Gauged Hamiltonian Floer homology I: Definition of the Floer homology groups

Xu, Guangbo

doi:10.1090/tran/6643

Gauged Hamiltonian Floer homology I: Definition of the Floer homology groups
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Trans. Amer. Math. Soc. 368 (2016), 2967-3015 Request permission

Abstract:

We construct the vortex Floer homology group $VHF\left ( M, \mu ; H\right )$ for an aspherical Hamiltonian $G$-manifold $(M, \omega , \mu )$ and a class of $G$-invariant Hamiltonian loops $H_t$, following a proposal of Cieliebak, Gaio, and Salamon (2000). This is a substitute for the ordinary Hamiltonian Floer homology of the symplectic quotient of $M$. The equation for connecting orbits is a perturbed symplectic vortex equation on the cylinder $\mathbb {R} \times S^1$. We achieve the transversality of the moduli space by the classical perturbation argument instead of the virtual technique, so the homology can be defined over $\mathbb {Z}$.

References

S. Agmon and L. Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math. 20 (1967), 207–229. MR 204829, DOI 10.1002/cpa.3160200106
Paul Biran and Octav Cornea, Quantum structures for Lagrangian submanifolds, arXiv: 0708.4221, 2007.
Paul Biran and Octav Cornea, A Lagrangian quantum homology, New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, vol. 49, Amer. Math. Soc., Providence, RI, 2009, pp. 1–44. MR 2555932, DOI 10.1090/crmp/049/01
Élie Cartan, La topologie des espaces représentatifs des groupes de Lie, L’Enseignement Mathématique 35 (1936), 177–200.
K. Cieliebak, A. Floer, and H. Hofer, Symplectic homology. II. A general construction, Math. Z. 218 (1995), no. 1, 103–122. MR 1312580, DOI 10.1007/BF02571891
K. Cieliebak, A. Floer, H. Hofer, and K. Wysocki, Applications of symplectic homology. II. Stability of the action spectrum, Math. Z. 223 (1996), no. 1, 27–45. MR 1408861, DOI 10.1007/PL00004267
Kai Cieliebak, A. Rita Gaio, Ignasi Mundet i Riera, and Dietmar A. Salamon, The symplectic vortex equations and invariants of Hamiltonian group actions, J. Symplectic Geom. 1 (2002), no. 3, 543–645. MR 1959059
Kai Cieliebak, Ana Rita Gaio, and Dietmar A. Salamon, $J$-holomorphic curves, moment maps, and invariants of Hamiltonian group actions, Internat. Math. Res. Notices 16 (2000), 831–882. MR 1777853, DOI 10.1155/S1073792800000453
S. K. Donaldson, Floer homology groups in Yang-Mills theory, Cambridge Tracts in Mathematics, vol. 147, Cambridge University Press, Cambridge, 2002. With the assistance of M. Furuta and D. Kotschick. MR 1883043, DOI 10.1017/CBO9780511543098
Stamatis Dostoglou and Dietmar A. Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. (2) 139 (1994), no. 3, 581–640. MR 1283871, DOI 10.2307/2118573
Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, DOI 10.1007/978-3-0346-0425-3_{4}
Andreas Floer, An instanton-invariant for $3$-manifolds, Comm. Math. Phys. 118 (1988), no. 2, 215–240. MR 956166
Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
Andreas Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575–611. MR 987770
A. Floer and H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212 (1993), no. 1, 13–38. MR 1200162, DOI 10.1007/BF02571639
A. Floer and H. Hofer, Symplectic homology. I. Open sets in $\textbf {C}^n$, Math. Z. 215 (1994), no. 1, 37–88. MR 1254813, DOI 10.1007/BF02571699
Andreas Floer, Helmut Hofer, and Dietmar Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80 (1995), no. 1, 251–292. MR 1360618, DOI 10.1215/S0012-7094-95-08010-7
Urs Adrian Frauenfelder, Floer homology of symplectic quotients and the Arnold-Givental conjecture, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Dr.sc.math.)–Eidgenoessische Technische Hochschule Zuerich (Switzerland). MR 2715556
Urs Adrian Frauenfelder, Floer homology of symplectic quotients and the Arnold-Givental conjecture, International Mathematics Research Notices 2004 (2004), no. 42, 2179–2269.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Floer theory on compact toric manifolds. I, Duke Math. J. 151 (2010), no. 1, 23–174. MR 2573826, DOI 10.1215/00127094-2009-062
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Floer theory on compact toric manifolds II: bulk deformations, Selecta Math. (N.S.) 17 (2011), no. 3, 609–711. MR 2827178, DOI 10.1007/s00029-011-0057-z
Kenji Fukaya and Kaoru Ono, Floer homology and Gromov-Witten invariant over integer of general symplectic manifolds—summary, Taniguchi Conference on Mathematics Nara ’98, Adv. Stud. Pure Math., vol. 31, Math. Soc. Japan, Tokyo, 2001, pp. 75–91. MR 1865088, DOI 10.2969/aspm/03110075
Kenji Fukaya and Kaoru Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), no. 5, 933–1048. MR 1688434, DOI 10.1016/S0040-9383(98)00042-1
Ana Rita Pires Gaio and Dietmar A. Salamon, Gromov-Witten invariants of symplectic quotients and adiabatic limits, J. Symplectic Geom. 3 (2005), no. 1, 55–159. MR 2198773
Mark J. Gotay, On coisotropic imbeddings of presymplectic manifolds, Proc. Amer. Math. Soc. 84 (1982), no. 1, 111–114. MR 633290, DOI 10.1090/S0002-9939-1982-0633290-X
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. MR 770935
H. Hofer and D. A. Salamon, Floer homology and Novikov rings, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 483–524. MR 1362838
Michael Hutchings and Michael Sullivan, Rounding corners of polygons and the embedded contact homology of $T^3$, Geom. Topol. 10 (2006), 169–266. MR 2207793, DOI 10.2140/gt.2006.10.169
Michael Hutchings and Clifford Henry Taubes, Gluing pseudoholomorphic curves along branched covered cylinders. I, J. Symplectic Geom. 5 (2007), no. 1, 43–137. MR 2371184
Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918
Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043, DOI 10.1017/CBO9780511543111
Gang Liu and Gang Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998), no. 1, 1–74. MR 1642105
Charles-Michel Marle, Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectique, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), no. 2, 227–251 (1986) (French, with English summary). MR 859857
Dusa McDuff and Dietmar Salamon, $J$-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2004. MR 2045629, DOI 10.1090/coll/052
Ignasi Mundet i Riera, Yang-Mills-Higgs theory for symplectic fibrations, Ph.D. thesis, Universidad Autonoma de Madrid, 1999.
Ignasi Mundet i Riera, Hamiltonian Gromov-Witten invariants, Topology 42 (2003), no. 3, 525–553. MR 1953239, DOI 10.1016/S0040-9383(02)00023-X
I. Mundet i Riera and G. Tian, A compactification of the moduli space of twisted holomorphic maps, Adv. Math. 222 (2009), no. 4, 1117–1196. MR 2554933, DOI 10.1016/j.aim.2009.05.019
Kaoru Ono, On the Arnol′d conjecture for weakly monotone symplectic manifolds, Invent. Math. 119 (1995), no. 3, 519–537. MR 1317649, DOI 10.1007/BF01245191
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
S. Piunikhin, D. Salamon, and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, Contact and symplectic geometry (Cambridge, 1994) Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 171–200. MR 1432464
Joel Robbin and Dietmar Salamon, The Maslov index for paths, Topology 32 (1993), no. 4, 827–844. MR 1241874, DOI 10.1016/0040-9383(93)90052-W
Joel Robbin and Dietmar Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1–33. MR 1331677, DOI 10.1112/blms/27.1.1
Joel W. Robbin and Dietmar A. Salamon, Asymptotic behaviour of holomorphic strips, Ann. Inst. H. Poincaré C Anal. Non Linéaire 18 (2001), no. 5, 573–612 (English, with English and French summaries). MR 1849689, DOI 10.1016/S0294-1449(00)00066-4
Dietmar Salamon, Lectures on Floer homology, Symplectic geometry and topology (Park City, UT, 1997) IAS/Park City Math. Ser., vol. 7, Amer. Math. Soc., Providence, RI, 1999, pp. 143–229. MR 1702944, DOI 10.1016/S0165-2427(99)00127-0
Stephen Schecter and Guangbo Xu, Morse theory for Lagrange multipliers and adiabatic limits, J. Differential Equations 257 (2014), no. 12, 4277–4318. MR 3268727, DOI 10.1016/j.jde.2014.08.018
Edward Witten, Supersymmetry and Morse theory, J. Differential Geometry 17 (1982), no. 4, 661–692 (1983). MR 683171
Chris T. Woodward, Quantum Kirwan morphism and Gromov-Witten invariants of quotients I, Transform. Groups 20 (2015), no. 2, 507–556. MR 3348566, DOI 10.1007/s00031-015-9313-1
Christopher T. Woodward, Gauged Floer theory of toric moment fibers, Geom. Funct. Anal. 21 (2011), no. 3, 680–749. MR 2810861, DOI 10.1007/s00039-011-0119-6
Guangbo Xu, The moduli space of twisted holomorphic maps with Lagrangian boundary condition: compactness, Adv. Math. 242 (2013), 1–49. MR 3055986, DOI 10.1016/j.aim.2013.04.011
Fabian Ziltener, A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane, Mem. Amer. Math. Soc. 230 (2014), no. 1082, vi+129. MR 3221852