Abstract
Two natural symplectic constructions, the Lagrangian suspension and Seidel’s quantum representation of the fundamental group of the group of Hamiltonian diffeomorphisms, Ham(M), with (M, ω) a monotone symplectic manifold, admit categorifications as actions of the fundamental groupoid Π(Ham(M)) on a cobordism category recently introduced in [BC14] and, respectively, on a monotone variant of the derived Fukaya category. We show that the functor constructed in [BC14] that maps the cobordism category to the derived Fukaya category is equivariant with respect to these actions.
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References
M. Akveld and D. Salamon, Loops of Lagrangian submanifolds and pseudoholomorphic discs, Geometric and Functional Analysis 11 (2001), 609–650.
V. Arnol’d, Lagrange and Legendre cobordisms. I, Akademiya Nauk SSSR. Funktsional’nyĭ Analiz i ego Prilozheniya 14 (1980), 1–13, 96.
V. Arnol’d, Lagrange and Legendre cobordisms. II, Akademiya Nauk SSSR. Funktsional’nyĭ Analiz i ego Prilozheniya 14 (1980), 8–17, 95.
M. Audin, Quelques calculs en cobordisme lagrangien, Université de Grenoble. Annales de l’Institut Fourier 35 (1985), 159–194.
P. Biran and O. Cornea, Quantum structures for Lagrangian submanifolds, Preprint (2007). http://arxiv.org/pdf/0708.4221.
P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geometry and Topology 13 (2009), 2881–2989.
P. Biran and O. Cornea, Lagrangian cobordism I, Journal of the American Mathematical Society (2013), 295–340.
P. Biran and O. Cornea, Lagrangian cobordism and Fukaya Categories, GAFA 24 (2014), 1731–1830.
F. Charette, Quelques propriétés des sous-variétés lagrangiennes monotones: Rayon de gromov et morphisme de seidel, Ph.D. thesis, University of Montreal, May 2012.
Y. Chekanov, Lagrangian embeddings and Lagrangian cobordism, in Topics in Singularity Theory, American Mathematical Society Transactions Series 2, Vol. 180, American Mathematical Society, Providence, RI, 1997, pp. 13–23.
Y. Eliashberg, Cobordisme des solutions de relations différentielles, South Rhone seminar on geometry, I (Lyon, 1983), Travaux en Cours, Hermann, Paris, 1984, pp. 17–31.
L. Haug, On the quantum homology of real Lagrangians in Fano toric manifolds, International Mathematics Research Notices (2013), no. 14, 3171–3220.
S. Hu and F. Lalonde, A relative Seidel morphism and the Albers map, Transactions of the American Mathematical Society 362 (2010), 1135–1168.
S. Hu, F. Lalonde and R. Leclercq, Homological Lagrangian monodromy, Geometry and Topology 15 (2011), 1617–1650.
C. Hyvrier, Lagrangian circle actions, Preprint, 2013; http://arxiv.org/abs/1307.8196.
R. Leclercq, Spectral invariants in lagrangian floer theory, Journal of Modern Dynamics 2 (2008), 249–286.
S. MacLane, Categories for the Working Mathematician, Graduate texts in mathematics, Springer, Berlin, 1998.
D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, Vol. 52, AmericanMathematical Society, Providence, RI, 2004.
D. McDuff and S. Tolman, Topological properties of Hamiltonian circle actions, IMRP. International Mathematics Research Papers (2006), 72826, 1–77. MR 2210662 (2007e:53115)
L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. MR 1826128 (2002g:53157)
P. Seidel, Homological mirror symmetry for the quartic surface, Memoirs of the American Mathematical Society, vol. 1116, 2015.
P. Seidel, π 1 of symplectic automorphism groups and invertibles in quantum homology rings, Geometric and Functional Analysis 7 (1997), 1046–1095.
P. Seidel, Fukaya categories and Picard-Lefschetz Theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008.
N. Sheridan, Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space, Inventiones mathematicae 199 (2015), 1–186.
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The first author was supported by a University of Montreal End of doctoral studies grant, and TIDY and Raymond and Beverly Sackler fellowships from Tel Aviv University.
The second author was supported by an NSERC Discovery grant and a FQRNT Group Research grant.
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Charette, F., Cornea, O. Categorification of Seidel’s representation. Isr. J. Math. 211, 67–104 (2016). https://doi.org/10.1007/s11856-015-1261-x
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DOI: https://doi.org/10.1007/s11856-015-1261-x