# nLab Lagrangian correspondences and category-valued TFT

This entry describes classes of examples of A-∞ category-valued FQFTs defined on a version of the symplectic category.

# Contents

## Overview

Let $\left(X.\omega \right)$ be a compact symplectic manifold. At least in good cases to this is associated a Fukaya category $\mathrm{Fuk}\left(X\right)$ of Lagrangian submanifolds and an enlarged version ${\mathrm{Fuk}}^{#}\left(X\right)$.

Write ${X}^{-}$ for the symplectiv manifold $\left(X,-\omega \right)$.

Now if $\left({X}_{j},{\omega }_{j}\right)$ for $j=0,1$ are two Lagrangian submanifolds and ${L}_{01}\subset {X}_{0}^{-}×{X}_{1}$ a Lagrangian correspondence then we get an A-∞ functor $\varphi \left({L}_{01}\right):{\mathrm{Fuk}}^{#}\left({X}_{0}\right)\to {\mathrm{Fuk}}^{#}\left({X}_{1}\right)$

###### Theorem

(Wehrheim, Woodward)

For ${L}_{01}\subset {X}_{0}^{-}×{X}_{1}$ and ${L}_{12}\subset {X}_{1}^{-}×{X}_{2}$ Lagrangian submanifolds, assuming monotonicity and Maslov conditions we have an ${A}_{\infty }$-homotopy

$\Phi \left({L}_{01}\right)\circ \Phi \left({L}_{12}\right)\simeq \Phi \left({L}_{01}\circ {L}_{12}\right)\phantom{\rule{thinmathspace}{0ex}},$\Phi(L_{01}) \circ \Phi(L_{12}) \simeq \Phi(L_{01} \circ L_{12}) \,,

where on the right we have a natural notion of composition of Lagrangian submanifolds.

This is the symplectic version of Mukai functors?.

###### Example

For ${X}_{0}$ and ${X}_{1}$ a compact Riemann surfaces and $M\left({X}_{0}\right),M\left({X}_{1}\right)$

their moduli spaces of fixed determinant rank $n$-bundles, and for ${Y}_{01}$ a cobordism (compact, oriented) from ${X}_{0}$ to ${X}_{1}$ then consider

$L\left({Y}_{01}\right):=\mathrm{Image}\left(M\left({Y}_{01}\right)\stackrel{\mathrm{restriction}}{\to }M\left({X}_{0}{\right)}^{-}×M\left({X}_{1}\right)\right)$L(Y_{01}) := Image( M(Y_{01}) \stackrel{restriction}{\to} M(X_0)^- \times M(X_1) )

If ${Y}_{01}$ is elementary in that there exists a Morse function $Y\to ℝ$ with $\le 1$ critical points then $L\left({Y}_{01}\right)$ is a Lagrangian correspondence.

###### Corollary

The assignment

${Y}_{01}↦\Phi \left(L\left({Y}_{01}\right)\right)$Y_{01} \mapsto \Phi(L(Y_{01}))

defines a 2+1-dimensional FQFT for connected cobordisms with values in A-∞ categories.

This is supposed to be the 2+1-dimensional part of Donaldson theory.

Other theories that fit into this framework:

1. symplectic Khovanov theory? (Seidel-Smith and Rezazodegab)

2. Heegard-Floer theory?

$X$ a surface $↦$ ${\mathrm{Fuk}}^{#}\mathrm{sym}X$

elementary cobordisms $↦$ $\Phi \left($vanishing cycle$\right)$

## Lagrangian correspondences

Write ${X}^{-}j=\left({X}_{j},-{\omega }_{j}\right)$ for a symplectic manifold with its symplectic form reversed.

###### Definition

For $\left({X}_{j},{\omega }_{j}\right)$ two symplectic manifolds, a Lagrangian correspondence is a Lagrangian submanifold of ${X}_{0}^{-}×{X}_{1}$, that is

$\iota :{L}_{0,1}↪{X}_{0}^{-}×{X}_{1}$\iota : L_{0,1} \hookrightarrow X^-_0 \times X_1

with $\mathrm{dim}\left({L}_{0,1}\right)=\frac{1}{2}\left(\mathrm{dim}\left({x}_{0}\right)+\mathrm{dim}\left({X}_{1}\right)\right)$

and

${\iota }^{*}\left(-{\pi }_{0}^{*}{\omega }_{0}+{\pi }_{1}^{*}{\omega }_{1}\right)=0\phantom{\rule{thinmathspace}{0ex}},$\iota^*(-\pi_0^* \omega_0 + \pi_1^* \omega_1) = 0 \,,

where ${\pi }_{i}$ are the two projections out of the product.

The composition of two Lagrangian submanifolds is

${L}_{01}\circ {L}_{12}:={\pi }_{02}\left({L}_{01}{×}_{{X}_{1}}{L}_{12}\right)$L_{01} \circ L_{12} := \pi_{02}(L_{01} \times_{X_1} L_{12})

which is a Lagrangian correspondence in ${X}_{0}^{-}×{X}_{2}$ if everything is suitably smoothly embedded by ${\pi }_{02}$.

###### Example

For $\varphi :{X}_{0}\to {X}_{1}$ a symplectomorphism we have

$\mathrm{graph}\left(\varphi \right)\subset {X}_{0}^{-}×{X}_{1}$ is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.

###### Example

Let $X$ be a manifold, $G=U\left(n\right)$ the unitary group, $P\to X$ a $G$-principal bundle and $D\to X$ a $U\left(1\right)$-bundle with connection.

Then there is the moduli space $M\left(X\right)=M\left(P,D\right)$ of connections on $P$ with central curvature and given determinant.

For example if $X$ has genus $g$ then

$M\left(X\right)=\left\{\left(A,B,\cdots ,{A}_{g},{B}_{g}\right)\in {G}^{2g}\right\}$M(X) = \{ (A,B, \cdots, A_g, B_g) \in G^{2g}\}

such that ${\prod }_{j=1}^{g}{A}_{j}{B}_{j}{A}_{j}^{-1}{B}_{j}^{-1}=\mathrm{diag}\left({e}^{2\pi id/}\right)/G$

Let ${Y}_{01}$ be a cobordism from ${X}_{0}$ to ${X}_{1}$ with extension

$L\left({Y}_{01}\right)=\mathrm{Image}\left(M\left({Y}_{01}\right)\stackrel{\mathrm{restr}.}{\to }M\left({X}_{0}{\right)}^{-}×M\left({X}_{1}\right)\right)$L(Y_{01}) = Image(M(Y_{01}) \stackrel{restr.}{\to} M(X_0)^- \times M(X_1) )

is a Lagrangian correspondence if ${Y}_{01}$ is sufficiently simple. Further assuming this we have for composition that

$L\left({Y}_{01}\circ {Y}_{12}\right)=L\left({Y}_{01}\right)\circ L\left({Y}_{12}\right)\phantom{\rule{thinmathspace}{0ex}}.$L(Y_{01} \circ Y_{12}) = L(Y_{01}) \circ L(Y_{12}) \,.

## References

Revised on September 10, 2013 14:57:27 by Urs Schreiber (82.113.99.141)