Moduli of Riemann surfaces, Hurwitz-type spaces, and their superanalogues.

*(English. Russian original)*Zbl 0948.32018
Russ. Math. Surv. 54, No. 1, 61-117 (1999); translation from Usp. Mat. Nauk 54, No. 1, 61-116 (1999).

From the author’s introduction: This survey is devoted to the description of the topological structure of the moduli spaces of Riemann surfaces, the moduli spaces of super Riemann surfaces, the spaces of holomorphic maps of Riemann surfaces, and the spaces of superholomorphic maps of super Riemann surfaces. We consider the case of hyperbolic surfaces only. For non-hyperbolic surfaces (for spheres with at least three punctures and for tori), the theory is much simpler but needs other constructions.

The topological description consists of the following two steps: 1) a description of the connected components of these spaces, 2) the representation of each of the components in the form \(T/\text{Mod}\), where \(T\) is a space homeomorphic to a vector space or a superanalogue and Mod is a discrete group.

In the classical case of Riemann surfaces, a description of the above type was first suggested by Fricke and Klein. In this case, a connected component is defined by the topological type of the surface. We prove that, in the case of supersurfaces, we must also choose the type of an induced Arf function (for \(N=1)\) or of a pair of Arf functions (for \(N=2)\). The connected components of spaces of holomorphic maps are defined by the conjugacy classes of subgroups of a free group. The connected components of spaces of superholomorphic maps are defined by equivalence classes of a pair of the form (a subgroup of a free group, an Arf function).

In the approach by Fricke and Klein, the space \(T\) arises as the space of conjugacy classes of special bases of Fuchsian groups. In this case, the group Mod arises as the group of changes of bases. Much later, Teichmüller suggested another description of the space \(T\), as the space of extremal quasiconformal maps of Riemann surfaces. Therefore, in our survey, \(T\) is called a Fricke-Klein-Teichmüller space.

There are many ways of parametrizing a Fricke-Klein-Teichmüller space. As in the original paper by Fricke and Klein, most of them use parameters related to the standard metric of constant curvature of a Riemann surface, namely, the lengths of the geodesics that correspond to the generators of a Fuchsian group, the lengths of the geodesics that correspond to the standard products of the generators, and the angles between them.

We use the parameters that describe the generators of a Fuchsian group via their fixed points and “parameters” of the shifts. This approach enables one readily to pass from a description of the moduli of Riemann surfaces to a description of the spaces of holomorphic maps and, after some modification, to a description of the superanalogues of these spaces. We explicitly indicate the range of these parameters for the standard generators of the Fuchsian groups. This range turns out to be a domain in a vector space (or a superanalogue) that is singled out by linear and quadratic inequalities. Hence, we can readily show that this range is homeomorphic to a (super)vector space.

We stress that the “parameters” of the shifts and their superanalogues are precisely the parameters that enter the formulae for the Mumford measure and its superanalogues. Thus, our parametrization of the moduli space can be useful for string theory. The above approach turns out to be convenient in the description of real algebraic curves and their superanalogues. In particular, this approach enables one to solve the Schottky problem for real algebraic curves and to describe the physically important solutions of the Kadomtsev-Petviashvili equations, to construct a quasiperiodic two-dimensional Schrödinger operator, and so on. However, in our opinion, this topic is worthy of a separate survey.

The topological description consists of the following two steps: 1) a description of the connected components of these spaces, 2) the representation of each of the components in the form \(T/\text{Mod}\), where \(T\) is a space homeomorphic to a vector space or a superanalogue and Mod is a discrete group.

In the classical case of Riemann surfaces, a description of the above type was first suggested by Fricke and Klein. In this case, a connected component is defined by the topological type of the surface. We prove that, in the case of supersurfaces, we must also choose the type of an induced Arf function (for \(N=1)\) or of a pair of Arf functions (for \(N=2)\). The connected components of spaces of holomorphic maps are defined by the conjugacy classes of subgroups of a free group. The connected components of spaces of superholomorphic maps are defined by equivalence classes of a pair of the form (a subgroup of a free group, an Arf function).

In the approach by Fricke and Klein, the space \(T\) arises as the space of conjugacy classes of special bases of Fuchsian groups. In this case, the group Mod arises as the group of changes of bases. Much later, Teichmüller suggested another description of the space \(T\), as the space of extremal quasiconformal maps of Riemann surfaces. Therefore, in our survey, \(T\) is called a Fricke-Klein-Teichmüller space.

There are many ways of parametrizing a Fricke-Klein-Teichmüller space. As in the original paper by Fricke and Klein, most of them use parameters related to the standard metric of constant curvature of a Riemann surface, namely, the lengths of the geodesics that correspond to the generators of a Fuchsian group, the lengths of the geodesics that correspond to the standard products of the generators, and the angles between them.

We use the parameters that describe the generators of a Fuchsian group via their fixed points and “parameters” of the shifts. This approach enables one readily to pass from a description of the moduli of Riemann surfaces to a description of the spaces of holomorphic maps and, after some modification, to a description of the superanalogues of these spaces. We explicitly indicate the range of these parameters for the standard generators of the Fuchsian groups. This range turns out to be a domain in a vector space (or a superanalogue) that is singled out by linear and quadratic inequalities. Hence, we can readily show that this range is homeomorphic to a (super)vector space.

We stress that the “parameters” of the shifts and their superanalogues are precisely the parameters that enter the formulae for the Mumford measure and its superanalogues. Thus, our parametrization of the moduli space can be useful for string theory. The above approach turns out to be convenient in the description of real algebraic curves and their superanalogues. In particular, this approach enables one to solve the Schottky problem for real algebraic curves and to describe the physically important solutions of the Kadomtsev-Petviashvili equations, to construct a quasiperiodic two-dimensional Schrödinger operator, and so on. However, in our opinion, this topic is worthy of a separate survey.

Reviewer: Viktor Z.Enol’skij (Kyïv)

##### MSC:

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

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |

14H42 | Theta functions and curves; Schottky problem |

58C50 | Analysis on supermanifolds or graded manifolds |

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

83E30 | String and superstring theories in gravitational theory |