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Algebraic geometry associated with matrix models of two dimensional gravity. (English) Zbl 0812.14017
Goldberg, Lisa R. (ed.) et al., Topological methods in modern mathematics. Proceedings of a symposium in honor of John Milnor’s sixtieth birthday, held at the State University of New York at Stony Brook, USA, June 14-June 21, 1991. Houston, TX: Publish or Perish, Inc. 235-269 (1993).
In this paper the author describes some conjectures in algebraic geometry associated with matrix models of two dimensional quantum gravity. The original conjecture of this type described by the author [in Proc. Conf., Cambridge 1990, Surv. Differ. Geom., Suppl. J. Differ. Geom. 1, 243-310 (1991; Zbl 0757.53049)] and proved by M. Kontsevich [Funct. Anal. Appl. 25, No. 2, 123-129 (1991); translation from Funkts. Anal. Prilozh. 25, No. 2, 50-57 (1991; Zbl 0742.14021)], involved intersection theory on the moduli space of Riemann surfaces. This time the conjecture is related with intersection theory on some covers of the moduli space of Riemann surfaces. The spaces he considers are covers \({\mathcal M}_{g,s}'\) of \({\mathcal M}_{g,s}\)= moduli space of complex Riemann surfaces of genus \(g\) and \(s\) punctures. They are defined by the data \((\Sigma, (x_ 1, \dots, x_ s), {\mathcal T})\), where \(\Sigma\) is a smooth Riemann surface of genus \(g\) with \(s\) marked points \((x_ 1, \dots, x_ s)\) labeled by integers \((m_ 1, \dots, m_ s)\) such that \(0 \leq m_ i \leq r - 1\), and \(r \geq 2\) fixed.
Let \(K\) be the canonical bundle of \(\Sigma\) and \({\mathcal S} = K \otimes_ i {\mathcal O} (x_ i)^{-m_ i}\). Then, if \(\deg {\mathcal S} = (2g-2 - \sum_ i m_ i)\) is divisible by \(r\), there are \(r^{2g}\) isomorphism classes of line bundles \({\mathcal T}\) such that \({\mathcal T}^{\otimes r} = {\mathcal S}\). The objects of principal interest are the intersection numbers \[ {1 \over r^ g} (\prod_{i = 1}^ s c_ 1 ({\mathcal L}_ i)^{n_ i} \cdot c_ D ({\mathcal V}), \overline {\mathcal M}_{g,s}') \tag{1} \] defined over a compactification \(\overline {\mathcal M}_{g,s}'\) of \({\mathcal M}_{g,s}'\). These numbers vanish unless \[ \sum n_ i + D = 3g - 3 + s.\tag{2} \] Here, \({\mathcal L}_ i\) is the line bundle whose fiber over \(\Sigma\) is the cotangent line \(T^* \Sigma/x_ i\). \({\mathcal V}\) is the vector bundle of fiber \(V = H^ 0 (\Sigma, K \otimes {\mathcal T}^{-1})\). This has dimension \(D = (g-1) (1-2/r) + (1/r) \sum_ i m_ i\), and \(c_ D ({\mathcal V})\) represents the top Chern class of \({\mathcal V}\).
The intersection numbers above must appear (in an appropriate quantum field theory) as expectation values of the operators \(\tau_{n,m} = \tau_ n (U_ m) = n^{th}\) gravitational descendant of a primary field \(U_ m\). Thus, denoting them as \(\langle \prod_{n,m} \tau_{n,m}^{d_{n,m}} \rangle\), where \(d_{n,m} \geq 0\), \(s = \sum d_{n,m}\) and \(d_{n,m}\) of the points \(x_ 1, \dots, x_ s\) are labeled by \((n_ i, m_ i) = (n,m)\). From this, one determines the genus \(g\) and \(D\) obeying (2) and sets \[ \langle \prod_{n,m} \tau_{n,m}^{d_{n, m}} \rangle = {1 \over r^ g} (\prod_{i = 1}^ s c_ 1({\mathcal L}_ i)^{n_ i} \cdot c_ D ({\mathcal V}), \overline {{\mathcal M}_{g,s}'}). \] The author introduces the variables \(t_{n,m}\), \(n \in N\), \(0 \leq m \leq r - 1\) and the free energy \(F(t_{0,0}, t_{0,1}, \dots) = \sum_{d_{n,m}} \langle \prod_{n,m} \tau_{n, m}^{d_{n,m}} \rangle \prod_{n,m} {\tau_{n,m}^{d_{n,m}} \over d_{n,m}!} = \sum_{g \geq 0} F_ g\), where \(F_ g\) is the contribution corresponding to genus \(g\). Set (3) \(v_ i = {\partial^ 2 F \over \partial t_{0,0} \partial t_{0,i}}\), \(0 \leq i \leq r - 2\), \(v_ i = - {r \over i + 1} \text{res} (Q^{(i+1)/r})\) and \(Q\) the differential operator \(Q = D^ r - \sum_{i = 0}^{r - 2} u_ i (t_{0,0}) D^ i\), \(D = {i \over \sqrt r} {\partial \over \partial t_{0,0}}\). (The \(u_ i\) are determined by the \(v_ i\) and viceversa). Then the author conjectures that the differential operator \(Q\) constructed with the \(v_ i\) as in (3) obey the Gelfand-Dikii equations \[ i{\partial Q \over \partial t_{n,m}} = [Q_ +^{n+(m+1)/r}, Q] \cdot {c_{n,m} \over \sqrt r}, \quad c_{n,m} = {(-1)^ nr^{n+1} \over (m+1) (r+m+1) \dots (nr + m + 1)}. \] In addition, \(F\) obeys the string equation \[ {\partial F \over \partial t_{0,0}} = {1 \over 2} \sum_{i,j = 0}^{r-2} \eta^{ij} t_{0,i} t_{0,j} + \sum_{n=1}^ \infty \sum_{m = 0}^{r - 2} t_{n + 1,m} {\partial F \over \partial t_{n,m}} \quad \eta^{ij} = \delta_{i + j, r - 2}. \] The author relates this conjecture with the original one and analyzes the genus zero case.
For the entire collection see [Zbl 0780.00031].

MSC:
14H15 Families, moduli of curves (analytic)
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.)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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