Abstract
Conformal defects describe the universal behaviors of a conformal field theory (CFT) in the presence of a boundary or more general impurities. The coupled critical system is characterized by new conformal anomalies which are analogous to, and generalize those of standalone CFTs. Here we study the conformal a- and c-anomalies of four dimensional defects in CFTs of general spacetime dimensions greater than four. We prove that under unitary defect renormalization group (RG) flows, the defect a-anomaly must decrease, thus establishing the defect a-theorem. For conformal defects preserving minimal supersymmetry, the full defect symmetry contains a distinguished U(1)R subgroup. We derive the anomaly multiplet relations that express the defect a- and c-anomalies in terms of the defect (mixed) ’t Hooft anomalies for this U(1)R symmetry. Once the U(1)R symmetry is identified using the defect a-maximization principle which we prove, this enables a non-perturbative pathway to the conformal anomalies of strongly coupled defects. We illustrate our methods by discussing a number of examples including boundaries in five dimensions and codimension-two defects in six dimensions. We also comment on chiral algebra sectors of defect operator algebras and potential conformal collider bounds on defect anomalies.
Highlights
Physics and leads to a reduction in the effective degrees of freedom
A particular feature of the 2d c-function is that it coincides with the conformal anomalies at the critical fixed points of the renormalization group (RG) flows, described by conformal field theories (CFT)
The conformal anomalies of conformal field theory (CFT) are present for even d and important physical observables that govern the CFT dynamics
Summary
We will prove the following defect a-theorem that constrains RG flows between conformal defects of dimension p = 4. Theorem 1 (Defect a-theorem) For a unitary defect RG flow between two unitary conformal defects DUV and DIR of dimension p = 4, the corresponding defect a-anomalies satisfy a(DUV) > a(DIR). One can formulate a stronger version of the defect a-theorem below (analogously for the strongest version), but a proof is beyond the scope of this work. Conjecture 1 Given a unitary CFT T of dimension d > 4, there exists a height function (a-function) a(λi) on the space of p = 4 unitary Poincaré invariant defects in T , parametrized by couplings {λi}, such that under a defect RG flow parametrized by scale μ, d. We emphasize that the a-function here is subject to the local condition, i.e. the μ dependence of a(λi) comes entirely from the running couplings λi
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