Can anyone explain what is definition of maslov index in Heegaard Floer homology? I am puzzled> Thank you.,



Maslov Index in  heegaard floer homology
Working in the Fukaya category (objects are Lagrangian submanifolds $L\_i$ of the symplectic 2n-manifold $M$ with some extra hypotheses), assume $2\cdot c\_1(M)=0$. Then we can trivialize $(\bigwedge^n\_\mathbb{C}TM)^2$ and choose a nowhere-vanishing section $\Omega\in\Gamma((\bigwedge^n\_\mathbb{C}TM)^2)$. This induces a "phase-squared map" $\phi:LGr\rightarrow S^1$, where $LGr$ is the Lagrangian Grassmannian bundle over $M$, which on Lagrangians is $\phi(L):(d\;vol\_L)^2/\Omega\rightarrow S^1$. Now $H^1(LGr)\cong\mathbb{Z}[\phi]$ so there exists a bundle $\tilde{LGr}\rightarrow LGr$ (fiber-wise universal cover). The *Maslov class* $\mu\in H^1(L)$ is the map induced by $\phi$, and $\mu=0$ iff $L$ lifts to $\tilde{LGr}$ (lift $l:L\rightarrow \tilde{LGr}$). Such a pair $(L,l)$ is a "graded Lagrangian".


For $x\in L\_0\pitchfork L\_1$ there exists a unique path $\gamma:[0,1]\rightarrow \tilde{LGr}$ (up to homotopy) connecting the lift of $x$ in $\tilde{L}\_0$ to the lift of $x$ in $\tilde{L}\_1$. Now let us work locally with $L\_i=\prod\_j\mathbb{R}$, and $\phi$ has components $\phi\_j$, and $\tilde{\phi}\_j$ is the lift of $\phi\_j$ to $\tilde{LGr}\rightarrow\mathbb{R}$. Then we have an index $i(x)=\sum\_j\ulcorner \tilde{\phi}\_j(x\in \tilde{L}\_1)-\tilde{\phi}\_j(x\in\tilde{L}\_0)\urcorner$ (ceiling notation).


The *Maslov index* of the operator $\bar{\partial}$ (defining a moduli space of J-holomorphic strips), is then $ind(\bar{\partial})=i(x)-\sum\_j i(x\_j)$, where $x$ is your ''incoming point'' on the $J$-holomorphic strip and the $x\_j$ are your ''outgoing points''.


We may then give the dimension of the moduli space $\mathcal{M}\_{d+1}$ of (d+1)-pointed disks as $ind(\bar{\partial})+d-2$.


