400Accesses
81Citations
3Altmetric
Abstract
LetX be a compact real analytic manifold, and letT*X be its cotangent bundle. In a recent paper with Zaslow (J Am Math Soc 22:233–286, 2009), we showed that the dg categoryShc(X) of constructible sheaves onX quasi-embeds into the triangulated envelopeF(T*X) of the Fukaya category ofT*X. We prove here that the quasi-embedding is in fact a quasi-equivalence. WhenX is a complex manifold, one may interpret this as a topological analogue of the identification of Lagrangian branes inT*X and regular holonomic\({{\mathcal D}_X}\) -modules developed by Kapustin (A-branes and noncommutative geometry, arXiv:hep-th/0502212) and Kapustin and Witten (Commun Number Theory Phys 1(1):1–236, 2007) from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians inT*X (with some modest homological assumptions) are equivalent in the Fukaya category to the zero section. In particular, this determines their (complex) cohomology ring and homology class inT*X, and provides a homological bound on their number of intersection points. An independent characterization of compact branes inT*X has recently been obtained by Fukaya et al. (Invent Math 172(1):1–27, 2008).
This is a preview of subscription content,log in via an institution to check access.
Access this article
Subscribe and save
- Get 10 units per month
- Download Article/Chapter or eBook
- 1 Unit = 1 Article or 1 Chapter
- Cancel anytime
Buy Now
Price includes VAT (Japan)
Instant access to the full article PDF.
Similar content being viewed by others
References
Audin, M., Lalonde, F., Polterovich, L.: Symplectic rigidity: Lagrangian submanifolds. In: Holomorphic Curves in Symplectic Geometry, Progr. Math, vol. 117, pp. 271–321. Birkhäuser, Basel (1994)
Beĭ linson, A.A.: Coherent sheaves onPn and problems in linear algebra (Russian). Funktsional. Anal. i Prilozhen.12 (1978), no. 3, 68–69; English translation: Functional Anal. Appl.12 (1978), no. 3, 214–216 (1979)
Bierstone E., Milman P.: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math.67, 5–42 (1988)
Buhovsky L.: Homology of Lagrangian submanifolds in cotangent bundles. Israel J. Math.143, 181–187 (2004)
Chen K.T.: Iterated integrals of differential forms and loop space homology. Ann. Math.97(2), 217–246 (1973)
Drinfeld V.: DG quotients of DG categories. J. Algebra272(2), 643–691 (2004)
Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory—anomaly and obstruction. Kyoto preprint Math 00-17 (2000)
Fukaya K., Oh Y.-G.: Zero-loop open strings in the cotangent bundle and Morse homotopy. Asian. J. Math.1, 96–180 (1997)
Fukaya K., Seidel P., Smith I.: Exact Lagrangian submanifolds in simply-connected cotangent bundles. Invent. Math.172(1), 1–27 (2008)
Ginsburg V.: Characteristic varieties and vanishing cycles. Invent. Math.84, 327–402 (1986)
Goresky M., MacPherson R.: Stratified Morse theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 14. Springer, Berlin (1988)
Harvey F.R., Lawson H.B. Jr: Finite volume flows and Morse theory. Ann. Math.153(1), 1–25 (2001)
Hori, K., Iqbal, A., Vafa, C.: D-branes and mirror symmetry. hep-th/0005247
Kapranov, M., Vasserot, E.: Vertex algebras and the formal loop space. Publ. Math. Inst. Hautes tudes Sci. No. 100, pp. 209–269 (2004)
Kapustin, A.: A-branes and noncommutative geometry. arXiv:hep-th/0502212
Kapustin A., Witten E.: Electric–magnetic duality and the geometric Langlands program. Commun. Number Theory Phys.1(1), 1–236 (2007)
Kashiwara M., Schapira P.: Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften, vol. 292. Springer, Berlin (1994)
Kasturirangan R., Oh Y.-G.: Floer homology of open sets and a refinement of Arnol’d’s conjecture. Math. Z.236, 151–189 (2001)
Kasturirangan, R., Oh, Y.-G.: Quantization of Eilenberg–Steenrod axioms via fary functors. RIMS preprint (1999)
Keller B.: On the cyclic homology of exact categories. J. Pure Appl. Algebra136(1), 1–56 (1999)
Keller, B.: On differential graded categories. International Congress of Mathematicians, vol. II, pp. 151–190. Eur. Math. Soc., Zürich (2006)
Khovanov, M., Rozansky, L.: Topological Landau–Ginzburg models on a world-sheet foam. arXiv:hep-th/0404189
Kontsevich, M.: Lectures at ENS, Paris, Spring 1998, notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet and H. Randriambololona
Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. Symplectic geometry and mirror symmetry (Seoul, 2000), pp. 203–263. World Sci. Publ., River Edge, NJ (2001)
Loday, J.-L.: Cyclic homology. Appendix E by María O. Ronco. Second edition. Chap. 13 by the author in collaboration with Teimuraz Pirashvili. Grundlehren der Mathematischen Wissenschaften, vol. 301. Springer, Berlin (1998)
Lalonde F., Sikorav J.-C.: Sous-variétés lagrangiennes et lagrangiennes exactes des fibrs cotangents. Comment. Math. Helv.66(1), 18–33 (1991)
Mau, S., Wehrheim, K., Woodward, C.T.:A∞-functors for Lagrangian correspondences (work in progress)
Nadler D., Zaslow E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc.22, 233–286 (2009)
Schmid W., Vilonen K.: Characteristic cycles of constructible sheaves. Invent. Math.124, 451–502 (1996)
Seidel, P.: Vanishing cycles and mutation. European Congress of Mathematics, vol. II (Barcelona, 2000), pp. 65–85, Progr. Math., vol. 202. Birkhauser, Basel (2001)
Seidel, P.: More about vanishing cycles and mutation. Symplectic geometry and mirror symmetry (Seoul, 2000), pp. 429–465. World Sci. Publ., River Edge, NJ (2001)
Seidel, P.: Exact Lagrangian submanifolds inT*Sn and the graded Kronecker quiver. Different faces of geometry, pp. 349–364. Int. Math. Ser. (NY), vol. 3. Kluwer/Plenum, New York (2004)
Seidel, P.: Fukaya categories and Picard-Lefschetz Theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008)
Sikorav, J.-C.: Some properties of holomorphic curves in almost complex manifolds. In: Holomorphic Curves in Symplectic Geometry, pp. 165–189. Birkhäuser, Basel (1994)
Smith, I.: Exact Lagrangian submanifolds revisited. Slides from talk at conference in honor of Dusa McDuff’s birthday, Stony Brook, October (2006)
van den Dries L., Miller C.: Geometric categories and o-minimal structures. Duke Math. J.84(2), 497–539 (1996)
Viterbo, C.: Generating functions, symplectic geometry, and applications. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2 (Zürich, 1994), pp. 537–547. Birkhäuser, Basel (1995)
Viterbo C.: Exact Lagrange submanifolds, periodic orbits and the cohomology of free loop spaces. J. Differ. Geom.47(3), 420–468 (1997)
Wehrheim, K., Woodward, C.T.: Functoriality for Lagrangian correspondences in Floer theory (2006, preprint)
Wehrheim, K., Woodward, C.T.: Orientations for pseudoholomorphic quilts (in preparation)
Author information
Authors and Affiliations
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA
David Nadler
- David Nadler
You can also search for this author inPubMed Google Scholar
Corresponding author
Correspondence toDavid Nadler.
Additional information
Dedicated to Paul S. Nadler
Rights and permissions
About this article
Cite this article
Nadler, D. Microlocal branes are constructible sheaves.Sel. Math. New Ser.15, 563–619 (2009). https://doi.org/10.1007/s00029-009-0008-0
Published:
Issue Date:
Share this article
Anyone you share the following link with will be able to read this content:
Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative