Abstract
We analyze renormalization group (RG) flows in two-dimensional quantum field theories in the presence of redundant directions. We use the operator picture in which redundant operators are total derivatives. Our analysis has three levels of generality. We introduce a redundancy anomaly equation which is analyzed together with the RG anomaly equation previously considered by H.Osborn [8] and D.Friedan and A.Konechny [7]. The Wess-Zumino consistency conditions between these anomalies yield a number of general relations which should hold to all orders in perturbation theory. We further use conformal perturbation theory to study field theories in the vicinity of a fixed point when some of the symmetries of the fixed point are broken by the perturbation. We relate various anomaly coefficients to OPE coefficients at the fixed point and analyze which operators become redundant and how they participate in the RG flow. Finally, we illustrate our findings by three explicit models constructed as current-current perturbations of SU(2)_k WZW model. At each generality level we discuss the geometric picture behind redundancy and how one can reduce the number of couplings by taking a quotient with respect to the redundant directions. We point to the special role of polar representations for the redundancy groups.
Highlights
The subject of two-dimensional quantum field theories (2d QFTs) has provided us with the richness of nonperturbative techniques such as the ones related to integrability and conformal symmetry, as well as with a number of powerful general results
Equation (3.22) shows that the commutator of the beta function vector field βwith the redundancy vector fields Ra closes again on the redundancy vector fields. This condition is crucial for the reduction of the renormalization group (RG) flow onto the quotient space in which we identify points on the orbits generated by the redundancy vector fields
What we have seen in the conformal perturbation theory analysis is that in the vicinity of fixed points with symmetry we can construct theories in which redundant operators originate from the broken symmetries
Summary
The subject of two-dimensional quantum field theories (2d QFTs) has provided us with the richness of nonperturbative techniques such as the ones related to integrability and conformal symmetry, as well as with a number of powerful general results. I.e. sigma models with vanishing beta functions, correspond to solutions to classical equations of motion for the string In this context the gradient formula has a special significance — it provides a string action principle. In particular there may be total derivative combinations of those operators φi which couple to the coupling constants parameterizing our QFTs. As any operator equation, formula (1.3) in general holds up to contact terms. The local operator that couples to such a coupling equals a total derivative up to the terms proportional to equations of motion, which are pure contact terms. To make this more explicit consider the following elementary example: a scalar field theory with action. The appendices contain some more technical details of the calculations
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