×

Stability conditions and the braid group. (English) Zbl 1179.53084

Summary: We find stability conditions [M. R. Douglas, in Ta Tsien Li (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. III: Invited lectures. Beijing: Higher Education Press. 395–408 (2002; Zbl 1008.81074), T. Bridgeland, Ann. Math. (2) 166, No. 2, 317–345 (2007; Zbl 1137.18008)] on some derived categories of differential graded modules over a graded algebra studied in [R. Rouquier and A. Zimmermann, Proc. Lond. Math. Soc., III. Ser. 87, No. 1, 197–225 (2003; Zbl 1058.18007), M. Khovanov and P. Seidel, J. Am. Math. Soc. 15, No. 1, 203–271 (2002; Zbl 1035.53122)]. This category arises in both derived Fukaya categories and derived categories of coherent sheaves. This gives the first examples of stability conditions on the A-model side of mirror symmetry, where the triangulated category is not naturally the derived category of an abelian category. The existence of stability conditions, however, gives many such abelian categories, as predicted by mirror symmetry.
In our examples in 2 dimensions we completely describe a connected component of the space of stability conditions as the universal cover of the configuration space \(C^0_{k+1}\) of \(k+1\) points in \(\mathbb C\) with centre of mass zero, with deck transformations the braid group action of [M. Khovanov and P. Seidel, loc. cit., P. Seidel and R. P. Thomas, Duke Math. J. 108, No. 1, 37–108 (2001; Zbl 1092.14025)]. This gives a geometric origin for these braid group actions and their faithfulness, and axiomatises the proposal for stability of Lagrangians in [R. P. Thomas, in K. Fukaya (ed.) et al., Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14–18, 2000. Singapore: World Scientific. 467–498 (2001; Zbl 1076.14525)] and the example proved by mean curvature flow in [R. P. Thomas and S.-T. Yau, Commun. Anal. Geom. 10, No. 5, 1075–1113 (2002; Zbl 1115.53054)].

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
18E30 Derived categories, triangulated categories (MSC2010)
20F36 Braid groups; Artin groups
PDFBibTeX XMLCite
Full Text: DOI arXiv