Worldline Formalism and It’s Application to AdS/CFT Correspondence

One of the interesting features of string theory is that, it reduces to quantum field theory in the infinite string tension limit. Consequently, in this limit the string scattering amplitudes should reduce to corresponding field theory amplitudes. First realization of this idea was the derivation of the one-loop β-function coefficient for the pure Yang-Mills theory from the partition function of an open string propagating in a Yang-Mills background [10]. Another investigation of the infinite string tension limit was done by Bern and Kosower [11] with the aim of finding useful results to be used for QCD calculations. The idea was to derive a parameter integral expression for gluon — gluon scattering amplitude from the string scattering amplitude in a superstring theory. In the infinite string tension limit all massive modes are suppressed and massless string modes survive, and one obtains a parameter integral representation of string scattering amplitude. This corresponds to the gauge boson scattering amplitude in some parameter (Schwinger parameter) integral representation in the field theory. The rules obtained in this approach for the calculation of field theory amplitudes are named as Bern-Kosower rules. In [1], Bern-Kosower rules have been derived in the field theory framework, as a continuation of this idea, without applying string theory. So there appeared a new formalism named worldline formalism. This formalism was applied to a number of field theory [2], and gravity problems [12]. A good pedagogical introduction to worldline formalism was given in [3]. In this lecture we intend to give an introduction to worldline formalism based on papers [1,2].