Abstract
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.
Similar content being viewed by others
References
Arinkin, D.: Fourier transform and middle convolution for irregular D-modules. (Preprint). arXiv:0808.0699
Bloch, S., Esnault, H.: Local Fourier transforms and rigidity for D-modules. Asian J. Math. 8, 587–606 (2004)
Dijkgraaf, R., Hollands, L., Sulkowski, P.: Quantum curves and D-modules. JHEP 0911, 047 (2009)
Fukuma, M., Kawai, H., Nakayama, R.: Explicit solution for p - q duality in two-dimensional quantum gravity. Commun. Math. Phys. 148, 101–116 (1992)
Graham-Squire A.: Calculation of local formal Fourier transforms. Ark. för Mat. 51, 71–84 (2013)
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)
Kac, V., Schwarz, A.: Geometric interpretation of the partition function of 2D gravity. Phys. Lett. B 257, 329–334 (1991)
Laumon G.: Transformation de Fourier, constantes d‘équations fonctionnelles et conjecture de Weil. Publ. Math. IHES 65, 131–210 (1987)
Lopez R.G.: Microlocalization and stationary phase. Asian J. Math 8, 747–768 (2004)
Liu, X., Schwarz, A.: Quantization of classical curves. Available at arXiv:1403.1000. (Preprint)
Luu, M., Schwarz, A.: Fourier duality of quantum curves. (Preprint)
Mulase, M.: Matrix integrals and integrable systems. In: Fukaya, K., et al. (eds.) Topology, Geometry and Field Theory, pp. 111–127, World Scientific (1994)
Sabbah, C.: An explicit stationary phase formula for the local formal Fourier-Laplace transform, In: Contemporary Math, vol. 474. AMS (2008)
Schwarz A.S.: On solutions to the string equation. Mod. Phys. Lett. A 6, 2713–2725 (1991)
Schwarz, A.S.: Quantum curves. Commun. Math. Phys. (2015). Available at arXiv:1401.1574
Varadarajan V.S.: Linear meromorphic differential equations: a modern point of view. Bull. Am. Math. Soc. 33, 1–42 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N. Reshetikhin
Rights and permissions
About this article
Cite this article
Luu, M.T. Duality of 2D Gravity as a Local Fourier Duality. Commun. Math. Phys. 338, 251–265 (2015). https://doi.org/10.1007/s00220-015-2380-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-015-2380-2