\documentclass[11 pt]{article} \usepackage{amsmath,amsthm,amsfonts,amssymb,titlesec} \usepackage{hyperref} \usepackage{tikz} \usepackage{verbatim} \usepackage{accents} \usepackage[citestyle=alphabetic,bibstyle=alphabetic,backend=bibtex]{biblatex} \usepackage{todonotes} \usepackage[american]{babel} \usepackage{fancyhdr} \hypersetup{colorlinks=false} \usetikzlibrary{calc, decorations.pathreplacing,shapes.misc} \usetikzlibrary{decorations.pathmorphing} \usepackage[left=1in,top=1in,right=1in]{geometry} \usepackage[capitalize]{cleveref} \newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}} \newcommand{\xxx}{T base with combinatorial potential data } \newcommand{\Xxx}{T base with combinatorial potential data } \newcommand{\xxxc}{combinatorial potential stratified space } \newcommand{\Xxxc}{combinatorial potential stratified space } \newcommand{\argument}{symplectic character } \newcommand{\arguments}{symplectic characters } \newcommand{\snip}[2]{#1} 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\Crefname{maincor}{Corollary}{Corollaries} \renewcommand*{\themainthm}{\Alph{mainthm}} \makeatletter \def\namedlabel#1#2{\begingroup \def\@currentlabel{#2}% \label{#1}\endgroup } \makeatother \fancypagestyle{firstpage}{% \fancyhf{} \renewcommand\headrulewidth{0pt} \fancyfoot[R]{Original text at \texttt{ \href{http://jeffhicks.net/snippets/index.php?tag=art_dehnTwists}{snippets/art\_dehnTwists}}} } \newcommand{\wt}{\widetilde} \newcommand{\wh}{\widehat} \newcommand{\wb}{\overline} \newcommand{\bb}{\mathbb} \newcommand{\scr}{\mathscr} \newcommand{\RR}{\mathbb R} \newcommand{\ZZ}{\mathbb Z} \newcommand{\CC}{\mathbb C} \newcommand{\TT}{\mathbb T} \newcommand{\NN}{\mathbb N} \newcommand{\PP}{\mathbb P} \newcommand{\LL}{\mathbb L} \newcommand{\II}{\mathbb I} \newcommand{\CP}{\mathbb{CP}} \newcommand{\del}{\nabla} \newcommand{\pp}{\mathbf{m}} \newcommand{\into}{\hookrightarrow} \newcommand{\emprod}{m} \newcommand{\tensor}{\otimes} \renewcommand{\Re}{\text{Re}} \renewcommand{\Im}{\text{Im}} 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\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\TropB}{TropB} \DeclareMathOperator{\weight}{wt} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Fuk}{Fuk} \DeclareMathOperator{\str}{star} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Bl}{Bl} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Arg}{\mathbf{M}}\begin{filecontents}{references.bib} @article{ballard2012hochschild, title={Hochschild dimensions of tilting objects}, author={Ballard, Matthew and Favero, David}, journal={International Mathematics Research Notices}, volume={2012}, number={11}, pages={2607--2645}, year={2012}, publisher={OUP} } @article{craw2007explicit, title={Explicit methods for derived categories of sheaves}, author={Craw, Alastair}, publisher={Citeseer} } 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We now describe a symplectomorphism called \emph{symplectic Dehn twist:} \[\tau_{S^n}: Y\to Y,\] described in \cite{seidel2003long}. \section{construction of the symplectic dehn twist} \label{con:dehnTwist} Fix the standard metric $g$ on $S^n$, and let $B_r^*S^{n}$ be the radius $r$ conormal ball of $S^n$. We first describe a symplectomorphism of $B_r^*S^n$. Let $\pi: B^*_rS^n\to S^n$ be projection to the base. Consider the function \begin{align*} f: B_r^*S^{n}\to& \RR\\ (q, p) \mapsto& |p|_g^2. \end{align*} The function $f$ is a smooth map on $B_r^*S^n$, and the Hamiltonian flow of $f$ is the geodesic flow. This is a smooth function on $B^*_rS^n\setminus S^n$. On the symplectic manifold $B^*_rS^n\setminus S^n$ the time $\pi$ flow of $\sqrt{f}$ is the antipodal map on the $S^n$ base (\cref{exr:dehnTwistAsSurgery}). We take a smooth function $\rho: \RR\to \RR$ with the property that $\rho \circ f = f$ when $f< \epsilon$, $\rho\circ f=\sqrt f$ when $f>r-\epsilon$, and $\rho$ is increasing. \[ \begin{tikzpicture} \draw[scale=0.5, domain=4:9, smooth, variable=\x, blue] plot ({\x}, {sqrt(\x)}); \draw[scale=0.5, domain=0:1, smooth, variable=\x, blue] plot ({\x}, {\x*\x}); \node at (0,0){}; \draw (0.5,0.5) .. controls (0.75,1) and (1,0.75) .. (2,1); \draw[->] (0,0) -- node[below]{$r$} (5,0); \draw[->] (0,0) --node[left]{$\rho$} (0,2); \end{tikzpicture}\] Let $H= \rho \circ f: B^*_rS^n\to \RR$, and let $\phi_H: B_r^*S^n\to B_r^*S^n$ be the time-one Hamiltonian isotopy of $H$. Finally, let $-\id: S^n\to S^n$ the antipodal map, which extends to a symplectomorphism $-\id: B_r^*S^n\to B_r^*S^n$. Define $-\phi_H:=-\id\circ \phi_H$. Observe that the map $-\phi_H: S^n\to S^n$ is a symplectomorphism of $B_r^*{S^n}$, which acts by the identity in a neighborhood of $\partial B_r^*S^{n}$. It acts by the antipodal map on the zero section. \begin{definition} \label{def:dehnTwist} Given a Lagrangian sphere $S^n\subset Y$, pick $r$ small enough to identify a Weinstein neighborhood $S^n\subset B_r^*S^n\subset X$. We define symplectic Dehn twist as the symplectomorphism: \[\tau_{S^n}(x):=\left\{\begin{array}{cc} x & \text{for $x\not\in B_r^*S^n$}\\ -\phi_H(x) & \text{for $x\in B_r^*S^n$} \end{array}\right.\] \end{definition} \begin{figure} \label{fig:symplecticDehnTwist} \centering \begin{tikzpicture} \begin{scope}[] \draw (-2.5,2.5) ellipse (1 and 0.5); \begin{scope}[shift={(0,-0.007)}] \begin{scope}[] \clip (-1.45,-2.55) rectangle (-3.55,-2); \draw[] (-2.5,-2) ellipse (1 and 0.5); \end{scope} \begin{scope}[] \clip (-1.45,-1.45) rectangle (-3.55,-2); \draw[ dashed] (-2.5,-2) ellipse (1 and 0.5); \end{scope} \end{scope}\begin{scope}[shift={(0,1.5)}] \begin{scope}[] \clip (-1.45,-2.55) rectangle (-3.55,-2); \draw[blue] (-2.5,-2) ellipse (1 and 0.5); \end{scope} \begin{scope}[] \clip (-1.45,-1.45) rectangle (-3.55,-2); \draw[blue, dashed] (-2.5,-2) ellipse (1 and 0.5); \end{scope} \end{scope}\begin{scope}[shift={(0,0.5)}] \draw[red] (-2.5,-2) .. controls (-2.5,-1.5) and (-3.5,-1.5) .. (-3.5,-1); \draw[red, dashed] (-3.5,-1) .. controls (-3.5,-0.5) and (-1.5,-0.5) .. (-1.5,0); \draw[red] (-1.5,0) .. controls (-1.5,0.5) and (-2.5,0.5) .. (-2.5,1); \draw[red] (-2.5,-3) -- (-2.5,-2) (-2.5,1) -- (-2.5,1.5); \end{scope} \draw (-3.5,2.5) -- (-3.5,-2); \draw (-1.5,2.5) -- (-1.5,-2); \end{scope} \begin{scope}[shift={(-5.5,0)}] \draw (-2.5,2.5) ellipse (1 and 0.5); \begin{scope}[shift={(0,-0.007)}] \begin{scope}[] \clip (-1.45,-2.55) rectangle (-3.55,-2); \draw[] (-2.5,-2) ellipse (1 and 0.5); \end{scope} \begin{scope}[] \clip (-1.45,-1.45) rectangle (-3.55,-2); \draw[ dashed] (-2.5,-2) ellipse (1 and 0.5); \end{scope} \end{scope}\begin{scope}[shift={(0,1.5)}] \begin{scope}[] \clip (-1.45,-2.55) rectangle (-3.55,-2); \draw[blue] (-2.5,-2) ellipse (1 and 0.5); \end{scope} \begin{scope}[] \clip (-1.45,-1.45) rectangle (-3.55,-2); \draw[blue, dashed] (-2.5,-2) ellipse (1 and 0.5); \end{scope} \end{scope} \draw (-3.5,2.5) -- (-3.5,-2); \draw (-1.5,2.5) -- (-1.5,-2); \draw[red] (-2.5,2) -- (-2.5,-2.5); \end{scope} \draw[->] (-6.5,0) -- (-4,0); \node[fill=white] at (-5.5,0) {$\tau_{S^1}$}; \end{tikzpicture}\caption{Performing a Dehn twist on the zero section of \(T^*S^1\)} \end{figure}\begin{theorem} \label{thm:seidelDehnTwist} Let $X$ be a symplectic manifold, and $S\subset X$ a Lagrangian sphere, and $L\subset X$ another Lagrangian submanifold. There is an exact triangle in the Fukaya category \[ \cdots \to \CF(S, L)\otimes S \to L \xrightarrow{\ev} \tau_S(L)\xrightarrow{[1]}\cdots.\] \end{theorem} Here, $\CF(S, L)\otimes S$ is a twisted complex. Recall that (as a vector space) $\CF(S, L)=\bigoplus_{x\in S\cap L} \Lambda\langle x \rangle$. The twisted complex $\CF(S, L)\otimes S$ is given by $\bigoplus_{x\in S\cap L} S\langle x \rangle$, which is to say that formal direct sum of copies of $S$ whose grading is determined by the intersection points $x$. The differential on a twisted complex is a collection of maps $\delta_{xy}^E\in \CF(S\langle x \rangle, S\langle y \rangle)$. The morphism we take is \[\delta_{xy}^E= \langle m^1(x), y\rangle \id.\] We now describe the map $\ev: \CF(S, L)\otimes S\to L$. Recall that a morphism of twisted complexes is a collection of maps. We must pick for each $S\langle x \rangle$ a morphism in $\hom(S\langle x \rangle , L)$. Fortunately, there is a canonical choice (which is $x$ itself). \printbibliography \end{document}