Abstract
We discuss Weyl anomaly and consistency conditions of local renormalization group in d=1+2 dimensional quantum field theories. We give a classification of the consistency conditions and ambiguities in most generality within the power-counting renormalization scheme. They provide many non-trivial constraints on possible forms of beta functions, anomalous dimensions and Weyl anomaly of general d=1+2 dimensional quantum field theories. We perform modest checks of our results in conformal perturbation theories, supersymmetric field theories and holographic computations.
Highlights
Studies of quantum field theories in curved space-time were originally developed in the context of gravitational physics, such as the probe in black hole geometry and the evolution in cosmology
The renormalization group with the space-time dependent cut-off (a.k.a local renormalization group) in the curved space-time and its relation to Weyl anomaly has been playing a significant role in revealing beautiful natures of the landscape of quantum field theories that are connected by the renormalization group flow [1][2]
Class 1 consistency condition comes from the general property of the local renormalization group operator ∆σ, and it does not depend on the specific form of the Weyl anomaly
Summary
Studies of quantum field theories in curved space-time were originally developed in the context of gravitational physics, such as the probe in black hole geometry and the evolution in cosmology. The starting point of the local renormalization group is to construct the generating functional for correlation functions (i.e. Schwinger vacuum energy functional [15]) by promoting coupling constants gI to space-time dependent background fields gI(x). When the covariant derivative acts on tensors, they must contain the additional space-time connection This compensated gauge invariance plays a significant role in understanding the importance of operator identities in the local renormalization group analysis [1][2]. The renormalization group equation for this Schwinger functional, whose study is the main goal of this paper, is known as the local renormalization group equation [1] because we perform the space-time dependent change of coupling constants as well as renormalization scale This has a huge advantage in discussing the conformal invariance (rather than merely scale invariance) because it directly provides the response to the non-constant Weyl transformation. The constant scale anomaly is weaker than the Weyl anomaly in such a situation (see e.g. [13] for a similar argument in relation to holography)
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