\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
example 0.0.1
We continue our discussion of the sphere from Missing Label (exm:CotangentBundleOfASphere)!.
The Lefschetz fibration \(\pi: T^*S^2\to S^2\) has two critical values, \(\{-1, 1\}\), whose critical points corresponding to the north and south pole of the sphere. We now look at the thimbles drawn in Missing Label (fig:VanishingPathsForTheCotangentBundleOfTheSphere)!.
The first example we consider is the Lagrangian thimble constructed from \(\gamma(t)=-1-t\), the real negative ray with endpoint on the critical value of the south pole.
The symplectic parallel transport along \(\gamma(t)\) is the negative gradient flow of the imaginary coordinate of \(\pi(z_0, z_1, z_2)=z_2\) from the critical point \((0,0,1)\).
The gradient flow of the imaginary coordinate is
\begin{align*}
\grad_{T^*S^2}(\Im(z_2))=&\text{proj}_{T(T^*S^2)}\grad_{\CC^3}(\Im(z_2))\\
=&\langle 0, 0, 1 \rangle- \cdot\frac{ \langle 0, 0, 1 \rangle\cdot \langle 2z_0, 2z_1, 2z_2 \rangle }{(2z_0)^2+(2z_1)^2+(2z_2)^2}\langle 2z_0, 2z_1, 2_2\rangle\\
=& h(z_0, z_1, z_2)\langle 0, 0, 1\rangle
\end{align*}
For some function \(h(z_0, z_1, z_2)\).
The space \(\{(ix_0, ix_1, 1+x_0^2+x_1^2)\}\) is a 2-dimensional Lagrangian subspace which contains \((0,0, 1)\) and is parallel to the \(\grad_{T^*S^2}(\Im(z_2))\), and therefore the Lagrangian thimble over \(\gamma(t)\).
This also corresponds to the cotangent fiber above the south pole, \(T^*_{sp}S^2\).
In this example, we can also consider the path \(\gamma(t)=1-2t\), which starts at the critical value for the south pole and ends at the critical value of the north pole.
This is a matching path, and therefore there is a Lagrangian \(S^2_{\gamma}\subset T^*S^2\) which lives above this path.
The zero section of the sphere, given by \(\{(x_0, x_1, x_2)\;|\; x_0^2+x_1^2+x_2^2=1, x_i\in \RR\}\) is a 2 dimensional submanifold of \(T^*S^2\) which lies parallel to \(\grad_{T^*S^2}(\Im(z_2))\).
The image of the zero section under \(\pi\) is the curve \(\gamma\), and the \(S^2\) zero section clearly contains the north and south pole.
Therefore, the Lagrangian sphere associated to the mapping path \(\gamma\) is exactly the zero section.