Geometry & Topology

The gerby Gopakumar–Mariño–Vafa formula

Dustin Ross and Zhengyu Zong

Full-text: Open access


We prove a formula for certain cubic n–Hodge integrals in terms of loop Schur functions. We use this identity to prove the Gromov–Witten/Donaldson–Thomas correspondence for local n–gerbes over 1.

Article information

Geom. Topol., Volume 17, Number 5 (2013), 2935-2976.

Received: 7 December 2012
Revised: 13 May 2013
Accepted: 23 June 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05E05: Symmetric functions and generalizations 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]

Gromov–Witten Donaldson–Thomas loop Schur


Ross, Dustin; Zong, Zhengyu. The gerby Gopakumar–Mariño–Vafa formula. Geom. Topol. 17 (2013), no. 5, 2935--2976. doi:10.2140/gt.2013.17.2935.

Export citation


  • D Abramovich, B Fantechi, Orbifold techniques in degeneration formulas
  • D Abramovich, T Graber, A Vistoli, Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008) 1337–1398
  • M Aganagic, A Klemm, M Mariño, C Vafa, The topological vertex, Comm. Math. Phys. 254 (2005) 425–478
  • M F Atiyah, R Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1–28
  • J Bryan, C Cadman, B Young, The orbifold topological vertex, Adv. Math. 229 (2012) 531–595
  • J Bryan, T Graber, R Pandharipande, The orbifold quantum cohomology of $\mathbb C\sp 2/Z\sb 3$ and Hurwitz–Hodge integrals, J. Algebraic Geom. 17 (2008) 1–28
  • C Cadman, R Cavalieri, Gerby localization, $Z_3$–Hodge integrals and the GW theory of $[\mathbb C^3/Z_3]$, Amer. J. Math. 131 (2009) 1009–1046
  • R Cavalieri, D Ross, Open Gromov–Witten theory and the crepant resolution conjecture, Michigan Math. J. 61 (2012) 807–837
  • Q Chen, S Marcus, H Úlfarsson, Very twisted stable maps, Comm. Anal. Geom. 18 (2010) 831–855
  • R Dijkgraaf, Mirror symmetry and elliptic curves, from: “The moduli space of curves”, (R Dijkgraaf, C Faber, G van der Geer, editors), Progr. Math. 129, Birkhäuser, Boston, MA (1995) 149–163
  • Ö N Eğecioğlu, J B Remmel, A combinatorial proof of the Giambelli identity for Schur functions, Adv. in Math. 70 (1988) 59–86
  • C Faber, R Pandharipande, Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000) 173–199
  • I B Frenkel, W Wang, Virasoro algebra and wreath product convolution, J. Algebra 242 (2001) 656–671
  • H Iritani, Ruan's conjecture and integral structures in quantum cohomology, from: “New developments in algebraic geometry, integrable systems and mirror symmetry”, (M-H Saito, S Hosono, K Yoshioka, editors), Adv. Stud. Pure Math. 59, Math. Soc. Japan, Tokyo (2010) 111–166
  • P D Johnson, Double Hurwitz number via the infinite wedge
  • P D Johnson, Equivariant Gromov–Witten theory of one dimensional stacks, PhD thesis, University of Michigan (2009)
  • T Lam, P Pylyavskyy, Total positivity in loop groups, I: Whirls and curls, Adv. Math. 230 (2012) 1222–1271
  • J Li, C-C M Liu, K Liu, J Zhou, A mathematical theory of the topological vertex, Geom. Topol. 13 (2009) 527–621
  • C-C M Liu, Localization in Gromov–Witten theory and orbifold Gromov–Witten theory
  • C-C M Liu, K Liu, J Zhou, A proof of a conjecture of Mariño–Vafa on Hodge integrals, J. Differential Geom. 65 (2003) 289–340
  • C-C M Liu, K Liu, J Zhou, A formula of two-partition Hodge integrals, J. Amer. Math. Soc. 20 (2007) 149–184
  • I G Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press (1995)
  • D Maulik, A Oblomkov, A Okounkov, R Pandharipande, Gromov–Witten/Donaldson–Thomas correspondence for toric $3$–folds, Invent. Math. 186 (2011) 435–479
  • K Nakada, $q$–hook formula of Gansner type for a generalized Young diagram, from: “21st International Conference on Formal Power Series and Algebraic Combinatorics”, (C Krattenthaler, V Strehl, M Kauers, editors), Discrete Math. Theor. Comput. Science Proc., Assoc. Discrete Math. Theor. Comput. Science, Nancy, France (2009) 685–696
  • A Okounkov, G Olshanski, Shifted Schur functions, II: The binomial formula for characters of classical groups and its applications, from: “Kirillov's seminar on representation theory”, (G I Olshanski, editor), Amer. Math. Soc. Transl. Ser. 2 181, Amer. Math. Soc. (1998) 245–271
  • A Okounkov, R Pandharipande, Hodge integrals and invariants of the unknot, Geom. Topol. 8 (2004) 675–699
  • A Okounkov, R Pandharipande, The equivariant Gromov–Witten theory of ${\bf P}^1$, Ann. of Math. 163 (2006) 561–605
  • A Okounkov, N Reshetikhin, C Vafa, Quantum Calabi–Yau and classical crystals, from: “The unity of mathematics”, (P Etingof, V Retakh, I M Singer, editors), Progr. Math. 244, Birkhäuser, Boston, MA (2006) 597–618
  • Z Qin, W Wang, Hilbert schemes of points on the minimal resolution and soliton equations, from: “Lie algebras, vertex operator algebras and their applications”, (Y-Z Huang, K C Misra, editors), Contemp. Math. 442, Amer. Math. Soc. (2007) 435–462
  • D Ross, Localization and gluing of orbifold amplitudes: The Gromov–Witten orbifold vertex, to appear in Trans. Amer. Math. Soc.
  • D Ross, The loop Murnaghan–Nakayama rule, to appear in J. Algebraic Combin.
  • Z Zong, Generalized Mariño–Vafa formula and local Gromov–Witten theory of orbi-curves (2012)