Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Mirror symmetry and $\mathbb{C}^\times$


Author: Nobuyoshi Takahashi
Journal: Proc. Amer. Math. Soc. 129 (2001), 29-36
MSC (2000): Primary 14N10; Secondary 05A15, 20B30
Published electronically: September 14, 2000
MathSciNet review: 1784014
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We show that counting functions of covers of $\mathbb{C}^\times$ are equal to sums of integrals associated to certain `Feynman' graphs. This is an analogue of the mirror symmetry for elliptic curves.


References [Enhancements On Off] (What's this?)

  • [D] Robbert Dijkgraaf, Mirror symmetry and elliptic curves, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 149–163. MR 1363055
  • [GJ1] I. P. Goulden and D. M. Jackson, A proof of a conjecture for the number of ramified coverings of the sphere by the torus, J. Combin. Theory Ser. A 88(1999), no. 2, 246-258. CMP 2000:04
  • [GJ2] I. P. Goulden and D. M. Jackson, The number of ramified coverings of the sphere by the double torus, and a general form for higher genera, J. Combin. Theory Ser. A 88 (1999), no. 2, 259-275. CMP 2000:04
  • [O] A. Okounkov, Toda equations for Hurwitz numbers, preprint (math.AG/0004128).
  • [SSV] B. Shapiro, M. Shapiro and A. Vainshtein, Ramified coverings of $S^2$ with one degenerate branching point and enumeration of edge-ordered graphs, Topics in Singularity Theory (A. Khovanski{\u{\i}}\kern.15em, A. Varchenko and V. Vassiliev eds.), Amer. Math. Soc. Transl. (2) 180 AMS (1997), 219-228.
  • [T1] N. Takahashi, Curves in the complement of a smooth plane cubic whose normalizations are $\mathbb{A}^1$, preprint(alg-geom/9605007).
  • [T2] N. Takahashi, Log mirror symmetry and local mirror symmetry, to appar in Commun. Math. Phys.
  • [V] R. Vakil, Recursions, formulas, and graph-theoretic interpretations of ramified coverings of the sphere by surfaces of genus 0 and 1, preprint(math.CO/9812105).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14N10, 05A15, 20B30

Retrieve articles in all journals with MSC (2000): 14N10, 05A15, 20B30


Additional Information

Nobuyoshi Takahashi
Affiliation: Department of Mathematics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Email: takahasi@math.sci.hiroshima-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-00-05901-3
Received by editor(s): March 15, 1999
Published electronically: September 14, 2000
Communicated by: Ron Donagi
Article copyright: © Copyright 2000 American Mathematical Society