We extend and refine recent results on Rényi entropy in two-dimensional con-formal field theories at large central charge. To do so, we examine the effects of higher spin symmetry and of allowing unequal left and right central charges, at leading and sub-leading order in large total central charge. The result is a straightforward generalization of previously derived formulae, supported by both gravity and CFT arguments. The preceding statements pertain to CFTs in the ground state, or on a circle at unequal left- and right-moving temperatures. For the case of two short intervals in a CFT ground state, we derive certain universal contributions to Rényi and entanglement entropy from Virasoro primaries of arbitrary scaling weights, to leading and next-to-leading order in the interval size; this result applies to any CFT. When these primaries are higher spin currents, such terms are placed in one-to-one correspondence with terms in the bulk 1-loop determinants for higher spin gauge fields propagating on handlebody geometries.
We extend and refine recent results on Rényi entropy in two-dimensional con-formal field theories at large central charge. To do so, we examine the effects of higher spin symmetry and of allowing unequal left and right central charges, at leading and sub-leading order in large total central charge. The result is a straightforward generalization of previously derived formulae, supported by both gravity and CFT arguments. The preceding statements pertain to CFTs in the ground state, or on a circle at unequal left- and right-moving temperatures. For the case of two short intervals in a CFT ground state, we derive certain universal contributions to Rényi and entanglement entropy from Virasoro primaries of arbitrary scaling weights, to leading and next-to-leading order in the interval size; this result applies to any CFT. When these primaries are higher spin currents, such terms are placed in one-to-one correspondence with terms in the bulk 1-loop determinants for higher spin gauge fields propagating on handlebody geometries.
In this paper we investigate the holographic Rényi entropy of two disjoint intervals on complex plane with small cross ratio x for conformal field theory with W symmetry in the ground state, which could be dual to a higher spin AdS3 gravity. We focus on the cases of W 3 and W 4 symmetries. In order to see the nontrivial contributions from the W fields, we calculate the Rényi entropy in the expansion of x to order x 8 in both the gravity and the CFT sides. In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields. In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from W fields, besides the ones from Virasoro Verma module. In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the Rényi entropy. The classical and 1-loop results in the gravity and the CFT sides are in exact match. This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of Rényi entanglement entropy, including the quantum correction, in both the AdS3 gravity and the higher spin AdS3 gravity.
Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.
Higher spin entanglement and W N WN conformal blocks
Two-dimensional conformal field theories with extended W -symmetry algebras have dual descriptions in terms of weakly coupled higher spin gravity in AdS3 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 Δ ∼ c. When two of the operators obey cΔ≪1 , we show by explicit computation that the vacuum conformal block is computed by a bulk Wilson line probing an asymptotically AdS3 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.
We calculate the contributions of a general non-vacuum conformal family to Rényi entropy in two-dimensional conformal field theory (CFT). The primary operator of the conformal family can be either non-chiral or chiral, and we denote its scaling dimension by Δ . For the case of two short intervals on a complex plane, we expand the Rényi mutual information by the cross ratio x to order x 2 Δ +2 . For the case of one interval on a torus with low temperature, we expand the Rényi entropy by q =exp(−2π β /L), with β being the inverse temperature and L being the spatial period, to order q Δ +2 . To make the result meaningful, we require that the scaling dimension Δ cannot be too small. For two intervals on a complex plane we need Δ > 1, and for one interval on a torus we need Δ > 2. We work in the small Newton constant limit on the gravity side and so a large central charge limit on the CFT side, and find matches of gravity and CFT results.
In this paper, we study the holographic R\'enyi entropy of a large interval on a circle at high temperature for the two-dimensional conformal field theory (CFT) dual to pure AdS3 gravity. In the field theory, the R\'enyi entropy is encoded in the CFT partition function on n-sheeted torus connected with each other by a large branch cut. As proposed by Chen and Wu [Large interval limit of R\'enyi entropy at high temperature, arXiv:1412.0763], the effective way to read the entropy in the large interval limit is to insert a complete set of state bases of the twist sector at the branch cut. Then the calculation transforms into an expansion of four-point functions in the twist sector with respect to ${e}^{\ensuremath{-}\frac{2\ensuremath{\pi}TR}{n}}$. By using the operator product expansion of the twist operators at the branch points, we read the first few terms of the R\'enyi entropy, including the leading and next-to-leading contributions in the large central charge limit. Moreover, we show that the leading contribution is actually captured by the twist vacuum module. In this case by the Ward identity the four-point functions can be derived from the correlation function of four twist operators, which is related to double interval entanglement entropy. Holographically, we apply the recipe in [T. Faulkner, The entanglement R\'enyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221] and [T. Barrella et al., Holographic entanglement beyond classical gravity, J. High Energy Phys. 09 (2013) 109] to compute the classical R\'enyi entropy and its one-loop quantum correction, after imposing a new set of monodromy conditions. The holographic classical result matches exactly with the leading contribution in the field theory up to ${e}^{\ensuremath{-}4\ensuremath{\pi}TR}$ and l6, while the holographical one-loop contribution is in exact agreement with next-to-leading results in field theory up to ${e}^{\ensuremath{-}\frac{6\ensuremath{\pi}TR}{n}}$ and l4 as well.
In this paper, we calculate the Rényi entropy of one single interval on a circle at finite temperature in 2D CFT. In the low temperature limit, we expand the thermal density matrix level by level in the vacuum Verma module, and calculate the first few leading terms in e −π/T L explicitly. On the other hand, we compute the same Rényi entropy holographically. After considering the dependence of the Rényi entropy on the temperature, we manage to fix the interval-independent constant terms in the classical part of holographic Rényi entropy. We furthermore extend the analysis in [9] to higher orders and find exact agreement between the results from field theory and bulk computations in the large central charge limit. Our work provides another piece of evidence to support holographic computation of Rényi entropy in AdS3/CFT2 correspondence, even with thermal effect.