SciPost Submission Page
Charged Quantum Fields in AdS$_2$
by Dionysios Anninos, Diego M. Hofman, Jorrit Kruthoff
This is not the current version.
Submission summary
As Contributors:  Jorrit Kruthoff 
Arxiv Link:  https://arxiv.org/abs/1906.00924v1 (pdf) 
Date submitted:  20190729 02:00 
Submitted by:  Kruthoff, Jorrit 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider quantum field theory near the horizon of an extreme Kerr black hole. In this limit, the dynamics is well approximated by a tower of electrically charged fields propagating in an $SL(2,\mathbb{R})$ invariant AdS$_2$ geometry endowed with a constant, symmetry preserving background electric field. At large charge the fields oscillate near the AdS$_2$ boundary and no longer admit a standard Dirichlet treatment. From the Kerr black hole perspective, this phenomenon is related to the presence of an ergosphere. We discuss a definition for the quantum field theory whereby we 'UV' complete AdS$_2$ by appending an asymptotically two dimensional Minkowski region. This allows the construction of a novel observable for the fluxcarrying modes that resembles the standard flat space Smatrix. We relate various features displayed by the highly charged particles to the principal series representations of $SL(2,\mathbb{R})$. These representations are unitary and also appear for massive quantum fields in dS$_2$. Both fermionic and bosonic fields are studied. We find that the free charged massless fermion is exactly solvable for general background, providing an interesting arena for the problem at hand.
Ontology / Topics
See full Ontology or Topics database.Current status:
Submission & Refereeing History
You are currently on this page
Reports on this Submission
Anonymous Report 3 on 2019914 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1906.00924v1, delivered 20190914, doi: 10.21468/SciPost.Report.1176
Strengths
1. explicit and thorough analysis of charged bosons and fermions in AdS2
2. identification of SL(2,R) principal series as wave functions that allow flux leaking through the horizon.
3. Smatrix observable for the states associated to principal series
Weaknesses
see requested changes
Report
Motivated by the near horizon geometry of an extremal Kerr black hole, the paper conducts a thorough analysis of scalar and fermionic fields charged under a U(1) gauge field in a fixed AdS2 background. The quantum mechanics of the system is naturally organized by the SL(2,R) symmetry of the background. In particular, the authors gave explicit formulae for the symmetry charges and wave functions in relation to SL(2,R) representation theory, where the "nonconventional" principal series has a physical interpretation as describing waves that carry flux through the Poincare horizon of AdS2. The authors then went on to define an Smatrixlike observable for such states. The paper is very concrete and wellwritten. The computations presented here will be useful for a variety future investigations (e.g. AdS2/CFT1, similar story in dS2), among which the connection to Kerr/CFT is particularly interesting. I recommend this paper for publication after the requested changes are considered.
Requested changes
1. The authors mentioned in section 2.1 that the rep theory for SL(2,R) and that for its universal cover differ. In fact, for Lorentzian AdS2, the latter is more natural since the killing vectors do not generate close loops (there's no compact U(1) subgroup). The authors should explain/justify why they restrict to SL(2,R) rep theory. (The authors actually consider representations of the universal cover of SL(2,R) later in section 6.1 ...)
2. The background gauge field in (3.4) is invariant under SL(2,R) only after a gauge transformation. Perhaps this is worth mentioning below (3.4).
3. In (3.7) and below (3.8), the constant part of $\alpha_\mu$ is ambiguous with the authors' definition. This ambiguity should be fixed by demanding the U(1) and SL(2,R) currents to be orthogonal.
4. Below (3.15), by "...for large enough values of r, the parameter λ becomes complex..." the authors probably meant " for small enough values of $r$..."
5. in section 4.2, the authors' procedure of defining the "Smatrix" for principal series involve gluing a flat space region to the boundary of AdS2, which is not a smooth procedure. It would be good to comment on how this observable would depend on regularizations of the gluing.
6. typo in the last paragraph before section 5.3, "... a particularly explicit treatment in encountered ..." should be "... a particularly explicit treatment encountered ..."
Anonymous Report 2 on 201998 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1906.00924v1, delivered 20190908, doi: 10.21468/SciPost.Report.1156
Report
The paper considers the quantization of charged fields in AdS2 in the presence of a background electric field. This problem comes from thinking about the nearhorizon geometry of an extremal Kerr black hole. The paper emphasizes the representationtheoretic significance of various states. In particular, states carrying nontrivial flux at the asymptotic boundary of AdS2 are shown to correspond to the principal series representations of SL(2,R). These representations are usually discarded on the basis of their energy being unbounded from below. However, the paper makes a convincing case that they should be considered in this setup. The paper also defines an interesting new quantity which corresponds to the Smatrix of these states.
The paper is very wellwritten and provides several new insights into the Kerr/CFT correspondence and quantum field theory in general. I recommend that it can be published in its current form.
Anonymous Report 1 on 201998 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1906.00924v1, delivered 20190908, doi: 10.21468/SciPost.Report.1155
Strengths
1 a novel UV completion of AdS2 gravity with an asymptotically flat region, which allows a definition of Smatrix for modes with complex conformal weight
2 it showed that the wave equation of a charged massless fermion on the general background is exactly solvable.
3 the quantization and the definition of vacua is discussed
4 It gives a nice review of SL(2) generators and their representations, and explicitly points out the relation between the principal series representation and the flux carrying modes.
Weaknesses
While the paper is generally well written, it would be better to make proper references to previous work in various places. In particular, the connection between the results in the paper and previous work on superradiance and geometries with SL(2)xU(1) isometries is not discussed.
Report
The authors study quantum field theory on the near horizon throat of extreme Kerr black holes, in the language of AdS2 gravity. In particular, the authors focus on the perturbative modes with complex conformal weight and carrying flux through the Poincare AdS2. Such modes were previously discussed in reference [10] and [15], where the connections with superradiance were observed. This paper provides an interesting new handle on the problem by embedding AdS2 in an asymptotically flat spacetime (4.13), which allows for a definition of the Smatrix, a novel observable for the fluxcarrying modes. It is also noticed that charged massless fermions are exactly solvable on the new background (4.13). Furthermore, the quantization of fermions and different vacua are discussed.
To conclude, the paper provides an interesting new approach to the problem of superradiance on extremal Kerr black holes, and contains a collection of interesting observations. I would like to recommend the paper for publication after the authors consider the suggestions in the "requested changes" section.
Requested changes
1. The discussion in the paper would be strengthened if there is a gravitational theory for which the general background (4.13) is a solution. This might be related to point 2 below. It would be good if the authors could comment on this.
2. Regarding the backreaction, the Maxwelldilatongravity in [16] seems to be relevant, as it is obtained from dimensional reduction of Einstein gravity on the background eq.(2.1). Besides, a detailed holographic dictionary for yet another Maxwelldilatongravity has been worked out in arXiv 1608.07018. The three models [16], [57] and arXiv 1608.07018, mainly differ in the coupling between the dilaton and the gauge field. As a result, the phase space of these models are different. As such difference might play a role in the discussion of backreaction, it would be good if the authors can comment on these other models as well.
3The authors defined two vacua. It would be helpful to comment on the connections to the FrolovThorne vacuum or the Unruh vacuum for the Kerr black holes.
4I think the authors should properly cite previous works on superradiance of extremal Kerr black holes. For example, Reference [11], [15] and arXiv 0908.3909.
5Solutions to classical wave equations for scalars, fermions, photons and gravitons on a finite temperature version of eq.(2.1) have been discussed previously in reference [15] and 0908.3909. It seems that the calculation in Poincare AdS2 (for example section 3.2) is equivalent to the near region calculation of the aforementioned papers, in the strictly extremal limit. It would be good to clarify the relations.
6I think [11] should be mentioned in the discussion of "Dirac sea and the ergosphere", where connections between Fermisea, superradiance and the ergosphere has been discussed for 5d extremal black holes.
7On page 30, it is mentioned that possible microscopic construction would be an SYK typemodel with global U(1) symmetry. A notable model with complex fermions were discussed in reference [5] of the paper and arXiv 1612.00849. It would be helpful if the authors could comment on whether this model might be relevant to the current paper.