Defect $a$-Theorem and $a$-Maximization

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.

Boundary conformal invariants and the conformal anomaly in five dimensions

“…The result is presented in Table 1. The values found for the Euler charge a agree with the earlier computations [14], [31].…”

A note on co-dimension 2 defects in N=4,d=7 gauged supergravity

“…Since our solution have a non-vanishing expectation value of the U (1) × U (1) R-symmetry currents we can interpret the defect as a homolomy defect for the R-symmetry currents. Examples of such defects have been constructed for free field theories [44][45][46][47][48]. For surface defects in four dimensional N = 4 SYM such defects can be are related to probe brane and fully back reacted LLM geometries [49][50][51] and some observables were matched in [35].…”