Some Relationships Between Dualities in String Theory
Paul S. Aspinwall
TL;DR
The paper investigates how various dualities in string theory and eleven-dimensional supergravity interconnect. It organizes dualities into $C$-, $T$-, $S$-, and $U$-duality, showing that nine-dimensional theories arise from compactifications on $T^2$ and that four-dimensional ${\cal N}=4$ theories can be understood through quotients of K3$\times T^2$ and their relation to CHL-like heterotic orbifolds, yielding a Teichmüller-like moduli space ${\cal T}$ with $k=22$ or $22-M$ and a duality group $U$ containing $O(6,22;\mathbb{Z}) \times Sl(2,\mathbb{Z})$. By analyzing boundary limits (weak coupling, large radius), the work recovers familiar dualities such as IIB $S$-duality, IIA mirror symmetry, and heterotic $S$-duality, and derives a triality among IIA, IIB, and heterotic descriptions. The study also explores speculative links between M-theory/eleven-dimensional supergravity and heterotic strings via squashed K3 geometries $\Xi_i$, suggesting deeper unification beyond perturbation theory, though the precise geometric realizations remain open. Overall, the paper provides a framework for deriving inter-theory dualities from higher-dimensional origins and for interpreting moduli spaces in terms of multiple equivalent descriptions.
Abstract
Some relationships between string theories and eleven-dimensional supergravity are discussed and reviewed. We see how some relationships can be derived from others. The cases of N=2 supersymmetry in nine dimensions and N=4 supersymmetry in four dimensions are discussed in some detail. The latter case leads to consideration of quotients of a K3 surface times a torus and to a possible peculiar relationship between eleven-dimensional supergravity and the heterotic strings in ten dimensions. Lecture given at "S-Duality and Mirror Symmetry", Trieste, June 1995.