Stability conditions on triangulated categories

By T. Bridgeland. 2002

1,023 citations recorded by the source corpus.

This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability conditions $\Stab(\T)$ on a fixed category $\T$ has a natural topology, thus defining a new invariant of triangulated categories. After setting up the necessary definitions I prove a deformation result which shows that the space $\Stab(\T)$ with its natural topology is a manifold, possibly infinite-dimensional.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/1d9eb70fadd0097c5025078c0a530a1d0493f685

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Citations in the corpus

Stability Manifold of P^1 (2004, background): “ar X iv :m at h/ 04 11 22 0v 3 [ m at h.”
Derived automorphism groups of K3 surfaces of Picard rank $1$ (2013, background): “he stability conditions ˙ i, the arguments of the complex numbers Z i(E) dier by at most ˇ. (c)The forgetful map Stab(X) !N(X) C sending a stability condition to the vector is a local homeomorphism [Bri07]. DERIVED AUTOMORPHISM GROUPS OF K3 SURFACES OF PICARD RANK 1 8 Let GL+ 2 (R) be the group of o”
Inducing stability conditions (2007, methodology): “Stability conditions on triangulated categories were introduced by Bridgeland in [ 6 ] following work of Douglas [14]) on �-stability in string theory.”
Birational models of moduli spaces of coherent sheaves on the projective plane (2016, background): “Among other results, the projectivity of moduli of Bridgeland stable objects is proved in [BMW].”
The Minimal Model Program for the Hilbert Scheme of Points on P^2 and Bridgeland Stability (2012, methodology): “s a wall in the stable base locus decomposition. 5. Preliminaries on Bridgelandstability conditions In this section, we review the basic facts concerning Bridgeland stability conditions introduced in [Br1] and recall several relevant constructions from [Br2] and [AB]. Let Db(coh(P2)) denote the boun”
Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture (2011, methodology): “We ﬁrst recall that, based on Bridgeland’s deformation theorem in [Bri07], we also showed the existence of a continuous family of stability conditions depending on real classes ω, B : be the subset of pairs of real classes ( ω, B ) for which ω is ample.”
The space of stability conditions on the local projective plane (2009, background): “), i.e., the accumulation point (µ,∆) is contained in S∞. APPENDIX B. BRIDGELAND’S STABILITY CONDITIONS In this section we give a brief review of stability conditions on derived categories, following [Bri07]. Let D be a triangulated category with good properties, e.g. the bounded derived category of”
Matrix factorizations and representations of quivers II: Type ADE case (2005, background): “(section 5). In section 4, we construct a stability condition, the notion of which is introduced by Bridgeland [ Bd ], for the triangulated category HMF gr”
Quadratic differentials as stability conditions (2013, background): “We refer the reader to [2, 3] for more details on this material.”
Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory (2008, background): “However, both Kontsevich-Soibelman and Joyce formulate their invariants over a larger space, the space of “Bridgeland stability conditions” [20].”
Self-similar solutions and translating solitons for Lagrangian mean curvature flow (2008, positive): “Bridgeland’s notion of stability condition on triangulated categories [ 4 ], one can state an improved (but still over-simplified) version of the Thomas–Yau conjecture as follows: for any Calabi–Yau n-fold M, there should exist a Bridgeland stability condition (Z, P) on DbFuk(M) depending on the hol”
Gluing Stability Conditions (2009, methodology): “We show in section 1 that all (locally finite) stability conditions considered in the works [2], [4], [5] and [11] are reasonable, so this does not seem to be much of a restriction.”
Stability conditions on Kuznetsov components (2017, background): “We also recall that the support property as defined here is equivalent to the one originally appearing in [Bri07]:”
Flat surfaces and stability structures (2014, neutral): “Comparison with the work of Bridgeland–Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .”
Bridgeland Stability Conditions on Fano Threefolds (2016, background): “ng the conjecture in [BMT14] by adding seems like a necessary step to understand Bridgeland stability on threefolds. In fact, Theorem 1.1 does still imply existence of Bridgeland stability conditions [Bri07] on all Fano threefolds exactly as in [BMT14]. When the Picard number of X is 1, Theorem 1.1 ”
MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations (2013, background): “v(E))| ≥ Ckv(E)k, for all Ethat are semistable with respect to P. We will write Z(E) instead of Z(v(E)) from now on. A stability condition is called algebraic if Im(Z) ⊂ Q⊕Q √ −1. The main theorem in [Bri07] shows that the set Stab(D) of stability conditions on D is a complex manifold; its dimension”