We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY3 triangulated categories defined using quivers with potential associated to triangulated surfaces. We relate the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition.
Accepted:
Online First:
Published online:
Tom Bridgeland 1; Ivan Smith 2
@article{PMIHES_2015__121__155_0,
author = {Tom Bridgeland and Ivan Smith},
title = {Quadratic differentials as stability conditions},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {155--278},
year = {2015},
publisher = {Springer Berlin Heidelberg},
address = {Berlin/Heidelberg},
volume = {121},
doi = {10.1007/s10240-014-0066-5},
language = {en},
url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-014-0066-5/}
}
TY - JOUR AU - Tom Bridgeland AU - Ivan Smith TI - Quadratic differentials as stability conditions JO - Publications Mathématiques de l'IHÉS PY - 2015 SP - 155 EP - 278 VL - 121 PB - Springer Berlin Heidelberg PP - Berlin/Heidelberg UR - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-014-0066-5/ DO - 10.1007/s10240-014-0066-5 LA - en ID - PMIHES_2015__121__155_0 ER -
%0 Journal Article %A Tom Bridgeland %A Ivan Smith %T Quadratic differentials as stability conditions %J Publications Mathématiques de l'IHÉS %D 2015 %P 155-278 %V 121 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-014-0066-5/ %R 10.1007/s10240-014-0066-5 %G en %F PMIHES_2015__121__155_0
Tom Bridgeland; Ivan Smith. Quadratic differentials as stability conditions. Publications Mathématiques de l'IHÉS, Volume 121 (2015), pp. 155-278. doi: 10.1007/s10240-014-0066-5
[1.] The space of stability conditions on the local projective plane, Duke Math. J., Volume 160 (2011), pp. 263-322 | DOI | Zbl | MR
[2.] Stability conditions on triangulated categories, Ann. Math., Volume 166 (2007), pp. 317-345 | DOI | Zbl | MR
[3.] Spaces of stability conditions, Algebraic Geometry—Seattle 2005. Part I, 1–21. Proc. Sympos. Pure Math. 80, Part 1 (2009)
[4.] Stability conditions on K3 surfaces, Duke Math. J., Volume 141 (2008), pp. 241-291 Duke Math. J. (2009) | DOI | Zbl | MR
[5.] Stability conditions on a non-compact Calabi-Yau threefold, Commun. Math. Phys., Volume 266 (2006), pp. 715-733 | DOI | Zbl | MR
[6.] Linear independence of cluster monomials for skew-symmetric cluster algebras, Compos. Math., Volume 149 (2013), pp. 1753-1764 | DOI | Zbl | MR
[7.] Quivers with potential associated to triangulated surfaces, Part III: Tagged triangulations and cluster monomials, Compos. Math., Volume 148 (2012), pp. 1833-1866 | DOI | Zbl | MR
[8.] Quivers with potential and their representations, I: Mutations, Sel. Math. New Ser., Volume 14 (2008), pp. 59-119 | DOI | Zbl | MR
[9.] Quasiconformal Maps and Teichmüller Theory (2007) (viii+189 pp.) | Zbl
[10.] Cluster algebras and triangulated surfaces, Part I: Cluster complexes, Acta Math., Volume 201 (2008), pp. 83-146 | DOI | Zbl | MR
[11.] S. Fomin and D. Thurston, Cluster algebras and triangulated surfaces, Part II: Lambda lengths. Preprint. Available at | arXiv
[12.] C. Geiss, D. Labardini-Fragoso and J. Schröer, The representation type of Jacobian algebras. Preprint. Available at | arXiv
[13.] Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys., Volume 299 (2010), pp. 163-224 | DOI | Zbl | MR
[14.] Wall-crossing, Hitchin systems and the WKB approximation, Adv. Math., Volume 234 (2013), pp. 239-403 | DOI | Zbl | MR
[15.] V. Ginzburg, Calabi-Yau algebras. Preprint. Available at | arXiv
[16.] W. Gu, Graphs with non-unique decomposition and their associated surfaces. Preprint. Available at | arXiv
[17.] D. Happel, I. Reiten and S. Smalo, Tilting in abelian categories and quasitilted algebras, Mem. Am. Math. Soc., 120 (1996).
[18.] Algebraic Geometry (1977) (xvi+496 pp.) | Zbl
[19.] The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc., Volume 55 (1987), pp. 59-126 | DOI | Zbl | MR
[20.] Complex Functions. An Algebraic and Geometric Viewpoint (1987) (xiv+342 pp.) | DOI | Zbl
[21.] On Differential Graded Categories (2006), pp. 151-190
[22.] On cluster theory and quantum dilogarithm identities, Representations of Algebras and Related Topics (2011), pp. 85-116 | DOI
[23.] Derived equivalences from mutations of quivers with potential, Adv. Math., Volume 226 (2011), pp. 2118-2168 | DOI | Zbl | MR
[24.] Moduli of representations of finite-dimensional algebras, Q. J. Math., Volume 45 (1994), pp. 515-530 | DOI | Zbl
[25.] A. D. King and Y. Qiu, Exchange graphs of acyclic Calabi-Yau categories. Preprint. Available at | arXiv
[26.] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. Preprint. Available at | arXiv
[27.] Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc., Volume 98 (2009), pp. 797-839 | DOI | Zbl | MR
[28.] D. Labardini-Fragoso, Quivers with potentials associated to triangulated surfaces, Part II: Arc Representations. Preprint. Available at | arXiv
[29.] D. Labardini-Fragoso, Quivers with potentials associated to triangulated surfaces, Part IV: Removing boundary assumptions. Preprint. Available at | arXiv
[30.] Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., Volume 53 (1986), pp. 307-314 | DOI | Zbl | MR
[31.] Multiple saddle trajectories on flat surfaces and the principal boundary of the moduli spaces of quadratic differentials, Geom. Funct. Anal., Volume 18 (2008), pp. 919-987 | DOI | Zbl | MR
[32.] Metric Spaces, Convexity and Nonpositive Curvature (2005) | Zbl
[33.] Braid group actions on derived categories of coherent sheaves, Duke Math. J., Volume 108 (2001), pp. 37-108 | DOI | Zbl | MR
[34.] Diffeomorphisms of the 2-sphere, Proc. Am. Math. Soc., Volume 10 (1959), pp. 621-626 | Zbl | MR
[35.] I. Smith, Quiver algebras as Fukaya categories. Preprint. Available at | arXiv
[36.] Quadratic Differentials (1984) | DOI | Zbl
[37.] T. Sutherland, The modular curve as the space of stability conditions of a CY3 algebra. Preprint. Available at | arXiv
[38.] T. Sutherland, Stability conditions and Seiberg-Witten curves, Ph.D. Thesis, University of Sheffield, 2014.
[39.] Moduli spaces of quadratic differentials, J. Anal. Math., Volume 55 (1990), pp. 117-171 | DOI | Zbl | MR
[40.] Stability conditions, torsion theories and tilting, J. Lond. Math. Soc., Volume 82 (2010), pp. 663-682 | DOI | Zbl | MR
[41.] On the problem of existence of algebraic functions of two variables possessing a given branch curve, Am. J. Math., Volume 51 (1929), pp. 305-328 | DOI | Zbl | MR
Cited by Sources: