There is another interesting framework due to Lotay and Pacini [57] [58], which makes the holomorphic curves appear on the forefront, and exhibits good analogy with the classical GIT picture, by enlarging the space of Lagrangians into the space of totally real submanifolds.

Let $(M,\omega,J)$ be a Kähler manifold. A real $n$-dimensional submanifold $L$ is called totally real, if at every point $T_{p}L$ is transverse to $JT_{p}L$. Lotay and Pacini introduce a formal principal bundle $\mathcal{P}\to\mathcal{T}$, where $\mathcal{P}$ is the space of totally real immersions $\iota:L\to X$ isotopic to a given immersion, and $\mathcal{T}$ is formally its quotient by the orientation preserving diffeomorphism group $Diff^{+}(L)$ of $L$.³⁹³⁹ 39 Lotay and Pacini did not worry about analytical issues involving the quotient; their picture is entirely formal. There is a horizontal distribution, which at each point $\iota\in\mathcal{P}$ assigns the transverse bundle $JT\iota(L)$, so gives a way to lift tangent vectors from $\mathcal{T}$ to $\mathcal{P}$. They then define a geodesic to be a curve $\iota_{t}$ in $\mathcal{P}$, such that the tangent vector field $\partial_{t}\iota_{t}$ is parallel with respect to the horizontal distribution. As a caveat, here the word ‘geodesic’ does not suggest a Riemannian metric. An alternative characterisation [57, Lem 2.2] of a geodesic, is a 1-parameter family of totally real submanifolds $\iota_{t}:L\to X$, such that there is a fixed vector field $Y\in\Gamma(L,TL)$, with

$$\frac{d}{dt}\iota_{t}=J\iota_{t*}Y,\quad[\iota_{t*}Y,J\iota_{t*}Y]=0.$$

Geometrically, one can imagine $(\iota_{t})_{0\leq t\leq 1}$ sweeps out an $(n+1)$-dimensional submanifold with boundary, foliated into complexified integral curves of $Y$, which are holomorphic curves inside $X$.

Lotay and Pacini also define the J-volume functional on $\mathcal{T}$. Pointwise on a totally real submanifold $L$, a real cotangent vector of $L$ corresponds to a $(1,0)$-form in $\Lambda^{(1,0)}X|_{L}$, so taking the $n$-th wedge power, we have a canonical isomorphism between $\Lambda^{n}L\otimes_{\mathbb{R}}\mathbb{C}$ and $K_{X}|_{L}$. Now the Kähler structure on $X$ induces a Hermitian metric on $K_{X}$, and pointwise on $L$ an element of $K_{X}$ with unit Hermitian norm uniquely specifies a volume form $dvol_{J}$ on $L$. Their $J$-volume functional is

$$\text{Vol}_{J}(L)=\int_{L}dvol_{J}.$$

It is easy to show $\text{Vol}_{J}$ provides a lower bound $\text{Vol}_{J}(L)\leq\text{Vol}(L)$ to the Riemannian volume of $L$, with equality precisely when $L$ is Lagrangian.

In case $X$ is Calabi-Yau, this construction is particularly transparent. The totally real condition is equivalent to the pointwise non-vanishing of $\Omega$ when restricted to $L$, and

$$|\int_{L}\Omega|\leq\text{Vol}_{J}(L)=\int_{L}e^{-i\theta}\Omega\leq\text{Vol}(L),$$

where the phase factor is chosen to make $e^{-i\theta}\Omega$ an orientation form on $L$. Consequently, any totally real submanifold in a Calabi-Yau manifold with $\theta=\hat{\theta}$ constant, with no need for the Lagrangian condition, is an absolute minimizer of the $\text{Vol}_{J}$ functional. This enormous space of critial points is closely related to the non-ellipticity of the critical point equation. The situation is somewhat better in a negative Kähler-Einstein ambient space, where the critical points coincide with minimal Lagrangians [57, section 5.5].

One of their main results is

There is also a formal GIT picture [57, section 6]: to some extent the space $\mathcal{P}$ can be viewed as the infinitesimal complexification of $Diff^{+}(L)$, with the space of orbits $\mathcal{T}$. The J-volume functional is formally a Kähler potential on $\mathcal{P}$, and the $Diff^{+}(L)$-moment map gives rise to critical points of the J-functional.