U-duality

U-duality is a symmetry in string theory and M-theory that combines S-duality and T-duality. It postulates a larger, unifying duality that relates different string theories and M-theory compactified on tori.

Specifically, U-duality acts on the moduli space of string theories and M-theory compactified on a torus T^d. The moduli space parameterizes the different possible shapes and sizes of the torus, as well as other background fields like the B-field. U-duality transformations map points in the moduli space to other points that are physically equivalent, meaning that the corresponding physical theories are identical.

Unlike T-duality, which typically acts on fields appearing in perturbative string theory, U-duality usually involves non-perturbative aspects of the theory, such as D-branes and other extended objects. This means that calculations involving U-duality are generally more complex than those involving T-duality.

Mathematically, U-duality is often described by discrete groups such as E_d(d) where d is the dimension of the torus. These groups are exceptional Lie groups and play a fundamental role in the underlying structure of M-theory. For example, when compactifying M-theory on a 7-torus, the U-duality group is E₇₍₇₎.

The study of U-duality is essential for understanding the landscape of string theory vacua and the relationships between different string theories. It also provides powerful tools for calculating non-perturbative effects and exploring the deeper structure of quantum gravity.