4-Manifold Invariants

Abstract

The concept of moduli space was introduced by Riemann in his study of the conformal (or equivalently, complex) structures on a Riemann surface. Let us consider the simplest non-trivial case, namely, that of a Riemann surface of genus 1 or the torus T ². The set of all complex structures $\mathcal{C}({T}^{2})$ on the torus is an infinite-dimensional space acted on by the infinite-dimensional group Diff(T ²). The quotient space

$$\mathcal{M}({T}^{2}) := \mathcal{C}({T}^{2})/\mathrm{Diff}({T}^{2})$$

is the moduli space of complex structures on T ². Since T ² with a given complex structure defines an elliptic curve, $\mathcal{M}({T}^{2})$ is, in fact, the moduli space of elliptic curves. It is well known that a point ω = ω₁ + iω₂ in the upper half-plane H (ω₂ > 0) determines a complex structure and is called the modulus of the corresponding elliptic curve. The modular group SL(2, Z) acts on H by modular transformations and we can identify $\mathcal{M}({T}^{2})$ with H ∕ SL(2, Z). This is the reason for calling $\mathcal{M}({T}^{2})$ the space of moduli of elliptic curves or simply the moduli space. The topology and geometry of the moduli space has rich structure. The natural boundary ω₂ = 0 of the upper half plane corresponds to singular structures. Several important aspects of this classical example are also found in the moduli spaces of other geometric structures. Typically, there is an infinite-dimensional group acting on an infinite-dimensional space of geometric structures with quotient a “nice space” (for example, a finite-dimensional manifold with singularities). For a general discussion of moduli spaces arising in various applications see, for example, [193].