Two-dimensional gravity and intersection theory on moduli space.

*(English)*Zbl 0757.53049
Proc. Conf., Cambridge/MA (USA) 1990, Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 243-310 (1991).

Summary: [For the entire collection see Zbl 0743.00053.]

These are notes based on two lectures given at the Conference on Geometry and Topology (Harvard University, April 1990). The first was mainly devoted to explaining a conjecture according to which stable intersection theory on moduli space of Riemann surfaces is governed by the KdV hierarchy of integrable equations. The second lecture was primarily an introduction to the “hermitian matrix model” of two-dimensional gravity, which is a crucial part of the background for the conjecture. Analogous but more general theories also exist and are sketched in these notes. The generalization in the first lecture involves considering intersection theory on the moduli space of pairs consisting of a Riemann surface \(\Sigma\) and a holomorphic map of \(\Sigma\) to a fixed Kähler manifold \(K\). The simplest analogous generalization in the second lecture involves a chain of hermitian matrices.

These are notes based on two lectures given at the Conference on Geometry and Topology (Harvard University, April 1990). The first was mainly devoted to explaining a conjecture according to which stable intersection theory on moduli space of Riemann surfaces is governed by the KdV hierarchy of integrable equations. The second lecture was primarily an introduction to the “hermitian matrix model” of two-dimensional gravity, which is a crucial part of the background for the conjecture. Analogous but more general theories also exist and are sketched in these notes. The generalization in the first lecture involves considering intersection theory on the moduli space of pairs consisting of a Riemann surface \(\Sigma\) and a holomorphic map of \(\Sigma\) to a fixed Kähler manifold \(K\). The simplest analogous generalization in the second lecture involves a chain of hermitian matrices.

##### MSC:

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14H15 | Families, moduli of curves (analytic) |

32G81 | Applications of deformations of analytic structures to the sciences |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

53C80 | Applications of global differential geometry to the sciences |

##### Keywords:

KdV hierarchy of integrable equations; hermitian matrix model; holomorphic map; Kähler manifold; chain of hermitian matrices
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\textit{E. Witten}, in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology, held at Harvard University, Cambridge, MA, USA, April 27-29, 1990. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 243--310 (1991; Zbl 0757.53049)