\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip:

example 0.0.1

Recall our running example \(\pi:\CC^2\to \CC\) from Missing Label (exm:LocalModelOfASingularity)!. We will prove that the symplectic parallel transport map preserves a class of Lagrangian submanifolds of the fiber. Consider the function \(H(z_1, z_2)= \frac{1}{2}\left(|z_1|^2-|z_2|^2\right)=\frac{1}{2}\left( x_1^2+y_1^2-x_2^2-y_2^2\right)\). The exterior derivative of this function, in local coordinates, is given by \[dH= x_1dx_1 +y_1dy_1 -x_2dx_2- y_2dy_2.\] We prove that \(H\) is invariant under the action of symplectic parallel transport along the fibration \(\pi:\CC^2\to \CC\). In this example, we can explicitly compute that \(H\) is invariant under vectors contained in \(\ker(d\pi)^{\omega_\bot}\). The kernel of \(d\pi=z_2dz_1+z_1dz_2\) at a point \((z_1, z_2)\) is the complex subspace generated by the vector \begin{align*} \ker_{(z_1, z_2)}(d\pi)=&\Span_\CC(\langle z_1, -z_2\rangle)\\ =&\Span_\RR(\langle x_1, y_1, -x_2, -y_2\rangle, \langle -y_1, x_1, y_2, -x_2\rangle ). \end{align*} In this setting, the symplectic complement is described by the orthogonal complement, and so \begin{align*} (\ker_{(z_1, z_2)}(d\pi))^{\omega\bot}=&\Span_\CC(\langle \bar z_2, \bar z_1\rangle)\\ =&\Span_\RR(\langle x_2, -y_2, x_1, -y_1\rangle, \langle y_2, x_2, y_1, x_1\rangle ). \end{align*} One then checks that \(dH\) vanishes on this by computing \(dH(v)=0\) for \(v\in (\ker_{(z_1, z_2)}(d\pi))^{\omega\bot}\) \begin{align*} \langle x_1, y_1, -x_2,- y_2\rangle\cdot \langle x_2, -y_2, x_1, -y_1\rangle=&0\\ \langle x_1, y_1, -x_2,-y_2\rangle\cdot \langle y_2, x_2, y_1, x_1 \rangle=&0 \end{align*} This means that the level sets of \(H\) are preserved under parallel transport. We use to this to describe some Lagrangian submanifolds of \(\CC^2\). If we take a level set of \(H\) and restrict to a fiber above the point \(re^{\jmath c}\), the level set \(H^{-1}(\lambda)\cap \pi^{-1}(re^{\jmath c})\) can be explicitly parameterized by \(S^1:=\theta\mapsto re^{\jmath c}\cdot(s e^{\jmath\theta}, s^{-1} e^{-\jmath\theta})\), where \(s\) is determined by \(r^2(s^2-s^{-2})=\lambda\). Simply because every curve is a Lagrangian submanifold of a \(\CC^*\), the level set of \(H\) restricted to a fiber of \(\pi\) is a Lagrangian submanifold. We can now apply Missing Label (prp:ParallelTransportOfLagrangianSubmanifold)! to obtain some new Lagrangian submanifolds of \(\CC^2\) from parallel transport of these level sets. Let \(\gamma:[0, 1]\to \CC\setminus 0\) be a closed curve, and \(\lambda\in \RR\) some value. Define the Lagrangian \(L_{\gamma, \lambda}\) to be the parallel transport of the \(\lambda\)-level set along the curve \(\gamma\). This already gives several interesting examples of Lagrangian submanifolds inside of \(\CC^2\). These Lagrangian submanifolds can also be characterized in the following way: \[L_{\gamma, \lambda}:=\{(z_1, z_2)\;|\; H(z_1, z_2)=\lambda, \pi(z_1, z_2)\in \Im(\gamma)\}.\] A good example of one of these Lagrangians is the product torus. Let \(\gamma_r=re^{\jmath\theta}\). Let \(s\) be the real value so that \(r^2(s^2-s^{-2})=\lambda\). Then the Lagrangian \(L_{\gamma_r, \lambda}\) is explicitly parameterized by: \[L_{\gamma_r, \lambda}=\{r(se^{\jmath\theta_1}, se^{\jmath\theta_2})\}\] This agrees with the definition of the product torus from (product torus).