On dilatation operator for a renormalizable theory

TL;DR

This paper constructs the dilatation operator for renormalizable theories, interpreting it as a Hamiltonian in the dual theory, and demonstrates its application in N=4 supersymmetric Yang-Mills theory at one-loop level.

Contribution

It provides a systematic method to derive the dilatation operator as a differential operator acting on composite operators, linking RG flow with dual Hamiltonian descriptions.

Findings

Reproduces known results in N=4 SYM at one-loop

Constructs the dilatation operator as a differential operator

Interprets the operator as a Hamiltonian in the dual theory

Abstract

Given a renormalizable theory we construct the dilatation operator, in the sense of generator of RG flow of composite operators. The generator is found as a differential operator acting on the space of normal symbols of composite operators in the theory. In the spirit of AdS/CFT correspondence, this operator is interpreted as the Hamiltonian of the dual theory. In the case of a field theory with non-abelian gauge symmetry the resulting system is a matrix model. The one-loop case is analyzed in details and it is shown that we reproduce known results from N=4 supersymmetric Yang-Mills theory.