Quantum Conformal Algebras and Closed Conformal Field Theory

TL;DR

Anselmi analyzes quantum conformal algebras in N=2 and N=1 supersymmetric gauge theories, introducing closed conformal field theories as those whose TT OPE closes on a finite set of conserved currents. Guided by the Nachtmann theorem and model-independent arguments, he shows that closed theories arise at strong coupling as the boundary of the open-algebra landscape, and that their structure is entirely fixed by two central charges $c$ and $a$. The paper provides a concrete finite-N_c N=2 example with $N_f=2N_c$ hypermultiplets, illustrating how the stress-tensor multiplet splits into ${\cal T}$ and ${\cal T}^*$ and how anomalous dimensions diverge in the strongly coupled limit, leaving a closed subalgebra. The OPE in the closed case reduces to the N=4 form when $c=a$ and is otherwise determined by $c$ and $a$, establishing a higher-dimensional analogue of 2D CFT and proposing deep links to AdS/CFT and strongly coupled gauge dynamics.

Abstract

We investigate the quantum conformal algebras of N=2 and N=1 supersymmetric gauge theories. Phenomena occurring at strong coupling are analysed using the Nachtmann theorem and very general, model-independent, arguments. The results lead us to introduce a novel class of conformal field theories, identified by a closed quantum conformal algebra. We conjecture that they are the exact solution to the strongly coupled large-N_c limit of the open conformal field theories. We study the basic properties of closed conformal field theory and work out the operator product expansion of the conserved current multiplet T. The OPE structure is uniquely determined by two central charges, c and a. The multiplet T does not contain just the stress-tensor, but also R-currents and finite mass operators. For this reason, the ratio c/a is different from 1. On the other hand, an open algebra contains an infinite tower of non-conserved currents, organized in pairs and singlets with respect to renormalization mixing. T mixes with a second multiplet T* and the main consequence is that c and a have different subleading corrections. The closed algebra simplifies considerably at c=a, where it coincides with the N=4 one.