Geometry & Topology

A mirror theorem for the mirror quintic

Yuan-Pin Lee and Mark Shoemaker

Full-text: Open access


The celebrated Mirror theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of its mirror dual orbifold. In this article, we establish a mirror-dual statement. Namely, the B model of the Fermat quintic threefold is shown to be equivalent to the A model of its mirror, and hence establishes the mirror symmetry as a true duality.

Article information

Geom. Topol., Volume 18, Number 3 (2014), 1437-1483.

Received: 5 November 2013
Accepted: 17 January 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]

mirror symmetry mirror theorem


Lee, Yuan-Pin; Shoemaker, Mark. A mirror theorem for the mirror quintic. Geom. Topol. 18 (2014), no. 3, 1437--1483. doi:10.2140/gt.2014.18.1437.

Export citation


  • D Abramovich, T Graber, A Vistoli, Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008) 1337–1398
  • V V Batyrev, Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994) 493–535
  • A Bertram, Another way to enumerate rational curves with torus actions, Invent. Math. 142 (2000) 487–512
  • R L Bryant, P A Griffiths, Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle, from “Arithmetic and geometry, Vol. II” (M Artin, J Tate, editors), Progr. Math. 36, Birkhäuser, Boston (1983) 77–102
  • P Candelas, X C de la Ossa, P S Green, L Parkes, A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991) 21–74
  • P Candelas, X de la Ossa, F Rodriguez-Villegas, Calabi–Yau manifolds over finite fields, I
  • W Chen, Y Ruan, Orbifold Gromov–Witten theory, from “Orbifolds in mathematics and physics” (A Adem, J Morava, Y Ruan, editors), Contemp. Math. 310, Amer. Math. Soc. (2002) 25–85
  • W Chen, Y Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004) 1–31
  • T Coates, A Corti, H Iritani, H-H Tseng, Computing genus-zero twisted Gromov–Witten invariants, Duke Math. J. 147 (2009) 377–438
  • T Coates, Y-P Lee, A Corti, H-H Tseng, The quantum orbifold cohomology of weighted projective spaces, Acta Math. 202 (2009) 139–193
  • D A Cox, S Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs 68, Amer. Math. Soc. (1999)
  • D A Cox, S Katz, Y-P Lee, Virtual fundamental classes of zero loci, from “Advances in algebraic geometry motivated by physics” (E Previato, editor), Contemp. Math. 276, Amer. Math. Soc. (2001) 157–166
  • P Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer (1970)
  • C Doran, B Greene, S Judes, Families of quintic Calabi–Yau $3$–folds with discrete symmetries, Comm. Math. Phys. 280 (2008) 675–725
  • A B Givental, Equivariant Gromov–Witten invariants, Internat. Math. Res. Notices (1996) 613–663
  • A B Givental, A mirror theorem for toric complete intersections, from “Topological field theory, primitive forms and related topics” (M Kashiwara, A Matsuo, K Saito, I Satake, editors), Progr. Math. 160, Birkhäuser (1998) 141–175
  • T Graber, R Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999) 487–518
  • B R Greene, M R Plesser, Duality in Calabi–Yau moduli space, Nuclear Phys. B 338 (1990) 15–37
  • P A Griffiths, On the periods of certain rational integrals, I, II, Ann. of Math. 90 (1969) 496–541
  • P A Griffiths (editor), Topics in transcendental algebraic geometry, Annals of Mathematics Studies 106, Princeton Univ. Press (1984)
  • H Iritani, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011) 2909–2958
  • B H Lian, K Liu, S-T Yau, Mirror principle, I, Asian J. Math. 1 (1997) 729–763
  • C-C M Liu, Localization in Gromov–Witten theory and orbifold Gromov–Witten theory, from “Handbook of Moduli, Vol. II”, Adv. Lect. Math. 25, Int. Press (2013) 353–425
  • H-H Tseng, Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol. 14 (2010) 1–81
  • E Witten, Mirror manifolds and topological field theory, from “Essays on mirror manifolds” (S-T Yau, editor), Int. Press (1992) 120–158