Aspects of duality in string theory

On compactification to six spacetime dimensions, the fundamental string admits as a soliton a dual string whose effective worldsheet action couples to the background fields of the dual formulation of six-dimensional supergravity. After compactification to four spacetime dimensions, the string/string duality implies the electric/magnetic duality. Different string theories in D = 6 are discussed. A dual heterotic string theory in D = 6 is constructed by simultaneously compactifying spacetime and worlvolume of the D = 10 heterotic fivebrane on K3. It is shown how to reproduce the Green-Schwarz spacetime anomaly corrections to the H field equations (a fundamental string one-loop effect) from the Chern-Simons worldsheet anomaly corrections to the {dollar}tilde H{dollar} Bianchi identities (a dual string tree-level effect), in accordance with the idea of interchanging the loop expansions. The quantum consistency of the dual string is discussed and the Kac-Moody algebras at higher levels for the dual string are built from the fivebrane functional loop-space algebra. Finally, a vector-like N = 2 theory, obtained by the compactification of the heterotic string on a torus, is considered. The strongly coupled regime for this theory corresponds to a weekly coupled Type IIA string compactified on K3, and it is shown to be a consequence of eleven dimensional membrane/fivebrane duality. The corrections to the 3-form field equations of the D = 10 Type IIA string (a mixture of tree-level and one-loop effects) are reproduced starting from the Chern-Simons corrections to the 7-form Bianchi identities of the D = 11 fivebrane (a purely tree-llevel effect). K3 compactification of the later then yields the familiar gauge and Lorentz Chern-Simons corrections to 3-form Bianchi identities of the heterotic string.