I'm currently reading Auroux's Mirror Symmetry and T-duality in the Complement of the Anticanonical Divisor and Special Lagrangian Fibrations, Wall-crossing, and Mirror Symmetry back and forth.
I'm having some difficulties with the moduli space of the pairs $(L,\nabla)$ consisting of special Lagrangian submanifold $L \subset X$ and a flat unitary connection $\nabla$ on the trivial complex line bundle over $L$, up to gauge equivalence. I will write $M$ for this moduli space.
Question 1 : What's the precise definition of "Cech dual" and how does it relate to $M$?
Auroux mentions that the dual torus $\check{T}$ of a torus $T$ is given by $$\check{T} = Hom(\pi_1(T),S^{1}).$$ Is this the general definition for "Cech dual"? So $\check{L} = Hom(\pi_1(L),S^{1})$?
If so, it seems that the author is identifying $\check{L}$ with $\mathcal{A}/\mathcal{G}$, where $\mathcal{A}$ is the (moduli) space of all flat connections on the trivial line bundle over $L$ and $\mathcal{G}$, gauge group. In general, we have
$$\mathcal{M}(\Sigma, G) \cong \frac{Hom(\pi_1(\Sigma),G)}{G},$$ where $\mathcal{M}(\Sigma, G)$ is the moduli space of all flat principal $G$-bundles over $\Sigma$, and the quotient on the right hand side is by conjugation.
Putting everything together, we have :
$$\mathcal{A}/\mathcal{G} \cong \mathcal{M}(L,\mathbb{C}) \cong \frac{Hom(\pi_1(L),\mathbb{C})}{\mathbb{C}} \cong Hom(\pi_1(L),S^{1}) = \check{L}$$ The first and third identification seems very wrong to me... (I guess I can just define the dual to be $\mathcal{A}/\mathcal{G}$ though).
Question 2 : The tangent space $T_{(L,\nabla)}M$ and the integrable complex structure of $M$
The author writes : the tangent space to $M$ at $(L,\nabla)$ is the set of all pairs $(v,\alpha)\in C^{\infty}(NL) \oplus \Omega^{1}(L,\mathbb{R})$ such that $v$ : infinitesimal special Lagrangian deformation, $\alpha$ : ($\psi$-)harmonic 1-form viewed as an infinitesimal deformation of the flat connection.
I understand how a section of the normal bundle $NL$ became the "vector" on the moduli space of special Lagrangians. But I can't wrap my head around a "real-valued 1-form being the vector on the moduli space flat connections". Here are some facts that I came across :
- A principal connection $\nabla$ on principal $G-$ bundle $P$ can be viewed as a $\mathfrak{g}$-valued 1-form on $P$, in the sense that for $p \in P$,
$$\nabla_{p} : T_{p}P \rightarrow \mathfrak{g}$$
- The moduli space $\mathcal{P}$ of principal connections is an affine space modeled by $\Omega^{1}(L;\mathfrak{g}_{P})$, where $\mathfrak{g}_{P}$ is the adjoint bundle $P \times_{Ad}\mathfrak{g}$, hence $T_{\nabla}\mathcal{P} = \Omega^{1}(L;\mathfrak{g}_{P})$.
In our situation, $G=\mathbb{C}$, $\mathfrak{g}=\mathbb{C}$, so
$$\nabla_{(p,z)} : T_{(p,z)}(L \times \mathbb{C}) \rightarrow \mathbb{C}, \in \Omega^{1}(L;\mathbb{C})$$
So somehow by taking account flatness and quotient by gauge equivalence, we should have $\Omega^{1}(L;\mathfrak{g}_{P}) = \Omega^{1}(L;\mathbb{R})$?
I'm quite new to this subject, any help would be appreciated! Thanks.