# Charged Quantum Fields in AdS$_2$

### Submission summary

 As Contributors: Jorrit Kruthoff Arxiv Link: https://arxiv.org/abs/1906.00924v2 (pdf) Date accepted: 2019-10-14 Date submitted: 2019-10-09 02:00 Submitted by: Kruthoff, Jorrit Submitted to: SciPost Physics Academic field: Physics Specialties: Gravitation, Cosmology and Astroparticle Physics High-Energy Physics - Theory 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 flux-carrying modes that resembles the standard flat space S-matrix. 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

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Published as SciPost Phys. 7, 054 (2019)

We thank the referees for their careful assessment of our manuscript and their comments and suggestions. Below we will respond to their requested changes.

Report 1 1. We thank the referee for the comment. The discussion in our paper is at the level of QFT on a fixed curved background. As such, we believe it would obscure the presentation to focus on the dynamical gravitational theory in main body of the text. The only place where we mention this briefly is in the discussion. We address this in point 2 below. 2. We appreciate the referee’s suggestion. We will include citations [16], [57], 1608.07018 in our discussion section. The level at which we discuss backreaction in the discussion section is schematic. We believe a detailed analysis of such models deserves a study in and of its own right. As such, we would prefer to defer this to future work. 3. We thank the referee for bringing up this point. The two fermionic vacua are defined on the global SL(2,R) invariant geometry and are shown to preserve the SL(2,R) symmetry. Thus, our fermionic vacua are not directly connected to vacua adapted to the presence of horizons. In particular they correspond to neither the Frolov-Thorne vacuum (which requires the introduction of a mirror), or the Unruh vacuum (whose stress tensor diverges at the past horizon). 4. As requested by the referee, we have added the necessary citations (hep-th/9905099, [11], 0906.1819, [15], and 0908.3909) at the appropriate point in the main body of the text (i.e. page 3 before “Another motivation....”) 5. These hypergeometric function have appeared repeatedly in the AdS/CFT literature, in the discussion of NHEK, and dS geometries for example. They are well known in the community and it would be impossible to compile all the relevant papers that have looked into these solutions. Having said that, we can confirm that it is indeed the case that the computation in the near region of the mentioned papers is equivalent to ours. 6. We thank the referee for mentioning this. We have added reference to [11] to the `Dirac Sea and ergosphere’ discussion right above section 6. 7. Reference [5] is indeed cited in the discussion of the microscopic models and we mention it presents a real weight for the fermion operator. Therefore, it does not obviously apply to our discussion. Other more promising models are discussed in this section.

Report 3 1. As the referee points out, it is the universal cover that we are mostly interested in. Upon consideration of the referee’s remark, and to make this more manifest, we have changed the wording in section 2.1 from “If one is interested...” to “We will mostly be interested...”. 2. We thank the referee for their remark. We added a comment below 3.4 where we state that the field strength is indeed SL(2,R) invariant. 3. We have added a footnote incorporating the referee’s comment. 4. We are grateful to the referee for picking this up. We have made the requested correction. Also we have changed r to calligraphic R, as we have used r for a coordinate elsewhere. 5. We appreciate the referee’s remark. In general the gluing can be made smooth as discussed near equation (4.13). For the specific case of the model in section 5, we actually address this issue in the “Closing the open quantum system” subsection. 6. We thank the referee for pointing out the typo. It has been fixed in the current version.