For the terminology and notation see McKean and Moll (1999, pp. 48–53).

The space of complex tori $\u2102/(\mathbb{Z}+\tau \mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers ${z}_{1}$ and ${z}_{2}$ are regarded as equivalent if there exist integers $m,n$ such that ${z}_{1}-{z}_{2}=m+\tau n$) is mapped into the projective space ${P}^{3}$ via the identification $z\to ({\theta}_{1}\left(2z\right|\tau ),{\theta}_{2}\left(2z\right|\tau ),{\theta}_{3}\left(2z\right|\tau ),{\theta}_{4}\left(2z\right|\tau ))$. Thus theta functions “uniformize” the complex torus. This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)).