This is a note (in Japanese) of lectures by the author at the Sophia University in Tokyo, Japan, on “The solution of Witten’s conjecture after Kontsevich” (which is the original title in Japanese).
In a previous paper [J. Differ. Geom., Suppl. 1, 243-310 (1991; Zbl 0808.32023)], E. Witten presented the conjecture that the partition functions arising from the matrix model and the topological model of 2-dimensional gravity coincide. It was proved by Kontsevich using the matrix Airy integral and slightly later by Witten using Kontsevich’s idea.
The aim is to review this work of M. Kontsevich [Commun. Math. Phys. 147, No. 1, 1-23 (1992; Zbl 0756.35081)=[K]] with more details on the theory of moduli stacks and Strebel’s theory which leads to the cellular decomposition of the moduli space.
The more detailed contents is as follows. The book consists of five chapters. The first one “Moduli stacks of stable curves” introduces the notion of algebraic stacks in the sense of P. Deligne and D. Mumford [Publ. Math., Inst. Hautes. Étud. Sci. 36, 75-110 (1969; Zbl 0181.48803)=[DM]] and the moduli spaces of (pointed) stable curves \({\mathcal M}_{g,n}\) as an example. The author tries to phrase the definition of algebraic stack down-to-earth, but it contains an erroneous statement. For an algebraic stack over the complex numbers \(\mathbb{C}\), a treatment of its cohomologies is given through simplicial schemes and cohomological descent.
In the second chapter “Theory of quadratic differentials,” the existence and uniqueness of Jenkins-Strebel quadratic differential is briefly proved. Then it is explained how such a quadratic differential provides a metrized ribbon graph when an \(n\)-pointed stable curve is given. More examples are desirable, the reviewer thinks.
The goal of the third chapter “Intersection theory of moduli stacks” is to state Kontsevich’s main identity which identifies a certain linear combination of intersection numbers on \({\mathcal M}_{g,n}\) with a summation over the set of equivalence classes of trivalent graphs with certain weight averaged by the automorphism group of the graph. Unfortunately, it is not explained and referred to the original paper of Kontsevich. In the course, the space \({\mathcal M}_{g,n}^{comb}\) of certain equivalence classes of connected metrized stable ribbon graphs is introduced. In order to define it, the author used the representation of a stack as a coequalizer \(\text{Coeq}(X_1\rightrightarrows X_0)\) in a certain sense. It is actually the same thing as the category of torsors under an algebraic groupoid in the sense of H. Gillet [J. Pure Appl. Algebra 34, 193-240 (1984; Zbl 0607.14004)]. Another remark is that the above Coeq is essentially the same as \(\text{Cosq}_1\) in the calculus of homotopy.
It should be noted that Kontsevich as well as the author used the intersection theory of singular cohomology theory for topological spaces. Its compatibility with the theory by Gillet and the theory of Vistoli is not explicitly written in the literature. The last one is used in the recent verification by Behrend of the axioms of Gromov-Witten invariants in the sense of Kontsevich-Manin.
The fourth chapter “Matrix model” is the key to relate the generating series of the intersection numbers with the \(\tau\)-function in the KP hierarchy. The physical background for matrix integrals is not reviewed but its computation in terms of (colored) ribbon graphs is explained. Together with Kontsevich’s main identity, the generating series of the intersection numbers of moduli spaces is identified with a matrix integral. Unfortunately, the reasoning for the matrix integral as an asymptotic expansion is not mentioned here.
Then its interpretation as a \(\tau\)-function in KdV hierarchy is done in the fifth chapter “The brief review of KP hierarchy”. After a brief review of Mikio Sato’s approach to a KP hierarchy, the matrix integral is identified with 2-reduced KP \(\tau\)-function, namely, a KdV \(\tau\)-function, with the help of a lemma on an integral over the hermitian matrices due to Harish-Chandra.
Throughout the whole book, computational aspects are worked out, but the historical background for Witten’s conjecture can be barely found. That seems to be contrary to the author’s aim. Readers might be benefited by looking into Looijenga’s survey [E. Looijenga, Sém. Bourbaki 1992/93, Exposé 768, Astérisque 216, 187-212 (1993; Zbl 0821.14005)] for more background and further generalization of Kontsevich’s result. It is also unfortunate that there are many misprints and that important references (e.g. [K],[DM]) are forgotten in the “references”.