Skip to main content

Duality of 2D Gravity as a Local Fourier Duality


The p–q duality is a relation between the (p, q) model and the (q, p) model of two-dimensional quantum gravity. Geometrically this duality corresponds to a relation between the two relevant points of the Sato Grassmannian. Kharchev and Marshakov have expressed such a relation in terms of matrix integrals. Some explicit formulas for small p and q have been given in the work of Fukuma-Kawai-Nakayama. Already in the duality between the (2, 3) model and the (3, 2) model the formulas are long. In this work a new approach to p–q duality is given: It can be realized in a precise sense as a local Fourier duality of D-modules. This result is obtained as a special case of a local Fourier duality between irregular connections associated to Kac–Schwarz operators. Therefore, since these operators correspond to Virasoro constraints, this allows us to view the p–q duality as a consequence of the duality of the relevant Virasoro constraints.

This is a preview of subscription content, access via your institution.


  1. Arinkin, D.: Fourier transform and middle convolution for irregular D-modules. (Preprint). arXiv:0808.0699

  2. Bloch, S., Esnault, H.: Local Fourier transforms and rigidity for D-modules. Asian J. Math. 8, 587–606 (2004)

  3. Dijkgraaf, R., Hollands, L., Sulkowski, P.: Quantum curves and D-modules. JHEP 0911, 047 (2009)

  4. Fukuma, M., Kawai, H., Nakayama, R.: Explicit solution for p - q duality in two-dimensional quantum gravity. Commun. Math. Phys. 148, 101–116 (1992)

  5. Graham-Squire A.: Calculation of local formal Fourier transforms. Ark. för Mat. 51, 71–84 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kharchev, S., Marshakov, A.V.: On p–q duality and explicit solutions in c ≤ 1 2D gravity models. Int. J. Mod. Phys. A 10, 1219–1236 (1995)

  7. Kac, V., Schwarz, A.: Geometric interpretation of the partition function of 2D gravity. Phys. Lett. B 257, 329–334 (1991)

  8. Laumon G.: Transformation de Fourier, constantes d‘équations fonctionnelles et conjecture de Weil. Publ. Math. IHES 65, 131–210 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Lopez R.G.: Microlocalization and stationary phase. Asian J. Math 8, 747–768 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Liu, X., Schwarz, A.: Quantization of classical curves. Available at arXiv:1403.1000. (Preprint)

  11. Luu, M., Schwarz, A.: Fourier duality of quantum curves. (Preprint)

  12. Mulase, M.: Matrix integrals and integrable systems. In: Fukaya, K., et al. (eds.) Topology, Geometry and Field Theory, pp. 111–127, World Scientific (1994)

  13. Sabbah, C.: An explicit stationary phase formula for the local formal Fourier-Laplace transform, In: Contemporary Math, vol. 474. AMS (2008)

  14. Schwarz A.S.: On solutions to the string equation. Mod. Phys. Lett. A 6, 2713–2725 (1991)

    Article  ADS  MATH  Google Scholar 

  15. Schwarz, A.S.: Quantum curves. Commun. Math. Phys. (2015). Available at arXiv:1401.1574

  16. Varadarajan V.S.: Linear meromorphic differential equations: a modern point of view. Bull. Am. Math. Soc. 33, 1–42 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Martin T. Luu.

Additional information

Communicated by N. Reshetikhin

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Luu, M.T. Duality of 2D Gravity as a Local Fourier Duality. Commun. Math. Phys. 338, 251–265 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Differential Operator
  • Companion Matrix
  • Laurent Series
  • Virasoro Constraint
  • Compositional Inverse