Just as explicit solutions of the Einstein and Hermitian-Yang-Mills equations exist only on spaces that are either low dimensional, non-compact and/or highly symmetric, so the equations for special Lagrangian (SLag) cycles, also important in physics, have the same properties. Physically there are also similarities in that we have two first order supersymmetric minimal energy equations (HYMs and SLag) implying the more standard second order equations (YMs and minimal volume equations).

There are powerful existence results of Calabi and Yau (and more recently Tian, Donaldson and others) for the Einstein equations, and of Donaldson-Uhlenbeck-Yau for the HYM equations, so long as we are on a Kähler (or projective) manifold; this often reduces an infinite dimensional problem in PDEs to a finite dimensional problem in linear algebra. Producing many Kähler-Einstein (e.g. Calabi-Yau) manifolds becomes trivial, and dealing with Hermitian-Yang-Mills connections requires only algebraic computations; in both cases the complicated role of the Kähler form and/or metric is almost removed. This can be thought of as possible because of the existence of some infinite dimensional geometry recasting the equations in terms of moment maps and symplectic reduction. A similar situation for SLags would therefore be highly desirable. In particular it might give a way of studying SLags using only Lagrangians and symplectic geometry, much as HYM connections are studied via stable bundles and algebraic geometry.

This paper explores the mirror symmetry of holomorphic bundles (on a Calabi-Yau $3$-fold $M$, often referred to here as ‘the complex side’) and Lagrangians (on the mirror Calabi-Yau 3-fold $W$, ’the symplectic side’, known as the Kähler side in the physics literature). Many people have worked and are still working on proving some kind of direct correspondence between such objects given an SYZ torus fibration [SYZ]; see for example [AP], [BBHM], [Ch], [Fu1], [Gr], [LYZ], [PZ], [Ty], and see [MMM] for a review of this and many many more issues in mirror symmetry. Here, however, we work purely formally without reference to a particular pair of mirror manifolds, without worrying about what mirror symmetry might rigorously mean, and we will not try to give any explicit correspondence. Using mirror symmetry merely as motivation, we point out some similar structures on both sides of the mirror map. Under some conditions (in some ‘large complex structure’ or ‘semi-classical’ or somesuch limit) these structures might be genuinely dual; again it does not matter if they are not in general. For instance, physics [MMMS], [DFR] predicts that one should consider not the HYM equations and slope but some perturbation of them away from the large complex structure limit; however these equations also come from a moment map and, conjecturally, a stability condition (for a discussion of such matters see [Le] or [T3]). So while the slope and phase of Lagrangians discussed below might not be exactly mirror to slope of bundles, it should be mirror to something with analogous properties and significance.

Loosely, we would like to think of submanifolds in a fixed homology class as mirror to connections on a fixed topological complex bundle (with Chern classes mirror to the homology class); then Lagrangians should correspond to holomorphic connections (i.e. integrable connections; those with no $(0,2)$-curvature) and special Lagrangians to those with HYM curvature. These last two conditions should be stability conditions for the group actions of hamiltonian deformations and complex gauge transformations, respectively. The full picture is much more complicated, involving triangulated categories and so forth, as envisaged some six years ago in the seminal conjecture of Kontsevich [K]; we can ignore this in only using mirror symmetry as motivation. It could be noted, however, that the functionals defined below are additive under exact sequences of holomorphic vector bundles and sums of Lagrangians, so should extend to the derived category of coherent sheaves and the derived Fukaya category of Lagrangians respectively.

First note that while the connections side has a complex structure and a complex gauge group involved, the Lagrangian side needs complexifying. So motivated by Kontsevich [K] and by physics (e.g. [SYZ]) we add in connections on the submanifolds (which will later reduce to flat connections on Lagrangians). The dictionary we are aiming towards, much of which is already standard, is the following in the 3-dimensional case; all the terms used will be defined in due course.

$$\begin{array}[]{|c|c|}\hline\cr\text{Complex side }M&\text{Symplectic side }W\\ \hline\cr\hline\cr\Omega=\Omega_{M}\in H^{3,0}&\omega=\omega_{W}\in H^{1,1}\\ \hline\cr H^{\mathrm{ev}}&H^{3}\\ \hline\cr\mathrm{Connections\ }A\mathrm{\ on\ a\ fixed\ }C^{\infty}\mathrm{\ complex}&\mathrm{Submanifolds/cycles\ }L\mathrm{\ in\ a\ fixed\ class}\\ \mathrm{bundle\ }E;\ \,v(E)=ch(E)\sqrt{\mathrm{Td\,}X}\in H^{\mathrm{ev}}&[L]\in H^{3},\mathrm{\ with\ a\ connection\ on\ }\mathbb{C}\,\times L\\ \hline\cr CS_{\mathbb{C}}\,(A=A_{0}+a)=&f_{\mathbb{C}}\,(A,L)=\int_{L_{0}}^{L}(F+\omega)^{2}\\ \hskip 17.07164pt\frac{1}{4\pi^{2}}\int_{M}\mathrm{tr}\left(\bar{\partial}_{A_{0}}a\wedge a+\frac{2}{3}a^{\wedge 3}\right)\wedge\Omega&\hskip 28.45274pt=\int_{L_{0}}^{L}(F^{2}+\omega^{2})+2\int_{L_{0}}^{L}\omega\wedge F\\ \hline\cr\mathrm{Critical\ points:\ }F_{A}^{0,2}=0&\mathrm{Critical\ points:\ }\omega\arrowvert_{L}=0,\ F_{A}=0\\ \mathrm{holomorphic\ bundles\ }&\mathrm{Lagrangians\ +\ flat\ line\ bundles\ }\\ \hline\cr\text{Holomorphic Casson invariant \cite[cite]{[\@@bibref{}{T1}{}{}]}}&\text{Counting SLags \cite[cite]{[\@@bibref{}{J}{}{}]}}\\ \hline\cr\mathrm{Gauge\ group}&U(1)\mathrm{\ gauge\ group\ on\ }L\\ \hline\cr\mathrm{Complexified\ gauge\ group}&\mathrm{Hamiltonian\ deformations}\\ \hline\cr\omega=\omega^{\ }_{M}\in H^{1,1}&\Omega=\Omega^{\ }_{W}\in H^{3,0}\\ \hline\cr\mathrm{Moment\ map\ \ }F_{A}\wedge\omega^{n-1}&\mathrm{Moment\ map\ \,}\,\mathrm{Im}\,\Omega\arrowvert_{L}\\ \hline\cr\mathrm{Stability,\ slope\ }\mu=\frac{1}{\mathrm{rk\,}E}\int\mathrm{tr}\,F_{A}\wedge\omega^{n-1}&\mathrm{Stability,\ slope\ }\mu=\frac{1}{\mathrm{vol\,}(L)}\int_{L}\,\mathrm{Im}\,\Omega\\ \hline\cr\end{array}$$

(In the fourth line, $v(E)$ is the Mukai vector of $E$; in the last line, vol$\,(L)$ is the cohomological volume measured with respect to Re$\,\Omega$.) A SLag cycle (of phase $\phi$) in a Calabi-Yau is a Lagrangian with Im $(e^{-i\phi}\Omega)\arrowvert_{L}\equiv 0$ [HL]; then Re $(e^{-i\phi}\Omega)\arrowvert_{L}$ is the Riemannian volume form on $L$ induced by the Ricci-flat metric on the Calabi-Yau. Obviously, rotating $\Omega$ by $e^{-i\phi}$ gives SLags in the more traditional sense of phase zero. The part of the theory to do with SLags will apply in all dimensions; it is only the functionals that are special to Calabi-Yau 3-folds.

We will partially justify the above table, though the symplectic structure and moment map give problems that will appear in due course. However we can derive enough to arrive at a conjecture about Lagrangians and SLags for which evidence will be given by using monodromy and mirror symmetry from [ST] to interpret an example of Lawlor and Joyce [J].