Supersymmetric Rényi entropy

158

Two-dimensional conformal field theories with extended W-symmetry algebras have dual descriptions in terms of weakly coupled higher spin gravity in AdS 3 at large central charge. Observables that can be computed and compared in the two descriptions include Rényi and entanglement entropies, and correlation functions of local operators. We develop techniques for computing these, in a manner that sheds light on when and why one can expect agreement between such quantities on each side of the duality. We set up the computation of excited state Rényi entropies in the bulk in terms of Chern-Simons connections, and show how this directly parallels the CFT computation of correlation functions. More generally, we consider the vacuum conformal block for general operators with ∆ ∼ c. When two of the operators obey ∆ c ≪ 1, we show by explicit computation that the vacuum conformal block is computed by a bulk Wilson line probing an asymptotically AdS 3 background with higher spin fields excited, the latter emerging as the effective bulk description of the excited state produced by the heavy operators. Among other things, this puts a previous proposal for computing higher spin entanglement entropy via Wilson lines on firmer footing, and clarifies its relation to CFT. We also study the corresponding computation in Toda theory and find that this provides yet another independent way to arrive at the same result.

Section: Branch Cutsmentioning

confidence: 91%

Higher spin entanglement and W N $$ {\mathcal{W}}_{\mathrm{N}} $$ conformal blocks

Boer

Castro

Hijano

et al. 2015

158

“…Following [40][41][42], who studied partition functions of three-dimensional N = 2 theories with a U(1) R symmetry on round spheres preserving four supercharges, much recent work has focused on squashed spheres [43][44][45][46][47][48][49][50][51], i.e. three-manifolds that are diffeomorphic to S 3 but carry a more general metric with less symmetry.…”

Section: Applicationsmentioning

confidence: 99%

“…The partition function on a round S 3 with four supercharges residing in SU(2|1) was computed in [40][41][42] using localization. This was subsequently generalized to various squashed spheres [43][44][45][46][47][48][49][50][51], all of which preserve at least two supercharges. 29 By looking at the explicit expressions for the partition functions in these examples, one is lead to the following observations: a) Z S 3 only depends on the squashing through a single complex parameter, usually called b, where b = 1 corresponds to the round S 3 of [40][41][42].…”

Section: Squashed Spheresmentioning

confidence: 99%

See 1 more Smart Citation

The geometry of supersymmetric partition functions

Closset

Dumitrescu

Festuccia

et al. 2014

172

415

We consider supersymmetric field theories on compact manifolds M and obtain constraints on the parameter dependence of their partition functions Z M . Our primary focus is the dependence of Z M on the geometry of M, as well as background gauge fields that couple to continuous flavor symmetries. For N = 1 theories with a U(1) R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. We find that Z M is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over M and Z M is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional N = 2 theories with a U(1) R symmetry, where the necessary geometric structure on M is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that Z M is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S 3 ×S 1 or S 2 ×S 1 , which are related to supersymmetric indices, and manifolds diffeomorphic to S 3 (squashed spheres). In examples where Z M has been calculated explicitly, our results explain many of its observed properties.

“…The authors argued that the same result can be obtained by considering a n-winding Wilson loop on the undeformed sphere and performing the derivative with respect to n, at n = 1. The justification of this correspondence relies on the Renyi entropy as obtained from a matrix model on a n-branched sphere [66], the explicit knowledge of the b dependence in the complicated matrix-model encoding the Wilson loop average [58] and the equivalence between the matrix model on the squashed and the branched spheres under the identification b = √ n. In our case it is tempting to propose a similar recipe, in spite of the fact that we do not have a solid argument to justify it (we do not have an expression for the matrix-model, if any, nor deep information about the behaviour of W B (ν) ν near ν = 1). Nevertheless, we assume that…”

Section: Jhep06(2014)123mentioning

confidence: 99%

BPS Wilson loops and Bremsstrahlung function in ABJ(M): a two loop analysis

Bianchi

Griguolo

Leoni

et al. 2014

193

We study a family of circular BPS Wilson loops in N = 6 super ChernSimons-matter theories, generalizing the usual 1/2-BPS circle. The scalar and fermionic couplings depend on two deformation parameters and these operators can be considered as the ABJ(M) counterpart of the DGRT latitudes defined in N = 4 SYM. We perform a complete two-loop analysis of their vacuum expectation value, discuss the appearance of framing-like phases and propose a general relation with cohomologically equivalent bosonic operators. We make an all-loop proposal for computing the Bremsstrahlung function associated to the 1/2-BPS cusp in terms of these generalized Wilson loops. When applied to our two-loop result it reproduces the known expression. Finally, we comment on the generalization of this proposal to the bosonic 1/6-BPS case.