nLab
Lagrangian correspondences and category-valued TFT

This entry describes classes of examples of A-∞ category-valued FQFTs.

Overview

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

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

Now if $(X_j,{\omega }_j)$ for $j=0,1$ are two Lagrangian submanifolds and $L_{01}\subset X_0^{-}\times X_1$ a Lagrangian correspondence then we get an A-∞ functor $\varphi (L_{01}):{\mathrm{Fuk}}^{\#}(X_0)\to {\mathrm{Fuk}}^{\#}(X_1)$

Theorem

(Wehrheim, Woodward)

For $L_{01}\subset X_0^{-}\times X_1$ and $L_{12}\subset X_1^{-}\times X_2$ Lagrangian submanifolds, assuming monotonicity and Maslov conditions we have an $A_{\mathrm{\infty }}$-homotopy

$$\Phi (L_{01})\circ \Phi (L_{12})\simeq \Phi (L_{01}\circ L_{12}),$$\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 functor?s.

Example For $X_0$ and $X_1$ a compact Riemann surfaces and $M(X_0),M(X_1)$

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

$$L(Y_{01}):=\mathrm{Image}(M(Y_{01})\stackrel{\mathrm{restriction}}{\to }M(X_0{)}^{-}\times M(X_1))$$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 $L(Y_{01})$ is a Lagrangian correspondence.

There should be a quantization of $L(Y_{01})$ that should give something like quantum Chern-Simons theory

Corollary

The assignment

$$Y_{01}\mapsto \Phi (L(Y_{01}))$$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 $\mapsto$ ${\mathrm{Fuk}}^{\#}\mathrm{sym}X$

   elementary cobordisms $\mapsto$ $\Phi ($vanishing cycle$)$

Lagrangian correspondences

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

Definition

For $(X_j,{\omega }_j)$ two symplectic manifolds, a Lagrangian correspondence is a Lagrangian submanifold of $X_0^{-}\times X_1$, that is

$$\iota :L_{0,1}\hookrightarrow X_0^{-}\times X_1$$\iota : L_{0,1} \hookrightarrow X^-_0 \times X_1

with $\dim (L_{0,1})=\frac{1}{2}(\dim (x_0)+\dim (X_1))$

and

$${\iota }^{*}(-{\pi }_0^{*}{\omega }_0+{\pi }_1^{*}{\omega }_1)=0,$$\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

$$L_{01}\circ L_{12}:={\pi }_{02}(L_{01}{\times }_{X_1}{L}_{12})$$L_{01} \circ L_{12} := \pi_{02}(L_{01} \times_{X_1} L_{12})

which is a Lagrangian correspondence in $X_0^{-}\times X_2$ if everything is suitably smoothly embedded by ${\pi }_{02}$.

Examples

1. For $\varphi :X_0\to X_1$ a symplectomorphism we have

   $\mathrm{graph}(\varphi )\subset X_0^{-}\times X_1$ is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.

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

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

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

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

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

   Let $Y_{01}$ be a cobordism from $X_0$ to $X_1$ with extension

   $$L(Y_{01})=\mathrm{Image}(M(Y_{01})\stackrel{\mathrm{restr}.}{\to }M(X_0{)}^{-}\times M(X_1))$$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

   $$L(Y_{01}\circ Y_{12})=L(Y_{01})\circ L(Y_{12}).$$L(Y_{01} \circ Y_{12}) = L(Y_{01}) \circ L(Y_{12}) \,.

References

• Katrin Wehrheim, Chris Woodward

  • Functoriality for Lagrangian correspondences in Floer theory (arXiv:0708.2851)

  • Floer Cohomology and Geometric Composition of Lagrangian Correspondences (arXiv:0905.1368)