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.
Similar content being viewed by others
Notes
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
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
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
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
Barot, M., Geiss, C.: Tubular cluster algebras I: categorification. Math. Z. 271, 1091–1115 (2012). arXiv:0905.0028
Bridgeland, T., Smith, I.: Quadratic differentials as stability conditions. Publ. Math. l’IHÉS 121(1), 155–278 (2015). arXiv:1302.7030
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
Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster-tilted algebras. Algebras Represent. Theory 9(4), 359–376 (2006). arXiv:math/0411238
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
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
Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations I: mutations. Sel. Math. 14(1), 59–119 (2008). arXiv:0704.0649
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
Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013). arXiv:0907.3982v2
Geiss, C., Labardini-Fragoso, D., Schröer, J.: The representation type of Jacobian algebras. arXiv:1308.0478
Labardini-Fragoso, D.: Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math. Soc. 98(3), 797–839 (2009). arXiv:0803.1328
Labardini-Fragoso, D.: Quivers with potentials associated to triangulated surfaces, part II: Arc representations. arXiv:0909.4100
Ladkani, S.: On Jacobian algebras from closed surfaces. arXiv:1207.3778
Mosher, L.: Tiling the projective foliation space of a punctured surface. Trans. Am. Math. Soc. 306, 1–70 (1988)
Nagao, K.: Mapping class group, Donaldson–Thomas theory and S-duality. http://www.math.nagoya-u.ac.jp/~kentaron/MCG_DT.pdf
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)
Schiffler, R.: A geometric model for cluster categories of type \(D_n\). J. Algebraic Comb. 27(1), 1–21 (2008). arXiv:math/0608264
Trepode, S., Valdivieso-Díaz, Y.: On finite dimensional Jacobian Algebras. arXiv:1207.1917
Xie, D.: Network, cluster coordinates and \(N=2\) theory I. arXiv:1203.4573
Zelevinsky, A.: Mutations for quivers with potentials: oberwolfack talk. arXiv:0706.0822
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
Corresponding author
Additional information
Dedicated in the memory of Professor Andrei Zelevinsky.
Rights and permissions
About this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-015-0188-8
Keywords
- Tagged triangulation
- Flip
- Pop
- Quiver with potential
- Mutation
- Right-equivalence
- Non-degeneracy
- Cluster algebra