arXiv subjects

Per Kraus

Publications and source records attributed to Per Kraus.

At least 37 records · Page 2Linked to original sources

arXiv2019

Renormalization of gravitational Wilson lines

We continue the study of the Wilson line representation of conformal blocks in two-dimensional conformal field theory; these have an alternative interpretation as gravitational Wilson lines in the context of the AdS$_3$/CFT$_2$ correspondence. The gravitational Wilson line involves a path-ordered exponential of the stress tensor, and its expectation value can be computed perturbatively in an expansion in inverse powers of the central charge $c$. The short-distance singularities which occur in the associated stress tensor correlators require systematic regularization and renormalization prescriptions, whose consistency with conformal Ward identities presents a subtle problem. The regularization used here combines dimensional regularization and analytic continuation. Representation theoretic arguments, based on SL(2,R) current algebra, predict an exact result for the Wilson line anomalous dimension and, by building on previous work, we verify that the perturbative calculations using our regularization and renormalization prescriptions reproduce the exact result to order $1/c^3$ included. We also discuss a related, but somewhat simpler, Wilson line in Wess-Zumino-Witten models that yields current algebra conformal blocks, and we emphasize the distinction between Wilson lines constructed out of non-holomorphic and purely holomorphic currents.

hep-th↗

arXiv2019

Late time Wilson lines

In the AdS$_3$/CFT$_2$ correspondence, physical interest attaches to understanding Virasoro conformal blocks at large central charge and in a kinematical regime of large Lorentzian time separation, $t\sim c$. However, almost no analytical information about this regime is presently available. By employing the Wilson line representation we derive new results on conformal blocks at late times, effectively resumming all dependence on $t/c$. This is achieved in the context of "light-light" blocks, as opposed to the richer, but much less tractable, "heavy-light" blocks. The results exhibit an initial decay, followed by erratic behavior and recurrences. We also connect this result to gravitational contributions to anomalous dimensions of double trace operators by using the Lorentzian inversion formula to extract the latter. Inverting the stress tensor block provides a pedagogical example of inversion formula machinery.

hep-th↗

arXiv2019

Light-state Dominance from the Conformal Bootstrap

We derive forms of light-state dominance for correlators in CFT$_d$, making precise the sense in which correlators can be approximated by the contribution of light operator exchanges. Our main result is that the four-point function of operators with dimension $Δ$ is approximated, with bounded error, by the contribution of operators with scaling dimension below $Δ_c > 2Δ$ in the appropriate OPE channel. Adapting an existing modular invariance argument, we use crossing symmetry to show that the heavy-state contribution is suppressed by a relative factor of $e^{2Δ-Δ_c}$. We extend this result to the first sheet and derivatives of the correlator. Further exploiting technical similarities between crossing and modular invariance, we prove analogous results for the $2d$ partition function along the way. We then turn to effective field theory in gapped theories and AdS/CFT, and make some general comments about the effect of integrating out heavy particles in the bulk. Combining our bounds with the Lorentzian OPE inversion formula we show that, under certain conditions, light-state dominance implies that integrating out heavy exchanges leads to higher-derivative couplings suppressed at large $Δ_{gap}$.

hep-th↗

arXiv2018

Cutoff AdS$_3$ versus the $T\bar{T}$ deformation

A recent proposal relates two dimensional holographic conformal field theories deformed by the integrable $T\bar{T}$ flow to AdS$_3$ with a finite radial cutoff. We investigate this proposal by studying perturbative correlation functions on the two sides. For low point correlators of the stress tensor, we successfully match the deformed CFT results at large central charge to bulk results obtained in classical pure gravity. The deformed CFT also provides definite predictions for loop corrections in the bulk. We then include matter fields in the bulk. To reproduce the classical bulk two-point function of a scalar operator we show that the deformed CFT needs to be augmented with double trace scalar operators, with the $T\bar{T}$ operator yielding corrections corresponding to loops in the bulk.

hep-th↗

arXiv2017

Black holes from CFT: Universality of correlators at large c

Two-dimensional conformal field theories at large central charge and with a sufficiently sparse spectrum of light states have been shown to exhibit universal thermodynamics. This thermodynamics matches that of AdS$_3$ gravity, with a Hawking-Page transition between thermal AdS and the BTZ black hole. We extend these results to correlation functions of light operators. Upon making some additional assumptions, such as large $c$ factorization of correlators, we establish that the thermal AdS and BTZ solutions emerge as the universal backgrounds for the computation of correlators. In particular, Witten diagrams computed on these backgrounds yield the CFT correlators, order by order in a large $c$ expansion, with exponentially small corrections. In pure CFT terms, our result is that thermal correlators of light operators are determined entirely by light spectrum data. Our analysis is based on the constraints of modular invariance applied to the torus two-point function.

hep-th↗

arXiv2017

Witten Diagrams for Torus Conformal Blocks

We give a holographic description of global conformal blocks in two dimensional conformal field theory on the sphere and on the torus. We show that the conformal blocks for one-point functions on the torus can be written as Witten diagrams in thermal AdS. This is accomplished by deriving a general conformal Casimir equation for global conformal blocks, and showing that Witten diagrams obey the same equation. We study the semi-classical limit of n-point conformal blocks, and show that these equal the action of a network of bulk world-lines obeying appropriate geodesic equations. We give an alternate description in the Chern-Simons formulation of 3D gravity, where the conformal blocks are described by networks of Wilson lines, and argue that these formulations are equivalent.

hep-th↗

arXiv2017

Anomalous dimensions from quantum Wilson lines

We study the self-energy of a gravitating point particle in AdS$_3$, and compare to operator dimensions in CFT$_2$. In particular, we compute the one and two loop diagram contributions to the expectation value of an open Wilson line in the SL(2,R)$\times$ SL(2,R) Chern-Simons formulation of AdS$_3$ gravity. This gives the two-point function of CFT primary operators to second order in a large $c$ expansion, and hence yields the scaling dimension $h(j,c)$ as a function of the SL(2,R) spin $j$. Comparison to CFT is made in the context of constructing Virasoro representations starting from representations of SL(2,R) current algebra. Our Wilson line computations follow the framework advanced recently by Fitzpatrick et. al., which is based on earlier work by H. Verlinde. We encounter some renormalization scheme ambiguities at the two-loop level which we are not able to fully resolve, hampering a definitive comparison with CFT expressions at this order.

hep-th↗

arXiv2016

Can a quantum critical state represent a blackbody?

The blackbody theory of Planck played a seminal role in the development of quantum theory at the turn of the past century. A blackbody cavity is generally thought to be a collection of photons in thermal equilibrium; the radiation emitted is at all wavelengths, and the intensity follows a scaling law, which is Planck's characteristic distribution law. These photons arise from non-interacting normal modes. Here we suggest that certain quantum critical states when heated emit "radiation" at all wavelengths and satisfy all the criteria of a blackbody. An important difference is that the "radiation" does not necessarily consist of non-interacting photons, but also emergent relativistic bosons or fermions. The examples we provide include emergent relativistic fermions at a topological quantum critical point. This perspective on a quantum critical state may be illuminating in many unforeseen ways.

cond-mat.str-el↗

arXiv2016

A Cardy Formula for Three-Point Coefficients: How the Black Hole Got its Spots

Modular covariance of torus one-point functions constrains the three point function coefficients of a two dimensional CFT. This leads to an asymptotic formula for the average value of light-heavy-heavy three point coefficients, generalizing Cardy's formula for the high energy density of states. The derivation uses certain asymptotic properties of one-point conformal blocks on the torus. Our asymptotic formula matches a dual AdS_3 computation of one point functions in a black hole background. This is evidence that the BTZ black hole geometry emerges upon course-graining over a suitable family of heavy microstates.

hep-th↗

arXiv2016

Holographic conformal blocks from interacting Wilson lines

We present a simple prescription for computing conformal blocks and correlation functions holographically in AdS$_3$ in terms of Wilson lines merging at a bulk vertex. This is shown to reproduce global conformal blocks and heavy-light Virasoro blocks. In the case of higher spin theories the space of vertices is in one-to-one correspondence with the space of ${\cal W}_N$ conformal blocks, and we show how the latter are obtained by explicit computations.

hep-th↗

arXiv2016

Nature abhors a horizon

The information paradox can be resolved if we recognize that the wavefunctional in gravity $Ψ[g]$ should be considered on the {\it whole} of superspace, the space of possible $g$. The largeness of the Bekenstein entropy implies a vast space of gravitational solutions, which are conjectured to be fuzzball configurations. In the WKB approximation, the wavefunctional for a collapsing shell is oscillatory in a small region of superspace, and the classical approximation picks out this part. But the wavefunctional will be damped (`under the barrier') in the remainder of this vast superspace. We perform a simple computation to show that at the threshold of black hole formation, the barrier is lowered enough to make the latter part oscillatory; this alters the classical evolution and avoids horizon formation.

hep-th↗

arXiv2015

General Results for Higher Spin Wilson Lines and Entanglement in Vasiliev Theory

We develop tools for the efficient evaluation of Wilson lines in 3D higher spin gravity, and use these to compute entanglement entropy in the hs$[λ]$ Vasiliev theory that governs the bulk side of the duality proposal of Gaberdiel and Gopakumar. Our main technical advance is the determination of SL(N) Wilson lines for arbitrary $N$, which, in suitable cases, enables us to analytically continue to hs$[λ]$ via $N \rightarrow -λ$. We apply this result to compute various quantities of interest, including entanglement entropy expanded perturbatively in the background higher spin charge, chemical potential, and interval size. This includes a computation of entanglement entropy in the higher spin black hole of the Vasiliev theory. These results are consistent with conformal field theory calculations. We also provide an alternative derivation of the Wilson line, by showing how it arises naturally from earlier work on scalar correlators in higher spin theory. The general picture that emerges is consistent with the statement that the SL(N) Wilson line computes the semiclassical $W_N$ vacuum block, and our results provide an explicit result for this object.

hep-th↗

arXiv2015

Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks

We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a "geodesic Witten diagram," which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.

hep-th↗

arXiv2015

Semiclassical Virasoro Blocks from AdS$_3$ Gravity

We present a unified framework for the holographic computation of Virasoro conformal blocks at large central charge. In particular, we provide bulk constructions that correctly reproduce all semiclassical Virasoro blocks that are known explicitly from conformal field theory computations. The results revolve around the use of geodesic Witten diagrams, recently introduced in arXiv:1508.00501, evaluated in locally AdS$_3$ geometries generated by backreaction of heavy operators. We also provide an alternative computation of the heavy-light semiclassical block -- in which two external operators become parametrically heavy -- as a certain scattering process involving higher spin gauge fields in AdS$_3$; this approach highlights the chiral nature of Virasoro blocks. These techniques may be systematically extended to compute corrections to these blocks and to interpolate amongst the different semiclassical regimes.

hep-th↗

arXiv2015

Worldline approach to semi-classical conformal blocks

We extend recent results on semi-classical conformal blocks in 2d CFT and their relation to 3D gravity via the AdS/CFT correspondence. We consider four-point functions with two heavy and two light external operators, along with the exchange of a light operator. By explicit computation, we establish precise agreement between these CFT objects and a simple picture of particle worldlines joined by cubic vertices propagating in asymptotically AdS$_3$ geometries (conical defects or BTZ black holes). We provide a simple argument that explains this agreement.

hep-th↗

arXiv2014

Higher Spin Entanglement and W_N Conformal Blocks

Two-dimensional conformal field theories with extended $\cal{W}$-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 $Δ\sim c\,$. When two of the operators obey ${Δ\over c} \ll 1\,$, 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.

hep-th↗

arXiv2014

A new spin on entanglement entropy

We argue that the usual notions of thermodynamic and entanglement entropy have novel analogs in the context of higher spin theories. In particular, the Wald and Ryu-Takayanagi formulas have natural higher spin extensions that we work out and study. On the CFT side, just as standard entanglement entropy in CFT_2 can be computed from twist field correlators, we demonstrate that by introducing corresponding operators carrying higher spin charge we can precisely reproduce our results from the bulk. We also show that the first law for entanglement entropy implies the linearized field equations for the metric and higher spin fields, generalizing recent work on deriving the linearized Einstein equations from the first law.

hep-th↗

arXiv2013

Probing higher spin black holes

We study the propagation of scalar fields on various backgrounds in three dimensional higher spin gravity. Our main emphasis is on obtaining the bulk-boundary propagator, which can be efficiently computed using group theory and higher spin gauge symmetry and from which we can extract scalar two-point functions in the dual CFT. As an illustration, we obtain a simple closed form expression for the propagator in a particular spin-3 deformation of AdS_3. In the case of higher spin black holes, we prove on general grounds that the propagator respects an imaginary time periodicity consistent with the thermal nature of the black hole; in doing so we make progress in understanding the group exponentiation of the higher spin Lie algebra hs[λ], and its center. We also explicitly compute the propagator in the black hole background at first order in the higher spin charge. Evaluated on the Lorentzian section, the result is consistent with an interpretation in which the black hole has two causally disconnected boundary components, as is the case for the BTZ black hole.

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