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:
and also versions for higher spin
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.