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.

Weyl anomalies of four dimensional conformal boundaries and defects

“…Indeed, in section 5 we use the results of sections 3 and 4 to compute several defect central charges in two classes of examples. First are monodromy defects of free, massless scalars our Dirac fermions in d = 6, which also provide non-trivial tests of the defect a-theorem [39]. We also use the results for monodromy defects in free field theories to find defect central charges for conical defects and orbifolds.…”

“…In ref. [39], Wang showed that a Σ obeys an a-theorem, for RG flows localized to the defect, and furthermore, if the defect preserves at least N = 1 four-dimensional SUSY, then a Σ obeys a defect version of a-maximization.…”

“…That is, in spite of the c-theorem for a Σ proven in ref. [39], and since no c-theorems are known for B-type anomalies, the presence of d 2 means eq. (4.38) is not necessarily monotonic under defect RG flows.…”

“…With an explicit expression for a Σ , we can test the p = 4 defect c-theorem of ref. [39] by studying a simple defect RG flow. Suppose only the ∆ = 2 − |α| mode, Ô− −α , is present in the defect OPE, i.e.…”

“…[36], for RG flows localized to the defect/boundary. Rigorous proofs of these c-theorems have appeared for p = 2 [37,38] and p = 4 [39], because in these cases the Weyl anomaly includes an Euler density term intrinsic to the defect/boundary, allowing the proofs of refs. [12,17] to be suitably adapted.…”