Skip to main content
Log in

Quivers with potentials associated to triangulated surfaces, part IV: removing boundary assumptions

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We prove that the quivers with potentials associated with triangulations of surfaces with marked points, and possibly empty boundary, are non-degenerate, provided the underlying surface with marked points is not a closed sphere with exactly five punctures. This is done by explicitly defining the QPs that correspond to tagged triangulations and proving that whenever two tagged triangulations are related to a flip, their associated QPs are related to the corresponding QP-mutation. As a by-product, for (arbitrarily punctured) surfaces with non-empty boundary, we obtain a proof of the non-degeneracy of the associated QPs which is independent from the one given by the author in the first paper of the series. The main tool used to prove the aforementioned compatibility between flips and QP-mutations is what we have called Popping Theorem, which, roughly speaking, says that an apparent lack of symmetry in the potentials arising from ideal triangulations with self-folded triangles can be fixed by a suitable right-equivalence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

Notes

  1. For the sphere with exactly six punctures, the proof of existence of the alluded \(\tau \) is omitted here, the reader can find such proof in the fourth arXiv version of this paper.

References

  1. Alim, M., Cecotti, S., Cordova, C., Espahbodi, S., Rastogi, A., Vafa, C.: BPS quivers and spectra of complete \(N=2\) quantum field theories. Commun. Math. Phys. 323(3), 1185–1227 (2013). arXiv:1109.4941

    Article  MATH  MathSciNet  Google Scholar 

  2. Alim, M., Cecotti, S., Cordova, C., Espahbodi, S., Rastogi, A., Vafa, C.: \(N=2\) quantum field theories and their BPS quivers. Adv. Theor. Math. Phys. 18(1), 27–127 (2014). arXiv:1112.3984

    Article  MATH  MathSciNet  Google Scholar 

  3. Assem, I., Brüstle, T., Charbonneau-Jodoin, G., Plamondon, P.-G.: Gentle algebras arising from surface triangulations. Algebra Number Theory 4(2), 201–229 (2010). arXiv:0903.3347

    Article  MATH  MathSciNet  Google Scholar 

  4. Barot, M., Geiss, C.: Tubular cluster algebras I: categorification. Math. Z. 271, 1091–1115 (2012). arXiv:0905.0028

    Article  MATH  MathSciNet  Google Scholar 

  5. Bridgeland, T., Smith, I.: Quadratic differentials as stability conditions. Publ. Math. l’IHÉS 121(1), 155–278 (2015). arXiv:1302.7030

    Article  MathSciNet  Google Scholar 

  6. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (\(A_n\)-case). Trans. Am. Math. Soc. 358(5), 1347–1364 (2006). arXiv:math/0401316

    MATH  MathSciNet  Google Scholar 

  7. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster-tilted algebras. Algebras Represent. Theory 9(4), 359–376 (2006). arXiv:math/0411238

    Article  MATH  MathSciNet  Google Scholar 

  8. Cecotti, S.: Categorical tinkertoys for \(N=2\) Gauge theories. Int. J. Mod. Phys. A 28(5 & 6), 1330006 (2013). doi:10.1142/S0217751X13300068. arXiv:1203.6734

    Article  MathSciNet  Google Scholar 

  9. Labardini-Fragoso, D., Cerulli Irelli, G.: Quivers with potentials associated to triangulated surfaces, part III: tagged triangulations and cluster monomials. Compos. Math. 148(06), 1833–1866 (2012). arXiv:1108.1774

    Article  MATH  MathSciNet  Google Scholar 

  10. Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations I: mutations. Sel. Math. 14(1), 59–119 (2008). arXiv:0704.0649

    Article  MATH  MathSciNet  Google Scholar 

  11. Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces, part I: cluster complexes. Acta Math. 201, 83–146 (2008). arXiv:math.RA/0608367

  12. Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013). arXiv:0907.3982v2

    Article  MATH  MathSciNet  Google Scholar 

  13. Geiss, C., Labardini-Fragoso, D., Schröer, J.: The representation type of Jacobian algebras. arXiv:1308.0478

  14. Labardini-Fragoso, D.: Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math. Soc. 98(3), 797–839 (2009). arXiv:0803.1328

    Article  MATH  MathSciNet  Google Scholar 

  15. Labardini-Fragoso, D.: Quivers with potentials associated to triangulated surfaces, part II: Arc representations. arXiv:0909.4100

  16. Ladkani, S.: On Jacobian algebras from closed surfaces. arXiv:1207.3778

  17. Mosher, L.: Tiling the projective foliation space of a punctured surface. Trans. Am. Math. Soc. 306, 1–70 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nagao, K.: Mapping class group, Donaldson–Thomas theory and S-duality. http://www.math.nagoya-u.ac.jp/~kentaron/MCG_DT.pdf

  19. Nagao, K.: Triangulated surface, mapping class group and Donaldson–Thomas theory. In: Proceedings of the Kinosaki Algebraic Geometry Symposium. http://www.math.sci.osaka-u.ac.jp/~kazushi/kinosaki2010/nagao.pdf (2010)

  20. Schiffler, R.: A geometric model for cluster categories of type \(D_n\). J. Algebraic Comb. 27(1), 1–21 (2008). arXiv:math/0608264

    Article  MATH  MathSciNet  Google Scholar 

  21. Trepode, S., Valdivieso-Díaz, Y.: On finite dimensional Jacobian Algebras. arXiv:1207.1917

  22. Xie, D.: Network, cluster coordinates and \(N=2\) theory I. arXiv:1203.4573

  23. Zelevinsky, A.: Mutations for quivers with potentials: oberwolfack talk. arXiv:0706.0822

Download references

Acknowledgments

I am sincerely grateful to Tom Bridgeland for his interest in this work and many very pleasant discussions. I thank him as well for informing me of Gaiotto–Moore–Neitzke’s use of the term “pop” (cf. [12, pp. 11–12, and Sections 5.6 and 5.8]) for the analogue in Geometry and Physics of what was called “swap” in a first version of this paper. Thanks are owed to the anonymous referee for a number of useful suggestions. I started considering the problem of compatibility between flips and QP-mutations when I was a PhD student at Northeastern University (Boston, MA, USA). As such, I profitted from numerous discussions with Jerzy Weyman and Andrei Zelevinsky. I am deeply grateful to both of them for their teachings. Professor Andrei Zelevinsky unfortunately passed away a few weeks before this paper was submitted. No words can express my gratitude toward him for all the teachings, guidance and encouragement I constantly received from him during my Ph.D. studies and afterward. My admiration for his way of doing Mathematics will always be of the deepest kind.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Daniel Labardini-Fragoso.

Additional information

Dedicated in the memory of Professor Andrei Zelevinsky.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Labardini-Fragoso, D. Quivers with potentials associated to triangulated surfaces, part IV: removing boundary assumptions. Sel. Math. New Ser. 22, 145–189 (2016). https://doi.org/10.1007/s00029-015-0188-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-015-0188-8

Keywords

Mathematics Subject Classification

Navigation