On the Rozansky–Witten weight systems

Roberts, Justin¹ ; Willerton, Simon²

¹Department of Mathematics, UC San Diego, 9500 Gilman Drive, La Jolla CA 92093, United States
²Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1455-1519

Résumé

Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold X gives rise to Vassiliev weight systems. In this paper we study these weight systems by using D(X), the derived category of coherent sheaves on X. The main idea (stated here a little imprecisely) is that D(X) is the category of modules over the shifted tangent sheaf, which is a Lie algebra object in D(X); the weight systems then arise from this Lie algebra in a standard way. The other main results are a description of the symmetric algebra, universal enveloping algebra and Duflo isomorphism in this context, and the fact that a slight modification of D(X) has the structure of a braided ribbon category, which gives another way to look at the associated invariants of links. Our original motivation for this work was to try to gain insight into the Jacobi diagram algebras used in Vassiliev theory by looking at them in a new light, but there are other potential applications, in particular to the rigorous construction of the (1+1+1)–dimensional Rozansky–Witten TQFT, and to hyperkähler geometry.

Détail

DOI : 10.2140/agt.2010.10.1455

Keywords: Rozansky–Witten invariants, derived category, universal enveloping algebra, Vassiliev invariants, weight systems, Kontsevich integral

Bibliographie

[1] M F Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957) 181

[2] B Bakalov, A Kirillov Jr., Lectures on tensor categories and modular functors, 21, Amer. Math. Soc. (2001)

[3] D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423

[4] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot, Israel J. Math. 119 (2000) 217

[5] D Bar-Natan, T T Q Le, D P Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. 7 (2003) 1

[6] A Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983)

[7] N Berline, E Getzler, M Vergne, Heat kernels and Dirac operators, 298, Springer (1992)

[8] A Căldăraru, The Mukai pairing. II. The Hochschild–Kostant–Rosenberg isomorphism, Adv. Math. 194 (2005) 34

[9] V Chari, A Pressley, A guide to quantum groups, Cambridge Univ. Press (1994)

[10] P Deligne, J W Morgan, Notes on supersymmetry (following Joseph Bernstein), from: "Quantum fields and strings : a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997)" (editors P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten), Amer. Math. Soc. (1999) 41

[11] V G Drinfel′D, Quasi-Hopf algebras, Algebra i Analiz 1 (1989) 114

[12] M Duflo, Caractères des groupes et des algèbres de Lie résolubles, Ann. Sci. École Norm. Sup. (4) 3 (1970) 23

[13] S I Gelfand, Y I Manin, Homological algebra, Springer (1999)

[14] M Gerstenhaber, S D Schack, Algebras, bialgebras, quantum groups, and algebraic deformations, from: "Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990)" (editors M Gerstenhaber, J Stasheff), Contemp. Math. 134, Amer. Math. Soc. (1992) 51

[15] D Guan, Examples of compact holomorphic symplectic manifolds which are not Kählerian. II, Invent. Math. 121 (1995) 135

[16] R Hartshorne, Algebraic geometry, 52, Springer (1977)

[17] V Hinich, A Vaintrob, Cyclic operads and algebra of chord diagrams, Selecta Math. (N.S.) 8 (2002) 237

[18] N Hitchin, Hyper-Kähler manifolds, from: "Séminaire Bourbaki, Vol. 1991/92", Astérisque 206, Soc. Math. France (1992) 3, 137

[19] N Hitchin, J Sawon, Curvature and characteristic numbers of hyper-Kähler manifolds, Duke Math. J. 106 (2001) 599

[20] M Kapranov, Rozansky–Witten invariants via Atiyah classes, Compositio Math. 115 (1999) 71

[21] M Kashiwara, P Schapira, Sheaves on manifolds, 292, Springer (1994)

[22] C Kassel, Quantum groups, 155, Springer (1995)

[23] M Kontsevich, Feynman diagrams and low-dimensional topology, from: "First European Congress of Mathematics, Vol. II (Paris, 1992)", Progr. Math. 120, Birkhäuser (1994) 97

[24] M Kontsevich, Rozansky–Witten invariants via formal geometry, Compositio Math. 115 (1999) 115

[25] M Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157

[26] T T Q Le, J Murakami, Parallel version of the universal Vassiliev-Kontsevich invariant, J. Pure Appl. Algebra 121 (1997) 271

[27] T T Q Le, J Murakami, T Ohtsuki, On a universal perturbative invariant of 3–manifolds, Topology 37 (1998) 539

[28] N Markarian, Poincaré–Birkhoff–Witt isomorphism and Riemann–Roch, Preprint (2000)

[29] J Murakami, T Ohtsuki, Topological quantum field theory for the universal quantum invariant, Comm. Math. Phys. 188 (1997) 501

[30] M Nieper-Wißkirchen, Chern numbers and Rozansky–Witten invariants of compact hyper-Kähler manifolds, World Scientific (2004)

[31] K G O’Grady, Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math. 512 (1999) 49

[32] K G O’Grady, A new six-dimensional irreducible symplectic variety, J. Algebraic Geom. 12 (2003) 435

[33] A C Ramadoss, The big Chern classes and the Chern character, Internat. J. Math. 19 (2008) 699

[34] J Roberts, Rozansky–Witten theory, from: "Topology and geometry of manifolds (Athens, GA, 2001)" (editors G Matić, C McCrory), Proc. Sympos. Pure Math. 71, Amer. Math. Soc. (2003) 1

[35] J Roberts, J Sawon, Generalisations of Rozansky–Witten invariants, from: "Invariants of knots and 3–manifolds (Kyoto, 2001)" (editors T Ohtsuki, T Kohno, T Le, J Murakami, J Roberts, V Turaev), Geom. Topol. Monogr. 4, Geom. Topol. Publ. (2002) 263

[36] L Rozansky, E Witten, Hyper-Kähler geometry and invariants of three-manifolds, Selecta Math. (N.S.) 3 (1997) 401

[37] J Sawon, Rozansky–Witten invariants of hyperkähler manifolds, PhD thesis, University of Cambridge (1999)

[38] R P Thomas, Derived categories for the working mathematician, from: "Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999)" (editors C Vafa, S T Yau), AMS/IP Stud. Adv. Math. 23, Amer. Math. Soc. (2001) 349

[39] V G Turaev, Quantum invariants of knots and 3–manifolds, 18, de Gruyter (1994)

[40] A Vaintrob, Vassiliev knot invariants and Lie S–algebras, Math. Res. Lett. 1 (1994) 579

[41] P Vogel, Algebraic structures on modules of diagrams, Preprint (1995)