Sphere Packings, Lattices and Groups

We study a two dimensional dilaton gravity system, recently examined by Almheiri and Polchinski, which describes near extremal black holes, or more generally, nearly $AdS_2$ spacetimes. The asymptotic symmetries of $AdS_2$ are all the time reparametrizations of the boundary. These symmetries are spontaneously broken by the $AdS_2$ geometry and they are explicitly broken by the small deformation away from $AdS_2$. This pattern of spontaneous plus explicit symmetry breaking governs the gravitational backreaction of the system. It determines several gravitational properties such as the linear in temperature dependence of the near extremal entropy as well as the gravitational corrections to correlation functions. These corrections include the ones determining the growth of out of time order correlators that is indicative of chaos. These gravitational aspects can be described in terms of a Schwarzian derivative effective action for a reparametrization.

Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space

The entanglement entropy is a fundamental quantity, which characterizes the correlations between sub-systems in a larger quantum-mechanical system. For two sub-systems separated by a surface the entanglement entropy is proportional to the area of the surface and depends on the UV cutoff, which regulates the short-distance correlations. The geometrical nature of entanglement-entropy calculation is particularly intriguing when applied to black holes when the entangling surface is the black-hole horizon. I review a variety of aspects of this calculation: the useful mathematical tools such as the geometry of spaces with conical singularities and the heat kernel method, the UV divergences in the entropy and their renormalization, the logarithmic terms in the entanglement entropy in four and six dimensions and their relation to the conformal anomalies. The focus in the review is on the systematic use of the conical singularity method. The relations to other known approaches such as ’t Hooft’s brick-wall model and the Euclidean path integral in the optical metric are discussed in detail. The puzzling behavior of the entanglement entropy due to fields, which non-minimally couple to gravity, is emphasized. The holographic description of the entanglement entropy of the blackhole horizon is illustrated on the two- and four-dimensional examples. Finally, I examine the possibility to interpret the Bekenstein–Hawking entropy entirely as the entanglement entropy.

/pdf/entanglement-entropy-of-black-holes-4zr6icmdts.pdf

Entanglement Entropy of Black Holes

We derive the general form for a three-dimensional scale-invariant field theory with $\mathcal{N}=6$ supersymmetry, $SU(4)$ R-symmetry and a $U(1)$ global symmetry. The results can be written in terms of a 3-algebra in which the triple product is not antisymmetric. For a specific choice of 3-algebra we obtain the $\mathcal{N}=6$ theories that have been recently proposed as models for M2-branes in an ${\mathbb{R}}^{8}/{\mathbb{Z}}_{k}$ orbifold background.

Three-algebras and N=6 Chern-Simons gauge theories

In the context of quantifying entanglement we study those functions of a multipartite state which do not increase under the set of local transformations. A mathematical characterization of these monotone magnitudes is presented. They are then related to optimal strategies of conversion of shared states. More detailed results are presented for pure states of bipartite systems. It is show that more than one measure are required simultaneously in order to quantify completely the non-local resources contained in a bipartite pure state, while examining how this fact does not hold in the so-called asymptotic limit. Finally, monotonicity under local transformations is proposed as the only natural requirement for measures of entanglement.

On the characterization of entanglement

We evaluate the one loop determinant of matter multiplet fields of N=4 supergravity in the near horizon geometry of quarter BPS black holes, and use it to calculate logarithmic corrections to the entropy of these black holes using the quantum entropy function formalism. We show that even though individual fields give non-vanishing logarithmic contribution to the entropy, the net contribution from all the fields in the matter multiplet vanishes. Thus logarithmic corrections to the entropy of quarter BPS black holes, if present, must be independent of the number of matter multiplet fields in the theory. This is consistent with the microscopic results. During our analysis we also determine the complete spectrum of small fluctuations of matter multiplet fields in the near horizon geometry.

/pdf/logarithmic-corrections-to-extremal-black-hole-entropy-from-4x5839zoxp.pdf

Logarithmic corrections to extremal black hole entropy from quantum entropy function

We compute logarithmic corrections to the entropy of supersymmetric extremal black holes in N=4 and N=8 supersymmetric string theories and find results in perfect agreement with the microscopic results. In particular these logarithmic corrections vanish for quarter BPS black holes in N=4 supersymmetric theories, but has a finite coefficient for 1/8 BPS black holes in the N=8 supersymmetric theory. On the macroscopic side these computations require evaluating the one loop determinant of massless fields around the near horizon geometry, and include, in particular, contributions from dynamical four dimensional gravitons propagating in the loop. Thus our analysis provides a test of one loop quantum gravity corrections to the black hole entropy, or equivalently of the AdS_2/CFT_1 correspondence. We also extend our analysis to N=2 supersymmetric STU model and make a prediction for the logarithmic correction to the black hole entropy in that theory.

/pdf/logarithmic-corrections-to-n-4-and-n-8-black-hole-entropy-a-17o2f0yk3c.pdf

Logarithmic Corrections to N=4 and N=8 Black Hole Entropy: A One Loop Test of Quantum Gravity

The original proposal of Dijkgraaf, Verlinde and Verlinde for the quarter BPS dyon partition function in heterotic string theory on T 6 is known to correctly produce the degeneracy of dyons of torsion 1, i.e. dyons for which gcd(Q Λ P)=1. We propose a generalization of this formula for dyons of arbitrary torsion. Our proposal satisfies the constraints coming from S-duality invariance, wall crossing formula, black hole entropy and the gauge theory limit. Furthermore using our proposal we derive a general wall crossing formula that is valid even when both the decay products are non-primitive half-BPS dyons.

Partition functions of torsion >1 dyons in heterotic string theory on T 6

We compute logarithmic corrections to the entropy of supersymmetric extremal black holes in $$ \mathcal{N} = {4} $$
 and $$ \mathcal{N} = {8} $$
 supersymmetric string theories and find results in perfect agreement with the microscopic results. In particular these logarithmic corrections vanish for quarter BPS black holes in $$ \mathcal{N} = {4} $$
 supersymmetric theories, but has a finite coefficient for 1/8 BPS black holes in the $$ \mathcal{N} = {8} $$
 supersymmetric theory. On the macroscopic side these computations require evaluating the one loop determinant of massless fields around the near horizon geometry, and include, in particular, contributions from dynamical four dimensional gravitons propagating in the loop. Thus our analysis provides a test of one loop quantum gravity corrections to the black hole entropy, or equivalently of the AdS
 2/CF T
 1 correspondence. We also extend our analysis to $$ \mathcal{N} = {2} $$
 supersymmetric STU model and make a prediction for the logarithmic correction to the black hole entropy in that theory.

/pdf/logarithmic-corrections-to-mathcal-n-4-and-mathcal-n-8-black-4wfpvajpbv.pdf

Logarithmic corrections to $$ \mathcal{N} = {4} $$ and $$ \mathcal{N} = {8} $$ black hole entropy: a one loop test of quantum gravity

In this paper we study the BMS symmetry of the celestial OPE of two positive helicity gravitons in Einstein theory in four dimensions. The celestial OPE is obtained by Mellin transforming the scattering amplitude in the (holomorphic) collinear limit. The collinear limit at leading order gives the singular term of the celestial OPE. We compute the first subleading correction to the OPE by analysing the four graviton scattering amplitude directly in Mellin space. The subleading term can be written as a linear combination of BMS descendants with the OPE coefficients determined by BMS algebra and the coefficient of the leading term in the OPE. This can be done by defining a suitable BMS primary state. We find that among the descendants, which appear at the first subleading order, there is one which is created by holomorphic supertranslation with simple pole on the celestial sphere.

/pdf/bms-symmetry-of-celestial-ope-31ziniu6ec.pdf

Shamik Banerjee

Papers

Logarithmic corrections to extremal black hole entropy from quantum entropy function

Logarithmic Corrections to N=4 and N=8 Black Hole Entropy: A One Loop Test of Quantum Gravity

Partition functions of torsion >1 dyons in heterotic string theory on T 6

Logarithmic corrections to $$ \mathcal{N} = {4} $$ and $$ \mathcal{N} = {8} $$ black hole entropy: a one loop test of quantum gravity

BMS symmetry of celestial OPE