nLab Weinstein symplectic category

Idea

When symplectic geometry is used to model mechanics in physics, then a symplectic manifold $(X,\omega)$ encodes the phase space of a mechanical system and a symplectomorphism

$$\phi \;\colon\; (X_1,\omega_1) \to (X_2, \omega_2)$$

encodes a process undergone by this system, for instance the time evolution induced by a Hamiltonian vector field.

However, this is too restrictive for a notion of a morphism. Indeed, even at the level of symplectic vector spaces, a symplectic morphism $\phi: (V,\omega)\to (W,\omega')$ is required to satisfy

$$\phi^*\omega'=\omega.$$

This implies that for any element $v\in\text{ker}(\phi)$, it holds that $\omega(v,w) = \omega'(f(v),f(w)) = 0$, and since $\omega$ is non-degenerate $\phi$ must be injective.

Now the graph of a symplectomorphism $\phi$ is a Lagrangian submanifold of the Cartesian product space $X_1 \times X_2$ regarded as a symplectic manifold with symplectic form $p_1^\ast \omega_1 - p_2^\ast \omega_2$. In other words, a symplectomorphism $\phi$ as above constitutes a Lagrangian correspondence between $(X_1,\omega_1)$ and $(X_2, \omega_2)$. See for instance (Cattaneo-Mnev-Reshetikhin 12) for a review.

This suggests that instead of the category whose objects are symplectic manifolds and whose morphisms are symplectomorphisms, one might consider a kind of category of correspondences whose objects are symplectic manifolds, and whose morphisms include Lagrangian correspondences, so that composition is given by forming the fiber product along adjacent legs of correspondences.

Alan Weinstein called this would-be category the symplectic category and suggested that it is the natural domain for geometric quantization.

However, take at face value, symplectic manifolds with Lagrangian correspondences between them do not quite form a category, since the usual 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).

Refinements

Prequantum correspondences

A refinement of the symplectic category to prequantum geometry is the following (see S 13).

Write $\mathbf{B}U(1)_{conn}$ for the moduli stack of smooth circle group-principal connections. Write Smooth∞Grpd for the cohesive (∞,1)-topos of smooth ∞-groupoids, and $Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}$ for the corresponding slice (∞,1)-topos. Finally write

$$Corr_1(Smooth\infty Grpd,\mathbf{B}U(1)_{conn})$$

for the (∞,1)-category of correspondences in $Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}$.

An object in here is a prequantum geometry $(X,\nabla)$ given by a map

$$\array{ X \\ \downarrow^{\mathrlap{\nabla}} \\ \mathbf{B}U(1)_{conn} } \,.$$

Under the curvature map $F_{(-)} \colon \mathbf{B}U(1)_{conn} \to \Omega^2_{cl}$ this maps to a presymplectic structure

$$(X,\omega) = (X, F_{\nabla}) \,.$$

If here $\omega$ is non-degenerate, this is a symplectic structure as in Weinstein’s symplectic category.

Moreover, a morphism $(X_1,\nabla_1) \to (X_2,\nabla_2)$ is a diagram of the form

$$\array{ && Z \\ & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}} \\ X_1 && \swArrow_{\eta} && X_2 \\ & {}_{\mathllap{\nabla_1}}\searrow && \swarrow_{\mathrlap{\nabla_2}} \\ && \mathbf{B}U(1)_{conn} } \,,$$

hence a correspondence space (smooth $\infty$-groupoid) $Z$ over $X$ and $Y$ together with an equivalence in an (∞,1)-category

$$\eta \colon i_2^\ast \nabla_2 \stackrel{\simeq}{\to} i_1^\ast \nabla_1 \,.$$

On the underlying curvatures this implies that

$$i_2^\ast \omega_2 = i_1^\ast \omega_1 \,.$$

Hence if $Z \to X \times Y$ is a maximal inclusion with this property, the above diagram is a prequantization of a morphism in the Weinstein symplectic category.

Motivic stabilization

Nitu Kitchloo defines the stable symplectic category $\mathbb{S}$, 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 $\mathbb{S}$, there is a canonical fiber functor $F : M \mapsto \mathbb{S}(pt, M)$, and one may consider the motivic Galois group $G$ of monoidal automorphisms of $F$. (Kitchloo 12, question 8.6, p. 19).

It turns out to have a natural subgroup which is isomorphic to the quotient of the Grothendieck-Teichmüller group.

Related concepts

• Fukaya category

• Lagrangian correspondences and category-valued TFT

References

The way that Lagrangian correspondences encode symplectomorphisms in symplectic geometry and hence evolution in mechanics is reviewed (and put in the broader context of BV-BRST formalism) in

• Alberto Cattaneo, Pavel Mnev, Nicolai Reshetikhin, Classical and quantum Lagrangian field theories with boundary (arXiv:1207.0239)

In his work on Fourier integral operators,

• Lars Hörmander, Fourier Integral Operators I., Acta Math. 127 (1971) 79–183.

  14

following

• Victor Maslov, Theory of Perturbations and Asymptotic Methods, (in Russian) Moskov. Gos. Univ., Moscow (1965).

observed that, under a transversality assumption, the set-theoretic composition of two Lagrangian submanifolds is again a Lagrangian submanifold, and that this composition is a “classical limit” of the composition of certain linear operators.

Shortly thereafter,

• Jędrzej Śniatycki, W.M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J. 22 (1972) 267-275

defined symplectic relations as isotropic submanifolds of products and showed that this class of relations was closed under “clean” composition. Following in part some (unpublished) ideas of Alan Weinstein,

• Victor Guillemin, Shlomo Sternberg, Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math. 101 (1979), 915–955 (JSTOR)

observed that the linear canonical relations (i.e., lagrangian subspaces of products of symplectic vector spaces) could be considered as the morphisms of a category, and they constructed a partial quantization of this category (in which lagrangian subspaces are enhanced by halfdensities.) The quantization of the linear symplectic category was part of a larger project of quantizing canonical relations (enhanced with extra structure, such as half-densities) in a functorial way, and this program was set out more formally

• Alan Weinstein, Symplectic Manifolds and Their Lagrangian Submanifolds, Advances in Math. 6 (1971) 329346 [doi:10.1016/0001-8708(71)90020-X]

• Alan Weinstein, Symplectic geometry, Bulletin Amer. Math. Soc. 5 (1981) 1-13 [doi:10.1090/S0273-0979-1981-14911-9]

See also:

• W.M.Tulczyjew, S.Zakrzewski, The category of Fresnel kernels, J. Geom. Phys. 1:3, 1984, 79–120 doi

Lecture notes reviewing these developments include

• Alan Weinstein, Symplectic Categories, proceedings of Geometry Summer School, Lisbon, July 2009 (arXiv:0911.4133)

from the introduction of which parts of the commented list of references above is taken. Further review includes

• Santiago Canez, Double Groupoids, Orbifolds, and the Symplectic Category (arXiv:1105.2592)

Further refinements in higher category theory:

• Katrin Wehrheim, Chris T. Woodward, Functoriality for Lagrangian correspondences in Floer theory, Quantum Topology 1 2 (2010) 129-170 [doi:10.4171/qt/4, 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:1212.6905]

• Damien Calaque, Three lectures on derived symplectic geometry and topological field theories, Indagationes Mathematicae 25 5 (2014) 926–947 [doi:10.1016/j.indag.2014.07.005]

• Rune Haugseng, Iterated spans and classical topological field theories, Mathematische Zeitschrift 289 3 (2018) 1427–1488 [arXiv, doi:10.1007/s00209-017-2005-x]

A closed symmetric monoidal category version of the symplectic category and the observation that this hence is a categorical semantics for quantum logic qua linear logic is in

• Sergey Slavnov, From proof-nets to bordisms: the geometric meaning of multiplicative connectives, Mathematical Structures in Computer Science 15:06 (2005) 1151–1178

• Sergey Slavnov, Geometrical semantics for linear logic (multiplicative fragment), Theoretical Computer Science 357, no. 1–3 (2006) 215–229 doi

Remarks about refinements to correspondences of smooth $\infty$-groupoids in the slice over prequantum moduli is in

• Urs Schreiber, Synthetic Quantum Field Theory, Talk at CMS Summer Meeting 2013

String diagrams for the linear and affine Weinstein category using graphical linear algebra

• Cole Comfort, Aleks Kissinger, A Graphical Calculus for Lagrangian Relations, In Proceedings ACT 2021. doi