Selective Categories and Linear Canonical Relations

David Li-Bland ; Alan Weinstein

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are “good”. We then apply this notion to the category $\mathbf{SLREL}$ of linear canonical relations and the result ${\rm WW}(\mathbf{SLREL})$ of our version of the WW construction, identifying the morphisms in the latter with pairs $(L,k)$ consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in $\mathbf{SLREL}$ itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts.

Détail

Keywords: symplectic vector space; canonical relation; rigid monoidal category; highly selective category; quantization.

Bibliographie

[1] Antoine J.-P., Trapani C., Partial inner product spaces. Theory and applications, Lecture Notes in Mathematics, 1986, Springer-Verlag, Berlin, 2009 | DOI | MR | Zbl

[2] Bakalov B., Kirillov A. (Jr.), Lectures on tensor categories and modular functors, University Lecture Series, 21, Amer. Math. Soc., Providence, RI, 2001 | MR | Zbl

[3] Benenti S., “The category of symplectic reductions”, Proceedings of the International Meeting on Geometry and Physics ({F}lorence, 1982), Pitagora, Bologna, 1983, 11–41 | MR

[4] Benenti S., Tulczyjew W., “Symplectic linear relations”, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 5 (1981), 71–140 | MR

[5] Chow W.-L., “On the geometry of algebraic homogeneous spaces”, Ann. of Math., 50 (1949), 32–67 | DOI | MR | Zbl

[6] Colombeau J.-F., New generalized functions and multiplication of distributions, North-Holland Mathematics Studies, 84, North-Holland Publishing Co., Amsterdam, 1984 | MR | Zbl

[7] Guillemin V., Sternberg S., “Some problems in integral geometry and some related problems in microlocal analysis”, Amer. J. Math., 101 (1979), 915–955 | DOI | MR | Zbl

[8] Hörmander L., “Fourier integral operators, I”, Acta Math., 127 (1971), 79–183 | DOI | MR

[9] Johnson-Freyd T., Li-Bland D., Weinstein A., A note on the quantization of the linear symplectic category, from a field theoretic approach, in preparation

[10] Kashiwara M., Schapira P., Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer-Verlag, Berlin, 1990 | DOI | MR | Zbl

[11] Landsman N. P., “Quantization as a functor”, Quantization, {P}oisson Brackets and Beyond ({M}anchester, 2001), Contemp. Math., 315, Amer. Math. Soc., Providence, RI, 2002, 9–24, arXiv: math-ph/0107023 | DOI | MR | Zbl

[12] MacLane S., Categories for the working mathematician, Graduate Texts in Mathematics, 5, Springer-Verlag, New York–Berlin, 1971 | MR

[13] Maslov V. P., Theory of perturbations and asymptotic methods, Moscow State University, 1965

[14] Saavedra Rivano N., Categories Tannakiennes, Lecture Notes in Math., 265, Springer, Heidelberg, 1972 | MR | Zbl

[15] Sabot C., “Electrical networks, symplectic reductions, and application to the renormalization map of self-similar lattices”, Fractal Geometry and Applications: a Jubilee of {B}enoît {M}andelbrot, v. 1, Proc. Sympos. Pure Math., 72, Amer. Math. Soc., Providence, RI, 2004, 155–205, arXiv: math-ph/0304015 | DOI | MR | Zbl

[16] Serre J.-P., Algèbre locale. {M}ultiplicités, Lecture Notes in Mathematics, 11, Springer-Verlag, Berlin–New York, 1965 | DOI | Zbl

[17] Tulczyjew W. M., Zakrzewski S., “The category of {F}resnel kernels”, J. Geom. Phys., 1 (1984), 79–120 | DOI | MR | Zbl

[18] Wehrheim K., Woodward C. T., “Functoriality for {L}agrangian correspondences in {F}loer theory”, Quantum Topol., 1 (2010), 129–170, arXiv: 0708.2851 | DOI | MR | Zbl

[19] Weinstein A., “Symplectic geometry”, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 1–13 | DOI | MR | Zbl

[20] Weinstein A., “The symplectic “category””, Differential Geometric Methods in Mathematical Physics ({C}lausthal, 1980), Lecture Notes in Math., 905, Springer, Berlin–New York, 1982, 45–51 | DOI | MR

[21] Weinstein A., “A note on the {W}ehrheim–{W}oodward category”, J. Geom. Mech., 3 (2011), 507–515, arXiv: 1012.0105 | DOI | MR | Zbl

[22] Weinstein A., “The {M}aslov cycle as a {L}egendre singularity and projection of a wavefront set”, Bull. Braz. Math. Soc. (N.S.), 44 (2013), 593–610, arXiv: 1207.0408 | DOI | MR | Zbl

[23] Zakrzewski S., “Quantum and classical pseudogroups. {I}: {U}nion pseudogroups and their quantization”, Comm. Math. Phys., 134 (1990), 347–370 | DOI | MR | Zbl