6. Five-dimensional null and timelike supersymmetric geometries Hyper-K¨ ahler four-dimensional base space.
It is well-known that **in** the time-like class the Killing vector can become null at some loci of the four-dimensional Hyper-K¨ ahler base manifold. Indeed, this is precisely what happens at the horizon of a **black** hole solution, or **in** the smooth five-dimensional solutions of [108] which will be considered **in** this chapter. A change **in** the causal character of the Killing vector is paralleled by a change of the supersymmetry conditions that the spinor satisfies. However, **in** all the solutions constructed so far this can happen only **in** some regions of codimension at most one of the space-time manifold, that are typically surfaces. Our goal **in** this chapter is to construct five-dimensional solutions **in** minimal Super- gravity whose Killing vector, which is generically time-like, becomes null at a point of the space-time manifold where all the derivatives of its norm vanish. The null condition is a closed condition, so if the norm of the spinor is a continuous function then the spinor can become null only on a closed subset of the manifold. Since all the derivatives of the norm vanish at the point where the Killing vector field becomes null, we conclude that the norm is not an analytic real function at this point, otherwise the Killing vector field would vanish on an open set, contradicting the fact that the null condition on the norm is a closed condition. Therefore, if we are able to construct a solution where the norm of the Killing vector field and all its derivatives vanish at some locus, we will know that the norm is not a real analytic function at those points. **In** either case, having a Killing vector whose norm has an infinite number of derivatives vanishing at a point is the closest scenario to having a null-spinor on an open set. These would be solutions that may mix, **in** a non-trivial way, the local classification that distinguishes between the time-like and the null classes. **In** this respect we consider smooth Supergravity solutions with a multi-center Gibbons-Hawking (GH) base manifold that asymptote to AdS 3 × S 2 . These can be gener-

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Unifying General Relativity and Quantum mechanics is not a theorist’s fantasy. Even if both frameworks have been experimentally tested with great precision **in** their domains of validity, the picture of the Universe remains scattered and their incompatibility gives rise to unsolved paradoxes. First, from observation, the matter well-described by those theories accounts only for 15% of the overall density and 68% of the energy allowing the expansion of the Universe to accelerate is unknown. Their names, dark matter and dark energy, are the only characteristics widely shared by the scientific community. Second, the understanding of our world is limited by the scales where the theoretical description breaks down. If the Standard model is naturally protected from divergences, General Relativity is not. General Relativity has two inherent singular behaviours delimiting the edge of the Universe as we know it today. First, the initial singularity, or big bang, corresponds to the furthest spacetime slice when the Universe was Planck size. Second, the **theory** contains also “**black**-hole-type” singularities where the curvature of spacetime diverges under strong deformations of compact masses. Those two singularities lie at the common theoretical border between General Relativity and Quantum Mechanics and must be resolved by a quantum **theory** of gravity. Because time travel back to the first nanoseconds of our Universe is not yet planned, **black** **holes** are the main theoretical and experimental laboratory for testing quantum theories of gravity such as **String** **Theory**.

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2.2.1 Percolation from the gravitational sector to the matter sector
While several quantum gravity proposals (**string** theories with extra-dimensions [100], non-commutative field **theory** [101] and Hoˇ rava gravity [77, 78] for instance) hint towards Lorentz violations appearing **in** the ultraviolet, i.e., at small scales, it is not completely ob- vious that we may freely break Lorentz invariance **in** the gravitational sector without messing things out **in** the matter sector. Indeed, one could reasonably argue that any gravitational Lo- rentz violation would percolate into particle physics, and since there are very tight constraints on Lorentz violations **in** the matter sector, such as those mentioned **in** sections 2.1 and 2.2, then one would conclude that gravitational Lorentz violations must also be small. There exist counter-examples to this argument, however, and infrared (i.e., at long distances) Lorentz invariance could still be preserved by different mechanisms, even if gravity presents Lorentz violations **in** the ultraviolet. For instance, it could be protected by supersymmetry [102], or it could be an emergent symmetry appearing at low energies [103, 104] for different reasons, such as a renormalization group flow of the system leading to infrared Lorentz invariance [102, 105], or it could even be an accidental symmetry as suggested by [106]. It has also been proposed that a classically Lorentz invariant matter sector may co-exist with a Lorentz violating gravity sector provided that their interaction is mediated by higher-dimensional operators and is suppressed by a high-energy scale [107]. We therefore conclude that testing Lorentz invariance **in** the gravitational sector is still a reasonable enterprise since there is an ample spectrum of mechanisms able to control Lorentz violating percolation, despite the stringent constraints from the matter sector.

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3.1 The **String** **Theory** dilaton
As explained **in** appendix ( A ), for **black** hole solutions of ungauged four-dimensional su- pergravity, the hyperscalars are truncated to a constant value. **In** principle, the dilaton belongs to the universal hypermultiplet, and therefore one may naively conclude that it should be constant for every **black** hole solution. However, **in** the process of obtaining the effective N = 2 four-dimensional supergravity **in** its standard form ( A.1 ), several rescal- ings and redefinitons are performed on the original ten-dimensional fields. **In** addition, since we are considering all the perturbative corrections to the Special Kähler sector, the corresponding N = 2 ungauged supergravity action is not the effective compactification **theory** of Type-IIA **String** **Theory** at tree level, so we should expect more intrincate redef- initions. Our purpose is to show now that Type-IIA quantum **black** **holes** have a constant ten-dimensional dilaton and a constant Calabi-Yau manifold volume, and we will do this **in** two steps:

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/
r 2 near the horizon, with all other components Oðr 0 Þ. V. CONCLUSION
**In** this paper we obtained a general expression for the action of the null Melvin twist, constructed a series of charged and rotating asymptotically Schro¨dinger **black** hole solutions of type IIB supergravity, found a 5D trunca- tion of the 10D IIB **theory** adapted to these solutions, and examined some of the salient features of their geometry and thermodynamics. Along the way we discovered that these space-times inherit extremal limits from their parent AdS spaces, but that some of their features—like the radii of curvature of the near-horizon regions—are not inherited, leading to a potentially interesting class of new solutions. The fact that these systems are explicitly embedded **in** IIB **string** **theory** allows us to identify the boundary CFT as the

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iosif.bena@cea.fr, hagen.triendl@cea.fr, bert.vercnocke@cea.fr
Abstract
We establish the relation between the structure governing supersymmetric and non- supersymmetric four- and five-dimensional **black** **holes** and multicenter solutions and Calabi- Yau flux compactifications of M-**theory** and type IIB **string** **theory**. We find that the known BPS and almost-BPS multicenter **black** hole solutions can be interpreted as GKP compact- ifications with (2,1) and (0,3) imaginary self-dual flux. We also show that the most general GKP compactification leads to new classes of BPS and non-BPS multicenter solutions. We explore how these solutions fit into N = 2 truncations, and elucidate how supersymmetry becomes camouflaged. As a necessary tool **in** our exploration we show how the fields **in** the largest N = 2 truncation fit inside the six-torus compactification of eleven-dimensional supergravity.

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8 Discussion
Understanding the dynamics of multiple M5 branes has been one of the most challenging and interesting issues **in** **string** **theory** for quite a number of years. There has been a huge effort **in** understanding how M5-brane theories can describe strongly-coupled gauge theories **in** four dimensions. Our purpose **in** this paper has been to study what should, perhaps, be one of the simplest avatars of the M5-brane field **theory**: The (1+1)-dimensional MSW CFT that comes from wrappings of an M5 brane on a very ample divisor of a Calabi-Yau manifold. This seemingly simple CFT remains enigmatic, almost twenty years after it was first shown to be able to encode microstate structure of four-dimensional **black** **holes** [ 15 ]. **In** this paper we have considered M5 branes wrapping 4-cycles **in** T 6 and T 2 ×K3, but our resulting M-**theory** solutions can be trivially extended to compactifications with a more general field content.

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Classification of elliptically fibered Calabi-Yau manifolds
It has been shown by Gross [30] that the number of distinct topological types of elliptically fibered Calabi-Yau threefolds is finite up to birational equivalence. Finiteness of the set of topologically distinct elliptically-fibered Calabi-Yau threefolds is shown **in** [6] using minimal surface **theory** and the fact that the Weierstrass form for an elliptic fibration over a fixed base has a finite number of possible distinct singularity structures. These arguments, however, do not give a clear picture of how such compactifications can be sys- tematically classified. A complete mathematical classification of elliptically fibered Calabi-Yau threefolds would be helpful **in** understanding the range of F-**theory** compactifications. The analogue of this question for four dimen- sions, while probably much more difficult, would be of even greater interest, since at this time we have very little handle on the scope of the space of four dimensional supergravity theories which can be realized through F-**theory** compactifications on Calabi-Yau fourfolds.

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K1R1, K2R1; Table 3 ).
**In** general, the larger the value of N BH , the lower the ﬁnal
total mass of the cluster. This is because higher N BH expands
the cluster more, and a more expanded cluster loses more mass due to galactic tides. The exact amount of expansion, mass loss, and the ﬁnal cluster mass also depends on the metallicity (Z) of the cluster, since the metallicity controls the wind-driven mass loss and the resulting mass of the BH population. Higher Z leads to the formation of lower-mass BHs, and as a result the overall expansion due to BH ejections is reduced. The adopted wind prescription yields a similar effect. While the strong wind prescription leads to a higher mass loss at early times compared to the weak wind prescription, the latter leads to the formation of more massive BHs **in** a cluster than the former. As a result, the same initial cluster under the weak wind assumption eventually undergoes more expansion when the energy production **in** the center is dominated by BH dynamics. This increased expansion leads to an increased rate of star loss **in** models with weak winds compared to those with strong winds, which is re ﬂected **in** the ﬁnal N **in** these clusters (Table 3 ).

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A fter exploring static or stationary non-singular **black** **holes** **in** the previous chapters, let us now turn to dynamical ones and describe the results obtained with P. Binétruy and A. Helou **in** [ 19 ]. Astrophysical **black** **holes** are by essence dynamical: they are formed after a gravitational collapse (contrarily to Schwarzschild’s eternal **black** **holes** and its maximal extension presented **in** Sec. 1 of Chap. 2 ), and are also thought to radiate via Hawking’s evaporation [ 66 ]. A few solutions to Einstein’s equations describe the dynamical formation of a **black** hole, such as Vaidya’s metric presented **in** Chap. 1 , but they remain classical. **In** this Chapter, we aim at studying effective metrics which incorporate quantum effects both near the outer horizon, to describe Hawking’s evaporation, and **in** the high curvature regime near r = 0 **in** order to avoid the singularity. A way of doing so consists **in** studying closed trapping horizons, whose dynamics can implement these two types of quantum effects. **In** Sec. 1 of this Chapter, we will start by deriving general conditions for the existence of singularity-free closed trapping horizons. We will then implement and study **in** Sec. 2 explicit models of closed trapping horizons inspired by the work of Hayward, Frolov and Bardeen, before discussing the behaviour of null geodesics **in** those models (Sec. 3 ). Finally, **in** Sec. 4 we solve Einstein’s equations **in** reverse to obtain the expression of the energy-momentum tensor for these models and analyse the weakest of energy conditions, the null energy condition (NEC). We find an explicit metric that recovers a null outgoing fluid mimicking Hawking radiation on I + , without having to make junctions. We

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• Naoki Yoshida and his research group for their hospitality during my internship at the University
of Tokyo. ありがとうございます
• The current and former PhD students at IAP, especially bureau 10 for the main motivation to come to IAP daily, Caterina, Siwei, and Alba for support **in** difficult times, and Melanie and Rebekka for help and support at the beginning.

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u(0) E , and the existence of a large family of solutions with the latter property follows from the construction of u .
We remark that Theorems 1 and 2 are completely independent from the behavior of linear waves at low frequency. **In** fact, we do not even use the boundedness **in** time of solutions for the wave equation, assuming merely that they grow at most exponentially (which is trivially true **in** our case); this suffices since O(h ∞ ) remainders overcome such growth for t = O(log(1/h)). If a boundedness statement is available, then our results can be extended to all times, though the corresponding rate of decay stays fixed for t log(1/h) because of the O(h ∞ ) term.

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‡
Institut de physique th´ eorique, Universit´ e Paris Saclay, CNRS, F-91191 Gif-sur-Yvette, France The monodromy relations **in** **string** **theory** provide a powerful and elegant formalism to understand some of the deepest properties of tree-level field **theory** amplitudes, like the color-kinematics duality. This duality has been instrumental **in** tremendous progress on the computations of loop amplitudes **in** quantum field **theory**, but a higher-loop generalisation of the monodromy construction was lacking. **In** this letter, we extend the monodromy relations to higher loops **in** open **string** **theory**. Our construction, based on a contour deformation argument of the open **string** diagram integrands, leads to new identities that relate planar and non-planar topologies **in** **string** **theory**. We write one and two-loop monodromy formulæ explicitly at any multiplicity. **In** the field **theory** limit, at one-loop we obtain identities that reproduce known results. At two loops, we check our formulæ by unitarity **in** the case of the four-point N = 4 super-Yang-Mills amplitude.

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2.1. The NOAR/TITAN and NLMC codes
TITAN is a code designed for warm media (T > a few 10 4 K) optically thick to Compton scattering. It computes the struc- ture of a plane-parallel slab of gas **in** thermal and ionization equilibrium, illuminated by a given spectrum on one or two sides of the slab (Dumont et al. 2000). It takes into account the returning flux using a two-stream approximation to solve the transfer **in** the lines (instead of the escape probability formal- ism). This code is coupled with a Monte Carlo code, NOAR, which takes into account Compton and inverse Compton dif- fusions **in** any geometry (Abrassart 2000). NOAR uses the lo- cal fractional ion abundances and the temperature provided by TITAN, while NOAR provides TITAN with the local Compton gains and losses **in** each layer. The Compton heating-cooling rate is indeed dominated by energy losses of photons at high energies ( >25 keV), not considered by TITAN. The coupling thus allows one to solve consistently both the global and the local energy balance. NOAR also allows computing the fluores- cence line profiles which are significantly Compton-broadened **in** the case of strong illumination, and the Comptonised reflec- tion spectrum above 25 keV.

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Abstract :
This work is devoted to the mathematical study of the Hawking effect for fermions **in** the setting of the collapse of a rotating charged star. We show that an observer who is located far away from the star and at rest with respect to the Boyer Lindquist coordinates observes the emergence of a thermal state when his proper time goes to infinity. We first introduce a model of the collapse of the star. We suppose that the space-time outside the star is given by the Kerr-Newman metric. The assumptions on the asymptotic behavior of the surface of the star are inspired by the asymptotic be- havior of certain timelike geodesics **in** the Kerr-Newman metric. The Dirac equation is then written using coordinates and a Newman-Penrose tetrad which are adapted to the collapse. This coordinate system and tetrad are based on the so called simple null geodesics. The quantization of Dirac fields **in** a globally hyperbolic space-time is described. We formulate and prove a theorem about the Hawking effect **in** this set- ting. The proof of the theorem contains a minimal velocity estimate for Dirac fields that is slightly stronger than the usual ones and an existence and uniqueness result for solutions of a characteristic Cauchy problem for Dirac fields **in** the Kerr-Newman space-time. **In** an appendix we construct explicitly a Penrose compactification of block I of the Kerr-Newman space-time based on simple null geodesics.

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Indeed, since the theory is local and since the parent bulk theory on the sphere determines already the operator product expansion of any two fields, the descendant field theory in the b[r]

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For example, social media users might be connected because they directly communicate with each other, or
– as is the case **in** this paper: they may just be linked because they follow each other.
3.3 Survival analysis
Assuming there are observations with variables (also known as covariates) **in** the dataset , and each observation (a pull request) is one row **in** , then describes the characteristics of observation . An observation contains information **in** a pull request: user who sends the pull request (sender), who receives it (receiver), their networks, repositories, organizations, and so on. There are

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n levels will be largely quenched at high density with exception of transitions from H-
and He-like ions.
We are presently involved **in** a systematic project to compute atomic parameters taking into account the plasma effects arising at high densities. **In** particular, Deprince et al. (2018a) have estimated the effects of the plasma environment on the atomic pa- rameters associated with K-vacancy states **in** He- and Li-like oxygen ions using multi- configuration Dirac–Fock and Breit–Pauli approaches. These computations have been carried out assuming a time-averaged Debye–Hückel potential for both the electron– nucleus and electron–electron interactions. Plasma effects on the ionization potential, excitation thresholds, transition wavelengths, and radiative emission rates are reported therein for O vi and O vii. The same methodology is also being applied to compute the plasma effects on the atomic structure and radiative and Auger decay rates of the K-lines of Fe xvii – Fe xxv (Deprince et al. 2018b).

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