Geometry & Topology

Quiver algebras as Fukaya categories

Ivan Smith

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We embed triangulated categories defined by quivers with potential arising from ideal triangulations of marked bordered surfaces into Fukaya categories of quasiprojective 3–folds associated to meromorphic quadratic differentials. Together with previous results, this yields nontrivial computations of spaces of stability conditions on Fukaya categories of symplectic six-manifolds.

Article information

Geom. Topol., Volume 19, Number 5 (2015), 2557-2617.

Received: 22 September 2013
Revised: 11 August 2014
Accepted: 4 November 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33]
Secondary: 16S38: Rings arising from non-commutative algebraic geometry [See also 14A22]

quiver algebra Fukaya category stability condition


Smith, Ivan. Quiver algebras as Fukaya categories. Geom. Topol. 19 (2015), no. 5, 2557--2617. doi:10.2140/gt.2015.19.2557.

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  • M Abouzaid, On the Fukaya categories of higher-genus surfaces, Adv. Math. 217 (2008) 1192–1235
  • M Abouzaid, I Smith, Exact Lagrangians in plumbings, Geom. Funct. Anal. 22 (2012) 785–831
  • M Akaho, Intersection theory for Lagrangian immersions, Math. Res. Lett. 12 (2005) 543–550
  • D Auroux, V Muñoz, F Presas, Lagrangian submanifolds and Lefschetz pencils, J. Symplectic Geom. 3 (2005) 171–219
  • T Bridgeland, I Smith, Quadratic differentials as stability conditions, Publ. Math. Inst. Hautes Études Sci. 121 (2015) 155–278
  • C H Clemens, Double solids, Adv. in Math. 47 (1983) 107–230
  • D-E Diaconescu, R Dijkgraaf, R Donagi, C Hofman, T Pantev, Geometric transitions and integrable systems, Nuclear Phys. B 752 (2006) 329–390
  • S Fomin, M Shapiro, D Thurston, Cluster algebras and triangulated surfaces, I: Cluster complexes, Acta Math. 201 (2008) 83–146
  • K Fukaya, Cyclic symmetry and adic convergence in Lagrangian Floer theory, Kyoto J. Math. 50 (2010) 521–590
  • K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory: Anomaly and obstruction, I, AMS/IP Studies Adv. Math. 46, Amer. Math. Soc. (2009)
  • K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory: Anomaly and obstruction, II, AMS/IP Studies Adv. Math. 46, Amer. Math. Soc. (2009)
  • K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J. 151 (2010) 23–174
  • D Galakhov, P Longhi, T Mainiero, G Moore, A Neitzke, Wild wall crossing and BPS giants
  • C Geiss, D Labardini-Fragoso, J Schröer, The representation type of Jacobian algebras
  • V Ginzburg, Calabi–Yau algebras
  • R Hind, Lagrangian spheres in $S\sp 2\times S\sp 2$, Geom. Funct. Anal. 14 (2004) 303–318
  • N J Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449–473
  • D Joyce, Y Song, A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 217, AMS (2012)
  • B Keller, Calabi–Yau triangulated categories, from: “Trends in representation theory of algebras and related topics”, (A Skowroński, editor), Eur. Math. Soc., Zürich (2008) 467–489
  • B Keller, Deformed Calabi–Yau completions, J. Reine Angew. Math. 654 (2011) 125–180
  • B Keller, D Yang, Derived equivalences from mutations of quivers with potential, Adv. Math. 226 (2011) 2118–2168
  • M Khovanov, P Seidel, Quivers, Floer cohomology and braid group actions, J. Amer. Math. Soc. 15 (2002) 203–271
  • M Kontsevich, Y Soibelman, Stability structures, motivic Donaldson–Thomas invariants and cluster transformations
  • M Kontsevich, Y Soibelman, Wall-crossing structures in Donaldson–Thomas invariants, integrable systems and mirror symmetry
  • D Labardini-Fragoso, Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. 98 (2009) 797–839
  • F Labourie, Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007) 1057–1099
  • S Ladkani, $2$–CY–tilted algebras that are not Jacobian
  • J Le, X-W Chen, Karoubianness of a triangulated category, J. Algebra 310 (2007) 452–457
  • D McDuff, D Salamon, $J$–holomorphic curves and symplectic topology, Amer. Math. Soc. Colloquium Publ. 52, Amer. Math. Soc. (2004)
  • T Perutz, Lagrangian matching invariants for fibred four-manifolds, I, Geom. Topol. 11 (2007) 759–828
  • Y Qiu, On the spherical twists on $3$–Calabi–Yau categories from marked surfaces
  • M Reid, Complete intersections of two or more quadrics, PhD thesis, University of Cambridge (1972) Available at \setbox0\makeatletter\@url {\unhbox0
  • E Segal, The $A\sb \infty$ deformation theory of a point and the derived categories of local Calabi–Yaus, J. Algebra 320 (2008) 3232–3268
  • P Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 (2003) 1003–1063
  • P Seidel, A biased view of symplectic cohomology, from: “Current developments in Math., $2006$”, (B Mazur, T Mrowka, W Schmid, R Stanley, S-T Yau, editors), International Press, Boston, MA (2008) 211–253
  • P Seidel, Fukaya categories and Picard–Lefschetz theory, Zürich Lectures in Advanced Math., Eur. Math. Soc., Zürich (2008)
  • P Seidel, Homological mirror symmetry for the genus-two curve, J. Algebraic Geom. 20 (2011) 727–769
  • P Seidel, Lagrangian homology spheres in $(A\sb m)$ Milnor fibres via $\mathbb C\sp *$–equivariant $A\sb \infty$–modules, Geom. Topol. 16 (2012) 2343–2389
  • P Seidel, Abstract analogues of flux as symplectic invariants, Mém. Soc. Math. Fr. 137, Soc. Math. France, Paris (2014)
  • N Sheridan, On the homological mirror symmetry conjecture for pairs of pants, J. Differential Geom. 89 (2011) 271–367
  • I Smith, Floer cohomology and pencils of quadrics, Invent. Math. 189 (2012) 149–250
  • I Smith, R P Thomas, S-T Yau, Symplectic conifold transitions, J. Differential Geom. 62 (2002) 209–242
  • K Strebel, Quadratic differentials, Ergeb. Math. Grenzgeb. 5, Springer, New York (1984)
  • B Szendrői, Sheaves on fibered threefolds and quiver sheaves, Comm. Math. Phys. 278 (2008) 627–641
  • R P Thomas, S-T Yau, Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 (2002) 1075–1113