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

Write X X^- for the symplectiv manifold (X,ω)(X,-\omega).

Now if (X j,ω j)(X_j, \omega_j) for j=0,1j = 0,1 are two Lagrangian submanifolds and L 01X 0 ×X 1L_{01} \subset X^-_0 \times X_1 a Lagrangian correspondence then we get an A-∞ functor ϕ(L 01):Fuk #(X 0)Fuk #(X 1)\phi(L_{01}) : Fuk^#(X_0) \to Fuk^#(X_1)

Theorem

(Wehrheim, Woodward)

For L 01X 0 ×X 1L_{01} \subset X^{-}_0 \times X_1 and L 12X 1 ×X 2L_{12} \subset X^{-}_1 \times X_2 Lagrangian submanifolds, assuming monotonicity and Maslov conditions we have an A A_\infty-homotopy

Φ(L 01)Φ(L 12)Φ(L 01L 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.

(Wehrheim-Woodward 07, section 3.2)

This is the symplectic version of Mukai functors?.

Example

For X 0X_0 and X 1X_1 a compact Riemann surfaces and M(X 0),M(X 1)M(X_0), M(X_1)

their moduli spaces of fixed determinant rank nn-bundles, and for Y 01Y_{01} a cobordism (compact, oriented) from X 0X_0 to X 1X_1 then consider

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

If Y 01Y_{01} is elementary in that there exists a Morse function YY \to \mathbb{R} with 1\leq 1 critical points then L(Y 01)L(Y_{01}) is a Lagrangian correspondence.

Corollary

The assignment

Y 01Φ(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?

    XX a surface \mapsto Fuk #symXFuk^# sym X

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

Lagrangian correspondences

Write X j=(X j,ω j)X^-j = (X_j , -\omega_j) for a symplectic manifold with its symplectic form reversed.

Definition

For (X j,ω j)(X_j, \omega_j) two symplectic manifolds, a Lagrangian correspondence is a Lagrangian submanifold of X 0 ×X 1X^-_0 \times X_1, that is

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

with dim(L 0,1)=12(dim(x 0)+dim(X 1))dim(L_{0,1}) = \frac{1}{2}(dim(x_0) + dim(X_1))

and

ι *(π 0 *ω 0+π 1 *ω 1)=0, \iota^*(-\pi_0^* \omega_0 + \pi_1^* \omega_1) = 0 \,,

where π i\pi_i are the two projections out of the product.

The composition of two Lagrangian submanifolds is

L 01L 12:=π 02(L 01× X 1L 12) L_{01} \circ L_{12} := \pi_{02}(L_{01} \times_{X_1} L_{12})

which is a Lagrangian correspondence in X 0 ×X 2X^-_0 \times X_2 if everything is suitably smoothly embedded by π 02\pi_{02}.

Example

For ϕ:X 0X 1\phi : X_0 \to X_1 a symplectomorphism we have

graph(ϕ)X 0 ×X 1graph(\phi) \subset X_0^- \times X_1 is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.

Example

Let XX be a manifold, G=U(n)G= U(n) the unitary group, PXP \to X a GG-principal bundle and DXD \to X a U(1)U(1)-bundle with connection.

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

For example if XX has genus gg then

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

such that j=1 gA jB jA j 1B j 1=diag(e 2πid/)/G\prod_{j=1}^g A_j B_j A_j^{-1} B_j^{-1} = diag(e^{2\pi i d/})/G

Let Y 01Y_{01} be a cobordism from X 0X_0 to X 1X_1 with extension

L(Y 01)=Image(M(Y 01)restr.M(X 0) ×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 01Y_{01} is sufficiently simple. Further assuming this we have for composition that

L(Y 01Y 12)=L(Y 01)L(Y 12). 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)