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Moduli spaces of curves and representation theory. (English) Zbl 0647.17010
Let \(d\) be the Lie algebra of holomorphic vector fields on \({\mathbb C}^{\times}\); it acts on the space \(V_ n\) of holomorphic differentials of degree \(n\) on \({\mathbb C}^{\times}\). This gives a homomorphism \(d\to a_{\infty}\), the Lie algebra of “restricted” endomorphisms of \(V_ n\). The universal central extension of \(a_{\infty}\) pulls back to \((6n^ 2-6n+1)\)-times that of \(d\).
On the other hand, if \(\pi: {\mathbb C}\to S\) is a family of genus \(g\) compact Riemann surfaces, \(\omega\) the relative dualizing sheaf of \(\pi\), \(\lambda_ n\) the determinant line bundle of \(\omega^ n\), then D. Mumford [Enseign. Math., II. Sér. 23, 39–110 (1977; Zbl 0363.14003)] has shown that \(c_ 1(\lambda_ n)=(6n^ 2-6n+1)c_ 1(\lambda_ 1).\)
The authors explain this coincidence by using Kodaira-Spencer deformation theory to produce a homomorphism \(d\to \text{Vect}(\widehat M_ g)\), the vector fields on the moduli space of curves of genus \(g\) with a basepoint \(p\) and a local parameter near \(p\). This implies that the tangent bundle \(T(\hat M_ g)\) is a quotient of the trivial bundle with fiber \(d\). This gives a homomorphism \(H^ 2(d)\to H^ 1(\Omega^ 1_{\hat M_ g})\) and from this one deduces the coincidence of the two formulas.
The authors further deduce an isomorphism \(H^ 2(d)\to H^ 2(M_ g,{\mathbb C})\) for \(g\geq 3\), where \(M_ g\) is the moduli space of curves of genus \(g\). Applications are also given to the cohomology of Teichmüller spaces.
Reviewer: A.N.Pressley

MSC:
17B66 Lie algebras of vector fields and related (super) algebras
58B25 Group structures and generalizations on infinite-dimensional manifolds
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
17B56 Cohomology of Lie (super)algebras
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[1] Alvarez-Gaumé, L., Gomez, C., Reina, C.: Loop groups, Grassmannians and string theory. Phys. Lett.190, 55-62 (1987)
[2] Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves, Vol. I. Berlin, Heidelberg, New York: Springer 1985, Vol. II (to appear) · Zbl 0559.14017
[3] Arbarello, E., Cornalba, M.: The Picard groups of the moduli spaces of curves. Topology26 (2), 153-171 (1987) · Zbl 0625.14014 · doi:10.1016/0040-9383(87)90056-5
[4] Beilinson, A.A., Manin, Yu.I.: The Mumford form and the Polyakov measure in string theory. Commun. Math. Phys.107, 359-376 (1986) · Zbl 0604.14016 · doi:10.1007/BF01220994
[5] Beilinson, A.A., Manin, Yu.I., Schechtman, V.V.: Sheaves of the Virasoro and Neveu-Schwarz algebras. Moscow University preprint, 1987
[6] Beilinson, A.A., Schechtman, V.V.: Determinant bundles and Virasoro algebras (preprint)
[7] Colombeau, J.-F.: Differential calculus and holomorphy. Amsterdam: North-Holland 1982
[8] Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Nonlinear integrable systems classical theory and quantum theory. Singapore: World Scientific 1983, pp. 39-119 · Zbl 0571.35098
[9] Harer, J.: The second homology group of the mapping class group of an orientable surface. Invent. Math.72, 221-239 (1983) · Zbl 0533.57003 · doi:10.1007/BF01389321
[10] Kac, V.G., Peterson, D.H.: Spin and wedge representations of infinite dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308-3312 (1981) · Zbl 0469.22016 · doi:10.1073/pnas.78.6.3308
[11] Kac, V.G.: Highest weight representations of conformal current algebras. In: Geometrical methods in field theory. Singapore: World Scientific 1986, pp. 3-15
[12] Kawamoto, N., Namikawa, Y., Tsuchiya, A., Yamada, Y.: Geometric realization of conformal field theory on Riemann surfaces. Commun. Math. Phys. (in press) · Zbl 0648.35080
[13] Knudsen, F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Math. Scand.39, 19-55 (1976) · Zbl 0343.14008
[14] Kontzevich, M.L.: Virasoro algebra and Teichmüller spaces. Funct. Anal. Appl.21, No. 2, 78-79 (1987)
[15] Manin, Yu.I.: Quantum string theory and algebraic curves. Berkeley I.C.M. talk, 1986
[16] Mumford, D.: Stability of projective varieties. Enseign. Math.23, 39-110 (1977) · Zbl 0363.14003
[17] Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Progress in Math., Vol. 36. Boston: Birkhäuser 1983 · Zbl 0554.14008
[18] Pressley, A., Segal, G.: Loop groups. Oxford: Oxford University Press 1986 · Zbl 0618.22011
[19] Ruckle, W.H.: Sequence spaces. London: Pitnam 1981 · Zbl 0491.46007
[20] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. I.H.E.S.61, 3-64 (1985) · Zbl 0592.35112
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