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

SIGMA 16 (2020), 013, 35 pages      arXiv:1901.04166      https://doi.org/10.3842/SIGMA.2020.013
Contribution to the Special Issue on Cluster Algebras

Cluster Structures and Subfans in Scattering Diagrams

Yan Zhou
Beijing International Center for Mathematical Research, Peking University, China

Received July 24, 2019, in final form March 01, 2020; Published online March 11, 2020

Abstract
We give more precise statements of Fock-Goncharov duality conjecture for cluster varieties parametrizing ${\rm SL}_{2}/{\rm PGL}_{2}$-local systems on the once punctured torus. Then we prove these statements. Along the way, using distinct subfans in the scattering diagram, we produce an example of a cluster variety with two non-equivalent cluster structures. To overcome the technical difficulty of infinite non-cluster wall-crossing in the scattering diagram, we introduce quiver folding into the machinery of scattering diagrams and give a quotient construction of scattering diagrams.

Key words: cluster varieties; Donaldson-Thomas transformations; Markov quiver; non-equivalent cluster structures; scattering diagrams; quiver folding.

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References

  1. 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.
  2. Carl M., Pumperla M., Siebert B., A tropical view of Landau-Ginzburg models, available at http://www.math.uni-hamburg.de/home/siebert/preprints/LGtrop.pdf.
  3. Dupont G., An approach to non-simply laced cluster algebras, J. Algebra 320 (2008), 1626-1661, arXiv:math.RT/0512043.
  4. Fock V., Goncharov A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1-211, arXiv:math.AG/0311149.
  5. Fomin S., Williams L., Zelevinsky A., Introduction to cluster algebras, Chapters 4-5, arXiv:1707.07190.
  6. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, arXiv:math.RT/0104151.
  7. Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112-164, arXiv:math.RA/0602259.
  8. Goncharov A., Shen L., Donaldson-Thomas transformations of moduli spaces of G-local systems, Adv. Math. 327 (2018), 225-348, arXiv:1602.06479.
  9. Gross M., Mirror symmetry for ${\mathbb P}^2$ and tropical geometry, Adv. Math. 224 (2010), 169-245, arXiv:0903.1378.
  10. Gross M., Hacking P., Keel S., Birational geometry of cluster algebras, Algebr. Geom. 2 (2015), 137-175, arXiv:1309.2573.
  11. Gross M., Hacking P., Keel S., Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65-168, arXiv:1106.4977.
  12. Gross M., Hacking P., Keel S., Kontsevich M., Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497-608, arXiv:1411.1394.
  13. Gross M., Siebert B., From real affine geometry to complex geometry, Ann. of Math. 174 (2011), 1301-1428, arXiv:math.AG/0703822.
  14. Keel S., Yu T.Y., The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus, arXiv:1908.09861.
  15. Keller B., Quiver mutation and combinatorial DT-invariants, arXiv:1709.03143.
  16. Kontsevich M., Soibelman Y., Affine structures and non-Archimedean analytic spaces, in The Unity of Mathematics, Progr. Math., Vol. 244, Birkhäuser Boston, Boston, MA, 2006, 321-385, arXiv:math.AG/0406564.
  17. Kontsevich M., Soibelman Y., Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
  18. Kontsevich M., Soibelman Y., Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry, in Homological Mirror Symmetry and Tropical Geometry, Lect. Notes Unione Mat. Ital., Vol. 15, Springer, Cham, 2014, 197-308.
  19. Lusztig G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498.
  20. Lusztig G., Total positivity in reductive groups, in Lie Theory and Geometry: In Honor of B. Kostant, Progr. Math., Vol. 123, Birkhäuser Boston, Boston, MA, 1994, 531-568.
  21. Mandel T., Theta bases and log Gromov-Witten invariants of cluster varieties, arXiv:1903.03042.
  22. Matherne J.P., Muller G., Computing upper cluster algebras, Int. Math. Res. Not. 2015 (2015), 3121-3149, arXiv:1307.0579.
  23. Nagao K., Donaldson-Thomas theory and cluster algebras, Duke Math. J. 162 (2013), 1313-1367, arXiv:1002.4884.
  24. Yu T.Y., Enumeration of holomorphic cylinders in log Calabi-Yau surfaces. I, Math. Ann. 366 (2016), 1649-1675, arXiv:1504.01722.
  25. Yu T.Y., Enumeration of holomorphic cylinders in log Calabi-Yau surfaces. II. Positivity, integrality and the gluing formula, arXiv:1608.07651.
  26. Zhou Y., Weyl groups and cluster structures of families of log Calabi-Yau surfaces, arXiv:1910.05762.

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