N is a 3-manifold with boundary \(\Sigma \) and \(g \in \mathcal {G} = C^\infty (\Sigma ,G).\) We have chosen extensions \(\mathbf A\) and \(\mathbf g\) of A and g (respectively) over the bounding 3-manifold N (the existence of \(\mathbf g\) relies on simple connectivity of G). It is shown in [7] that the Chern–Simons cocycle \(\Theta (A, g)\) is independent of the choice of these extensions. We define a line bundle \(\mathcal {L}\) over \(\mathcal {A_F}/\mathcal {G}\) as a \(\mathcal {G}\)-equivariant bundle over the space of flat connections \(\mathcal {A}_F\), where \(g \in \mathcal {G}\) acts on \(\mathcal {A} \times {\mathbb C}\) by

$$\begin{aligned} g: (A,z) \mapsto (A^g, \Theta (A, g) z). \end{aligned}$$

The definition of \(\mathcal {L}\) is

$$\begin{aligned} \mathcal {L}= \mathcal {A}_F \times _{\mathcal {G}} {\mathbb C}. \end{aligned}$$

The symplectic form \(\hat{\Omega }\) on \(\mathcal {A}\) is defined by (see [7], p. 412):

$$\begin{aligned} \hat{\Omega }(a,b) = \frac{i}{2\pi } \int _\Sigma \mathrm{Trace} (a \wedge b) \end{aligned}$$
(1)

for \(a,b \in \Omega ^1(\Sigma , \mathfrak {g})\). Notice that on the affine space \(\mathcal {A}\), the symplectic form is a constant quadratic form; it does not depend on choosing a point in \(\mathcal {A}. \)

3 Degree of the Chern–Simons line bundle in genus 1

Let N be a 3-manifold with boundary \(\Sigma \).

The symplectic form on \(\mathcal {A}\) from (1) descends to a 2-form \(\Omega \) on \(\mathcal {A}_F/\mathcal {G}\) (the space of flat connections), which is symplectic when restricted to the subspace \(\mathcal {A}_F^s \subset \mathcal {A}_F\) of irreducible flat connections.

The authors of [7] exhibit a unitary connection \(\hat{\omega }\) on the prequantum line bundle over \(\mathcal {A}_F\):

$$\begin{aligned} \hat{\omega }(a)= \frac{i}{4 \pi } \int _\Sigma \mathrm{Trace} (A \wedge a) \end{aligned}$$
(2)

whose curvature is \(\hat{\Omega }\). This is done on p. 412 of [7]. The proof uses the fact that the derivative of the Chern–Simons function is

$$\begin{aligned} dCS_A(v) = \frac{1}{4 \pi } \left( \int _{N} 2 \mathrm{Trace} (v \wedge F_A) - \int _\Sigma \mathrm{Trace} (A\wedge v)\right) \end{aligned}$$

for \(v \in T_A \mathcal {A} = \mathcal {A} = \Omega ^1(N, \mathfrak {g})\). This follows from a straightforward calculation using Stokes’ theorem. The above expression restricts on \(\mathcal {A}_F\) to

$$\begin{aligned} dCS_A(v) = -\frac{1}{4 \pi } \int _\Sigma \mathrm{Trace} (A\wedge v) = i \hat{\omega }(v) \end{aligned}$$

(recalling (2)). It is shown on p. 412 of [7] (second paragraph) that \(\hat{\omega }\) is the pullback of a connection \(\omega \) on \(\mathcal {A}_F \times _{\mathcal {G}} {\mathbb C}.\) This is demonstrated by introducing a vertical vector field Y for the action of \(\mathcal {G}\) and showing that

$$\begin{aligned} i_Y \hat{\omega } = L_Y \hat{\omega } = 0 \end{aligned}$$

so \(\hat{\omega }\) is basic and therefore descends to a 1-form on \(\mathcal {A}_F/\mathcal {G}\).

For the rest of the section, we restrict to \(G=SU(2)\). Let x, y be the flat coordinates on the genus 1 surface (see the proof of Lemma 3.1 for more details). Inside the space \(\mathcal {A}\) we can consider the space \(\mathcal {W}\) of all connections of the form \(a\, \mathrm{d}x + b \,\mathrm{d}y\) where \(a, b \in \mathrm{Lie}(T)\) and \(T = \left\{ \mathrm{diag } ( \lambda , \lambda ^{-1}) \,:\, \lambda \in S^1\right\} \) is the diagonal maximal torus of SU(2). Note that \( \mathrm{Lie}(T) = \{xX \,:\, x\in {\mathbb R}\}\), where \(X = \mathrm{diag } (i,-i) \in su(2).\)

Now \(\mathcal {W}\) is a subspace of \(\mathcal {A}\) so the bundle \(\mathcal {L}\) restricts to \(\mathcal {W}\) as a bundle with connection. This bundle is invariant under that part of the gauge group that preserves \(\mathcal {W}\). This consists of \( ({\mathbb Z}\times {\mathbb Z}) \ltimes {\mathbb Z}_2.\) Here \((m,n) \in {\mathbb Z}\times {\mathbb Z}\) is identified with the gauge transformation \( (e^{ix}, e^{iy}) \mapsto e^{imx} e^{iny} \), and \( {\mathbb Z}_2 = \{\pm 1\}\) is the Weyl group of SU(2).

Taking the quotient by \({\mathbb Z}\times {\mathbb Z}\) we get a bundle \(\mathcal {L}'\) on \(T \times T = \mathcal {W}/ \left( {\mathbb Z}\times {\mathbb Z}\right) \) with a connection \(\omega '\). We will show, via a direct computation (Lemma 3.1), that the curvature \(\Omega '\) of this connection has integral equal to \(-4 \pi i \), and using Chern–Weil theory we will be able to conclude that \(\mathcal {L}'\) has degree 2, while \(\mathcal {L}\) has degree 1.

The computation goes as follows.

Lemma 3.1

We have

$$\begin{aligned} \int _{T\times T} \Omega ' = - 4 \pi i . \end{aligned}$$

Proof

As above, let \(X = \mathrm{diag } (i,-i) \in su(2).\) Then \(\mathrm{Trace}(X^2) = -2. \)

Let \(\Gamma \) be a fundamental domain for the action of \({\mathbb Z}\times {\mathbb Z}\) on \(\mathrm{Lie}(T) \oplus \mathrm{Lie}(T)\). Parameterize \(\Gamma \) by \((x,y)\in [0, 2\pi ] \times [0, 2\pi ]\). Under the exponential map

$$\begin{aligned} \exp : \mathrm{Lie}(T) \rightarrow T, \end{aligned}$$

the vector fields \(\frac{\partial }{\partial x}\) and \(\frac{\partial }{\partial y}\) on \(\Gamma \) are identified with the constant vector field X on T, so with the above formula (1) for \(\hat{\Omega }\) we have that

$$\begin{aligned} \Omega '_{(x,y)}:= & {} \Omega '_{(x,y)}(\frac{\partial }{\partial x},\frac{\partial }{\partial y}) = \hat{\Omega }_{(x,y)} (\frac{\partial }{\partial x},\frac{\partial }{\partial y}) = \frac{i}{2 \pi } \int _0^{2\pi }\int _0^{2\pi } \mathrm{Trace}(X^2) \mathrm{d}x \mathrm{d}y\\= & {} -4 \pi i. \end{aligned}$$

The value of \(\Omega '_{(x,y)}\) is independent of x and y, so the integral of \(\Omega '\) over \(T\times T\) (integrating using an area form of total area 1) is also \(-4\pi i\). \(\square \)

Proof of Theorem 1.1 in genus 1. By Lemma 3.1, the cohomology class \([\Omega ']\in H^2 (T\times T; \mathbb {R})\) associated with \(\Omega '\) is \(-4\pi i \alpha \), where

$$\begin{aligned} \alpha = \frac{1}{(2\pi )^2} [\mathrm{d}x\wedge \mathrm{d}y] \in H^2 (T\times T; \mathbb {R}) \end{aligned}$$

denotes the fundamental class. By Chern–Weil theory,Footnote 2 we have

$$\begin{aligned}{}[\Omega '] = -2\pi i \,c_1 (\mathcal {L}'), \end{aligned}$$

so \(c_1 (\mathcal {L}') = \frac{2}{(2\pi )^2} [\mathrm{d}x\wedge \mathrm{d}y]= 2 \alpha \), and \(\mathcal {L}'\) has degree two.

The generator of \(W = \mathbb {Z}/2\mathbb {Z}\) acts on \(T\times T\) by complex conjugation on each factor, inducing a quotient map \(f: T\times T \rightarrow (T\times T)/W\), and we have a homeomorphism \((T\times T)/W \cong S^2\). Our bundle \(\mathcal {L}'\) is \({\mathbb Z}_2\)-equivariant, and it descends to the bundle \(\mathcal {L}\) on \((T \times T)/ W\) (note here that if \((z,w)\in T\times T\) is fixed by W, then the action of W on the fiber of \(\mathcal {L}'\) over (z, w) is trivial: this action is defined in terms of the cocycle \(\Theta (A, g)\), which is zero whenever g fixes A). So \(f^*(\mathcal {L}) = \mathcal {L}'\), and hence, \(\deg (\mathcal {L}) =\frac{1}{\deg (f)} \deg (\mathcal {L}')\). An elementary calculation (e.g., using a \({\mathbb Z}_2\)-equivariant CW complex structure on \(T\times T\)) shows that \(\deg (f) = 2\), completing the proof. \(\Box \)

The key point is that we have computed the degree on \(T\times T\), which has a canonical smooth manifold structure, so we can use Chern–Weil theory. The proof of Theorem 1.1 in higher genus is given in Sect. 5.

4 The conjecture of Lawton and Ramras on the Chern–Simons line bundle

Let \(\Sigma \) be a closed oriented 2-manifold of genus \(g>0\). Let \({\mathcal G}_{SU(n)} = {\mathcal G}_{SU(n)} (\Sigma )\) and \({\mathcal G}_{U(n)} = {\mathcal G}_{U(n)} (\Sigma )\) denote the gauge groups of the trivial SU(n) and U(n)-bundles on \(\Sigma \) (respectively), and let \(\mathcal {A}^{SU(n)}_F(\Sigma )\) and \(\mathcal {A}^{U(n)}_F(\Sigma )\) denote the spaces of flat SU(n)- and U(n)-connections on these bundles. Define

$$\begin{aligned} \mathcal {M}_U (\Sigma )= {{\mathrm{colim}}}_n \mathcal {A}^{U(n)}_F(\Sigma )/{\mathcal G}_{U(n)} \text { and } \mathcal {M}_{SU} (\Sigma ) = {{\mathrm{colim}}}_n \mathcal {A}^{SU(n)}_F(\Sigma )/{\mathcal G}_{SU(n)}. \end{aligned}$$

We refer to these as the stable moduli spaces of flat unitary (or special unitary) connections over \(\Sigma \). Let \(\mathcal {L}_n \rightarrow \mathcal {A}^{SU(n)}_F(\Sigma )/{\mathcal G}_{SU(n)}\) denote the prequantum line bundle. As n varies, these bundles are compatible with the inclusions \(SU(n) \hookrightarrow SU(n+1)\) and hence induce a line bundle \(\mathcal {L}_\infty \rightarrow \mathcal {M}_{SU}(\Sigma )\), which we call the stable prequantum line bundle.

The homotopy types of the stable moduli spaces were determined in [5, 8]:

$$\begin{aligned} \mathcal {M}_{U}(\Sigma ) \simeq {\mathbb C}P^\infty \times (S^1)^{2g}, \,\,\,\, \mathcal {M}_{SU}(\Sigma ) \simeq {\mathbb C}P^\infty . \end{aligned}$$
(3)

These results are computational and rely on the uniqueness of Eilenberg–MacLane spaces; that is, no explicit homotopy equivalences between these spaces have been constructed. Here we offer bundle-theoretic descriptions of these homotopy equivalences.

Remark 4.1

The proof that \(\mathcal {M}_{U}(\Sigma )\) and \(\mathcal {M}_{SU}(\Sigma )\) have the homotopy types stated above relies on an independent result showing that these spaces have the homotopy types of CW complexes. In the unitary case, this is proven in [8, Lemma 5.7], and the same argument works in the special unitary case.

Theorem 4.2

The classifying map

$$\begin{aligned} \mathcal {M}_{SU}(\Sigma ) \longrightarrow {\mathbb C}P^\infty \end{aligned}$$

for the stable prequantum line bundle \(\mathcal {L}_\infty \) is a homotopy equivalence.

Proof

Writing \(\Sigma = \Sigma ' \# T\), where T is a torus, the quotient map induces a homeomorphism \(\Sigma \rightarrow \Sigma /\Sigma ' \cong T\). Together with the inclusions \(SU(2)\hookrightarrow SU(n)\), this induces a map

$$\begin{aligned} \mathcal {A}^{SU(2)}_F(T)/{\mathcal G}_{SU(2)} (T)\longrightarrow \mathcal {M}_{SU}(\Sigma ), \end{aligned}$$

which induces an isomorphism on \(H^2 ( -; {\mathbb Z})\) by [5, Theorem 5.3]. We have shown in the previous section that the classifying map for the bundle

$$\begin{aligned} \mathcal {L}_2 \rightarrow \mathcal {A}^{SU(2)}_F(T)/{\mathcal G}_{SU(2)}(T) \end{aligned}$$

induces an isomorphism on \(H^2(- ; \mathbb {Z})\). Since the classifying map for \(\mathcal {L}_\infty \) restricts to a classifying map for \(\mathcal {L}_2\), we find that the classifying map for \(\mathcal {L}_\infty \) must also induce an isomorphism on \(H^2 (-; \mathbb {Z})\). But up to homotopy, this is a self-map of \({\mathbb C}P^\infty \), and any self-map of \({\mathbb C}P^\infty \) that induces an isomorphism on \(H^2 (-; \mathbb {Z})\) is a homotopy equivalence. \(\square \)

We now turn to the unitary case. Fix generators \(\alpha _i\), \(\beta _i\) (\(i=1, \ldots , g\)) for \(\pi _1 (\Sigma )\) (with \(\prod _i [\alpha _i, \beta _i] = 1\)). The determinant map

$$\begin{aligned} \det : \mathcal {M}_{U}(\Sigma ) \rightarrow (S^1)^{2g} \end{aligned}$$

is defined by

$$\begin{aligned}{}[A] \mapsto (\det (\rho _A (\alpha _1)), \det (\rho _A (\beta _1)), \ldots , \det (\rho _A (\alpha _g)), \det (\rho _A (\beta _g))), \end{aligned}$$

where \(A\in \mathcal {A}^{U(n)}_F(\Sigma )\) and \(\rho _A : \pi _1 (\Sigma ) \rightarrow U(n)\) is its holonomy representation.

Corollary 4.3

There is a line bundle \(\mathcal {L}^U \rightarrow \mathcal {M}_{U}(\Sigma )\) that restricts to

$$\begin{aligned} \mathcal {L}_\infty \rightarrow \mathcal {M}_{SU}(\Sigma ), \end{aligned}$$

and if \(\alpha \) is a classifying map for \(\mathcal {L}^U\), then the map

$$\begin{aligned} \mathcal {M}_{U}(\Sigma ) \xrightarrow {(\det , \alpha )} (S^1)^{2g}\times {\mathbb C}P^\infty \end{aligned}$$

is a homotopy equivalence.

Proof

Let \(f: \mathcal {M}_{SU}\mathop {\longrightarrow }\limits ^{\simeq } {\mathbb C}P^\infty \) be a classifying map for \(\mathcal {L}_\infty \), and choose a homotopy equivalence \(\phi : \mathcal {M}_{U}(\Sigma ) \mathop {\longrightarrow }\limits ^{\simeq } {\mathbb C}P^\infty \times (S^1)^{2g}\), as in (3). Let \({\mathbb C}P^\infty \times (S^1)^{2g}\mathop {\longrightarrow }\limits ^{p_1} {\mathbb C}P^\infty \) be the projection, and let \(i: \mathcal {M}_{SU}(\Sigma ) \hookrightarrow \mathcal {M}_{U}(\Sigma )\) be the inclusion. By [5, Theorem 5.3], i induces an isomorphism on \(\pi _2\), so the composite

$$\begin{aligned} i_1 := p_1 \circ \phi \circ i : \mathcal {M}_{SU}(\Sigma ) \longrightarrow {\mathbb C}P^\infty \end{aligned}$$

induces an isomorphism on \(\pi _2\) and hence is a homotopy equivalence. Define

$$\begin{aligned} \alpha := f \circ i_1^{-1} \circ p_1 \circ \phi : \mathcal {M}_{U}(\Sigma ) \rightarrow {\mathbb C}P^\infty , \end{aligned}$$

where \(i_1^{-1}\) is a homotopy inverse to \(i_1\). Then we have a homotopy commutative diagram

and we define \(\mathcal {L}^U\) to be the pullback, under \(\alpha \), of the universal bundle over \({\mathbb C}P^\infty \). It remains to show that \((\det , \alpha )\) is a homotopy equivalence. Since \(\det \) is split by the inclusion of \((S^1)^{2g} \cong \,\mathrm{Hom}\,(\pi _1 (\Sigma ), U(1))/U(1)\) into \(\mathcal {M}_{U}(\Sigma )\), we see that on fundamental groups, \(\det _*\) is a surjection between free abelian groups of rank 2g, hence an isomorphism. Since \(\alpha _*\) is an isomorphism on \(\pi _2\), the result follows from the Whitehead theorem (and Remark 4.1) \(\square \)

5 The degree of the line bundle in higher genus

Let \(\mathcal {L}_g\) denote the prequantum line bundle on the moduli space \(\mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}\) of flat connections on a trivial SU(2)-bundle over the genus g surface \(\Sigma ^g\). We now show that \(\mathcal {L}_g\) has degree 1 for every genus g surface (\(g>0\)), not just \(g=1\) (thereby completing the proof of Theorem 1.1). This statement is meaningful, since we have:

Lemma 5.1

For any \(g\ge 1\), we have \(H^2 (\mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}; \mathbb {Z}) \cong \mathbb {Z}\).

Proof

In [5], it was proven that \(\mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}\) is simply connected (and a second proof of this fact was given in [1]). Now, triviality of \(H_1 (\mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}; \mathbb {Z})\) implies that \(H^2 (\mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}; \mathbb {Z})\) is torsion free (by the universal coefficient theorem), and a simple direct analysis of the Poincaré polynomial of \(\mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}\), as determined by Cappell–Lee–Miller [2], shows that \(H^2 (\mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}; \mathbb {Z})\) has rank 1. \(\square \)

A map \(f: \Sigma ^g \rightarrow \Sigma ^h\) induces a map

$$\begin{aligned} f^\#: \mathcal {A}_F^{SU(2)}(\Sigma ^h)/{\mathcal G}\rightarrow \mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}, \end{aligned}$$

and as noted in [7, Remark 3, p. 412], if f has degree 1 then \((f^\#)^*(\mathcal {L}_g) = \mathcal {L}_h\). This implies that

$$\begin{aligned} (f^\#)^* (c_1(\mathcal {L}_g)) = c_1 (\mathcal {L}_h), \end{aligned}$$

and taking \(h = 1\) we find that \((f^\#)^* (c_1(\mathcal {L}_g)) = c_1 (\mathcal {L}_1) = 1\) (by the result in Sect. 3). Now Lemma 5.1 implies that \(c_1(\mathcal {L}_g)\) is a generator of \(H^2 (\mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}; \mathbb {Z})\). This completes the proof of Theorem 1.1. \(\square \)

Remark 5.2

The results in [5] in fact show that the map

$$\begin{aligned} H^2 (\mathcal {M}_{SU}(\Sigma ^g); \mathbb {Z}) \longrightarrow H^2 (\mathcal {M}_{SU}(\Sigma ^1); \mathbb {Z}) \end{aligned}$$

is determined by

$$\begin{aligned} f^*: H^2 (\Sigma ^1) \rightarrow H^2 (\Sigma ^g). \end{aligned}$$

Thus, a choice of generator in \(H^2 (\mathcal {A}_F^{SU(2)}(\Sigma ^1)/{\mathcal G}; \mathbb {Z})\) and a choice of orientations on \(\Sigma ^1\) and \(\Sigma ^g\) give a choice of generator in \(H^2 (\mathcal {A}_F^{SU(2)}(\Sigma ^g)/{\mathcal G}; \mathbb {Z})\), and the above discussion shows that this generator coincides with \(c_1 (\mathcal {L}_g)\).