nLab
Weinstein symplectic category

Idea

Weinstein suggested that geometric quantization should yield a representation of a “category” whose

• objects are symplectic manifolds $(X,\omega )$;

• morphisms $(X_1,{\omega }_1)\to (X_2,{\omega }_2)$ are Lagrangian correspondences: Lagrangian submanifolds of $(X_1,{\omega }_1)\times (X_2,-{\omega }_2)$;

and composition is given by taking fiber products of these Lagrangian submanifolds.

However, this is not actually quite a category, since composition is only well-defined when the intersection of $L_1\times L_2\cap X_1\times \Delta (X_2)\times X_3$ is transverse.

Proposals for how to rectify this are in (Wehrheim-Woodward) and in (Kitchloo) (by turning this into an (infinity,1)-category).

Kitchloo approach

Kitchloo defines the stable symplectic category $𝕊$, which has as objects symplectic manifolds, and morphisms are certain Thom spectra associated to Lagrangian correspondences $\overline{M}\times N$, where $\overline{M}$ denotes the conjugate with symplectic form $-\omega$. One can view this as a category of symplectic motives.

Considering an oriented version of the category $𝕊$, there is a canonical fiber functor $F:M\mapsto 𝕊(\mathrm{pt},M)$, and one may consider the motivic Galois group? $G$ of monoidal automorphisms of $F$. It turns out to have a natural subgroup which is isomorphic to the quotient of the Grothendieck-Teichmüller group.

Related concepts

• Fukaya category

References

• Katrin Wehrheim and Chris Woodward, Functoriality for Lagrangian correspondences in Floer theory, arXiv:0708.2851

• Nitu Kitchloo, The Stable Symplectic Category and Geometric Quantization (arXiv:1204.5720)

• Nitu Kitchloo, Jack Morava, The Grothendieck–Teichmüller group and the stable symplectic category, 2012. (arxiv)