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arXiv:1611.02485

On co-dimension two defect operators

Jiang Long

Published Nov 8, 2016• 8 citations
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Abstract

Conformal symmetry is broken by a flat or spherical defect operator $\mathcal{D}$. We show that this defect operator, may be identified as a pair of twist operators which are inserted at the tips of its causal diamond. Any $k-$point correlation function in a flat or spherical defect CFT is equivalent to a $(k+2)-$point correlation function. We reproduce one point correlation functions and also solve two point correlation functions in defect CFTs . Mutual Rényi entropy is computed and agrees with previous result in a certain limit. We conjecture there may be universal terms in general co-dimension two defect CFTs.

DOI
10.48550/arxiv.1611.02485

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Conformal symmetry is broken by a flat or spherical defect operator D\mathcal{D}D. We show that this defect operator, may be identified as a pair of twist operators which are inserted at the tips of its causal diamond. Any k−k-k−point correlation function in a flat or spherical defect CFT is equivalent to a (k+2)−(k+2)-(k+2)−point correlation function. We reproduce one point correlation functions and also solve two point correlation functions in defect CFTs . Mutual Rényi entropy is computed and agrees with previous result in a certain limit. We conjecture there may be universal terms in general co-dimension two defect CFTs.
We propose a field theoretic framework for calculating the dependence of Rényi entropies on the shape of the entangling surface in a conformal field theory. Our approach rests on regarding the corresponding twist operator as a conformal defect and in particular, we define the displacement operator which implements small local deformations of the entangling surface. We identify a simple constraint between the coefficient defining the two-point function of the displacement operator and the conformal weight of the twist operator, which consolidates a number of distinct conjectures on the shape dependence of the Rényi entropy. As an example, using this approach, we examine a conjecture regarding the universal coefficient associated with a conical singularity in the entangling surface for CFTs in any number of spacetime dimensions. We also provide a general formula for the second order variation of the Rényi entropy arising from small deformations of a spherical entangling surface, extending Mezei’s results for the entanglement entropy.
We give a recipe for computing correlation functions of the displacement operator localized on a spherical or planar higher dimensional twist defect using AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT. Such twist operators are typically used to construct the nnnth Renyi entropies of spatial entanglement in conformal field theories and are holographically dual to black holes with hyperbolic horizons. The displacement operator then tells us how the Renyi entropies change under small shape deformations of the entangling surface. We explicitly construct the bulk to boundary propagator for the displacement operator insertion as a linearized metric fluctuation of the hyperbolic black hole and use this to extract the coefficient of the displacement operator two point function CD{C}_{D}CD​ in any dimension. The $n\ensuremath{\rightarrow}1$ limit of the twist displacement operator gives the same bulk response as the insertion of a null energy operator in vacuum, which is consistent with recent results on the shape dependence of entanglement entropy and modular energy.
Abstract We study co-dimension two monodromy defects in theories of conformally coupled scalars and free Dirac fermions in arbitrary d dimensions. We characterise this family of conformal defects by computing the one-point functions of the stress-tensor and conserved current for Abelian flavour symmetries as well as two-point functions of the displacement operator. In the case of d = 4, the normalisation of these correlation functions are related to defect Weyl anomaly coefficients, and thus provide crucial information about the defect conformal field theory. We provide explicit checks on the values of the defect central charges by calculating the universal part of the defect contribution to entanglement entropy, and further, we use our results to extract the universal part of the vacuum Rényi entropy. Moreover, we leverage the non-supersymmetric free field results to compute a novel defect Weyl anomaly coefficient in a d = 4 theory of free N \mathcal{N} N N = 2 hypermultiplets. Including singular modes in the defect operator product expansion of fundamental fields, we identify notable relevant deformations in the singular defect theories and show that they trigger a renormalisation group flow towards an IR fixed point with the most regular defect OPE. We also study Gukov-Witten defects in free d = 4 Maxwell theory and show that their central charges vanish.
Abstract This paper is designed to be a practical tool for constructing and investigating two-point correlation functions in defect conformal field theory, directly in physical space, between any two bulk primaries or between a bulk primary and a defect primary, with arbitrary spin. Although geometrically elegant and ultimately a more powerful approach, the embedding space formalism gets rather cumbersome when dealing with mixed symmetry tensors, especially in the projection to physical space. The results in this paper provide an alternative method for studying two-point correlation functions for a generic d-dimensional conformal field theory with a flat p-dimensional defect and d − p = q co-dimensions. We tabulate some examples of correlation functions involving a conserved current, an energy momentum tensor and a Maxwell field strength, while analysing the constraints arising from conservation and the equations of motion. A method for obtaining bulk-to-defect correlators is also explained. Some explicit examples are considered: free scalar theory on ℝp× (ℝq/ℤ2) and a free four dimensional Maxwell theory on a wedge.
The entanglement entropy in three-dimensional conformal field theories (CFTs) receives a logarithmic contribution characterized by a regulator-independent function a(θ) when the entangling surface contains a sharp corner with opening angle θ. In the limit of a smooth surface (θ → π), this corner contribution vanishes as a(θ) = σ(θ − π)2. In arXiv:1505.04804 , we provided evidence for the conjecture that for any d = 3 CFT, this corner coefficient σ is determined by C T , the coefficient appearing in the two-point function of the stress tensor. Here, we argue that this is an instance of a much more general relation connecting the analogous corner coefficient σ n appearing in the nth Rényi entropy and the scaling dimension h n of the corresponding twist operator. In particular, we find the simple relation h n /σ n = (n − 1)π. We show how it reduces to our previous result as n → 1, and explicitly check its validity for free scalars and fermions. With this new relation, we show that as n → 0, σ n yields the coefficient of the thermal entropy, c S . We also reveal a surprising duality relating the corner coefficients of the scalar and the fermion. Further, we use our result to predict σ n for holographic CFTs dual to four-dimensional Einstein gravity. Our findings generalize to other dimensions, and we emphasize the connection to the interval Rényi entropies of d = 2 CFTs.
We study twist operators in higher dimensional CFT’s. In particular, we express their conformal dimension in terms of the energy density for the CFT in a particular thermal ensemble. We construct an expansion of the conformal dimension in power series around n =1, with n being replica parameter. We show that the coefficients in this expansion are determined by higher point correlations of the energy-momentum tensor. In particular, the first and second terms, i.e. the first and second derivatives of the scaling dimension, have a simple universal form. We test these results using holography and free field theory computations, finding agreement in both cases. We also consider the ‘operator product expansion’ of spherical twist operators and finally, we examine the behaviour of correlators of twist operators with other operators in the limit n → 1.