We study whether the relations between the Weyl anomaly, entanglement entropy (EE), and thermal entropy of a two-dimensional (2D) conformal field theory (CFT) extend to 2D boundaries of 3D CFTs, or 2D defects of D ≥ 3 CFTs. The Weyl anomaly of a 2D boundary or defect defines two or three central charges, respectively. One of these, b, obeys a c-theorem, as in 2D CFT. For a 2D defect, we show that another, d2, interpreted as the defect's "conformal dimension," must be non-negative if the Averaged Null Energy Condition (ANEC) holds in the presence of the defect. We show that the EE of a sphere centered on a planar defect has a logarithmic contribution from the defect fixed by b and d2. Using this and known holographic results, we compute b and d2 for 1/2-BPS surface operators in the maximally supersymmetric (SUSY) 4D and 6D CFTs. The results are consistent with b's c-theorem. Via free field and holographic examples we show that no universal "Cardy formula" relates the central charges to thermal entropy.

We use entanglement entropy to define a central charge associated to a twodimensional defect or boundary in a conformal field theory (CFT). We present holographic calculations of this central charge for several maximally supersymmetric CFTs dual to eleven-dimensional supergravity in Anti-de Sitter space, namely the M5-brane theory with a Wilson surface defect and three-dimensional CFTs related to the M2-brane theory with a boundary. Our results for the central charge depend on a partition of N M2-branes ending on M M5-branes. For the Wilson surface, the partition specifies a representation of the gauge algebra, and we write our result for the central charge in a compact form in terms of the algebra’s Weyl vector and the representation’s highest weight vector. We explore how the central charge scales with N and M for some examples of partitions. In general the central charge does not scale as M 3 or N 3/2, the number of degrees of freedom of the M5- or M2-brane theory at large M or N , respectively.

We compute the holographic entanglement entropy contribution from planar two-dimensional defects in six-dimensional N = (2, 0) superconformal field theory, holographically dual to probe M2-and M5-branes in AdS 7 × S 4 . In particular, we test the viability of the universal contribution of the defect to entanglement entropy as a candidate C-function. We find that this coefficient is not monotonic under defect renormalization group flows triggered by the vacuum expectation value of a marginal operator. Another candidate C-function, the on-shell action inside the entanglement wedge, monotonically decreases under the flows we study.

Abstract:We use holographic duality to study the entanglement entropy (EE) of Conformal Field Theories (CFTs) in various spacetime dimensions d, in the presence of various deformations: a relevant Lorentz scalar operator with constant source, a temperature T , a chemical potential µ, a marginal Lorentz scalar operator with source linear in a spatial coordinate, and a circle-compactified spatial direction. We consider EE between a strip or sphere sub-region and the rest of the system, and define the "entanglement density" (ED) as the change in EE due to the deformation, divided by the sub-region's volume. Using the deformed CFTs above, we show how the ED's dependence on the strip width or sphere radius, L, is useful for characterizing states of matter. For example, the ED's small-L behavior is determined either by the dimension of the perturbing operator or by the first law of EE. For Lorentz-invariant renormalization group (RG) flows between CFTs, the "area theorem" states that the coefficient of the EE's area law term must be larger in the UV than in the IR. In these cases the ED must therefore approach zero from below as L → ∞. However, when Lorentz symmetry is broken and the IR fixed point has different scaling from the UV, we find that the ED often approaches the thermal entropy density from above, indicating area theorem violation.

For a perturbation of the state of a Conformal Field Theory (CFT), the response of the entanglement entropy is governed by the so-called "first law" of entanglement entropy, in which the change in entanglement entropy is proportional to the change in energy. Whether such a first law holds for other types of perturbations, such as a change to the CFT Lagrangian, remains an open question. We use holography to study the evolution in time t of entanglement entropy for a CFT driven by a t-linear source for a conserved U (1) current or marginal scalar operator. We find that although the usual first law of entanglement entropy may be violated, a first law for the rates of change of entanglement entropy and energy still holds. More generally, we prove that this first law for rates holds in holography for any asymptotically (d + 1)-dimensional Anti-de Sitter metric perturbation whose t dependence first appears at order z d in the Fefferman-Graham expansion about the boundary at z = 0. CONTENTS