nLab Witten conjecture




There are many conjectures due to Edward Witten; but by the Witten conjecture one usually refers to the statement proven in (Kontsevich 92) with new proofs due to Mirzakhani 07a, Mirzakhani 07b and others.

This is about a generating function for intersection pairing numbers of Mumford-Morita-Miller stable classes on the compactification of a moduli space of punctured curves. The conjecture relates them to the Korteweg-de Vries integrable hierarchy. The conjecture was formulated by Edward Witten based on the conjectured equivalence of the partition function of two models of 2-dimensional quantum gravity (i.e. string worldsheet field theories), which are not manifestly equivalent.


Reviews include

  • Claude Itzykson in MR93e:32027

  • Ali Ulas Ozgur Kisisel, Integrable systems and Gromov-Witten theory, p. 135–161 in: Topics in cohomological studies of algebraic varieties, Impanga lecture notes. Edited by Piotr Pragacz. Trends Math., Birkhäuser, Basel, 2005, MR2006d:14066

  • Robbert Dijkgraaf, Edward Witten, Developments in Topological Gravity (arXiv:1804.03275)

The original proof is due to

A celebrated new proof is due to

  • Maryam Mirzakhani, Simple Geodesics and Weil-Petersson Volumes Of Moduli Spaces of Bordered Riemann Surfaces, Invent. Math. 167 (2007) 179-222

  • Maryam Mirzakhani, Weil-Petersson Volumes And Intersection Theory On The Moduli Space Of Curves, Journal of the American Mathematical Society 20 (2007) 1-23

Some of the other newer approaches to the proof:

  • M. E. Kazarian, S. K. Lando, An algebro-geometric proof of Witten’s conjecture

and also versions for higher spin

  • Tyler J.Jarvis, Takashi Kimura, Arkady Vaintrob, Moduli spaces of higher spin curves and integrable hierarchies, Compositio Math. 126 (2001), no. 2, 157–212.

Gregory Naber’s lectures on Witten conjecture (this is another Witten conjecture, about Seiberg-Witten invariants) can be found at his homepage.

See also

Last revised on May 27, 2021 at 18:51:15. See the history of this page for a list of all contributions to it.