AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Applications of Polyfold Theory I: The Polyfolds of Gromov–Witten Theory
About this Title
H. Hofer, Institute for Advanced Study, USA, K. Wysocki, Penn State University and E. Zehnder, ETH-Zurich,Switzerland
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 248, Number 1179
ISBNs: 978-1-4704-2203-5 (print); 978-1-4704-4060-2 (online)
DOI: https://doi.org/10.1090/memo/1179
Published electronically: March 20, 2017
Keywords: sc-smoothess,
polyfolds,
polyfold Fredholm sections,
GW-invariants
MSC: Primary 58B99, 58C99, 57R17
Table of Contents
Chapters
- 1. Introduction and Main Results
- 2. Recollections and Technical Results
- 3. The Polyfold Structures
- 4. The Nonlinear Cauchy-Riemann Operator
- 5. Appendices
Abstract
In this paper we start with the construction of the symplectic field theory (SFT). As a general theory of symplectic invariants, SFT has been outlined in Introduction to symplectic field theory (2000), by Y. Eliashberg, A. Givental and H. Hofer who have predicted its formal properties. The actual construction of SFT is a hard analytical problem which will be overcome be means of the polyfold theory due to the present authors. The current paper addresses a significant amount of the arising issues and the general theory will be completed in part II of this paper. To illustrate the polyfold theory we shall use the results of the present paper to describe an alternative construction of the Gromov-Witten invariants for general compact symplectic manifolds.- F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888. MR 2026549, DOI 10.2140/gt.2003.7.799
- Kai Cieliebak and Klaus Mohnke, Symplectic hypersurfaces and transversality in Gromov-Witten theory, J. Symplectic Geom. 5 (2007), no. 3, 281–356. MR 2399678
- P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240
- Simon Donaldson, Riemann surfaces, Oxford Graduate Texts in Mathematics, vol. 22, Oxford University Press, Oxford, 2011. MR 2856237
- 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}
- Halldór I. Elĭasson, Geometry of manifolds of maps, J. Differential Geometry 1 (1967), 169–194. MR 226681
- J. Fish and H. Hofer, Lectures on Polyfold Constructions in Symplectic Geometry, in preparation..
- J. Fish and H. Hofer, The Polyfolds of Symplectic Field Theory, in preparation..
- Andreas Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813. MR 948771, DOI 10.1002/cpa.3160410603
- Andreas Floer, A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 4, 393–407. MR 933228, DOI 10.1002/cpa.3160410402
- 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
- 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
- Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2553465
- Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2548482
- M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
- H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515–563. MR 1244912, DOI 10.1007/BF01232679
- H. Hofer, A general Fredholm theory and applications, Current developments in mathematics, 2004, Int. Press, Somerville, MA, 2006, pp. 1–71. MR 2459290
- H. Hofer, Polyfolds and a General Fredholm Theory, Lectures in Geometry, Oxford University Press, 2017, pp. 87-158.
- H. Hofer, Deligne–Mumford type spaces with a View Towards Symplectic Field Theory, Lecture notes in preparation..
- H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 3, 337–379 (English, with English and French summaries). MR 1395676, DOI 10.1016/s0294-1449(16)30108-1
- H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisation. IV. Asymptotics with degeneracies, Contact and symplectic geometry (Cambridge, 1994) Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 78–117. MR 1432460
- H. Hofer, K. Wysocki, and E. Zehnder, A general Fredholm theory. I. A splicing-based differential geometry, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 841–876. MR 2341834, DOI 10.4171/JEMS/99
- Helmut Hofer, Krzysztof Wysocki, and Eduard Zehnder, A general Fredholm theory. II. Implicit function theorems, Geom. Funct. Anal. 19 (2009), no. 1, 206–293. MR 2507223, DOI 10.1007/s00039-009-0715-x
- Helmut Hofer, Kris Wysocki, and Eduard Zehnder, A general Fredholm theory. III. Fredholm functors and polyfolds, Geom. Topol. 13 (2009), no. 4, 2279–2387. MR 2515707, DOI 10.2140/gt.2009.13.2279
- H. Hofer, K. Wysocki, and E. Zehnder, Integration theory on the zero sets of polyfold Fredholm sections, Math. Ann. 346 (2010), no. 1, 139–198. MR 2558891, DOI 10.1007/s00208-009-0393-x
- H. Hofer, K. Wysocki, and E. Zehnder, Integration theory on the zero sets of polyfold Fredholm sections, Erratum, in preparation..
- H. Hofer, K. Wysocki, and E. Zehnder, Finite energy cylinders of small area, Ergodic Theory Dynam. Systems 22 (2002), no. 5, 1451–1486. MR 1934144, DOI 10.1017/S0143385702001013
- Helmut Hofer, Kris Wysocki, and Eduard Zehnder, sc-smoothness, retractions and new models for smooth spaces, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 665–788. MR 2644764, DOI 10.3934/dcds.2010.28.665
- H. Hofer, K. Wysocki, and E. Zehnder, Polyfold and Fredholm Theory I: Basic Theory in M-Polyfolds, arXiv 1407.3185..
- H. Hofer, K. Wysocki, and E. Zehnder, Polyfold and Fredholm Theory, a Reference Book in preparation..
- Eugene Lerman, Orbifolds as stacks?, Enseign. Math. (2) 56 (2010), no. 3-4, 315–363. MR 2778793, DOI 10.4171/LEM/56-3-4
- Christoph Hummel, Gromov’s compactness theorem for pseudo-holomorphic curves, Progress in Mathematics, vol. 151, Birkhäuser Verlag, Basel, 1997. MR 1451624
- Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic $4$-manifolds (Irvine, CA, 1996) First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 47–83. MR 1635695
- Robert B. Lockhart and Robert C. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 409–447. MR 837256
- Dusa McDuff, Groupoids, branched manifolds and multisections, J. Symplectic Geom. 4 (2006), no. 3, 259–315. MR 2314215
- Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616
- 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
- D. McDuff and K. Wehrheim, Smooth Kuranishi structures with trivial isotropy, arXiv:1208.1340..
- Ieke Moerdijk, Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 205–222. MR 1950948, DOI 10.1090/conm/310/05405
- I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. MR 2012261
- Joel W. Robbin and Dietmar A. Salamon, A construction of the Deligne-Mumford orbifold, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 4, 611–699. MR 2262197, DOI 10.4171/JEMS/101
- Yongbin Ruan, Topological sigma model and Donaldson-type invariants in Gromov theory, Duke Math. J. 83 (1996), no. 2, 461–500. MR 1390655, DOI 10.1215/S0012-7094-96-08316-7
- Yongbin Ruan, Symplectic topology on algebraic $3$-folds, J. Differential Geom. 39 (1994), no. 1, 215–227. MR 1258920
- J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher; Notes on Mathematics and its Applications. MR 0433481
- Gang Tian, Quantum cohomology and its associativity, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 361–401. MR 1474981