Special Lagrangian with phase

Let $(X,J,\omega ,\mathrm{\Omega })$ be almost-Calabi–Yau data of complex dimension $n$, let $\theta \in \mathbb{R}$, and let $i:L\hookrightarrow X$ be an oriented embedded smooth real $n$-manifold without boundary. It is special Lagrangian of phase $\theta$ when $i^{\ast }\omega =0,\mathrm{Im}(e^{-i\theta }{i}^{\ast }\mathrm{\Omega })=0,$ and $\mathrm{Re}(e^{-i\theta }{i}^{\ast }\mathrm{\Omega })$ is a positive real $n$-form for the given orientation. Positivity means its value on every positively oriented basis is positive. A submanifold of a symplectic real $2n$-manifold is Lagrangian when its tangent space at each point equals its symplectic orthogonal, equivalently when it has real dimension $n$ and the symplectic form restricts to zero. Thus the first condition and the specified dimension make $L$ Lagrangian; the second and third fix the phase and orientation.

The phase can be replaced by $\theta +2\pi r$ for $r\in \mathbb{Z}$. Compactness, normalization of $\mathrm{\Omega }$, calibration, minimality, and volume minimization are not implicit. Immersed, singular, weak, and brane-with-local-system variants require separate definitions.

Almost-Calabi–Yau data for special Lagrangians

Symplectic and almost-complex curve data