×

Microlocal branes are constructible sheaves. (English) Zbl 1197.53116

The main result of this paper is beautifully summarized in its title: the microlocalization functor is an \(A_\infty\)-quasi-equivalence between the differential graded category \(Sh_c(X)\) of constructible sheaves on a compact real analytic manifold \(X\) and the triangulated envelope \(F(T^*X)\) of the Fukaya category of \(T^*X\).
The microlocalization functor \(\mu_X:Sh_c(X)\to F(T^*X)\) was introduced in a joint work of the author with E. Zaslow [J. Am. Math. Soc. 22, 233–286 (2009)], where it was also proved that \(\mu_X\) is an \(A_\infty\)-quasi-embedding, i.e., that the induced functor in cohomology is a fully faithful embedding of the bounded derived category of constructible complexes of sheaves on \(X\) into the derived Fukaya category. In the present paper, this result is refined showing that \(H(\mu_X)\) is actually an equivalence of triangulated categories. This can be viewed as a categorification of the fact that the characteristic cycle homomorphism is an isomorphism from constructible functions to conical Lagrangian cycles. Moreover, when \(X\) is a complex manifold, this result can also be interpreted as a topological analogue of the identification of Lagrangian branes in \(T^*X\) and regular holonomic \(\mathcal{D}_X\)-modules developed by A. Kapustin and E. Witten [Commun. Number Theory Phys. 1, No. 1 (2007; Zbl 1128.22013); arXiv:hep-th/0502212].
As a concrete geometrical application, the author shows that under mild homological assumptions, compact connected Lagrangians \(L\subset T^*X\) which are exact and have trivial Maslov class are equivalent in the Fukaya category to a shift of the zero section of \(T^*X\). In particular, this implies that \([L]=\pm[X]\) as homology classes in \(H_{\dim X}(T^*X,\mathbb{C})\), the existence of a ring isomorphism \(H^*(L,\mathbb{C})\simeq H^*(X,\mathbb{C})\) and the following homological lower bound for the (possibly infinite) number of intersection points of two such Lagrangians: \(\#(L\cap L')\geq \sum_k\dim H^k(X,\mathbb{C})\).
An independent characterization of compact branes in \(T^*X\) has been obtained in [K.  Fukaya, P. Seidel and I. Smith, Invent. Math. 172, No. 1, 1–27 (2008; Zbl 1140.53036)].

MSC:

53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
53D40 Symplectic aspects of Floer homology and cohomology
57R56 Topological quantum field theories (aspects of differential topology)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 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)
[2] Beĭ linson, A.A.: Coherent sheaves on P n 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)
[3] Bierstone E., Milman P.: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. 67, 5–42 (1988) · Zbl 0674.32002 · doi:10.1007/BF02699126
[4] Buhovsky L.: Homology of Lagrangian submanifolds in cotangent bundles. Israel J. Math. 143, 181–187 (2004) · Zbl 1090.53063 · doi:10.1007/BF02803498
[5] Chen K.T.: Iterated integrals of differential forms and loop space homology. Ann. Math. 97(2), 217–246 (1973) · Zbl 0227.58003 · doi:10.2307/1970846
[6] Drinfeld V.: DG quotients of DG categories. J. Algebra 272(2), 643–691 (2004) · Zbl 1064.18009 · doi:10.1016/j.jalgebra.2003.05.001
[7] Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory–anomaly and obstruction. Kyoto preprint Math 00-17 (2000) · Zbl 1181.53003
[8] Fukaya K., Oh Y.-G.: Zero-loop open strings in the cotangent bundle and Morse homotopy. Asian. J. Math. 1, 96–180 (1997) · Zbl 0938.32009
[9] Fukaya K., Seidel P., Smith I.: Exact Lagrangian submanifolds in simply-connected cotangent bundles. Invent. Math. 172(1), 1–27 (2008) · Zbl 1140.53036 · doi:10.1007/s00222-007-0092-8
[10] Ginsburg V.: Characteristic varieties and vanishing cycles. Invent. Math. 84, 327–402 (1986) · Zbl 0598.32013 · doi:10.1007/BF01388811
[11] Goresky M., MacPherson R.: Stratified Morse theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 14. Springer, Berlin (1988)
[12] Harvey F.R., Lawson H.B. Jr: Finite volume flows and Morse theory. Ann. Math. 153(1), 1–25 (2001) · Zbl 1001.58005 · doi:10.2307/2661371
[13] Hori, K., Iqbal, A., Vafa, C.: D-branes and mirror symmetry. hep-th/0005247
[14] Kapranov, M., Vasserot, E.: Vertex algebras and the formal loop space. Publ. Math. Inst. Hautes tudes Sci. No. 100, pp. 209–269 (2004) · Zbl 1106.17038
[15] Kapustin, A.: A-branes and noncommutative geometry. arXiv:hep-th/0502212 · Zbl 1156.14319
[16] Kapustin A., Witten E.: Electric–magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(1), 1–236 (2007) · Zbl 1128.22013
[17] Kashiwara M., Schapira P.: Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften, vol. 292. Springer, Berlin (1994) · Zbl 0709.18001
[18] Kasturirangan R., Oh Y.-G.: Floer homology of open sets and a refinement of Arnol’d’s conjecture. Math. Z. 236, 151–189 (2001) · Zbl 0985.53039 · doi:10.1007/PL00004822
[19] Kasturirangan, R., Oh, Y.-G.: Quantization of Eilenberg–Steenrod axioms via fary functors. RIMS preprint (1999)
[20] Keller B.: On the cyclic homology of exact categories. J. Pure Appl. Algebra 136(1), 1–56 (1999) · Zbl 0923.19004 · doi:10.1016/S0022-4049(97)00152-7
[21] Keller, B.: On differential graded categories. International Congress of Mathematicians, vol. II, pp. 151–190. Eur. Math. Soc., Zürich (2006) · Zbl 1140.18008
[22] Khovanov, M., Rozansky, L.: Topological Landau–Ginzburg models on a world-sheet foam. arXiv:hep-th/0404189 · Zbl 1137.81041
[23] Kontsevich, M.: Lectures at ENS, Paris, Spring 1998, notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet and H. Randriambololona
[24] 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) · Zbl 1072.14046
[25] 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)
[26] Lalonde F., Sikorav J.-C.: Sous-variétés lagrangiennes et lagrangiennes exactes des fibrs cotangents. Comment. Math. Helv. 66(1), 18–33 (1991) · Zbl 0759.53022 · doi:10.1007/BF02566634
[27] Mau, S., Wehrheim, K., Woodward, C.T.: A functors for Lagrangian correspondences (work in progress)
[28] Nadler D., Zaslow E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22, 233–286 (2009) · Zbl 1227.32019 · doi:10.1090/S0894-0347-08-00612-7
[29] Schmid W., Vilonen K.: Characteristic cycles of constructible sheaves. Invent. Math. 124, 451–502 (1996) · Zbl 0851.32011 · doi:10.1007/s002220050060
[30] Seidel, P.: Vanishing cycles and mutation. European Congress of Mathematics, vol. II (Barcelona, 2000), pp. 65–85, Progr. Math., vol. 202. Birkhauser, Basel (2001) · Zbl 1042.53060
[31] 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) · Zbl 1079.14529
[32] Seidel, P.: Exact Lagrangian submanifolds in T* S n and the graded Kronecker quiver. Different faces of geometry, pp. 349–364. Int. Math. Ser. (NY), vol. 3. Kluwer/Plenum, New York (2004) · Zbl 1070.53049
[33] Seidel, P.: Fukaya categories and Picard-Lefschetz Theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008) · Zbl 1159.53001
[34] Sikorav, J.-C.: Some properties of holomorphic curves in almost complex manifolds. In: Holomorphic Curves in Symplectic Geometry, pp. 165–189. Birkhäuser, Basel (1994)
[35] Smith, I.: Exact Lagrangian submanifolds revisited. Slides from talk at conference in honor of Dusa McDuff’s birthday, Stony Brook, October (2006)
[36] van den Dries L., Miller C.: Geometric categories and o-minimal structures. Duke Math. J. 84(2), 497–539 (1996) · Zbl 0889.03025 · doi:10.1215/S0012-7094-96-08416-1
[37] 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) · Zbl 0838.53030
[38] Viterbo C.: Exact Lagrange submanifolds, periodic orbits and the cohomology of free loop spaces. J. Differ. Geom. 47(3), 420–468 (1997) · Zbl 0946.37017
[39] Wehrheim, K., Woodward, C.T.: Functoriality for Lagrangian correspondences in Floer theory (2006, preprint) · Zbl 1206.53088
[40] Wehrheim, K., Woodward, C.T.: Orientations for pseudoholomorphic quilts (in preparation) · Zbl 1339.53084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.