Abstract
We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder [Fr1]. The resulting cohomology theory, which we call quasimap Floer cohomology, is an obstruction to displaceability of Lagrangians in the symplectic quotient. We use the theory to reproduce the results of Fukaya–Oh–Ohta–Ono [FuOOO3,1] and Cho–Oh [CO] on non-displaceability of moment fibers of not-necessarily-Fano toric varieties and extend their results to toric orbifolds, without using virtual fundamental chains. Finally, we describe a conjectural relationship with Floer cohomology in the quotient.
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References
M. Abouzaid, Framed bordism and Lagrangian embeddings of exotic spheres, arXiv.org:0812.4781
M. Abouzaid, A topological model for the Fukaya categories of plumbings, arXiv.org:0904.1474
P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz Morphism and Two Comparison Homomorphisms in Floer Homology, Int. Math. Res. Not. IMRN (4) (2008).
G. Alston, Lagrangian Floer homology of the Clifford torus and real projective space in odd dimensions, 2009; arXiv.org:0902.0197
G. Alston, L. Amorim, Floer cohomology of torus fibers and real Lagrangians in Fano toric manifolds, arXiv:1003.3651
D. Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, in “Geometry, Analysis, and Algebraic Geometry: Forty Years of the Journal of Differential Geometry”, Surv. Differ. Geom. 13, Int. Press, Somerville, MA (2009), 1–47.
Auroux D., Katzarkov L., Orlov D.: Mirror symmetry for weighted projective planes and their noncommutative deformations. Ann. of Math. (2) 167(3), 867–943 (2008)
P. Biran, O. Cornea, Quantum structures for Lagrangian submanifolds, arXiv:0708.4221
J.M. Boardman, R.M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces, Springer Lect. Notes in Math. 347 (1973).
L. Buhovsky, The Maslov class of Lagrangian tori and quantum products in Floer cohomology, arXiv:math/0608063
Cho C.-H.: Products of Floer cohomology of torus fibers in toric Fano manifolds. Comm. Math. Phys. 260(3), 613–640 (2005)
Cho C.-H., Oh Y.-G.: Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds. Asian J. Math. 10(4), 773–814 (2006)
Cieliebak K., Rita Gaio A., Mundeti Riera I., Salamon D.A.: The symplectic vortex equations and invariants of Hamiltonian group actions. J. Symplectic Geom. 1(3), 543–645 (2002)
Entov M., Polterovich L.: Rigid subsets of symplectic manifolds. Compos. Math. 145(3), 773–826 (2009)
Floer A.: Morse theory for Lagrangian intersections. J. Differential Geom. 28(3), 513–547 (1988)
Floer A., Hofer H., Salamon D.: Transversality in elliptic Morse theory for the symplectic action. Duke Math. J. 80(1), 251–292 (1995)
U. Frauenfelder, Floer homology of symplectic quotients and the Arnold– Givental conjecture, PhD Thesis, ETH Zurich, 2003.
Frauenfelder U.: The Arnold–Givental conjecture and moment Floer homology. Int. Math. Res. Not. 42, 2179–2269 (2004)
K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian Floer theory on compact toric manifolds II : Bulk deformations (2008), arXiv.org:0810.5654
K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction, AMS/IP Studies in Advanced Mathematics 46, American Mathematical Society, Providence, RI (2009).
Fukaya K., Oh Y.-G., Ohta H., Ono K.: Lagrangian Floer theory on compact toric manifolds. I, Duke Math. J. 151(1), 23–174 (2010)
Rita Pires Gaio A., Salamon D.A.: Gromov–Witten invariants of symplectic quotients and adiabatic limits. J. Symplectic Geom. 3(1), 55–159 (2005)
Givental A.B.: Equivariant Gromov–Witten invariants. Internat. Math. Res. Notices 13, 613–663 (1996)
E. Gonzalez, C. Woodward, Area-dependence in gauged Gromov–Witten theory, arXiv:0811.3358
E. Gonzalez, C. Woodward, Gauged Gromov–Witten theory for small spheres, arXiv:0907.3869
E. Gonzalez, C. Woodward, Deformations of symplectic vortices, Ann. of Global Anal. and Geom., to appear; arXiv:0811.3711
V.W. Guillemin, S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer-Verlag, Berlin, 1999; (with an appendix containing two reprints by H. Cartan [MR 13,107e; MR 13,107f].
K. Hori, M. Herbst, D. Page, Phases of N=2 theories in 1 + 1 dimensions with boundary, arXiv:0803.2045v1
K. Hori, C. Vafa, Mirror symmetry, hep-th/0002222
L. Hörmander, The Analysis of Linear Partial Differential Operators, III. Springer-Verlag, Berlin, 1994. (Pseudo-differential operators, corrected reprint of the 1985 original).
Katić J., Milinković D.: Coherent orientation of mixed moduli spaces in Morse-Floer theory. Bull. Braz. Math. Soc. (N.S.) 40(2), 253–300 (2009)
Kirwan F.C.: Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes 31. Princeton Univ. Press, Princeton (1984)
M. Kontsevich, Y. Soibelman, Homological mirror symmetry and torus fibrations, in “Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ (2001), 203–263.
E. Lerman, Y. Karshon, Non-compact toric manifolds, arXiv:0907.2891
Lerman E., Tolman S.: Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Amer. Math. Soc. 349(10), 4201–4230 (1997)
Lockhart R.B., McOwen R.C.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12(3), 409–447 (1985)
S. Ma’u, Gluing pseudoholomorphic quilted disks, arxiv:0909.339
Ma’u S., Woodward C.: Geometric realizations of the multiplihedron and its complexification. Compos. Math. 146, 1002–1028 (2010)
S. Mau, K. Wehrheim, C.T. Woodward, A∞-functors for Lagrangian correspondences, in preparation.
D. McDuff, Displacing Lagrangian toric fibers via probes, arXiv:0904.1686
D. McDuff, D. Salamon, J-Holomorphic Curves and Symplectic Topology, American Mathematical Society Colloquium Publications 52, American Mathematical Society, Providence, RI (2004).
Oh Y.-G.: Floer cohomology of Lagrangian intersections and pseudoholomorphic disks. I, Comm. Pure Appl. Math. 46(7), 949–993 (1993)
Oh Y.-G.: Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings. Internat. Math. Res. Notices 7, 305–346 (1996)
Pandharipande R., Solomon J., Walcher J.: Disk enumeration on the quintic 3-fold. J. Amer. Math. Soc. 21(4), 1169–1209 (2008)
S. Piunikhin, D. Salamon, M. Schwarz, Symplectic Floer–Donaldson theory and quantum cohomology, in “Contact and Symplectic Geometry (Cambridge, 1994)”, Publ. Newton Inst. 8, Cambridge Univ. Press, Cambridge (1996), 171–200.
M. Poźniak, Floer homology, Novikov rings and clean intersections, in “Northern California Symplectic Geometry Seminar”, Amer. Math. Soc. Transl. Ser. (2) 196, Amer. Math. Soc., Providence, RI (1999), 119–181.
Royden H.L.: Real Analysis. The Macmillan Co., New York (1963)
Schwarz M.: Morse Homology, Progress in Math. 111. Birkhäuser Verlag, Basel (1993)
P. Seidel, Fukaya categories and deformations, in “Proceedings of the International Congress of Mathematicians II (Beijing, 2002)”, Higher Ed. Press, Beijing (2002), 351–360.
P. Seidel, Fukaya Categories and Picard–Lefschetz Theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008.
P. Seidel, Homological mirror symmetry for the genus two curve, arXiv:0812.1171
Seidel P.: Suspending Lefschetz fibrations, with an application to mirror symmetry. Comm. Math. Phys. 297, 515–528 (2010)
Seidel P.: A∞-subalgebras and natural transformations, Homology. Homotopy Appl. 10(2), 83–114 (2008)
J. Stasheff, H-Spaces from a Homotopy Point of View, Springer Lect. Notes in Math. 161 (1970).
J. Wehrheim, Vortex invariants and toric manifolds, arXiv:0812.0299
Wehrheim K., Woodward C.: Functoriality for Lagrangian correspondences in Floer theory. Quantum Topology 1, 129–170 (2010)
K. Wehrheim, C. Woodward, Pseudoholomorphic quilts, arXiv:0905.1369
K. Wehrheim, C. Woodward, Orientations for pseudoholomorphic quilts, preprint (2009).
Witten E.: Phases of N = 2 theories in two dimensions. Nuclear Phys. B 403(1-2), 159–222 (1993)
C. Woodward, F. Ziltener, Functoriality for Gromov–Witten invariants under symplectic quotients, preprint (2008).
Ziltener F.: The invariant symplectic action and decay for vortices. J. Symplectic Geom. 7(3), 357–376 (2009)
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Partially supported by NSF grant DMS0904358
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Woodward, C.T. Gauged Floer Theory Of Toric Moment Fibers. Geom. Funct. Anal. 21, 680–749 (2011). https://doi.org/10.1007/s00039-011-0119-6
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DOI: https://doi.org/10.1007/s00039-011-0119-6