Consider the space $\mathcal{A}$ of $(0,1)$-connections $A$ on a fixed complex bundle on a Calabi-Yau 3-fold $M$. This infinite dimensional space has a natural complex structure, with respect to which it admits a holomorphic functional, Witten’s holomorphic Chern-Simons functional [W1], [DT],

$$CS_{\mathbb{C}}\,(A=A_{0}+a)=\frac{1}{4\pi^{2}}\int_{M}\mathrm{tr}\left(\bar{\partial}_{A_{0}}a\wedge a+\frac{2}{3}a\wedge a\wedge a\right)\wedge\Omega,$$

where $\Omega$ is the holomorphic (3,0)-form. It is infinitesimally gauge-invariant (gauge transformations not homotopic to the identity can give periods to $CS_{\mathbb{C}}\,$) and its gradient is $F_{A}^{0,2}$, with zeros the integrable connections. That is, after dividing by gauge equivalence (under which grad$\,CS_{\mathbb{C}}\,$ is invariant), the critical points of $CS_{\mathbb{C}}\,$ form the space of holomorphic bundles of the same topological type. As critical points of a functional, moduli of holomorphic bundles have virtual dimension zero, and one might try to make sense of counting them – a holomorphic Casson invariant [T1]. This is independent of deformations of the complex structure, but can have wall-crossing changes as the Kähler form varies. (This is because we count only stable bundles, and the notion of stability depends on a Kähler form.)

On the other hand, on a different Calabi-Yau 3-fold $W$ (for instance the mirror, in some situation where this makes sense), Lagrangians are the critical points of a functional too, on the space of all 3-dimensional submanifolds (or cycles):

$$f_{\mathbb{R}}(L)=\int_{L_{0}}^{L}\omega\wedge\omega,$$

where $\omega$ is the symplectic form on $W$. Here $L_{0}$ is a fixed cycle in the same homology class, and we integrate over a 4-chain with boundary $L-L_{0}$; the functional $f_{\mathbb{R}}$ is invariant under the choice of different, homologous, 4-chains (picking non-homologous 4-chains can give periods to $f_{\mathbb{R}}$). It is invariant under deformations of $L$ pulled back from hamiltonian deformations of $W$ (deformations generated by vector fields $v$ on $W$ whose contraction with $\omega$ is exact $v{\mbox{\,}\_\hskip-4.2679pt\shortmid\hskip 1.5pt}\omega=dh$ at each point in time) as $\int_{L}\omega\wedge dh=0$, and its gradient is $\omega|_{L}$. Thus its critical points are Lagrangian submanifolds. We would like to think of $f_{\mathbb{R}}$ as mirror to $CS_{\mathbb{C}}\,$, but to do so we must complexify it.

Thus we work on the space $\mathcal{A}$ of submanifolds $L$ of $W$ with $U(1)$ connections $A$ on the trivial bundle $\mathbb{C}\,\times L$ on $L$. Notice these submanifolds are not parameterised by a map of a real 3-manifold into $W$; we are only interested in the image $L$. From now on we shall restrict attention to smooth Lagrangian submanifolds. Formally, we consider the tangent space to $\mathcal{A}$ at a point $(A,L\subset W)$ to be

$$\Omega^{1}(L;\mathbb{R})\oplus\Omega^{1}(L;\mathbb{R}),$$

(2.1)

at least for those $L$ with no $J$-invariant subspaces of its tangent spaces ($J$ is the complex structure on $W$, and this is reasonable since we are looking for Lagrangian submanifolds after all). The first factor is the obvious tangent space to the connections on $L$; the second gives deformations of $L$ via the vector fields produced by contracting with the Kähler form $\omega$ on $W$. That is, we use the metric on $W$ to map $\Omega^{1}(L)$ to $\Omega^{1}(W)|_{L}$, then use the isomorphism provided by $\omega$ to get a vector field along $L$. Equivalently, using the metric on $W$, we may think of one-forms on $L$ as tangent vectors to $L$, then apply the complex structure $J$ on $W$ to give $W$-vector fields on $L$. We denote this map from one-forms to normal vector fields by

$$\Omega^{1}(L)\to TW\arrowvert_{L},\qquad\sigma\mapsto\sigma{\mbox{\,}\_\hskip-4.2679pt\shortmid\hskip 1.5pt}\omega^{-1}.$$

(2.2)

Connections on $L$ are carried along by the vector field to connections on nearby cycles, and we are identifying the space of $U(1)$ connections with $i\Omega^{1}(L;\mathbb{R})$.

There is a natural almost complex structure on $\mathcal{A}$, acting as

$$J=\left(\begin{array}[]{cc}0&1\\ \!-1&0\end{array}\right),$$

with respect to the splitting (2.1) of the tangent spaces. With respect to this we claim to have the following holomorphic functional

$$f_{\mathbb{C}}\,(A,L)=\int_{L_{0}}^{L}(F+\omega)^{2}=\int_{L_{0}}^{L}(F^{2}+\omega^{2})\ +\ 2\int_{L_{0}}^{L}\omega\wedge F.$$

Here we have extended $A$ to a connection on the trivial bundle on the whole of $W$ (restricting to a fixed connection $A_{0}$ on $L_{0}$, and to $A$ on $L$) and taken its curvature form $F$. We have again picked a 4-cycle bounding $L-L_{0}$; because $F$ and $\omega$ are closed the resulting functional is independent of different homologous choices of the 4-cycle, and in general well defined up to the addition of some discrete periods. It is also (again) independent of hamiltonian isotopies of $L$. Notice that the $\int_{L_{0}}^{L}F^{2}$ term is just the real Chern-Simons functional $CS_{\mathbb{R}}$ of the connection $A$ on $L$, whose critical points are well known to be flat connections. As pointed out to me by Eric Zaslow, the real and complex Chern-Simons functionals already appear in [W1] and [Va] as possible mirror partners (this is partially justified in [LYZ]), but without the terms in the symplectic form (and including instanton corrections from holomorphic discs which we are ignoring for our rough analogy). Asking for a real function to be equal to a complex one is possible when one restricts attention to a real slice such as the space of Lagrangian submanifolds in $\mathcal{A}$; deforming within this space the imaginary part of $f_{\mathbb{C}}\,$ remains constant and it reduces to $CS_{\mathbb{R}}$. But allowing the imaginary counterparts to these real deformations the right functional to consider is $f_{\mathbb{C}}\,$. Notice also that if $\omega/2\pi$ is integral, so that we can pick a connection $B$ with curvature $-i\omega$, then the action functional can be written in the more familiar looking Chern-Simons form

$$f_{\mathbb{C}}\,(A,L)=\int_{L}(B+iA)\wedge d(B+iA)=\int_{L}CdC$$

for the ‘complexified connection’ $C=B+iA$ (a $\mathbb{C}\,^{\!\times}$-connection, instead of a $U(1)$-connection.) This makes more contact with the physics literature and allows one to extend the identification of $CS_{\mathbb{R}}$ and $CS_{\mathbb{C}}\,$ in [LYZ] to non Lagrangian sections, giving complex values. Tian has informed me that he and Chen have also considered the functional $f_{\mathbb{C}}\,(A,L)$ [Ch].

Mirror symmetry should relate Lagrangians not just to bundles but the whole derived category. For Riemann surfaces $C\subset M$, for instance, there is a functional in [DT], [W2] rather like $f_{\mathbb{R}}$ above:

$$\int_{C_{0}}^{C}\Omega$$

is formally holomorphic and has as critical points the holomorphic curves $C$. Similarly for four-manifolds $S\subset M$ with connections on them the following functional (formally similar to $f_{\mathbb{C}}\,$)

$$\int_{S_{0}}^{S}\mathrm{tr}\,F\wedge\Omega$$

has critical points the holomorphic surfaces with flat connection on them. Alternatively, as $CS_{\mathbb{C}}\,$ is additive under extensions of bundles it does extend to the derived category. (Whether these two approaches are compatible; i.e. whether or not the functional associated to a curve or surface is the same as $CS_{\mathbb{C}}\,$ applied to a locally free resolution of its structure sheaf, up to a constant, seems to not have been worked out.)

That $f_{\mathbb{C}}\,$ is holomorphic follows from the computation that the derivative of $f_{\mathbb{C}}\,$ down $a\in\Omega^{1}(L)\oplus 0$ (that only changes the connection $A\mapsto A+\delta a$) is $\int_{L}2F\wedge ia+2\omega\wedge ia$, while the derivative down $-Ja\in 0\oplus\Omega^{1}(L)$, i.e. down the vector field $a{\mbox{\,}\_\hskip-4.2679pt\shortmid\hskip 1.5pt}\omega^{-1}$, is $\int_{L}2\omega\wedge a+2a\wedge F$. The second expression is $-i$ times the first, so the derivative is complex linear and $f$ is holomorphic. Equivalently we are saying that $df_{\mathbb{C}}\,$ is the 2-form

$$2i\,(F+\omega)\ \oplus\ 2\,(F+\omega),$$

which pairs with the tangent space (2.1) by integration over $L$ to give a form of type (1,0) on (2.1).

Thus critical points of the functional are Lagrangian cycles with flat line bundles on them: exactly the basic building blocks of the objects proposed in [K] to be mirror dual to the holomorphic bundles that are the critical points of $CS$. So this ties in three well known moduli problems of virtual dimension zero (i.e. with deformation theories whose Euler characteristic vanishes) – flat bundles on 3-manifolds, holomorphic bundles on Calabi-Yau 3-folds, and Lagrangians (up to hamiltonian deformation) in symplectic 6-manifolds.

So as mirror to [T1] one would like to count Lagrangians (up to hamiltonian deformations) plus flat line bundles on them, and this is what Joyce’s work [J] has begun to tackle (in the rigid case of $L$ being a homology sphere). Mirroring precisely the behaviour of the holomorphic Casson invariant this count appears to be independent of deformations of the Kähler form and to have wall-crossing changes as the complex structure varies.