2.1.1 Spaces of connections and Sobolev completions

We let $\mathscr{A}_E$ denote the space of U(2) connections A on E which induce the fixed connection $\lambda$ on its determinant $\Lambda$ . Informally, we refer to A as a ‘unitary’ or U(2) connection with ‘fixed determinant’.

What we are really interested in is the space $\mathscr{A}_{E}^c$ of $GL(2,\mathbb{C})$ connections $\mathbb{A}$ on E inducing the fixed connection $\lambda$ on its determinant $\Lambda$ . Using the unitary structure on E, such a connection $\mathbb{A}$ can be decomposed as

$$\mathbb{A} = A + i \Phi,$$

where $A \in \mathscr{A}_E$ is a unitary connection with fixed determinant and $\Phi \in \Omega^1 (Y , \mathfrak{g}_E )$ is a $\mathfrak{g}_E$ -valued 1-form on Y. Here $\mathfrak{g}_E = \mathfrak{su}(E)$ is the adjoint bundle consisting of traceless, skew-adjoint endomorphisms of E. Thus, the space of $GL(2,\mathbb{C})$ connections with fixed determinant is $\mathscr{A}_{E}^c = \mathscr{A}_E \times \Omega^1 (Y,\mathfrak{g}_E )$ , and we typically refer to the connection $\mathbb{A}$ by its corresponding pair $(A, \Phi )$ . Informally, we shall refer to $\mathbb{A}$ as a ‘complex’ or $GL(2,\mathbb{C})$ connection on E with ‘fixed determinant’.

Let $\mathscr{G}_E$ be the group of U(2) bundle automorphisms of E which induce the identity on its determinant $\Lambda$ , and $\mathscr{G}_{E}^c$ be the group of $GL(2,\mathbb{C})$ bundle automorphisms of E which induce the identity on $\Lambda$ (again, we refer to these as ‘unitary’ or ‘complex’ gauge transformations with ‘fixed determinant’). The groups $\mathscr{G}_{E}$ and $\mathscr{G}_{E}^c$ act on $\mathscr{A}_E$ and $\mathscr{A}_{E}^c$ , respectively, in a natural way (our convention is that they act on the left): for example, a unitary gauge transformation u with fixed determinant acts on the complex connection $(A, \Phi )$ by

\begin{align*}u\cdot (A, \Phi ) = (A - d_{A}u \cdot u^{-1} , u \Phi u^{-1} ) .\end{align*}

For later purposes, it will be convenient to make $\mathscr{A}_{E},\mathscr{A}_{E}^c$ and $\mathscr{G}_E , \mathscr{G}_{E}^c$ into Hilbert manifolds and Hilbert Lie groups, respectively, by considering their completions with respect to suitable Sobolev norms. Namely, from now on we will only consider unitary or complex connections with regularity in $L^{2}_{k}$ , and unitary or complex gauge transformations with regularity in $L^{2}_{k+1}$ . For our purposes, choosing any $k \geq 2$ will suffice. We denote Sobolev-completed spaces simply by $\mathscr{A}_E , \mathscr{A}_{E}^c , \mathscr{G}_E , \mathscr{G}_{E}^c$ unless we need to make reference to the Sobolev structure, in which case we shall use the notation $\mathscr{A}_{E,k}, \mathscr{A}_{E,k}^c$ , $\mathscr{G}_{E,k+1}, \mathscr{G}_{E,k+1}^c$ .

If the determinant $(\Lambda, \lambda )$ is trivial as a bundle with connection, then E has structure group SU(2) and $\mathscr{A}_E, \mathscr{A}_{E}^c$ consist of the SU(2) or $SL(2,\mathbb{C})$ connections. We allow for non-trivial $(\Lambda, \lambda )$ in order to also study also the $PSU(2) = SO(3)$ connections and $PSL(2,\mathbb{C})$ connections, via the following standard construction.

2.1.2 The ‘U(2) model’ for SO(3) and $PSL(2,\mathbb{C})$ connections

The U(2) connections on E can be regarded as $PSU(2) = SO(3)$ connections by the following procedure (see [Reference Braam and DonaldsonBD95, Part II] and [Reference DonaldsonDon02, § 5.6] for further details).

Associated to a U(2) bundle E there is a $PSU(2) = SO(3)$ bundle $P = E \times_{U(2)} SO(3)$ , whose corresponding rank-3 real vector bundle over Y is the adjoint bundle $\mathfrak{g}_E = \mathfrak{su}(E)$ . Conversely, any SO(3) bundle P lifts to a U(2) bundle E which is unique up to replacing E by its tensor product $E \otimes L$ with a U(1) bundle L. (Indeed, on a 3-manifold the U(2) bundles are classified by $c_1 (E) = c_1 (\Lambda ) \in H^2 (Y , \mathbb{Z} )$ and the SO(3) bundles by $w_2 (P) \in H^2 (Y , \mathbb{Z}/2)$ ; a choice of lift of P corresponds to an integral lift of $w_2 (P)$ , and this is unique modulo even classes in $H^2(Y, \mathbb{Z})$ .) Of course, a similar relationship holds if we replace U(2) and SO(3) by their complexifications $GL(2,\mathbb{C})$ and $PSL(2,\mathbb{C})$ .

A U(2) connection with fixed determinant on E induces an SO(3) connection on P, and conversely an SO(3) connection on P lifts to a U(2) connection on E with fixed determinant which is unique up to tensoring (E, A) by a flat real line bundle. This gives a bijection between $\mathscr{A}_P / \mathscr{G}_P$ and the orbits of the action of $H^1(Y, \mathbb{Z}/2)$ on $\mathscr{A}_E /\mathscr{G}_E$ . The same relationship holds between the $GL(2,\mathbb{C})$ connections with fixed determinant and the $PSL(2,\mathbb{C})$ connections.

This gives a fairly concrete model for studying the spaces of SO(3) (or $PSL(2,\mathbb{C})$ ) connections, effectively translating the complexity in the SO(3) stabilizers to that of the SU(2) stabilizers and the stabilizers of the action of $H^1 (Y, \mathbb{Z}/2 )$ on $\mathscr{A}_E / \mathscr{G}_E$ .

2.2.1 The Kähler structure of $\mathscr{A}_{E}^c$

The space $\mathscr{A}_{E}^c$ carries a canonical complex structure,

(2.1) \begin{align}J (\dot{A}, \dot{\Phi} ) = (-\dot{\Phi } ,\dot{A} ), \end{align}

and a canonical symplectic form $\omega_J$ ,

(2.2) \begin{align}\omega_J ((\dot{A}_1 , \dot{\Phi}_1) , (\dot{A}_2, \dot{\Phi}_2) ) = - \int_Y \mathrm{Tr}( \dot{A}_1 \wedge \ast \dot{\Phi}_2 -\dot{\Phi_1}\wedge \ast \dot{A}_2). \end{align}

The pair $(\omega_J , J )$ is compatible, making $\mathscr{A}_{E}^c$ into a Hilbert Kähler manifold. The complex gauge group $\mathscr{G}_{E}^c$ is a Hilbert complex Lie group which acts holomorphically on $(\mathscr{A}_{E}^c , J )$ . In turn, the unitary gauge group $\mathscr{G}_E$ preserves the whole Kähler structure.

2.2.2 Moment maps and stability

Furthermore, the unitary gauge group $\mathscr{G}_E$ acts on $(\mathscr{A}_{E}^c , \omega_J )$ in a Hamiltonian fashion, with moment map given by

$$\mathscr{A}_{E}^c \rightarrow \mathrm{Lie}\mathscr{G}_E = \Omega^0 (Y, \mathfrak{g}_E ), \quad (A, \Phi ) \mapsto d_{A}^\ast \Phi . $$

Indeed, the vector field $X_\xi$ given by the infinitesimal action of $\xi \in \Omega^0 (Y, \mathfrak{g}_E)$ is $X_\xi = (-d_{A}\xi , [\xi, \Phi])$ and we have

\begin{align*}\omega_J ( X_\xi , (\dot{A}, \dot{\Phi})) = -\langle \dot{\Phi}, d_{A}\xi \rangle_{L^2} - \langle \dot{A},[\xi,\Phi]\rangle_{L^2} = \langle \ast [ \dot{A} , \ast \Phi ] - d_{A}^\ast \dot{\Phi} , \xi \rangle_{L^2},\end{align*}

where $ \ast [ \dot{A} , \ast \Phi ] - d_{A}^\ast \dot{\Phi}$ is the linearization of $-d_{A}^\ast \Phi$ at $(A, \Phi)$ . We may thus attempt to carry out the Kähler quotient construction, albeit in infinite dimensions, of the Hilbert Kähler manifold $(\mathscr{A}_E , \omega_J , J )$ by the $\mathscr{G}_E$ -action.

The zeros of the moment map (i.e., $d_{A}^\ast \Phi = 0$ ) have the following interpretation.

In this case, the U(2) bundle with connection $(E, \mathbb{A})$ splits as a sum $(L,a) \oplus (L^\perp , a^\perp )$ of U(1) bundles with connection (such that $(\Lambda,\lambda ) = (L , a ) \otimes (L^\perp , a^\perp )$ ), and $\Phi$ acts diagonally with respect to this decomposition. Hence, the $\mathscr{G}_E$ -stabilizer of $\mathbb{A} = A + i \Phi$ contains a U(1) subgroup. Assertion (2) follows.

A complex connection $\mathbb{A} $ satisfying the conclusion of Proposition 2.1(1) is said to be polystable (or completely reducible, or reductive), whereas a complex connection $\mathbb{A} $ satisfying the conclusion of Proposition 2.1(2) is said to be stable (or irreducible)). By a recent result by Wu and Zhang [Reference Wu and ZhangWZ23], the converse to Proposition 2.1 is known (but we do not need this result): $\mathbb{A}$ is polystable if and only if there exists a complex gauge transformation $u \in \mathscr{G}_{E}^c$ such that $u\cdot \mathbb{A} = A_u + i \Phi_u$ solves $d_{A_u}^\ast \Phi_u = 0$ (in the case when $\mathbb{A}$ is flat, this is a fundamental result by Donaldson [Reference DonaldsonDon87b] and Corlette [Reference CorletteCor88]); and $\mathbb{A}$ and is stable if and only if, in addition, the $\mathscr{G}_E$ -stabilizer of $u \cdot \mathbb{A}$ is $\{\pm I\}$ . In light of this, and in an abuse of terminology, we make the following definition.

The first step towards carrying out the infinite-dimensional Kähler quotient construction is the following result.

Proposition 2.3. The space of stable configurations $\mathscr{C}_{E,k}^\ast \subset \mathscr{A}_{E,k}^c$ is a Hilbert submanifold. Its tangent space is

$$T_{(A, \Phi )}\mathscr{C}_{E, k}^\ast = \{ (\dot{A}, \dot{\Phi} ) \in (\Omega^1 \oplus \Omega^1)(Y, \mathfrak{g}_E )_k \, | \, \ast [ \dot{A} , \ast \Phi ] - d_{A}^\ast \dot{\Phi} = 0 \} . $$

(Here, and likewise subsequently, the subscript k stands for the Sobolev $L^{2}_k$ completion of the space of sections shown.)

Before proving this result we make some general observations. Consider, for each $i = 1, 2$ and $0 \leq j \leq k$ , the pair of bounded linear operators

\begin{align*} \delta_{(A, \Phi)}^i : & \Omega^0 (Y, \mathfrak{g}_E )_{j+1} \rightarrow (\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )_{j},\\ (\delta_{(A, \Phi)}^i)^\ast : & (\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )_{j} \rightarrow \Omega^0 (Y, \mathfrak{g}_E )_{j-1},\end{align*}

given by

\begin{align*} \delta_{(A, \Phi)}^1 (\dot{\xi}) & = (- d_A \dot{\xi} , [\dot{\xi} , \Phi ] ), \\ (\delta_{(A, \Phi)}^1)^\ast (\dot{A} , \dot{\Phi}) & = - d_{A}^\ast \dot{A} - \ast [ \dot{\Phi} , \ast \Phi ],\\ \delta_{(A, \Phi)}^2 (\dot{\xi}) & = ( -[\dot{\xi} , \Phi ] , - d_A \dot{\xi}), \\ (\delta_{(A, \Phi)}^2)^\ast (\dot{A} , \dot{\Phi}) &= \ast [ \dot{A} , \ast \Phi ] - d_{A}^\ast \dot{\Phi}.\end{align*}

It is easily seen that $(\delta_{(A, \Phi )}^i)^\ast $ is the formal $L^2$ adjoint of $\delta_{(A, \Phi )}^i$ . Let us also introduce the following notation:

(2.3) \begin{align}\mathscr{T}_{(A, \Phi), j}^i & := \mathrm{Im} \delta_{(A, \Phi )}^i \subset (\Omega^1 \oplus \Omega^1) (\mathfrak{g}_E )_j ,\end{align}

(2.4) \begin{align}\mathscr{K}_{(A, \Phi ) , j}^i &:= \mathrm{Ker} (\delta_{(A, \Phi )}^i )^\ast \subset (\Omega^1 \oplus \Omega^1) (\mathfrak{g}_E )_j. \end{align}

The significance of these operators is as follows. Identifying $(\Omega^0 \oplus \Omega^0 )(Y, \mathfrak{g}_E)_{k+1} \cong \mathrm{Lie} \mathscr{G}_{E,k+1}^c$ , the map sending

(2.5) \begin{align}(\dot{\xi}_1 , \dot{\xi}_2 ) \in (\Omega^0 \oplus \Omega^0 )(Y, \mathfrak{g}_E)_{k+1} \mapsto \delta_{(A, \Phi )}^1 (\dot{\xi}_1) + \delta_{(A, \Phi)}^2 (\dot{\xi}_2) = \delta_{(A, \Phi )}^1 (\dot{\xi}_1) + J\delta_{(A, \Phi)}^1 (\dot{\xi}_2) \end{align}

can be identified with the infinitesimal action of the complex gauge group $\mathscr{G}_{E,k+1}^c$ on the space of complex connections.

The bounded linear operator $\delta_{(A, \Phi )}^i$ is associated to a first-order differential operator with injective symbol, and as a matter of Hodge theory we thus have that for $0 \leq j \leq k$ the subspace $\mathrm{Ker}\delta_{(A, \Phi ) }^i \subset \Omega^0 (Y, \mathfrak{g}_E )_{j+1}$ is finite-dimensional (and independent of j), and there are $L^2$ -orthogonal decompositions

(2.6) \begin{align}\Omega^0 (Y, \mathfrak{g}_E )_{j+1} & = \mathrm{Ker}\delta_{(A, \Phi)}^i \oplus \mathrm{Im}(\delta_{(A, \Phi)}^i )^\ast , \end{align}

(2.7) \begin{align}(\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )_j & = \mathrm{Im}\delta_{(A, \Phi)}^i \oplus \mathrm{Ker}(\delta_{(A, \Phi)}^i)^\ast = \mathscr{T}_{(A, \Phi ) , j }^i \oplus \mathscr{K}_{(A, \Phi ) , j }^i.\end{align}

In particular, Proposition 2.3 asserts that, at a stable configuration,

$$T_{(A, \Phi )}\mathscr{C}_{E , k}^\ast = \mathscr{K}_{(A, \Phi ) , k}^2 . $$

Observe also that the composition $(\delta_{(A, \Phi )}^1)^\ast \circ \delta_{(A,\Phi)}^2 $ vanishes at a polystable configuration $(A, \Phi )$ :

$$(\delta_{(A, \Phi )}^1)^\ast \circ \delta_{(A,\Phi)}^2 (\dot{\xi}) = - \ast d_A [\dot{\xi}, \ast \Phi] + \ast [d_{A}\dot{\xi}, \Phi] = [\dot{\xi}, d_{A}^\ast \Phi ] = 0, $$

where the last equality used $d_{A}^\ast \Phi = 0$ . In particular, it follows from this that the decomposition (2.7) restricts to $L^2$ -orthogonal decompositions for $0\leq j \leq k$ :

(2.8) \begin{align}\mathscr{K}_{(A, \Phi ) , j}^2 = \mathscr{T}_{(A, \Phi ) , j}^1 \oplus ( \mathscr{K}_{(A, \Phi ) , j}^1 \cap \mathscr{K}_{(A, \Phi ) , j}^2 ) , \end{align}

(2.9) \begin{align}\mathscr{K}_{(A, \Phi ) , j}^1 = \mathscr{T}_{(A, \Phi ) , j}^2 \oplus ( \mathscr{K}_{(A, \Phi ) , j}^1 \cap \mathscr{K}_{(A, \Phi ) , j}^2 ) .\end{align}

Proof of Proposition 2.3. As noted above, the linearization of the equation $d_{A}^\ast \Phi = 0$ is $-\delta_{2}^\ast (\dot{A}, \dot{\Phi} ) = 0$ . By the implicit function theorem for Hilbert spaces, it suffices to establish the surjectivity of $\delta_{2}^\ast : (\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )_{k} \rightarrow \Omega^0 (Y, \mathfrak{g}_E )_{k-1}$ whenever $(A, \Phi) \in \mathscr{C}_{E,k}^\ast $ . By the decomposition (2.6) with $i = 2$ on the $L^{2}_{k-1}$ spaces, this is equivalent to $\mathrm{Ker} \delta_{(A, \Phi )}^2 = 0$ . Indeed, $\mathrm{Ker} \delta_{(A, \Phi )}^2$ is identified as the (rotation by J) of the tangent space to the $\mathscr{G}_{E, k}$ -stabilizer of $(A, \Phi )$ , which vanishes when $(A, \Phi )$ is stable.

The latter assertion can be verified directly as follows. Suppose for a contradiction that $[\dot{\xi} , \Phi] $ and $d_{A} \dot{\xi}$ vanish identically, where $0 \neq \dot{\xi} \in \Omega^0 (Y, \mathfrak{g}_E )_{k-1}$ . Elliptic regularity ensures that $\dot{\xi}$ has regularity $L^{2}_{k+1}$ (and, in particular, $\dot{\xi}$ is $C^0$ ). Since $\dot{\xi}$ is A-parallel (i.e., $d_A \dot{\xi} = 0$ ) and non-trivial, the eigenspaces of $\dot{\xi}$ induce a bundle decomposition $E = L \oplus L^{-1}\Lambda$ which is preserved by the connection A, for some U(1) bundle L. The equation $[\dot{\xi} , \Phi] = 0$ says that $\Phi$ is diagonal with respect to this decomposition. Altogether, it follows that the $\mathscr{G}_{E, k+1}$ -stabilizer of $(A, \Phi )$ contains a U(1) subgroup, which is a contradiction.

2.2.3 Local structure of the stable orbit space

Formally, the Kähler quotient construction yields the space of stable orbits $\mathscr{B}_{E,k}^\ast = \mathscr{C}_{E,k}^\ast / \mathscr{G}_{E,k+1}$ . For the moment, we regard it as a topological space with the quotient topology. We now describe how to equip $\mathscr{B}_{E,k}^\ast $ with a Hilbert manifold structure by constructing slices for the action of $\mathscr{G}_{E,k+1}$ on $\mathscr{C}_{E,k}^\ast $ .

Definition 2.4. Let $(A, \Phi ) \in \mathscr{C}_{E,k}^\ast $ be a stable configuration. The stable Coulomb slice at $(A, \Phi )$ is the subspace $\mathcal{S}_{(A, \Phi ),k} \subset \mathscr{C}^{\ast }_{E,k}$ given by

$$\mathcal{S}_{(A, \Phi ),k} = ( (A, \Phi ) + \mathscr{K}_{(A, \Phi ) , k}^1) \cap \mathscr{C}^{\ast }_{E,k} . $$

Proof. To prove assertion (1) it suffices, by the implicit function theorem for Hilbert spaces, to show that the operator

\begin{align*}T_{(A, \Phi )}\mathscr{C}_{E,k}^\ast = \mathscr{K}_{(A, \Phi ) , k }^2 \xrightarrow{(\delta^{1}_{(A, \Phi)})^\ast } \Omega^0 (Y, \mathfrak{g}_E )_{k-1}\end{align*}

is surjective. But the operator $(\delta_{(A,\Phi )}^1)^\ast : (\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E)_{k} \rightarrow \Omega^0 (Y, \mathfrak{g}_E )_{k-1}$ is surjective (this follows by the same argument as for $(\delta_{(A,\Phi )}^2)^\ast $ from the proof of Proposition 2.3, which uses the fact that $(A,\Phi )$ is stable). Thus, the assertion follows from this, the vanishing of $(\delta^{1}_{(A, \Phi )})^\ast \delta_{(A, \Phi )}^2$ (using $d_{A}^\ast \Phi = 0$ ) and the decomposition (2.7) with $i = 2$ .

Assertion (2) follows now from the decomposition (2.8), noting that the tangent spaces at $(A, \Phi )$ to $\mathscr{C}_{E,k}^\ast $ , to the $\mathscr{G}_{E,k+1}$ -orbit of $(A, \Phi )$ and to $\mathcal{S}_{(A, \Phi ),k}$ are given by $\mathscr{K}_{(A, \Phi ) , k}^2$ , $\mathscr{T}_{(A, \Phi ),k}^1$ and $\mathscr{K}_{(A, \Phi ) , k}^1 \cap \mathscr{K}_{(A, \Phi ) , k}^2$ , respectively.

We now show assertion (3). In fact, we show that for an arbitrary pair $(A^\prime, \Phi^\prime) \in \mathscr{A}_{E,k}^c$ , not necessarily in $\mathscr{C}_{E,k}$ , there exists a unitary gauge transformation u such that

(2.10) \begin{align}(\delta^{1}_{(A, \Phi )})^\ast ( u\cdot (A^\prime , \Phi^\prime ) - (A, \Phi ) ) = 0 .\end{align}

Writing $(A^\prime, \Phi^\prime ) = (A, \Phi ) + (a, \phi ) $ , we seek, for sufficiently small $(a, \phi)$ , a unitary gauge transformation of the form $u = \exp \xi$ where $\xi \in \Omega^0 (Y, \mathfrak{g}_E )_{k+1}$ , such that (2.10) holds. Note that

$$ u \cdot (A^\prime , \Phi^\prime ) - (A, \Phi ) = (-d_A \xi + (\exp \xi)a (\exp \xi)^{-1} ,(\exp \xi)\Phi (\exp \xi)^{-1} +(\exp \xi)\phi (\exp \xi)^{-1} ). $$

By the implicit function theorem for Hilbert spaces, it suffices to show that the linearization of (2.10) in the $\xi$ direction is surjective at $(\xi , a , b ) = (0,0,0)$ , and this is the operator given by

(2.11) \begin{align}\dot{\xi} \in \Omega^0 (Y, \mathfrak{g}_E )_{k+1} \mapsto (\delta_{(A,\Phi)}^1 )^\ast \delta_{(A,\Phi)}^1 \dot{\xi} \in \Omega^0 (Y, \mathfrak{g}_E )_{k-1}. \end{align}

By (2.6) and (2.7) the operator (2.11) surjects onto the complement of $\mathrm{Ker}\delta_{(A,\Phi )}^1$ . But $\mathrm{Ker}\delta_{(A,\Phi )}^1 = 0$ by stability of $(A, \Phi )$ (this follows by the same argument we used for $\delta_{(A,\Phi)}^2$ in the proof of Proposition 2.3), and hence the assertion follows.

By Proposition 2.5, the map $\mathcal{S}_{(A, \Phi )} \rightarrow \mathscr{B}_{E,k}^\ast = \mathscr{C}_{E,k}^\ast /\mathscr{G}_{E,k+1}$ is a local homeomorphism at $(A, \Phi )$ , and the Hilbert manifold structure on the neighborhood of $(A, \Phi ) \in \mathcal{S}_{(A, \Phi ),k}$ provides Hilbert manifold charts for the stable orbit space $\mathscr{B}_{E,k}^\ast $ . The tangent spaces of $\mathscr{B}_{E,k}^\ast $ are thus given by

(2.12) \begin{align}T_{[(A, \Phi )]} \mathscr{B}_{E,k}^\ast & = T_{(A, \Phi )}\mathcal{S}_{(A, \Phi ),k} =\mathscr{K}_{(A, \Phi ) , k}^1 \cap \mathscr{K}_{(A, \Phi ) , k}^2\nonumber \\& = \{(\dot{A}, \dot{\Phi}) \in (\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E)_{k} \, | \, -d_{A}^\ast \dot{A} - \ast [\dot{\Phi},\ast \Phi] = \ast [\dot{A}, \ast \Phi] - d_{A}^\ast \dot{\Phi} = 0 \}.\end{align}

The projection $\mathscr{C}_{E,k}^\ast \rightarrow \mathscr{B}_{E,k}^\ast $ then becomes a submersion of Hilbert manifolds. Noting that the $\mathscr{G}_{E,k+1}$ -action on $\mathscr{A}_{E,k}^c$ is by isometries of the flat $L^2$ metric, there is an induced $L^2$ metric on $\mathscr{B}_{E,k}^\ast $ making this projection a Riemannian submersion. By Proposition 2.5(2), in the description (2.12) of $T_{(A, \Phi ) }\mathscr{B}_{E,k}^\ast $ the $L^2$ metric coincides with that induced from the standard $L^2$ inner product.

Furthermore, recall that $\mathscr{G}_{E,k+1}$ acts on the Hilbert Kähler manifold $(\mathscr{A}_{E,k}^c , \omega_J , J ) $ preserving the Kähler structure and in a Hamiltonian fashion. By the infinite-dimensional version of the Kähler quotient theorem, the complex structure J and the symplectic form $\omega_J$ from (2.1) to (2.2) thus descend to give an (integrable) complex structure and symplectic form on the stable orbit space $\mathscr{B}_{E,k}^\ast $ , compatible with the $L^2$ metric on $\mathscr{B}_{E,k}^\ast $ . In the description (2.12) of $T_{[(A,\Phi)]}\mathscr{B}_{E,k}^\ast $ , these two structures are given by the expressions as equations (2.1) and (2.2). Summing up, we have the following result.

2.2.4 The variational viewpoint

We have obtained the stable orbit space $\mathscr{B}_{E}^\ast $ as the Kähler quotient of $(\mathscr{A}_{E}^c , \omega_J , J )$ by the $\mathscr{G}_E$ -action. There is an alternative viewpoint in which the ‘polystability’ condition $d_{A}^\ast \Phi = 0$ arises from a minimization problem. We only mention this briefly, referring to [Reference CorletteCor88] for the details.

Namely, consider a complex orbit $\mathscr{G}_{E}^c \cdot (A_0 , \Phi_0 )\subset \mathscr{A}_{E}^c$ and the restriction of the squared $L^2$ -norm functional $\| \Phi \|_{L^2}^2$ onto the orbit, that is, the $\mathscr{G}_E$ -invariant functional

(2.13) \begin{align}\mathscr{G}_{E}^c \cdot (A_0 , \Phi_0 ) \ni (A,\Phi ) \mapsto \| \Phi \|_{L^2}^2 . \end{align}

Then $(A, \Phi) \in \mathscr{G}_{E}^c \cdot (A_0 , \Phi_0 )$ satisfies $d_{A}^\ast \Phi = 0$ if and only if $(A, \Phi )$ is a critical point of (2.13). Furthermore, when the $\mathscr{G}_E$ -stabilizer of $(A,\Phi )$ is smallest possible, the function (2.13) induces a convex function on $\mathscr{G}_{E}^c / \mathscr{G}_E$ , and hence if a critical point on the orbit exists then it is unique and attains the minimum of $\| \Phi \|_{L^2}^2$ on that orbit.

2.3.1 The complex Chern–Simons functional

Throughout, we make a choice of a ‘basepoint’ in $ \mathscr{A}_{E , k }^c$ , that is, we fix a $GL(2,\mathbb{C} )$ connection on E with fixed determinant $(\Lambda, \lambda )$ , which we denote by $\mathbb{A}_0$ .

Definition 2.7. The (complex) Chern–Simons functional is the function $CS : \mathscr{A}_{E , k }^c \rightarrow \mathbb{C}$ defined at a connection $\mathbb{A} = \mathbb{A}_0 + \alpha$ by

\begin{align*}CS (\mathbb{A} ) = \int_Y \mathrm{Tr}( \alpha \wedge ( (F_{\mathbb{A}} + F_{\mathbb{A}_0}) - \tfrac{1}{6}[\alpha , \alpha ])).\end{align*}

The Chern–Simons functional has the following property. Suppose M is an oriented and compact 4-manifold with boundary Y equipped with an extension of E to M (as a bundle with fixed determinant) and extensions of the connection $\mathbb{A}$ and the basepoint connection $\mathbb{A}_0$ (as connections with fixed determinant). Then, using Stokes’ theorem, one can show that

(2.14) \begin{align}CS( \mathbb{A} ) = \int_M \mathrm{Tr}(F_{\mathbb{A}}\wedge F_{\mathbb{A}}) - \mathrm{Tr}(F_{\mathbb{A}_0} \wedge F_{\mathbb{A}_0}) . \end{align}

We can use (2.14) to understand the periods of $CS(\mathbb{A})$ under the action of complex gauge transformations with fixed determinant. Namely, for a given $u \in \mathscr{G}_{E , k+1}^c$ choose any path of connections $\mathbb{A}(t) \in \mathscr{A}_{E,k}^c$ from $\mathbb{A}(0) = \mathbb{A}$ to $\mathbb{A}(1) = u \cdot \mathbb{A}$ . This path can be regarded as a connection $\underline{\mathbb{A}}$ on the bundle over $[0,1]\times Y$ obtained as the trivial extension of $E \rightarrow Y$ . We extend the basepoint connection $\mathbb{A}_0$ trivially to $[0,1]\times Y$ . Denote by $E_u \rightarrow S^1 \times Y$ the bundle obtained by clutching with u. By (2.14) and Chern–Weil theory we have

(2.15) \begin{align}CS (u \cdot \mathbb{A} ) - CS (\mathbb{A}) & = \int_{[0,1]\times Y} \mathrm{Tr}(F_{\underline{\mathbb{A}}}^2) = 8 \pi^2 ( c_2 (E_u ) - \tfrac{1}{2}c_1 (E_u )^2 ) \cdot [S^1 \times Y] \nonumber\\& = 8 \pi^2 c_2 (E_u )\cdot [S^1 \times Y]\in 8 \pi^2 \mathbb{Z}, \end{align}

where we note that $c_{1}(E_u )^2 = 0$ because u acts as the identity on $\Lambda = \Lambda^2 E$ .

In particular, $\mathrm{Re}CS$ has periods in the lattice $8 \pi^2 \mathbb{Z} \subset \mathbb{R}$ . In turn, $\mathrm{Im}CS$ has no periods; furthermore, it is independent of the chosen basepoint connection $\mathbb{A}_0$ and is given by the following formula: in terms of the decomposition $\mathbb{A} = A + i \Phi$ ,

(2.16) \begin{align}\mathrm{Im}CS (A + i \Phi ) = 2 \int_{Y} \mathrm{Tr}(\Phi\wedge(F_A - \tfrac{1}{6}[\Phi , \Phi])). \end{align}

2.3.2 Stable flat connections

The variation of the Chern–Simons functional $CS : \mathscr{A}_{E, k}^c \rightarrow \mathbb{C}$ is given by

(2.17) \begin{align}CS (\mathbb{A} + \dot{\mathbb{A}} ) = CS(\mathbb{A} ) = 2 \int_Y \mathrm{Tr} ( \dot{\mathbb{A}} \wedge F_{\mathbb{A}} ) + \int_Y \mathrm{Tr}(\dot{\mathbb{A}} \wedge d_{\mathbb{A}}\dot{\mathbb{A}}) + \frac{1}{3}\int_{Y}\mathrm{Tr}(\dot{\mathbb{A}}\wedge [\dot{\mathbb{A}}, \dot{\mathbb{A}}]).\end{align}

From (2.17) one sees that CS is a smooth (in fact, holomorphic) function on the $L^{2}_k$ connections (as long as $k \geq 1$ ), and its critical points are the connections $\mathbb{A}$ whose curvature has vanishing traceless part (since the ‘Killing form’ $(X,Y) \mapsto - \mathrm{Tr}(XY)$ on the Lie algebra $\mathfrak{gl}(2, \mathbb{C} )$ has kernel $Z( \mathfrak{gl}(2,\mathbb{C})) = \mathbb{C}\cdot I $ ):

$$F_{\mathbb{A}} - \frac{I}{2} F_{\lambda} = 0. $$

Said differently, these are the $GL(2,\mathbb{C})$ connections $\mathbb{A}$ on E with fixed determinant $(\Lambda , \lambda )$ which are projectively flat, in the sense that the connection that $\mathbb{A}$ induces on the $PSL(2,\mathbb{C})$ bundle $P^c = E \times_{U(2)} PSL(2,\mathbb{C} )$ is flat. Informally, we might often refer to such connections plainly as ‘flat’. In terms of the decomposition $\mathbb{A} = A + i \Phi$ , this equation reads

(2.18) \begin{align}F_A - \frac{I}{2} F_{\lambda} - \frac{1}{2} [\Phi , \Phi ] & = 0, \end{align}

(2.19) \begin{align}d_{A}\Phi &= 0. \end{align}

Definition 2.8. A polystable (projectively) flat $GL(2,\mathbb{C})$ connection with fixed determinant is a pair $(A, \Phi ) \in \mathscr{A}_{E,k}^c$ which solves (2.18) and (2.19) and

(2.20) \begin{align}d_{A}^\ast \Phi & = 0 .\end{align}

That is, $A + i \Phi$ is flat and $(A, \Phi ) $ lies in the polystable locus $\mathscr{C}_{E,k}$ . The group $\mathscr{G}_{E,k+1}$ of unitary gauge transformations fixing the determinant preserves the solutions to (2.18)–(2.20). If, in addition, $(A, \Phi )$ has stabilizer $\pm I$ under the $\mathscr{G}_E$ -action then $(A, \Phi )$ is stable.

We denote by $\mathscr{M}_{E,k} \subset \mathscr{B}_{E,k}$ (respectively, $\mathscr{M}_{E,k}^\ast \subset \mathscr{B}_{E,k}^\ast $ ) the topological spaces of $\mathscr{G}_{E,k+1}$ -orbits of solutions of (2.18)–(2.20) (respectively, those which are stable). It follows from Remark 2.9 that every such $\mathscr{G}_{E,k+1}$ -orbit contains a smooth solution and the topological spaces $\mathscr{M}_{E,k} , \mathscr{M}_{E,k}^\ast $ do not depend on the chosen k. Thus, we subsequently drop the subscript k from our notation.

Remark 2.9 (Ellipticity). Equations (2.18)–(2.20) do not form an elliptic system, even modulo $\mathscr{G}_E $ . However, this can be achieved if we ‘augment’ the equations by introducing two additional fields $\Psi_1 , \Psi_2 \in \Omega^0 (Y , \mathfrak{g}_E )_{k}$ and considering the equations

\begin{align*}F_A - \frac{I}{2} F_\lambda - \frac{1}{2} [\Phi , \Phi] & = \ast d_{A} \Psi_1 + \ast [\Psi_2 , \Phi],\\d_{A} \Phi & = \pm \ast ( [ \Phi , \Psi_1 ] + d_{A}\Psi_2 ),\\d_{A}^\ast \Phi & = [\Psi_1 , \Psi_2],\end{align*}

on whose space of solutions $\mathscr{G}_{E, k+1}$ acts (by $u \Psi_i u^{-1}$ on the additional fields $\Psi_i$ ). Above, we mean that there are two possible systems of equations according to the choice of sign $\pm $ . (Both have been studied in [Reference Nagy and OliveiraNO20]; for the $-$ sign and $\Psi _2 = 0$ the equations are known as the extended Bogomolny equations [Reference He and MazzeoHM19, Reference He and MazzeoHM20].) Both of these augmented systems are elliptic modulo $\mathscr{G}_E$ , in the usual sense.

2.3.3 Hessians

From now on we fix an angle $\theta \in [0 , 2 \pi)$ , and consider the functional $S = \mathrm{Re}(e^{- i \theta} CS ) : \mathscr{A}_{E , k }^c \rightarrow \mathbb{R}$ with CS as in (2.14). For simplicity, when performing calculations and displaying formulas we shall assume $\theta = \pi/2$ (i.e., use the imaginary part of the Chern–Simons functional (2.16), which has the advantage of having no periods under the action of $\mathscr{G}_{ E , k+1}^c$ ). However, the discussion that follows goes through for all $\theta$ with little modification.

The formal gradient of $S : \mathscr{A}_{E , k }^c \rightarrow \mathbb{R}$ with respect to the flat $L^2$ metric makes sense as a smooth section, denoted by $\mathrm{grad}S$ , of the $L^{2}_{k-1}$ completion of the tangent bundle $T \mathscr{A}_{E, k-1}^c \rightarrow \mathscr{A}_{E,k}^c$ . It is given by

(2.21) \begin{align}\mathrm{grad}S = 2 \cdot \bigg(\! \ast d_{A}\Phi , \ast \bigg(\!F_A -\frac{I}{2} F_{\lambda} - \frac{1}{2}[\Phi , \Phi]\bigg) \!\bigg) . \end{align}

Since S is the imaginary part of the holomorphic function CS, the zeros of $\mathrm{grad}S$ agree with the critical points of CS (i.e., the flat connections). By the following proposition, the critical points of the restriction $S|_{\mathscr{C}_{E , k }^\ast }$ of S to the stable locus are the stable flat connections.

Proof. Using the Bianchi identity $d_A F_A = 0$ , we compute that $(\delta_{(A, \Phi )}^2)^\ast (\mathrm{grad}S )$ is given by

\begin{align*}\ast [\ast d_{A}\Phi , \ast \Phi] - d_{A}^\ast \bigg(\!\ast \bigg(\!F_A - \frac{I}{2}F_{\lambda} - \frac{1}{2}[\Phi ,\Phi]\bigg)\!\bigg) = \ast [d_{A}\Phi , \Phi ] - \ast [d_A \Phi ,\Phi] = 0.\end{align*}

Thus, $\mathrm{grad}S$ gives a section of $T \mathscr{C}_{E , k-1}^\ast \rightarrow \mathscr{C}_{E , k }^\ast $ . This section is $\mathscr{G}_{E , k+1}$ -equivariant; hence, it descends to a well-defined ‘vector field’ on the stable orbit space $\mathscr{B}_{E , k }^\ast $ . More precisely, we obtain the following smooth section $[\mathrm{grad}S]$ of the Hilbert vector bundle.

(2.22)

Equivalently, the section $[\mathrm{grad}]$ can be regarded as the formal gradient of $S : \mathscr{B}_{E, k}^\ast \to \mathbb{R}$ with respect to the $L^2$ metric. In what follows we show that the section (2.22) is Fredholm.

We first introduce the notation $\mathscr{K}_{(A, \Phi), j } = \mathscr{K}_{(A, \Phi ) , j }^1 \cap \mathscr{K}_{(A, \Phi ) , j }^2$ for $0 \leq j \leq k$ (cf. Definitions (2.3) and (2.4) and the $L^2$ -orthogonal decompositions (2.6)–(2.9)). As $(A, \Phi )$ varies in the stable configuration space, these subspaces form $\mathscr{G}$ -equivariant Hilbert bundles $\mathscr{K}_{j}\rightarrow \mathscr{C}_{E , k }^\ast $ (and similarly for $\mathscr{K}_{j}^1$ and $\mathscr{K}_{j}^2$ ). In particular, $\mathscr{K}_{j}$ is the pullback of the tangent bundle of $\mathscr{B}_{E, j }^\ast $ along the projection $\mathscr{C}_{E, j}^\ast \rightarrow \mathscr{B}_{E , j }^\ast $ and $\mathscr{K}_{j}^2$ is the tangent bundle of $\mathscr{C}_{E, j}^\ast $ .

Now, the $L^2$ Riemannian metric on $\mathscr{B}_{E, k}^\ast $ provides the Levi-Civita connection on the Hilbert vector bundle (2.22), and we use this connection to define the vertical derivative of the section $[\mathrm{grad}S]$ . This is a vector bundle map

$$\mathscr{D} := \mathcal{D}[\mathrm{grad}S] : \mathscr{K}_k \rightarrow \mathscr{K}_{k-1}. $$

Formally, $\mathscr{D}_{(A, \Phi )}$ is the covariant Hessian of S at $[(A, \Phi )]$ with respect to the $L^2$ metric. The following proposition gives a more explicit description.

Proposition 2.12. We have

(2.23) \begin{align}\mathscr{D} = (\Pi_{\mathscr{K}_{k-1}} \circ \mathrm{Hess}S)|_{\mathscr{K}_k} : \mathscr{K}_{k}\rightarrow \mathscr{K}_{ k-1} , \end{align}

where:

• – $\Pi_{\mathscr{K}_{j}} : (\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )_j \rightarrow \mathscr{K}_{j}$ is the $L^2$ -orthogonal projection (for $0\leq j \leq k$ ); here we are using the following $L^2$ -orthogonal decomposition obtained from (2.7) to (2.8) at a stable configuration $(A, \Phi ) \in \mathscr{C}_{(A, \Phi) , k }^\ast $ ,

  (2.24) \begin{align}(\Omega^1 \oplus \Omega^1)(Y, \mathfrak{g}_E )_j = \mathscr{T}_{(A, \Phi ), j}^1 \oplus \mathscr{T}_{(A, \Phi ) , j}^2 \oplus \mathscr{K}_{(A, \Phi ) , j} ;\end{align}

• – $\mathrm{Hess}S : (\Omega^1 \oplus \Omega^1 )(Y,\mathfrak{g}_E )_k \rightarrow (\Omega^1 \oplus \Omega^1 )(Y,\mathfrak{g}_E )_{k-1}$ is the Hessian of $S : \mathscr{A}_{E, k }^c\rightarrow \mathbb{R}$ for the flat $L^2$ metric; this is given by

  (2.25) \begin{align}\mathrm{Hess}S = 2 \cdot \begin{pmatrix} \ast [\Phi , \cdot ] & \ast d_A \\\ast d_A & - \ast [\Phi , \cdot ]\end{pmatrix}\! ._{\vphantom{\Big(}} \end{align}

Proof. By definition, the projection $\mathscr{C}_{E , k }^\ast \rightarrow \mathscr{B}_{E, k}^\ast $ is a Riemannian submersion with respect to the $L^2$ metric on $\mathscr{B}_{E, k }^\ast $ and the restriction to $\mathscr{C}_{E , k}^\ast \subset \mathscr{A}_{E , k }^c$ of the flat $L^2$ metric. As a matter of general principle, the covariant Hessian $\mathscr{D}$ of S on $\mathscr{B}_{E}^\ast $ is then given by

(2.26) \begin{align}\mathscr{D} = (\Pi_{\mathscr{K}_{k-1}}^{\mathscr{K}_{k-1}^2} \circ \mathscr{{D}}^\prime )|_{\mathscr{K}_k} : \mathscr{K}_{k} \rightarrow \mathscr{K}_{k-1} ,\end{align}

where:

• – $\mathscr{D}^\prime : \mathscr{K}^{2}_k \to \mathscr{K}^{2}_{k-1}$ denotes the covariant Hessian of $S|_{\mathscr{C}_{E , k }^\ast }$ with respect to the restriction of the flat $L^2$ metric; and

• – $\Pi_{\mathscr{K}_{k-1}}^{\mathscr{K}_{k-1}^2} :\mathscr{K}_{k-1}^2 \to \mathscr{K}_{k-1}$ is the $L^2$ -orthogonal projection in (2.8)).

Again by general principle, because $\mathrm{grad}S$ is tangent to the stable locus $\mathscr{C}_{E , k}^\ast \subset \mathscr{A}_{E , k}^c$ (Proposition 2.11), the operator $\mathscr{D}^\prime$ is related to the flat Hessian by

(2.27) \begin{align} \mathscr{D}^\prime = (\Pi_{\mathscr{K}_{k-1}^2} \circ \mathrm{Hess}S )|_{\mathscr{K}_{k}^2} : \mathscr{K}_{k}^2 \rightarrow \mathscr{K}_{k-1}^2 ,\end{align}

where $\Pi_{\mathscr{K}_{k-1}^2} :(\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )_{k-1} \rightarrow \mathscr{K}_{k-1}^2$ is the $L^2$ -orthogonal projection in (2.7). Putting together (2.26) and (2.27) gives the required result. (Recall that $(\delta^1 )^\ast \circ \delta^2 = 0$ at stable flat connections, which implies that $\Pi_{\mathscr{K}_{k-1}}$ agrees with $\Pi_{\mathscr{K}_{k-1}}^{\mathscr{K}^{2}_{k-1}} \circ \Pi_{\mathscr{K}^{2}_{k-1}}$ .)

Proof. We use an argument similar to [Reference Kronheimer and MrowkaKM07, Proposition 12.3.1] to establish the Fredholm property of $\mathscr{D}$ . For a configuration $(A, \Phi )$ we consider the extended Hessian operator

$$( \widehat{\mathrm{Hess}}S )_{(A, \Phi ) , k } : (\Omega^1 \oplus \Omega^1 \oplus \Omega^0 \oplus \Omega^0 )(Y, \mathfrak{g}_E )_{k} \rightarrow (\Omega^1 \oplus \Omega^1 \oplus \Omega^0 \oplus \Omega^0 )(Y, \mathfrak{g}_E )_{k-1} ,$$

given in block form by

(2.28) \begin{equation}( \widehat{\mathrm{Hess}}S )_{(A, \Phi ) , k } = \begin{pmatrix} 2 \ast [\Phi , \cdot ]\;\;\;\;\; & 2 \ast d_A\;\;\;\;\; & -d_{A}\;\;\;\;\; & - [\cdot , \Phi] \\2 \ast d_A\;\;\;\;\; & - 2 \ast [\Phi , \cdot ]\;\;\;\;\; & [\cdot , \Phi ]\;\;\;\;\; & -d_{A} \\-d_{A}^\ast \;\;\;\;\; & - \ast [ \cdot , \ast \Phi]\;\;\;\;\; & 0\;\;\;\;\; & 0\\\ast [ \cdot , \ast \Phi]\;\;\;\;\; & -d_{A}^\ast \;\;\;\;\; & 0\;\;\;\;\; & 0\end{pmatrix} \!. \end{equation}

The extended Hessian is a Fredholm operator of index zero, since it is a first-order self-adjoint elliptic operator. The $L^2$ -orthogonal decomposition (2.24) induces a further decomposition when $(A, \Phi ) \in \mathscr{C}_{E,k}^\ast $ is stable: for $0\leq j \leq k$ ,

(2.29) \begin{align}(\Omega^1 \oplus \Omega^1 \oplus \Omega^0 \oplus \Omega^0 )(Y, \mathfrak{g}_E )_{j} = \mathscr{T}_{j}^1 \oplus \mathscr{T}_{j}^2 \oplus \mathscr{K}_{j} \oplus \Omega^0 (Y, \mathfrak{g}_E )_j \oplus \Omega^0 (Y, \mathfrak{g}_E )_j .\end{align}

With respect to the latter decomposition the extended Hessian has the block form

\begin{align*} ( \widehat{\mathrm{Hess}}S )_{(A, \Phi ) , k } = \begin{pmatrix} 0\;\;\;\;\; & 0\;\;\;\;\; & x_1\;\;\;\;\; & \delta^1\;\;\;\;\; & 0 \\ 0\;\;\;\;\; & 0\;\;\;\;\; & x_2\;\;\;\;\; & 0\;\;\;\;\; & \delta^2 \\ x_{1}^\ast \;\;\;\;\; & x_{2}^\ast \;\;\;\;\; & \mathscr{D}\;\;\;\;\; & 0\;\;\;\;\; & 0 \\ (\delta^1)^\ast \;\;\;\;\; & 0\;\;\;\;\; & 0\;\;\;\;\; & 0\;\;\;\;\; & 0 \\ 0\;\;\;\;\; & (\delta^2)^\ast \;\;\;\;\; & 0\;\;\;\;\; & 0\;\;\;\;\; & 0 \end{pmatrix}\!,\end{align*}

where

\begin{align*}x_i = ((1 - \Pi_{\mathscr{K}_{k-1}^i} )\circ \mathrm{Hess}S )|_{\mathscr{K}_k} .\end{align*}

We claim that $x_i$ is a compact operator (and hence so is its adjoint $x_{i}^\ast $ ). Indeed, the operator $ (1 - \Pi_{\mathscr{K}_{k-1}^i} )$ (equivalently, the projection to $\mathscr{T}_{k-1}^i$ in (2.7)) is given by

$$(1 - \Pi_{\mathscr{K}_{k-1}^i} ) = \delta^i \circ ((\delta^i)^\ast \delta^i )^{-1} \circ (\delta^i )^{\ast }, $$

and we then observe that:

• – $\delta^i \circ ((\delta^i)^\ast \delta^i )^{-1}$ is a pseudo-differential operator of order $-1$ (i.e., a smoothing operator of order 1);

• – $\delta_{i}^\ast \circ \mathrm{Hess}S $ is a first-order differential operator; in particular, it defines a bounded linear operator $L^{2}_k \rightarrow L^{2}_{k-1}$ .

The second item is verified by direct calculation; for example, as a differential operator $(\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E ) \rightarrow \Omega^0 ( Y , \mathfrak{g}_E )$ we have

$$(\delta^{2})^\ast \mathrm{Hess}S = 2 \cdot \bigg( \bigg[\ast \bigg(F_A - \frac{I}{2}F_\lambda - \frac{1}{2}[\Phi , \Phi]\bigg), \cdot \bigg] , - \ast [d_{A}\Phi , \cdot ] \bigg) . $$

At this point, we have that the operator obtained from $\widehat{\mathrm{Hess}}S$ by dropping the entries $x_{i} , x_{i}^\ast $ is also Fredholm. Furthermore, this operator contains $\mathscr{D}$ as a summand, hence the Fredholm property of $\mathscr{D}$ follows.

Assertion (1) holds for the extended Hessian, which is a self-adjoint elliptic operator of order 1. Hence, it also holds for its compact perturbation obtained by dropping the entries $x_{i}, x_{i}^\ast $ . Since the latter operator contains $\mathscr{D}$ as a summand, assertion (1) follows. Assertion (2) follows from the fact that $x_i$ vanishes at a critical point.

The fact that the complex Chern–Simons functional CS is holomorphic manifests itself in the symmetries of the Hessian of $S = \mathrm{Im}CS $ . To give some context, we recall that on a finite-dimensional complex manifold (M,J) the real part $f = \mathrm{Re}F$ of a holomorphic function $F : M \to \mathbb{C}$ is pluriharmonic: $d d^c f = 0$ . If (M, J) admits a Kähler metric g, then we obtain from g the covariant Hessian $\mathrm{Hess}f : TM \to TM$ , and pluriharmonicity of f becomes equivalent to the vanishing of the ‘complex average’ of its Hessian: for all $x \in M$ , $v,w \in T_x M$ ,

$$\tfrac{1}{2} ( \langle (\mathrm{Hess} f )_x v , w \rangle + \langle (\mathrm{Hess}f)_x Jc , Jw\rangle ) = 0 . $$

Proposition 2.14. For any stable connection $(A, \Phi )$ (not necessarily flat), the Hessian $\mathscr{D}_{(A, \Phi ) }: \mathscr{K}_{(A, \Phi ) , k }\to \mathscr{K}_{(A, \Phi ) , k-1}$ satisfies the following identity: for all $v\in \mathscr{K}_{(A, \Phi ) , k }$ and $w \in \mathscr{K}_{(A, \Phi ) , k-1}$ ,

$$ \langle \mathscr{D} v , w \rangle_{L^2} + \langle \mathscr{D}(J v) , J w \rangle_{L^2} = 0 . $$

Proof. The required identity holds for the flat Hessian defined in (2.25):

$$ \langle (\mathrm{Hess}S )_{(A, \Phi )} v , w \rangle_{L^2} + \langle (\mathrm{Hess}S )_{(A, \Phi )} J v , Jw \rangle_{L^2} = 0 . $$

The claim now follows since $\mathscr{K}_{k}$ is a J-linear subspace of $(\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )$ .

In particular, we have that the kernel of $\mathscr{D}_{(A, \Phi )}$ is a complex-linear subspace of $\mathscr{K}_{(A, \Phi ) , k}$ .

2.3.4 Kuranishi models

An important point that is used later is the existence of finite-dimensional models for S, which readily follows from Theorem 2.13.

Let $(A, \Phi )$ be a stable flat connection. Recall that the Hilbert space $\mathscr{K}_{(A, \Phi ) , k }$ is identified as the tangent space to the stable configuration space $\mathscr{B}_{E, k}^\ast $ . Let $U \subset \mathscr{K}_{(A, \Phi ) , k}$ (respectively, $U^\prime \subset \mathscr{K}_{(A, \Phi ), k-1}$ ) be the closed subspace of $\mathscr{K}_{(A, \Phi ) , k}$ (respectively, $\mathscr{K}_{(A, \Phi ) , k-1}$ ) given as the $L^2$ -orthogonal complement of $\mathscr{H}_{(A, \Phi)}^1 $ . Then the Hessian $\mathscr{D}_{(A, \Phi )}$ restricts to a bounded linear isomorphism from U to $U^\prime$ , since $\mathscr{D}_{(A, \Phi )}$ is a self-adjoint operator.

Corollary 2.16. Let $(A, \Phi )$ be a stable flat connection. Then there exist:

• – a diffeomorphism G from an open neighborhood of $[(A, \Phi )] \in \mathscr{B}_{E, k }^\ast $ to an open neighborhood of (0,0) in $U \oplus \mathscr{H}_{A, \Phi}^1$ , such that $G ([(A, \Phi )]) = (0,0)$ ; and

• – a smooth function $S^{\prime} $ on a neighborhood of 0 in $ \mathscr{H}_{(A, \Phi )}^1$ with $(dS^{\prime})_0 = 0$ and $(\mathrm{Hess}S^{\prime})_0 = 0$ ;

such that on the codomain of G we have

$$S(G^{-1}(u,h)) = \tfrac{1}{2}\langle \mathscr{D}_{(A, \Phi )} u , u \rangle_{L^2} + S^{\prime}(h). $$

In particular, a neighborhood of $[(A, \Phi)]$ in the moduli space of stable flat connections $\mathscr{M}_{E}^\ast $ is homeomorphic to a neighborhood of 0 in the critical locus of $S^\prime : \mathscr{H}^{1}_{(A, \Phi ) } \rightarrow \mathbb{R}$ .

Proof. The argument closely follows [Reference DonaldsonDon02, Proposition 2.5]. At a connection $(A+a , \Phi + \phi )$ the gradient $\mathrm{grad}S$ vector field is tangent to the subspace $\mathscr{K}_{(A+a , \Phi +\phi ),k-1}$ . The $L^2$ -orthogonal projection $\Pi_{\mathscr{K}_{(A, \Phi ) , k-1}}$ from $(\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )_{k-1}$ onto $\mathscr{K}_{(A, \Phi ) , k-1}$ thus restricts to an isomorphism from $\mathscr{K}_{(A+a , \Phi +\phi ),k-1}$ to $\mathscr{K}_{(A, \Phi ) , k-1}$ for $(a,\phi )$ in a small neighborhood of (0,0). Consider the following smooth map defined along the stable Coulomb slice (cf. Proposition 2.5):

$$f : (A+ a, \Phi + \phi ) \in \mathcal{S}_{(A, \Phi ) , k} \mapsto \Pi_{\mathscr{K}_{(A, \Phi ) , k-1}}( (\mathrm{grad}S)_{(A+a , \Phi + \phi )}) \in \mathscr{K}_{(A, \Phi ) , k-1} . $$

Now, the function $\Pi_{U^\prime} \circ f : \mathcal{S}_{(A, \Phi ),k} \to U^\prime$ has the property that its derivative at $(a, \phi ) = (0,0)$ is surjective when restricted to directions in $U \subset \mathscr{K}_{(A, \Phi ) , k} = T_{(A,\Phi )}\mathcal{S}_{(A, \Phi)}$ . Thus, a change of coordinates linearizes $\Pi_U \circ f$ . Namely, using the inverse function theorem for Hilbert spaces, one obtains a diffeomorphism F from an open neighborhood of $(A, \Phi )$ in $ \mathcal{S}_{(A, \Phi ) , k}$ onto an open neighborhood of (0,0) in $U \oplus \mathscr{H}^{1}_{(A, \Phi )}$ , sending $(A, \Phi )$ to (0,0), together with a smooth function $f^\prime $ from an open neighborhood of (0,0) in $ U \oplus \mathscr{H}^{1}_{(A, \Phi )}$ to an open neighborhood of 0 in $\mathscr{H}^{1}_{(A, \Phi )}$ , $f^\prime (0,0) = 0$ , and such that

$$f( F^{-1}(u,h) ) = ( \mathscr{D}_{(A, \Phi )} u , f^\prime (u,h) ) . $$

Consider now the family of smooth functions $S_h : U \rightarrow \mathbb{R}$ obtained by restricting $S\circ F^{-1}$ onto $U \times \{ h \}$ . By the above, it follows that each of these functions has a critical point at $0 \in U$ and this is a non-degenerate critical point for $h = 0$ . Thus, for all h sufficiently close to 0, the function $f_h$ will have a non-degenerate critical point at 0. Then by the parametric version of the Morse–Palais lemma [Reference LangLan85], a further coordinate change ensures that each $f_h$ is given by a quadratic function on U plus a constant. A further change of coordinates by a family of linear isomorphisms of U ensures that the quadratic functions are independent of h, and the constants then give the required function $S^\prime (h)$ .

For future reference, we include here the following definition.

3.2.1 Covariant derivatives

We first describe how the exterior covariant derivatives $d^A : \Omega^n (Y, \mathfrak{g}_E ) \to \Omega^{n+1}(Y, \mathfrak{g}_E )$ , $n\geq 0$ , interact with the Seifert fibration $\pi: Y \rightarrow C$ .

By pullback, we have the subbundle $\pi^\ast (T^\ast C) \subset T^\ast Y$ . (This is the annihilator of the vertical subbundle $\mathbb{R} = \mathrm{Ker}(d\pi ) \subset TY$ .) Denote its nth exterior power by $\Lambda^{n}_H := \Lambda^n (\pi^\ast (T^\ast C) ) $ , and the space of sections of $\Lambda^{n}_H$ by $\Omega_{H}^n (Y, \mathfrak{g}_E )$ . Using the connection $\mu = i \eta $ on N, we obtain the splitting $T^\ast Y = \mathbb{R} \oplus \pi^\ast (T^\ast C)$ dual to (3.2), which induces a splitting for all $n\geq 0$ :

(3.3)

We consider the following differential operators:

• – the adiabatic covariant derivative $\nabla_{\zeta}^A : \Omega^n (Y, \mathfrak{g}_E ) \to \Omega^n (Y, \mathfrak{g}_E ) $ along the vertical vector field $\zeta$ that generates the Seifert $S^1$ -action;

• – the horizontal exterior covariant derivative $d_{A}^H : \Omega_{H}^n (Y , \mathfrak{g}_E ) \rightarrow \Omega_{H}^{n + 1} (Y, \mathfrak{g}_E )$ defined by $(d_{A}^H \gamma )(V_1 , \ldots , V_{n+1}) = (d_A \gamma )(V_{1}^{H} , \ldots , V_{n +1}^{H})$ , where $V^H$ stands for the projection of a vector field V on Y onto the horizontal subspace of (3.2);

• – the formal $L^2$ -adjoint of $d_{A}^H$ , given by $(d_{A}^H)^\ast = - \ast _H d_{A}^H \ast _H : \Omega^n (Y, \mathfrak{g}_E ) \to \Omega^{n-1}(Y, \mathfrak{g}_E )$ , where $\ast _H : \Lambda^{\ast }_H \xrightarrow{\cong} \Lambda^{2-\ast }_H$ is the bundle isomorphism induced by the Hodge star operator $\ast _C $ associated to the orbifold surface $(C,g_C)$ .

We need the following result.

Lemma 3.3. With respect to the splitting (3.3), the exterior covariant derivative $d_{A} : \Omega^{n}(Y, \mathfrak{g}_E ) \rightarrow \Omega^{n+1}(Y, \mathfrak{g}_E )$ is given in block form by

\begin{equation*}d_A = \begin{pmatrix}- d_{A}^H\;\;\;\;\; & \nabla_{\zeta}^A \\2 \xi (\pi^\ast \omega_C ) \wedge \cdot\;\;\;\;\; & d_{A}^H\end{pmatrix}\end{equation*}

and its formal $L^2$ -adjoint $d_{A}^\ast : \Omega^n (Y, \mathfrak{g}_E ) \rightarrow \Omega^{n -1} (Y, \mathfrak{g}_E )$ by

\begin{equation*}d_{A}^\ast = \begin{pmatrix}- (d_{A}^H)^\ast \;\;\;\;\; & 2\xi (\pi^\ast \omega_C \wedge \cdot)^\ast \\- \nabla_{\zeta}^A\;\;\;\;\; & (d_{A}^H)^\ast \end{pmatrix}\!.\end{equation*}

Proof. Fix a local orthonormal frame and coframe as in Lemma 3.2. We recall the standard formula expressing the exterior derivative $d_A$ in terms of the Levi-Civita connection

$$d_A = \eta \wedge \widehat{\nabla}_{\zeta}^A + \eta^1 \wedge \widehat{\nabla}_{\zeta_1}^A + \eta^2 \wedge \widehat{\nabla}_{\zeta_2}^A . $$

Let $\Psi \in \Omega^{n}_H (Y, \mathfrak{g}_E )$ . Then the horizontal exterior covariant derivative is given by

(3.4) \begin{align}d_{A}^H \Psi = \eta^1 \wedge \nabla_{\zeta_1}^A \Psi + \eta^2 \wedge \nabla_{\zeta_2}^A \Psi .\end{align}

By Lemma 3.2 when $n = 1$ , and hence by the Leibniz rule for general n, we have

\begin{align*}\eta \wedge \widehat{\nabla}_{\zeta}^B \beta + \eta^1 \wedge \widehat{\nabla}_{\zeta_1}^B \Psi + \eta^2 \wedge \widehat{\nabla}_{\zeta_2}^B \Psi = \eta \wedge \nabla_{\zeta}^B \Psi + \eta^1 \wedge \nabla_{\zeta_1}^B \Psi + \eta^2 \wedge \nabla_{\zeta_2}^B \Psi.\end{align*}

Putting everything together, we have $d_A \Psi = \eta \wedge \nabla_{\zeta}^A \Psi + d_{A}^H \Psi $ , which gives the second column in the block matrix for $d_A$ . By this formula and the Leibniz rule we also have

\begin{align*}d_A (\eta \wedge \Psi ) = d\eta \wedge \Psi - \eta \wedge d_{A}\Psi = 2 \xi \pi^\ast (\omega_C ) \wedge \Psi - \eta \wedge d_{A}^H \Psi,\end{align*}

which gives the second column of the block matrix for $d_A$ . The formula for $d_{A}^\ast $ follows by taking the adjoint of the block matrix for $d_A$ .

3.2.2 A Bochner–Weitzenböck formula

We decompose the imaginary part $\Phi \in \Omega^1 (Y, \mathfrak{g}_E )$ of a $GL(2, \mathbb{C})$ connection $A + i \Phi$ according to (3.3) as

(3.5) \begin{equation} \Phi = \eta \cdot \alpha + \Psi ,\end{equation}

where $\alpha \in \Omega^0 (Y, \mathfrak{g}_E )$ and $ \Psi \in \Omega_{H}^1 (Y, \mathfrak{g}_E )$ .

Proposition 3.4. Let $A + i \Phi$ be a $GL(2,\mathbb{C})$ connection on E with fixed determinant, and let $\alpha , \Psi$ be as in (3.5). Then the following pointwise identity holds:

\begin{align*}\langle ( d_{A}d_{A}^\ast + d_{A}^\ast d_{A})\Phi , \Phi \rangle = & \langle -(\nabla_{\zeta}^A )^2 \alpha , \alpha \rangle + \langle (\ast _H \nabla_{\zeta}^B)^2 \Psi , \Psi \rangle \\& + \langle (d_{A}^H)^\ast d_{A}^H \alpha , \alpha \rangle + \langle d_{A}^H(d_{A}^H)^\ast \Psi , \Psi \rangle + \langle [\iota_{\zeta}(F_A ) , \alpha] , \Psi \rangle \\& + \langle (d_{A}^H)^\ast d_{A}^H \Psi , \Psi \rangle + 4\xi \langle \ast _H d_{A}^H \Psi , \alpha \rangle + 4\xi^2 \cdot | \alpha|^2.\end{align*}

(Here $\iota_\zeta F_A = ( F_A )(\zeta , \cdot )$ denotes contraction of $F_A$ with the vertical vector field $\zeta$ .)

Proof. Consider the self-adjoint differential operator $D_A : (\Omega^0 \oplus \Omega^1)(Y, \mathfrak{g}_E ) \rightarrow (\Omega^0 \oplus \Omega^1)(Y, \mathfrak{g}_E )$ given by the odd signature operator coupled with the connection A on $\mathfrak{g}_E$ :

\begin{align*}D_A & = \begin{pmatrix} 0 & - d_{A}^\ast \\ - d_A & \ast d_A\end{pmatrix} \!.\end{align*}

This operator squares to give

(3.6) \begin{align}(D_A)^2 &= \begin{pmatrix} d_A d_{A}^\ast + d_{A}^\ast d_A & \ast [F_A , \cdot ] \\ -\ast [F_A , \cdot ] & d_A d_{A}^\ast + d_{A}^\ast d_A \end{pmatrix}\!. \end{align}

We further decompose $(\Omega^0 \oplus \Omega^1 )(Y, \mathfrak{g}_E ) = (\Omega^0 \oplus \Omega^0 \oplus \Omega_{H}^1 )(Y , \mathfrak{g}_E )$ using the splitting (3.3) for $n = 1$ . By Lemma 3.3 the operator $D_A$ is given by $D_A = D_{\zeta}^A + D_{H}^A$ , where

\begin{align*}D_{\zeta}^A = \begin{pmatrix} 0 & \nabla_{\zeta}^A & 0 \\- \nabla_{\zeta}^A & 2\xi & 0 \\0 & 0 & \ast _H \nabla_{\zeta}^A \end{pmatrix}\!, \quad D_{H}^A = \begin{pmatrix} 0 & 0 & - (d_{A}^H)^\ast \\ 0 & 0 & \ast _H d_{A}^H \\- d_{A}^H & - \ast _H d_{A}^H & 0 \end{pmatrix}\!.\end{align*}

We have the identity

(3.7) \begin{equation}(D_A)^2 = (D_{\zeta}^A)^2 + (D_{H}^A)^2 + (D_{\zeta}^{A}D_{H}^A + D_{H}^A D_{\zeta}^A) \end{equation}

and in what follows now we compute the three operators on the right-hand side of (3.7) one by one:

(3.8) \begin{align}(D_{\zeta}^A)^2 & = \begin{pmatrix} - (\nabla_{\zeta}^A)^2 & 2\xi\nabla_{\zeta}^A & 0 \\[2pt]-2\xi \nabla_{\zeta}^A & - (\nabla_{\zeta}^A)^2 + 4\xi^2 & 0 \\[2pt]0 & 0 & (\ast _H \nabla_{\zeta}^{A} )^2\end{pmatrix}\!, \end{align}

(3.9) \begin{align}(D_{H}^A)^2 & = \begin{pmatrix} (d_{A}^H)^\ast d_{A}^H & \ast _H (d_{A}^H)^2 & 0 \\[2pt] -\ast _H (d_{A}^H)^2 & (d_{A}^H)^\ast d_{A}^H & 0 \\[2pt] 0 & 0 & d_{A}^H (d_{A}^H)^\ast + (d_{A}^H)^\ast d_{A}^H\end{pmatrix}\!, \end{align}

and $( D_{\zeta}^{B}D_{H}^B + D_{H}^B D_{\zeta}^B) $ is given by

$$\!\!\begin{pmatrix} 0 & 0 & \nabla^{A}_{\zeta} \ast _H d_{A}^H - \ast _H d_{A}^H \nabla_{\zeta}^A\\[2pt]0 & 0 & \nabla_{\zeta}^A (d_{A}^H)^\ast - (d_{A}^H)^\ast \nabla_{\zeta}^A + 2 \xi \ast _H d_{A}^H{\kern-2pt} \\[2pt]- \ast _H \nabla_{\zeta}^A d_{A}^H + \ast _H d_{A}^H \nabla_{\zeta}^A &- \ast _H \nabla_{\zeta}^A \ast _H d_{A}^H - d_{A}^H \nabla_{\zeta}^A -2 \xi \ast _H d_{A}^H & 0\!\!\! \!\!\!\end{pmatrix}\!\!.$$

The entries in the last matrix can be described more succinctly. For this, we fix a local orthonormal frame $\zeta_0 (= \zeta ) , \zeta_2 , \zeta_2$ and dual coframe $\eta^0 (= \eta ), \eta^1 , \eta^2$ as in Lemma 3.2. By definition of the adiabatic connection, we have $\nabla_{\zeta}\eta^1 = \nabla_{\zeta}\eta^2 = 0$ . We then compute as follows, for an auxiliary $s \in \Omega^0 (Y, \mathfrak{g}_E )$ :

\begin{align*}(- \ast _H \nabla_{\zeta}^A \ast _H d_{A}^H - d_{A}^H \nabla_{\zeta}^A )s & = - \ast _H \nabla_{\zeta}^A (\eta^2 \wedge \nabla_{\zeta_1}^A s - \eta^1 \wedge \nabla_{\zeta_2}^A s ) - ( \eta^1 \wedge \nabla_{\zeta_1}^A \nabla_{\zeta}^A s + \eta^2 \wedge \nabla_{\zeta_2}^A \nabla_{\zeta}^A s )\\[2pt]& = - \ast _H ( \eta^2 \wedge \nabla_{\zeta}^A \nabla_{\zeta_1}^A s - \eta^1 \wedge \nabla_{\zeta}^A \nabla_{\zeta_2}^A s ) \\[2pt]&\quad - ( \eta^1 \wedge \nabla_{\zeta_1}^A \nabla_{\zeta}^A s + \eta^2 \wedge \nabla_{\zeta_2}^A \nabla_{\zeta}^A s )\\[2pt]& = e_1 [\nabla_{\zeta}^A , \nabla_{e_1}^A ] + e_2 [\nabla_{\zeta}^A , \nabla_{e_2}^A] )s = \tfrac{1}{2} [ \iota_{\zeta} F_B ,s ],\\[2pt](- \ast _H \nabla_{\zeta}^A d_{A}^H + \ast _H d_{A}^H \nabla_{\zeta}^A )s & = -\ast _H \nabla_{\zeta}^A (\eta^1 \wedge \nabla_{\zeta_1}^A s + \eta^2 \wedge \nabla_{\zeta_2}^A s ) + \ast _H d_{A}^H \nabla_{\zeta}^A s \\[2pt]& = -\ast _H ( \eta^1 \wedge \nabla_{\zeta}^A \nabla_{\zeta_1}^A s + \eta_2 \wedge \nabla_{\zeta}^A \nabla_{\zeta_2}^A s ) \\[2pt]&\quad + \ast _H ( \eta_1 \wedge \nabla_{\zeta_1}^A \nabla_{\zeta}^A s + \eta_2 \wedge \nabla_{\zeta_2}^A \nabla_{\zeta}^A s ) \\[2pt]& = -\ast _H ( \eta_1 \wedge [\nabla_{\zeta}^B , \nabla_{\zeta_1}^B] + \eta_2 \wedge [\nabla_{\zeta}^B , \nabla_{\zeta_2}^B])s = -\ast _H \tfrac{1}{2} [\iota_\zeta F_B ,s ] .\end{align*}

By the self-adjointness of $(D_{\zeta}^{A}D_{H}^A + D_{H}^A D_{\zeta}^A)$ , the remaining entries in the matrix can be determined from the previous two. Putting everything together we obtain

(3.10) \begin{equation}(D_{\zeta}^{A}D_{H}^A + D_{H}^A D_{\zeta}^A) = \begin{pmatrix} 0 & 0 & -\ast _H \frac{1}{2} [\iota_\zeta F_A , \cdot ]\\[2pt]0 & 0 & \ast _H \frac{1}{2} [\iota_\zeta F_A , \ast _H \cdot ] + 2 \xi \ast _H d_{A}^H \\[2pt]-\ast _H \frac{1}{2} [\iota_\zeta F_A , \cdot ] & \frac{1}{2} [\iota_{\zeta} F_A , \cdot ] - 2\xi \ast _H d_{A}^H & 0 \end{pmatrix} \!. \end{equation}

Finally, the required identity follows by applying both sides of (3.7) to $(0 , \alpha , \Psi )$ , inserting the expressions (3.6), (3.8)–(3.10), and then taking the pointwise inner product against $(0 , \alpha , \Psi)$ .

We have now arrived at the crucial vanishing result.

Corollary 3.5. Let $A + i \Phi$ be a $GL(2,\mathbb{C})$ connection on E with fixed determinant, and let $\alpha , \Psi$ be as in (3.5). Suppose that $A+i\Phi$ is projectively flat (cf. (2.18) and (2.19)) and polystable (cf. (2.20)). Then the following quantities vanish identically:

\begin{align*}\nabla_{\zeta}^A \alpha = 0 , \quad \nabla_{\zeta}^A \Psi = 0, \quad d_{A}^H \alpha = 0 , \quad [\alpha , \Psi] = 0,\end{align*}

and we have

\begin{align*}F_A - \frac{I}{2} F_\lambda - \frac{1}{2}[\Psi , \Psi] & = 0,\\d_{A}^H \Psi & = 0,\\(d_{A}^H )^\ast \Psi & = 0.\end{align*}

Furthermore, if either the orbifold degree $\deg N$ is non-zero or $A + i \Phi$ is stable, then $\alpha = 0$ also.

Proof. By (2.18) we have $\iota_\zeta F_A = [\alpha , \Psi ] + \frac{I}{2} \iota_\zeta F_\lambda$ , and thus

$$\langle [\iota_\zeta F_A , \alpha ] , \Psi \rangle = \langle [[\alpha , \Psi],\alpha],\Psi \rangle = | [\alpha , \Psi]|^2 . $$

By (2.19) and (2.20) the left-hand side of the identity in Proposition 3.4 vanishes identically. Integrating that identity over Y and integrating by parts yields the following identity:

\begin{align*}0 = & \| \nabla_{\zeta}^A \alpha \|_{L^2 (Y)}^2 + \| \nabla_{\zeta}^A \Psi \|_{L^2(Y)}^2 + \| d_{A}^H \alpha \|_{L^2 (Y)}^2 + \| (d_{A}^H )^\ast \Psi \|_{L^2 (Y)}^2+ \| [\alpha , \Psi ]\|_{L^2 (Y)}^2\\& + \|\ast _H d_{A}^H \Psi + 2 \xi \alpha \|_{L^2 (Y)}^2 .\end{align*}

Thus,

\begin{align*}\nabla_{\zeta}^A \alpha &= 0 , \quad \nabla_{\zeta}^A \Psi = 0 , \quad d_{A}^H \alpha = 0,\\ (d_{A}^H )^\ast \Psi &= 0 , \quad [\alpha , \Psi ] = 0 , \quad \ast _H d_{A}^H \Psi + 2 \xi \alpha = 0 . \end{align*}

Applying $d_{A}^H$ to the last identity above and using $d_{A}^H \alpha = 0$ gives $d_{A}^H \ast _H d_{A}^H \Psi = 0$ . Integration by parts then gives $d_{A}^H \Psi = 0$ and hence also $\xi \alpha = 0$ .

Thus, if $\xi \neq0$ (equivalently, if $\deg N\neq 0$ ) then $\alpha = 0$ . Finally, suppose that $\xi = 0$ but $A + i \Phi$ is stable. Then from $\nabla_{\zeta}^A \alpha = 0$ , $d_{A}^H \alpha = 0$ and Lemma 3.3 we have $d_A \alpha =0$ . This, combined with $[\alpha , \Phi ]= [\alpha , \Psi ] = 0$ , implies that $\alpha$ vanishes by stability. With this, the proof is complete.

We also need an ‘infinitesimal’ analog of Corollary 3.5. Let $A + i \Phi$ be a stable projectively flat $GL(2,\mathbb{C})$ connection with fixed determinant. Then the Zariski tangent space $\mathscr{H}^{1}_{(A, \Phi )} = \mathrm{Ker}\mathscr{D}_{(A, \Phi )}$ to the moduli space $\mathscr{M}_{E}^\ast $ at $[(A, \Phi )]$ is given by the space of $(\dot{A}, \dot{\Phi} ) \in (\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )$ such that

(3.11) \begin{align}- d_{A}^\ast \dot{A} - \ast [\dot{\Phi}, \ast \Phi ] & = 0, \nonumber\\\ast [\dot{A} , \ast \Phi] - d_{A}^\ast \Phi & = 0, \nonumber\\\ast [\dot{A} , \Phi ] + \ast d_{A}\dot{\Phi} & = 0, \nonumber\\\ast d_{A}\dot{A} - \ast [\dot{\Phi} , \Phi] & = 0. \end{align}

Decompose $\dot{A} = \eta \cdot \dot{a} + \dot{B}$ and $\dot{\Phi} = \eta \cdot \dot{\alpha} + \dot{\Psi}$ according to the splitting (3.3).

Proposition 3.6. Suppose that $A + i \Phi$ is a polystable projectively flat $GL(2, \mathbb{C} )$ connection on E with fixed determinant, and let $\alpha , \Psi$ be as in (3.5). Suppose that $(\dot{A} , \dot{\Phi} ) \in (\Omega^1 \oplus \Omega^1 )(Y, \mathfrak{g}_E )$ solves equations (3.11). Then the following quantities vanish identically:

\begin{align*}\nabla_{\zeta} \dot{a} &= 0 , \nabla_{\zeta}^A \dot{B} = 0 , [\dot{a}, \alpha]= 0 , [\dot{a}, \Psi] = 0 , [\dot{B},\alpha] = 0,\\\nabla_{\zeta}\dot{\alpha} &= 0, \quad \nabla_{\zeta}^A \dot{\Psi} = 0 , [\dot{\alpha},\alpha] = 0 , [\dot{\alpha} , \Psi] = 0 , [\dot{\Psi},\alpha] = 0,\end{align*}

and we have

\begin{align*}- (d_{A}^H )^\ast \dot{B} - \ast _H [ \dot{\Psi} , \ast _H \Psi ] & = 0,\\ \ast _H [ \dot{B}, \ast _H \Psi ] - (d_{A}^H )^\ast \dot{\Psi}& = 0,\\\ast _H [ \dot{B}, \Psi ] + \ast _H d_{A}^H \dot{\Psi} & = 0,\\\ast _H d_{A}^H \dot{B} - \ast _H [\dot{\Psi}, \Psi] &= 0.\end{align*}

Furthermore, if either the orbifold degree $\deg N$ is non-zero or $A+ i \Phi$ is stable, then $\dot{a} = 0$ and $\dot{\alpha} = 0$ also.

Proof. Equations (3.11) can be decomposed into vertical and horizontal parts using Lemma 3.3:

(3.12) \begin{align}- (d_{A}^H )^\ast \dot{B} - \ast _H [ \dot{\Psi} , \ast _H \Psi ] & = -\nabla_{\zeta}^A \dot{a} + [\dot{\alpha} , \alpha ], \nonumber\\ \ast _H [ \dot{B}, \ast _H \Psi ] - (d_{A}^H )^\ast \dot{\Psi}& = -\nabla_{\zeta}^A \dot{\alpha} - [\dot{a} , \alpha ],\nonumber\\\ast _H [ \dot{B}, \Psi ] + \ast _H d_{A}^H \dot{\Psi} & = -2\xi \dot{\alpha}, \nonumber\\- \ast _H d_{A}^H \dot{\alpha} + \ast _H \nabla_{\zeta}^A \dot{\Psi} + \ast _H [\dot{a}, \Psi] - \ast _H [\dot{B},\alpha] & = 0, \nonumber\\\ast _H d_{A}^H \dot{B} - \ast _H [\dot{\Psi}, \Psi] &= -2\xi \dot{a},\nonumber\\-\ast _H d_{A}^H \dot{a} + \ast _H \nabla_{\zeta}^A \dot{B} - \ast _H [\dot{\alpha}, \Psi] + \ast _H [\dot{\Psi},\alpha] & = 0.\end{align}

We first establish the vanishing of the quantities $\nabla_{\zeta}\dot{\alpha}, d_{A}^H \dot{\alpha},[\dot{\alpha},\alpha] , [\dot{\alpha} , \Psi] $ . For this, we express the second and third equations in (3.11) jointly in terms of the operator $D_A$ introduced in the proof of Proposition 3.4, as

\begin{align*}D_A \begin{pmatrix} 0 \\ \dot{\Phi}\end{pmatrix} + \begin{pmatrix} \ast [\dot{A}, \ast \Phi ] \\ \ast [\dot{A} , \Phi] \end{pmatrix} = 0 .\end{align*}

We then apply $D_A$ to this identity and take the inner product against $(0 , \eta \dot{\alpha} )$ :

(3.13) \begin{equation} \bigg\langle D_{A}^2 \begin{pmatrix} 0 \\ \dot{\Phi}\end{pmatrix}\!, \begin{pmatrix} 0 \\ \eta \dot{\alpha} \end{pmatrix}\!\!\bigg\rangle + \bigg\langle D_A \begin{pmatrix} \ast [\dot{A}, \ast \Phi ] \\ \ast [\dot{A} , \Phi] \end{pmatrix}\!, \begin{pmatrix} 0 \\ \eta \dot{\alpha }\end{pmatrix}\!\!\bigg\rangle = 0.\end{equation}

We now compute each term in (3.13). For the first term, by (3.6), (3.8), (3.9), and the vanishing of the terms involving $\iota_\zeta F_A $ in (3.10) (since $[\alpha , \Psi] = 0$ by Corollary 3.5), we have

(3.14) \begin{align}\bigg\langle D_{A}^2 \begin{pmatrix} 0 \\ \dot{\Phi}\end{pmatrix}\!, \begin{pmatrix} 0 \\ \eta \dot{\alpha} \end{pmatrix}\!\!\bigg\rangle & = \langle - \nabla_{\zeta}^A \dot{\alpha} , \dot{\alpha}\rangle + 4\xi^2 |\dot{\alpha}|^2 + \langle (d_{A}^H )^\ast d_{A}^H \dot{\alpha}, \dot{\alpha}\rangle + 2 \xi \ast \langle \ast _H d_{A}^H \dot{\Psi}, \dot{\alpha}\rangle \nonumber \\& = \langle - \nabla_{\zeta}^A \dot{\alpha} , \dot{\alpha}\rangle + \langle (d_{A}^H )^\ast d_{A}^H \dot{\alpha}, \dot{\alpha}\rangle - 2\xi \langle \ast _H [\dot{B}, \Psi], \dot{\alpha}\rangle, \end{align}

where in the last line we have used the third identity in (3.12). For the second term in (3.13), we use Lemma 3.3 and the vanishing of $\nabla_{\zeta}^A \alpha , \nabla_{\zeta}^A \Psi, d_{A}^H \alpha $ and $(d_{A}^H )^\ast \Psi $ to obtain

(3.15) \begin{align}& \bigg\langle D_A \begin{pmatrix} \ast [\dot{A}, \ast \Phi ] \\ \ast [\dot{A} , \Phi] \end{pmatrix}\!, \begin{pmatrix} 0 \\ \eta \dot{\alpha }\end{pmatrix}\!\!\bigg\rangle \nonumber \\&\quad = \langle \, - [\nabla_{\zeta}^A \dot{a} , \alpha] - \ast _H [\nabla_{\zeta}^A \dot{B}, \ast _H \Psi] + \ast _H [d_{A}^H \dot{a} , \ast _H \Psi] + [ (d_{A}^H )^\ast \dot{B} , \alpha] + 2 \xi \ast _H [\dot{B},\Psi] , \dot{\alpha} \, \rangle \nonumber \\&\quad = \langle \, - [\ast _H [\dot{\Psi} , \ast _H \Psi],\alpha] - [[\dot{\alpha},\alpha],\alpha] - \ast _H [[\dot{\alpha},\Psi],\ast _H \Psi] + \ast _H [[\dot{\Psi},\alpha],\ast _H \Psi] + 2\xi \ast _H [\dot{B},\Psi] , \dot{\alpha} \, \rangle \nonumber \\&\quad = |[\dot{\alpha},\alpha]|^2 + |[\dot{\alpha}, \Psi]|^2 + 2\xi \langle \ast _H [\dot{B},\Psi],\dot{\alpha}\rangle, \end{align}

where to get the penultimate line we used the first and last identities in (3.12), and to get the last line we took adjoints and used the vanishing of the terms

$$ - [\ast _H [\dot{\Psi} , \ast _H \Psi],\alpha] + \ast _H [[\dot{\Psi},\alpha],\ast _H \Psi] = \ast _H [[\Psi , \alpha], \ast _H \dot{\Psi}] = 0, $$

due to $[\Psi , \alpha] = 0$ (by Corollary 3.5).

Altogether, combining (3.13)–(3.15), canceling terms, and integrating by parts gives the identity

\begin{align*}0 = \| \nabla_{\zeta}^A \dot{\alpha}\|_{L^2 (Y)}^2 + \| d_{A}^H \dot{\alpha}\|_{L^2 (Y)}^2 + \| [\dot{\alpha}, \alpha]\|_{L^2 (Y)}^2 + \| [\dot{\alpha}, \Psi]\|_{L^2 (Y)}^2 ,\end{align*}

and, hence, the vanishing of $\nabla_{\zeta}\dot{\alpha}, d_{A}^H \dot{\alpha},[\dot{\alpha},\alpha] , [\dot{\alpha} , \Psi]$ .

An analogous argument gives the identity

$$0 = \| \nabla_{\zeta}^A \dot{a}\|_{L^2 (Y)}^2 + \| d_{A}^H \dot{a}\|_{L^2 (Y)}^2 + \| [\dot{a}, \alpha]\|_{L^2 (Y)}^2 + \| [\dot{a}, \Psi]\|_{L^2 (Y)}^2 ,$$

and, hence, the vanishing of $\nabla_{\zeta}\dot{a}, d_{A}^H \dot{a},[\dot{a},\alpha] , [\dot{a} , \Psi]$ .

By the vanishing of all these quantities, (3.11) now read

(3.16) \begin{align}- (d_{A}^H )^\ast \dot{B} - \ast _H [ \dot{\Psi} , \ast _H \Psi ] & = 0, \nonumber\\ \ast _H [ \dot{B}, \ast _H \Psi ] - (d_{A}^H )^\ast \dot{\Psi}& = 0, \nonumber\\\ast _H [ \dot{B}, \Psi ] + \ast _H d_{A}^H \dot{\Psi} & = -2\xi \dot{\alpha}, \nonumber\\ \ast _H \nabla_{\zeta}^A \dot{\Psi} - \ast _H [\dot{B},\alpha] & = 0, \nonumber\\\ast _H d_{A}^H \dot{B} - \ast _H [\dot{\Psi}, \Psi] &= -2\xi \dot{a},\nonumber\\ \ast _H \nabla_{\zeta}^A \dot{B} + \ast _H [\dot{\Psi},\alpha] & = 0. \end{align}

Taking $\nabla_{\zeta}^A$ of the fourth equation of (3.16), using $\nabla_{\zeta}^A \alpha = 0$ (cf. Corollary 3.5), then inserting the last equation of (3.16), and finally integrating by parts yields

$$\| \nabla_{\zeta}^A \dot{\Psi} \|_{L^2 (Y)}^2 + \| [\dot{\Psi }, \alpha ]\|_{L^2 (Y)}^2 = 0 . $$

This says the quantities $\nabla_{\zeta}^A \dot{\Psi}, [\dot{\Psi} , \alpha ]$ vanish. An analogous argument gives the vanishing of $\nabla_{\zeta}^A \dot{B} , [\dot{B} , \alpha]$ . Finally, observe that $\xi \dot{\alpha} = 0$ , which follows easily by taking the inner product of the third equation in (3.16) with $\dot{\alpha}$ , integrating by parts and using the vanishing of $[\Psi , \dot{\alpha}] , d_{A}^H \dot{\alpha}$ . Likewise, $\xi \dot{a} = 0$ .

At this point, we have established all but the final assertion in Proposition 3.6. The required conclusion for $\dot{\alpha}$ readily follows from the vanishing of $\xi\dot{\alpha} , [\Psi , \dot{\alpha}] , [\alpha , \dot{\alpha}],d_{A}\dot{\alpha}$ , and similarly for $\dot{a}$ . This concludes the proof.

3.3.1 Stable flat connections and Higgs pairs on C

Let $E_0 \to C$ be an orbifold U(2) bundle with fixed determinant $(\Lambda_0 = \Lambda^2 E_0 , \lambda_0 )$ over the closed oriented Riemannian 2-orbifold $(C, g_C )$ , where $\lambda_0$ is a fixed U(1) connection on $\Lambda_0$ . We now gather some basic results about the moduli space of stable projectively flat connections on the bundle $E_0 \to C$ , which are discussed in further detail in [Reference HitchinHit87, Reference DonaldsonDon87b, Reference Furuta and SteerFS92, Reference Nasatyr and SteerNS95].

The discussion from § 2.2 applies also to the Riemannian 2-orbifold $(C, g_C )$ , and we obtain the spaces $\mathscr{B}_{E_0 , k }(C), \mathscr{B}_{E_0 , k }^\ast (C)$ of polystable or stable orbits of $GL(2,\mathbb{C} )$ connections $B + i \Psi$ with fixed determinant $\lambda_0$ and regularity $L^{2}_{k}$ under the action of the group $\mathscr{G}_{E_0 , k+1}(C)$ of $L^{2}_{k+1}$ gauge transformations. As before, ‘polystable’ here means $d_{B}^\ast \Psi = 0$ , and ‘stable’ means that, in addition, the stabilizer of $B + i \Psi$ under the $\mathscr{G}_{E_0 , k+1}$ -action is $\pm I$ . Choosing any $k\geq 2$ will suffice for our purposes, and $\mathscr{B}_{E_0 , k} (C)$ then becomes a Hilbert Kähler manifold when equipped with the complex structure J and symplectic form $\omega_J$ given in (2.1) and (2.2).

The moduli space $\mathscr{M}_{E_0}^\ast (C,g_C )$ of polystable projectively flat $GL(2,\mathbb{C})$ orbifold connections on $E_0 \to C$ with fixed determinant $(\Lambda_0 , \lambda_0 )$ is given by the $\mathscr{G}_{E_0}$ -orbits of configurations $\mathbb{B} = B + i \Phi $ solving

(3.17) \begin{align} F_B - \frac{I}{2}F_{\lambda_0} - \frac{1}{2} [\Psi , \Psi ] &= 0, \end{align}

(3.18) \begin{align} d_{B} \Psi & = 0, \end{align}

(3.19) \begin{align}d_{B}^\ast \Psi & = 0. \end{align}

The moduli space $\mathscr{M}_{E_0}^\ast (C) \subset \mathscr{B}_{E_0}^\ast (C) $ arises as the zero locus of a holomorphic Fredholm section, denoted by s, of a holomorphic Hilbert vector bundle, denoted by $\mathscr{V}_{k-1} \rightarrow \mathscr{B}_{E_0}^\ast (C)$ . Namely, $\mathscr{V}_{k-1}$ is the bundle induced from the $\mathscr{G}_{E_0 , k+1}$ -equivariant vector bundle over the stable configuration space $\mathscr{C}_{E_0 , k}^\ast (C)$ with fiber $\Omega^2 (C, \mathfrak{g}_E ) \oplus \Omega^2 (C , \mathfrak{g}_E )$ at every point, and s is induced from the equivariant section given the left-hand side of equations (3.17) and (3.18).

As a matter of general principle, the moduli space $\mathscr{M}_{E_0}^\ast $ is naturally a complex-analytic space with complex structure J, with local charts provided by finite-dimensional Kuranishi models. In fact, more is true in this case.

For the formula for the dimension of $\mathscr{M}_{E_0}^\ast (C)$ in terms of the isotropy data of $E_0$ (which we shall not need) we refer to [Reference Nasatyr and SteerNS95, Theorem 3.5(1)]. The proof of Proposition 3.7 appears in [Reference HitchinHit87, § 5] in a slightly different guise (and in the case when C has no orbifold points). We now briefly outline a version of this argument, for completeness.

Proof of Proposition 3.7. At a stable flat connection $(B, \Psi )$ the vertical derivative of s is the operator

\begin{align*}(\mathcal{D} s)_{(B, \Psi )} : \mathscr{K}_{(B, \Psi )} &= \mathrm{Ker}(\delta^{1}_{(B, \Psi )})^\ast \cap \mathrm{Ker}(\delta^{2}_{(B, \Psi )})^\ast \subset (\Omega^1 \oplus \Omega^1 )(C, \mathfrak{g}_E ) \to (\Omega^2 \oplus \Omega^2 )(C, \mathfrak{g}_E ) ,\\(\mathcal{D} s)_{(B, \Psi )} (\dot{B} , \dot{\Psi} ) &= ( d_{B}\dot{B} - [\dot{\Psi},\Psi] , [\dot{B}, \Psi] + d_{B}\dot{\Psi}),\end{align*}

where $\delta^1$ and $\delta^2$ are the corresponding versions on $(C,g_C )$ of the operators defined in § 2.2.2. Thus, an element $(\dot{F} , \dot{G} )$ which is $L^2$ -orthogonal to the image of $\mathcal{D}s$ satisfies the following identity coming from taking adjoints: for any $(\cdot{B}, \cdot{\Psi}) \in \mathscr{K}$ ,

(3.20) \begin{equation}0 = \langle ( d_{B}^\ast \dot{F} - \ast [\Psi , \ast \dot{G}] , \ast [\Psi , \ast \dot{F}] + d_{B}^\ast \dot{G} ) , (\dot{B} , \dot{\Psi} ) \rangle_{L^2 (C)}. \end{equation}

One can then show that the element $( d_{B}^\ast \dot{F} - \ast [\Psi , \ast \dot{G}] , \ast [\Psi , \ast \dot{F}] + d_{B}^\ast \dot{G})$ lies in the kernel of $(\delta^1)^\ast $ and $(\delta^2 )^\ast $ when $(B, \Psi )$ is flat and, hence, (2.7) and (3.20) imply that

$$d_{B}^\ast \dot{F} - \ast [\Psi , \ast \dot{G}] = 0 , \quad \ast [\Psi , \ast \dot{F}] + d_{B}^\ast \dot{G} = 0 . $$

Taking $d_B$ on both of the two identities, then integrating the first against $\dot{F}$ and the second against $\dot{G}$ , and finally adding everything up leads to $0 = \| d_{B}^\ast \dot{F} \|_{L^2 (C) } + \| d_{B}^\ast \dot{G} \|_{L^2 (C) }$ and, hence, $\dot{F} = \dot{G} = 0$ as required.

It is well known that if ${\dim}C = 2$ , then $\mathscr{M}^{\ast }_{E_0}(C,g_C )$ is naturally a hyperkähler manifold and admits a holomorphic description as a moduli space of stable Higgs pairs. We briefly recall this now (see [Reference HitchinHit87, Reference Nasatyr and SteerNS95] for details).

The Hodge star operator of the metric $g_C$ takes 1-forms to 1-forms, hence gives an additional complex structure on $\mathscr{A}_{E_0 , k}^c $ denoted by $I (\dot{B} , \dot{\Psi } ) = (\ast _C \dot{B} , \ast _C \dot{\Psi} )$ . The complex structures I, J anticommute, and hence we have a third complex structure $K = IJ$ (altogether, I,J,K satisfy the quaternion relations). All three complex structures are compatible with the flat Riemannian metric G on $\mathscr{A}_{E_0 , k}^c$ obtained by the $L^2$ inner product, and we obtain three corresponding Kähler (symplectic) forms $\omega_I , \omega_J , \omega_K$ . For example, $\omega_J$ was given in (2.2) and $\omega_I$ is given by

(3.21) \begin{equation}\omega_I ( (\dot{B}_1 , \dot{\Psi}_1 ) , (\dot{B}_2 , \dot{\Psi}_2 ) ) = G ( I (\dot{B}_1 , \dot{\Psi}_1 ) , (\dot{B}_2 , \dot{\Psi}_2 ) ) = - \int_C \mathrm{Tr}( \dot{B}_1 \wedge \dot{B}_2 + \dot{\Psi}_1 \wedge \dot{\Psi}_2 ) . \end{equation}

Now, the $\mathscr{G}_{E_0 , k+1}$ -action is Hamiltonian for each of the three symplectic forms. With respect to $\omega_J$ , the moment map is given by the left-hand side of (3.19), as we saw also in the three-dimensional case (this holds in any dimension). The moment maps for $\omega_I$ and $\omega_K$ can be identified with the left-hand sides in (3.17) and (3.18), respectively. Thus, $\mathscr{M}_{E_0}^\ast (C)$ arises as an infinite-dimensional hyperkähler reduction of $(\mathscr{A}_{E_0 , k}^c , G , I , J , K )$ by the $\mathscr{G}_{E_0 , k +1}$ -action.

In turn, the complex gauge group $\mathscr{G}_{E_0 , k+1}^c$ preserves the structures I and J. By infinite-dimensional versions of the Kempf–Ness theorem due to Hitchin [Reference HitchinHit87] and Donaldson [Reference DonaldsonDon87b] (see [Reference Nasatyr and SteerNS95] for the orbifold case) the moduli space $\mathscr{M}_{E_0}^{\ast }(C,g_C )$ then admits a holomorphic interpretation in the complex structures I and J.

• – With the complex structure I, $\mathscr{M}_{E_0}^{\ast }(C,g_C ) $ is identified with the moduli space of stable Higgs pairs on the orbifold bundle $E_0 \to C$ with fixed determinant ([Reference HitchinHit87, Reference Nasatyr and SteerNS95]):

  $$ ( \mathscr{M}_{E_0}^{\ast }(C,g_C ) , I ) \cong \mathscr{M}_{E_0}^{\mathrm{Higgs}, \ast } (C). $$
  We recall the definition of $ \mathscr{M}_{E_0}^{\mathrm{Higgs}, \ast } (C)$ . For this, we use the orbifold complex structure on C defined by $g_C$ , and the orbifold holomorphic structure on $\Lambda_0$ induced by $\lambda_0$ . Then $ \mathscr{M}_{E_0}^{\mathrm{Higgs}, \ast } (C)$ is given by the isomorphism classes of pairs $(\mathscr{E} , \theta )$ where $\mathscr{E}$ is an orbifold holomorphic structure $\mathscr{E}$ on $E_0$ with fixed determinant and $\theta \in H^0 ( C , T^\ast C \otimes \mathfrak{sl}(\mathscr{E} ))$ (i.e., $\theta$ is a holomorphic section of the orbifold holomorphic bundle $T^\ast C \otimes \mathfrak{sl}(\mathscr{E} )$ of $T^\ast C$ -valued trace-free endomorphisms of $\mathscr{E}$ ), subject to the requirement that $(\mathscr{E} , \theta )$ is stable: if $\mathscr{L} \subset \mathscr{E}$ is a $\theta$ -invariant proper non-trivial holomorphic subbundle, then
  $$\deg \mathscr{L} \lt \tfrac{1}{2}\deg \mathscr{E}. $$
  The correspondence between stable (projectively) flat connections and stable Higgs pairs is given by
  $$\mathbb{B} = B + i \Psi \mapsto (\mathscr{E} , \theta ) = ( \overline{\partial}_B , - i \Psi^{1,0} ) . $$

• – Even though we do not need it, we mention for context that with the complex structure J the moduli space $\mathscr{M}_{E_0}^{\ast }(C,g_C ) $ is biholomorphic to the smooth quasi-affine variety consisting of conjugacy classes of irreducible representations of the orbifold fundamental group of C in $SL(2, \mathbb{C})$ , suitably compatible with the isotropy data of $E_0$ ([Reference DonaldsonDon87b, Reference Nasatyr and SteerNS95]):

  $$( \mathscr{M}_{E_0}^{\ast }(C,g_C ) , J ) \cong \mathrm{Hom}_{E_0}^\ast ( \pi_{1}^{\mathrm{orb}} (C) , SL(2, \mathbb{C} ) )/SL(2, \mathbb{C} ) . $$

3.3.2 From stable flat connections on Y to stable flat connections on C

Consider a U(2) bundle E over a Seifert-fibered 3-manifold Y with fixed determinant $\Lambda$ . We now describe the moduli space $\mathscr{M}_{E}^\ast (Y)$ in terms of the moduli spaces on C.

If E satisfies any of the three equivalent conditions from the next lemma, we say that E ‘pulls back from C up to tensoring’.