A Calabi–Yau $m$-fold is a quadruple $(M,J,g,\Omega)$ such that $(M,J)$ is an $m$-dimensional complex manifold, $g$ is a Kähler metric on $(M,J)$ with Kähler form $\omega$, and $\Omega$ is a holomorphic $(m,0)$-form on $(M,J)$ satisfying

Then $g$ is Ricci-flat and its holonomy group is a subgroup of $\mathop{\rm SU}(m)$. We do not require $M$ to be compact, or $g$ to have holonomy $\mathop{\rm SU}(m)$, although many authors make these restrictions.

If $(M,J,g,\Omega)$ is a Calabi–Yau $m$-fold with Kähler form $\omega$, then $(M,\omega)$ is a symplectic manifold. A Lagrangian $L$ in $M$ is a real $m$-dimensional submanifold (embedded or immersed) with $\omega|_{L}=0$.

Let $L$ be a Lagrangian in $M$. Then $\Omega|_{L}$ is a complex $m$-form on $L$. Equation (2.1) implies that $\big|\Omega|_{L}\big|=1$, where $|\,.\,|$ is computed using the Riemannian metric $g|_{L}$. Suppose $L$ is oriented. Then we have a volume form ${\rm d}V_{L}$ on $L$ defined using the metric $g|_{L}$ and orientation with $|{\rm d}V_{L}|=1$, so $\Omega|_{L}=\Theta_{L}\cdot{\rm d}V_{L}$, where $\Theta_{L}:L\rightarrow{\rm U}(1)$ is a unique smooth function, and ${\rm U}(1)=\{z\in{\mathbin{\mathbb{C}}}:|z|=1\}$.

There is an induced morphism of cohomology groups $\Theta_{L}^{*}:H^{1}({\rm U}(1),{\mathbin{\mathbb{Z}}})\rightarrow H^{1}(L,{\mathbin{\mathbb{Z}}})$. The Maslov class $\mu_{L}\in H^{1}(L;{\mathbin{\mathbb{Z}}})$ of $L$ is the image under $\Theta_{L}^{*}$ of the generator of $H^{1}({\rm U}(1),{\mathbin{\mathbb{Z}}})\cong{\mathbin{\mathbb{Z}}}$. If $H^{1}(M,{\mathbin{\mathbb{R}}})=0$ then $\mu_{L}$ depends only on $(M,\omega),L$ and not on $g,J,\Omega$. We call $L$ Maslov zero if $\mu_{L}=0$.

A grading or phase function of an oriented Lagrangian $L$ is a smooth function $\theta_{L}:L\rightarrow{\mathbin{\mathbb{R}}}$ with $\Theta_{L}=\exp(i\theta_{L})$, so that $\Omega|_{L}=e^{i\theta_{L}}{\rm d}V_{L}$. That is, $i\theta_{L}$ is a continuous choice of logarithm for $\Theta_{L}$. Gradings exist if and only if $L$ is Maslov zero. If $L$ is connected then gradings are unique up to addition of $2\pi n$ for $n\in{\mathbin{\mathbb{Z}}}$. A graded Lagrangian $(L,\theta_{L})$ in $M$ is an oriented Lagrangian $L$ with a grading $\theta_{L}$. Usually we refer to $L$ as the graded Lagrangian, leaving $\theta_{L}$ implicit.

An oriented Lagrangian $L$ in $M$ is called almost calibrated if $(\cos\phi\,\mathop{\rm Re}\Omega-\sin\phi\,\mathop{\rm Im}\Omega)|_{L}$ is a positive $m$-form on $L$ for some $\phi\in{\mathbin{\mathbb{R}}}$. Then $L$ admits a unique grading $\theta_{L}$ taking values in $(\phi-\frac{\pi}{2},\phi+\frac{\pi}{2})$. If a graded Lagrangian $L$ has phase variation less than $\pi$, then it is almost calibrated.

An oriented Lagrangian $L$ in $M$ is called special Lagrangian with phase $e^{i\phi}$ if $\Theta_{L}$ is constant with value $e^{i\phi}\in{\rm U}(1)$. If we do not specify a phase, we usually mean phase 1. We will write SL for special Lagrangian, and SL $m$-fold for special Lagrangian submanifold. SL $m$-folds with phase $e^{i\phi}$ are Maslov zero, and graded with phase function $\theta_{L}=\phi$. They are minimal submanifolds in $(M,g)$. Compact SL $m$-folds are volume-minimizing in their homology class.