Abstract
Let \({\mathcal{A}}\) be the category of modules over a complex, finite-dimensional algebra. We show that the space of stability conditions on \({\mathcal{A}}\) parametrises an isomonodromic family of irregular connections on ℙ1 with values in the Hall algebra of \({\mathcal{A}}\). The residues of these connections are given by the holomorphic generating function for counting invariants in \({\mathcal{A}}\) constructed by D. Joyce (in Geom. Topol. 11, 667–725, 2007).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abe, E.: Hopf Algebras. Cambridge Tracts in Mathematics, vol. 74. Cambridge University Press, Cambridge (1980)
Balser, W., Jurkat, W.B., Lutz, D.A.: Birkhoff invariants and Stokes multipliers for meromorphic linear differential equations. J. Math. Anal. Appl. 71, 48–94 (1979)
Boalch, P.P.: Symplectic manifolds and isomonodromic deformations. Adv. Math. 163, 137–205 (2001)
Boalch, P.P.: Stokes matrices, Poisson Lie groups and Frobenius manifolds. Invent. Math. 146(3), 479–506 (2001)
Boalch, P.P.: G-bundles, isomonodromy, and quantum Weyl groups. Int. Math. Res. Not. 2002, 1129–1166 (2002)
Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. 100, 317–346 (2007)
Bridgeland, T.: Spaces of stability conditions. In: Algebraic Geometry—Seattle 2005. Part 1. Proc. Sympos. Pure Math., vol. 80, pp. 1–21. AMS, Providence (2009)
Bridgeland, T., Toledano Laredo, V.: Stokes factors and multilogarithms. arXiv:1006.4623
Douglas, M.R.: Dirichlet branes, homological mirror symmetry, and stability. In: Proceedings of the International Congress of Mathematicians, Beijing, 2002, vol. III, pp. 395–408. Higher Ed. Press, Beijing (2002)
Humphreys, J.E.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21. Springer, Berlin (1975)
Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. Physica D 2, 306–352 (1981)
Joyce, D.: Configurations in abelian categories. II. Ringel–Hall algebras. Adv. Math. 210, 635–706 (2007)
Joyce, D.: Holomorphic generating functions for invariants counting coherent sheaves on Calabi–Yau 3-folds. Geom. Topol. 11, 667–725 (2007)
Joyce, D.: Configurations in abelian categories. IV. Invariants and changing stability conditions. Adv. Math. 217, 125–204 (2008)
Kaplansky, I.: Bialgebras. Lecture Notes in Mathematics. Department of Mathematics, University of Chicago (1975)
Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316, 565–576 (2000)
King, A.D.: Moduli of representations of finite-dimensional algebras. Q. J. Math. Oxf. Ser. 45, 515–530 (1994)
Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson–Thomas invariants and cluster transformations. arXiv:0811.2435
Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc. 4, 365–421 (1991)
MacPherson, R.D.: Chern classes for singular algebraic varieties. Ann. Math. 100, 423–432 (1974)
Reineke, M.: The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli. Invent. Math. 152, 349–368 (2003)
Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs. New Series. Oxford Science Publications, vol. 7. The Clarendon Press, Oxford University Press, Oxford (1993)
Riedtmann, C.: Lie algebras generated by indecomposables. J. Algebra 170, 526–546 (1994)
Ringel, C.M.: Hall algebras and quantum groups. Invent. Math. 101, 583–591 (1990)
Ringel, C.M.: Lie algebras arising in representation theory. In: Representations of Algebras and Related Topics, Kyoto, 1990. London Math. Soc. Lecture Note Ser., vol. 168, pp. 284–291. Cambridge University Press, Cambridge (1992)
Schiffmann, O.: Lectures on Hall algebras. math/0611617
Schofield, A.: Quivers and Kac–Moody Lie algebras. Unpublished manuscript
Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Dover, New York (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Graeme Segal, whose approach to mathematics has been an inspiration to both of us
T.B. supported by a Royal Society University Research Fellowship.
V.T.L. supported in part by NSF grants DMS-0707212 and DMS-0635607.
Rights and permissions
About this article
Cite this article
Bridgeland, T., Toledano Laredo, V. Stability conditions and Stokes factors. Invent. math. 187, 61–98 (2012). https://doi.org/10.1007/s00222-011-0329-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-011-0329-4