Abstract:
Conformal field theory is a formalism encountered in many branches of
physics, such as String Theory and condensed matter physics. Given the
wide range of its applicability it has become a subject of extensive study
and research. In such field theoretic descriptions we are usually interested
in computing observables called correlators. Two dimensional CFTs are important
not only because they are simplified by the presence of an infinite
dimensional symmetry algebra, thereby making it easier to compute correlators,
but also because they play a very important role in the Polyakov String
action.
Liouville theory emerges when one couples a conformally invariant field to a
two dimensional quantized gravitational background. The gravity sector of
Liouville theory matches that of non-critical string theory, hence assigning it
more importance.
In this project we first try to understand the conceptual and computational
aspects of two dimensional conformal field theories. Thereafter, the discussion
will move onto Liouville theory, which is an example of an irrational
conformal Field theory. This will include the study of how Liouville theory
emerges from two-dimensional quantum gravity plus a conformal Field theory,
and studying the DOZZ proposal: The conjectural formula for the three
point structure constant in Liouville theory.
