# Conformal Field Theory

Conformal field theories are quantum field theories that are invariant under conformal transformations. These are transformations of the underlying space-time that leave angles unchanged, but not necessarily lengths. In particular these theories are invariant under a rescaling of all distances.

Conformal field theories arise for example on the two-dimensional world-sheet of strings (one dimension for the spatial extension of the string, and one for the time), where the fields are interpreted as the target space-time coordinates of the string's position. The invariance under conformal transformations is a remnant of the invariance under a reparameterisation of the string: it is only the movement in the target space-time that counts, and not the way we describe it.

One can go one step further: it is only the conformal field theory that counts, and not its geometric interpretation. This means that for string theory one takes all conformal field theories (with certain properties) into account, and not only those that correspond to strings moving in a nice geometric background. In this way one can abandon geometry from the fundamental description, and instead one can rederive it as an effective geometry in certain limits.

Therefore, the classification and analysis of two-dimensional conformal field theories is also conceptually of great importance in string theory. In our group we study the space of such theories, its properties, renormalisation group flows between different theories (corresponding to dynamical, time-dependent processes in string theory), boundary conditions (corresponding to open strings attached to D-branes), and defects or interfaces that can separate different conformal field theories, and that might be connected to generalised symmetries in string theory.

Conformal field theories also arise as duals of gravitational theories on space-times that asymptotically look like Anti-de Sitter space-time. In this duality, a gravity theory on a (d+1)-dimensional space-time has an equivalent conformal field theory description on a d-dimensional space-time. Such a duality is conjectured for string theory, and for theories that extend gravity by higher-spin gauge fields. In our group we analyse three-dimensional higher-spin theories and their two-dimensional dual conformal field theories.