Abstract
In this chapter we present a circle of ideas introduced by Witten leading to a conjecture, bearing his name, regarding the intersection numbers of tautological classes on \(\overline{M}_{g,n}\). As conjectured by Witten and first proved by Kontsevich, the generating series F of these numbers satisfies the Virasoro differential equations L n (e F)=0 for n≥−1. After a self-contained presentation of Feynman diagrams and matrix models, we proceed to give Kontsevich’s proof of Witten’s conjecture. Following a brief review of equivariant cohomology, we then present Harer and Zagier’s computation of the virtual Euler–Poincaré characteristics of moduli spaces of smooth curves. We end the chapter with a very quick tour of Gromov–Witten invariants.
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© 2011 Springer-Verlag Berlin Heidelberg
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Arbarello, E., Cornalba, M., Griffiths, P.A. (2011). Intersection theory of tautological classes. In: Geometry of Algebraic Curves. Grundlehren der mathematischen Wissenschaften, vol 268. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69392-5_12
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DOI: https://doi.org/10.1007/978-3-540-69392-5_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42688-2
Online ISBN: 978-3-540-69392-5
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