Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 005, 13 pages      arXiv:1301.4541      https://doi.org/10.3842/SIGMA.2013.005
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

Upper Bounds for Mutations of Potentials

John Alexander Cruz Morales a and Sergey Galkin b, c, d, e
a) Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-037, Japan
b) Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
c) Independent University of Moscow, 11 Bolshoy Vlasyevskiy per., 119002, Moscow, Russia
d) Moscow Institute of Physics and Technology, 9 Institutskii per., Dolgoprudny, 141700, Moscow Region, Russia
e) Universität Wien, Fakultät für Mathematik, Garnisongasse 3/14, A-1090 Wien, Austria

Received May 31, 2012, in final form January 16, 2013; Published online January 19, 2013

Abstract
In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52].

Key words: cluster algebras; Laurent phenomenon; mutation of potentials; mirror symmetry.

pdf (439 kb)   tex (22 kb)

References

  1. Akhtar M., Coates T., Galkin S., Kasprzyk A.M., Minkowski polynomials and mutations, SIGMA 8 (2012), 094, 707 pages, arXiv:1212.1785.
  2. Auroux D., Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007), 51-91, arXiv:0706.3207.
  3. Berenstein A., Fomin S., Zelevinsky A., Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 1-52, math.RT/0305434.
  4. Berenstein A., Zelevinsky A., Quantum cluster algebras, Adv. Math. 195 (2005), 405-455, math.QA/0404446.
  5. Cruz Morales J.A., Galkin S., Quantized mutations of potentials and their upper bounds, in preparation.
  6. Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930, math.AG/0311245.
  7. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, math.RT/0104151.
  8. Fukaya K., Oh Y.G., Ohta H., Ono K., Lagrangian Floer theory and mirror symmetry on compact toric manifolds, arXiv:1009.1648.
  9. Galkin S., Usnich A., Mutations of potentials, Preprint IPMU 10-0100, 2010.
  10. Kontsevich M., Noncommutative identities, arXiv:1109.2469.
  11. Zelevinsky A., Quantum cluster algebras: Oberwolfach talk, February 2005, math.QA/0502260.

Previous article  Next article   Contents of Volume 9 (2013)