Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities

Ikuo Satake[1]; Atsushi Takahashi[2]

  • [1] Faculty of Education, Kagawa University, 1-1 Saiwai-cho Takamatsu Kagawa, 760-8522, Japan
  • [2] Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka Osaka, 560-0043, Japan

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 7, page 2885-2907
  • ISSN: 0373-0956

Abstract

top
We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov–Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.

How to cite

top

Satake, Ikuo, and Takahashi, Atsushi. "Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities." Annales de l’institut Fourier 61.7 (2011): 2885-2907. <http://eudml.org/doc/275655>.

@article{Satake2011,
abstract = {We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov–Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.},
affiliation = {Faculty of Education, Kagawa University, 1-1 Saiwai-cho Takamatsu Kagawa, 760-8522, Japan; Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka Osaka, 560-0043, Japan},
author = {Satake, Ikuo, Takahashi, Atsushi},
journal = {Annales de l’institut Fourier},
keywords = {a mirror symmetry; simple elliptic singularities; Frobenius manifolds; Gromov–Witten theory; weighted projective line; primitive forms; the invariant theory; an elliptic Weyl group; an eta product; mirror symmetry; Gromov-Witten theory; eta product},
language = {eng},
number = {7},
pages = {2885-2907},
publisher = {Association des Annales de l’institut Fourier},
title = {Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities},
url = {http://eudml.org/doc/275655},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Satake, Ikuo
AU - Takahashi, Atsushi
TI - Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 7
SP - 2885
EP - 2907
AB - We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov–Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.
LA - eng
KW - a mirror symmetry; simple elliptic singularities; Frobenius manifolds; Gromov–Witten theory; weighted projective line; primitive forms; the invariant theory; an elliptic Weyl group; an eta product; mirror symmetry; Gromov-Witten theory; eta product
UR - http://eudml.org/doc/275655
ER -

References

top
  1. D. Abramovich, T. Graber, A. Vistoli, Gromov–Witten theory of Deligne–Muford stacks, Amer. J. Math. 130 (2008), 1337-1398 Zbl1193.14070MR2450211
  2. W. Chen, Y. Ruan, Orbifold Gromov–Witten Theory, Orbifolds in mathematics and physics (Madison, WI, 2001) 310 (2002), 25-85, Amer. Math. Soc., Providence, RI Zbl1091.53058MR1950939
  3. B. Dubrovin, Geometry of 2d topological field theories, Integrable systems and quantum groups (Montecatini Terme, 1993) 1620 (1996), 120-348, Springer, Berlin Zbl0841.58065MR1397274
  4. B. Dubrovin, Y. Zhang, Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation, Commun. Math. Phys. 198 (1998), 311-361 Zbl0923.58060MR1672512
  5. B. Dubrovin, Y. Zhang, Frobenius manifolds and Virasoro constraints, Selecta Math. (N.S.) 5 (1999), 423-466 Zbl0963.81066MR1740678
  6. W. Ebeling, A. Takahashi, Strange duality of weighted homogeneous polynomials, Compositio Math. 147 (2011), 1413-1433 Zbl1238.14029MR2834726
  7. H. Fan, T. Jarvis, Y. Ruan, The Witten equation, mirror symmetry and quantum singularity theory Zbl1310.32032
  8. M. Krawitz, Y. Shen, Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic Orbifold ℙ 1  
  9. T. Milanov, Y. Ruan, Gromov-Witten theory of elliptic orbifold ℙ 1 and quasimodular forms 
  10. D. Mumford, Tata Lectures on Theta I, 28 (1983), Birkhäuser Zbl0509.14049MR688651
  11. M. Noumi, Y. Yamada, Notes on the flat structures associated with simple and simply elliptic singularities, Integrable Systems and Algebraic Geometry (1998), 373-383, World-Scientific Zbl0964.32025MR1672069
  12. Y. Ohyama, Differential relations of theta functions, Osaka J. Math. 32 (1995), 431-450 Zbl0864.34001MR1355752
  13. Y. Ohyama, Differential equations for modular forms of level three, Funkcial. Ekvac. 44 (2001), 377-389 Zbl1145.11310MR1893938
  14. P. Rossi, Gromov-Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of Seifert fibrations Zbl1235.14053
  15. K. Saito, Primitive forms for a universal unfolding of a function with an isolated critical point, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1982), 775-792 Zbl0523.32015MR656053
  16. K. Saito, Period mapping associated to a primitive form, Publ. RIMS, Kyoto Univ. 19 (1983), 1231-1264 Zbl0539.58003MR723468
  17. K. Saito, Extended Affine Root System II, Publ. RIMS, Kyoto Univ. 26 (1990), 15-78 Zbl0713.17014MR1053908
  18. K. Saito, Duality for regular systems of weights, Asian J. Math. 2 (1998), 983-1047 Zbl0963.32023MR1734136
  19. K. Saito, A. Takahashi, From Primitive Forms to Frobenius manifolds, 78 (2008), 31-48 Zbl1161.32013MR2483747
  20. I. Satake, Flat Structure and the Prepotential for the Elliptic Root System of Type D 4 ( 1 , 1 ) , Topological field theory, primitive forms and related topics 160 (1998), 427-452, Birkhäuser Zbl0933.11026MR1653034
  21. I. Satake, Frobenius manifolds for elliptic root systems, Osaka J. Math. 17 (2010), 301-330 Zbl1191.53057MR2666136
  22. I. Strachan, Simple elliptic singularities: A note on their G-function Zbl1256.53056
  23. A. Takahashi, Weighted projective lines associated to regular systems of weights of dual type, Adv. Stud. Pure Math. 59 (2010), 371-388 Zbl1213.14075MR2683215
  24. K. Ueda, Homological mirror symmetry and simple elliptic singularities Zbl06291393
  25. E. Verlinde, N. P. Warner, Topological Landau-Ginzburg matter at c = 3 , Physics Letters B 269 (1991), 96-102 MR1134056

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.