24,143 research outputs found

### Convexity, translation invariance and subadditivity for $g$-expectations and related risk measures

Under the continuous assumption on the generator $g$, Briand et al.
[Electron. Comm. Probab. 5 (2000) 101--117] showed some connections between $g$
and the conditional $g$-expectation
$({\mathcal{E}}_g[\cdot|{\mathcal{F}}_t])_{t\in[0,T]}$ and Rosazza Gianin
[Insurance: Math. Econ. 39 (2006) 19--34] showed some connections between $g$
and the corresponding dynamic risk measure $(\rho^g_t)_{t\in[0,T]}$. In this
paper we prove that, without the additional continuous assumption on $g$, a
$g$-expectation ${\mathcal{E}}_g$ satisfies translation invariance if and only
if $g$ is independent of $y$, and ${\mathcal{E}}_g$ satisfies convexity (resp.
subadditivity) if and only if $g$ is independent of $y$ and $g$ is convex
(resp. subadditive) with respect to $z$. By these conclusions we deduce that
the static risk measure $\rho^g$ induced by a $g$-expectation ${\mathcal{E}}_g$
is a convex (resp. coherent) risk measure if and only if $g$ is independent of
$y$ and $g$ is convex (resp. sublinear) with respect to $z$. Our results extend
the results in Briand et al. [Electron. Comm. Probab. 5 (2000) 101--117] and
Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19--34] on these subjects.Comment: Published in at http://dx.doi.org/10.1214/105051607000000294 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org

### Higher Spin Entanglement Entropy

In this paper, we develop a perturbation formulation to calculate the single
interval higher spin R$\acute{e}$nyi and entanglement entropy for two
dimensional conformal field theory with $\mathcal{W}_{\infty}(\lambda)$
symmetry. The system is at finite temperature and is deformed by higher spin
chemical potential. We manage to compute higher spin R$\acute{e}$nyi entropy
with various spin deformations up to order $\mathcal{O}(\mu^2)$. For spin 3
deformation, we calculate exact higher spin R$\acute{e}$nyi entropy up to
$\mathcal{O}(\mu^4)$. When $\lambda=3$, in the large $c$ limit, we find perfect
match with tree level holographic higher spin entanglement entropy up to order
$\mu^4$ obtained by the Wilson line prescription. We also find quantum
corrections to higher spin entanglement entropy which is beyond tree level
holographic results. The quantum correction is universal at order $\mu^4$ in
the sense that it is independent of $\lambda$. Our computation relies on a
multi-valued conformal map from $n$-sheeted Riemann surface $\mathcal{R}_n$ to
complex plane and correlation functions of primary fields on complex plane. The
method can be applied to general conformal field theories with $\mathcal{W}$
symmetry.Comment: 49 pages,1 figure, to be published in JHE

### Real-time Correlators and Hidden Conformal Symmetry in Kerr/CFT Correspondence

In this paper, we study the real-time correlators in Kerr/CFT, in the low
frequency limit of generic non-extremal Kerr(-Newman) black holes. From the low
frequency scattering of Kerr-Newman black holes, we show that for the uncharged
scalar scattering, there exists hidden conformal symmetry on the solution
space. Similar to Kerr case, this suggests that the Kerr-Newman black hole is
dual to a two-dimensional CFT with central charges $c_L=c_R=12J$ and
temperatures $T_L=\frac{(r_++r_-)-Q^2/M}{4\pi a}, T_R=\frac{r_+-r_-}{4\pi a}$.
Using the Minkowski prescription, we compute the real-time correlators of
charged scalar and find perfect match with CFT prediction. We further discuss
the low-frequency scattering of photons and gravitons by Kerr black hole and
find that their retarded Green's functions are in good agreement with CFT
prediction. Our study supports the idea that the hidden conformal symmetry in
the solution space is essential to Kerr/CFT correspondence.Comment: 15 pages, Latex; typos corrected, references updated; minor
correction, published versio

### Classical static final state of collapse with supertranslation memory

The Kerr metric models the final classical black hole state after
gravitational collapse of matter and radiation. Any stationary metric which is
close to the Kerr metric has been proven to be diffeomorphic to it. Now, finite
supertranslation diffeomorphisms are symmetries which map solutions to
inequivalent solutions as such diffeomorphisms generate conserved superrotation
charges. The final state of gravitational collapse is therefore parameterized
by its mass, angular momentum and supertranslation field, signaled by its
conserved superrotation charges.
In this paper, we first derive the angle-dependent energy conservation law
relating the asymptotic value of the supertranslation field of the final state
to the details of the collapse and subsequent evolution of the system. We then
generate the static solution with an asymptotic supertranslation field and we
study some of its properties. Up to a caveat, the deviation from the
Schwarzschild metric could therefore be predicted on a case-by-case basis from
accurate modeling of the angular dependence of the ingoing and outgoing energy
fluxes leading to the final state.Comment: 35 pages, 7 figures, published version (only refs updated with
respect to v2

### Vacua of the gravitational field

The Poincar\'e invariant vacuum is not unique in quantum gravity. The BMS
supertranslation symmetry originally defined at null infinity is spontaneously
broken and results in inequivalent Poincar\'e vacua. In this paper we construct
the unique vacua which interpolate between past and future null infinity in BMS
gauge and which are entirely characterized by an arbitary Goldstone boson
defined on the sphere which breaks BMS invariance. We show that these vacua
contain a defect which carries no Poincar\'e charges but which generically
carries superrotation charges. We argue that there is a huge degeneracy of
vacua with multiple defects. We also present the single defect vacua with its
canonically conjugated source which can be constructed from a Liouville boson
on the stereographic plane. We show that positivity of the energy forces the
stress-tensor of the boson to vanish as a boundary condition. Finite
superrotations, which turn on the sources, are therefore physically ruled out
as canonical transformations around the vacua. Yet, infinitesimal
superrotations are external symplectic symmetries which are associated with
conserved charges which characterize the Goldstone boson.Comment: Accepted in JHEP, comments added, 34 page

### Hidden Conformal Symmetry and Quasi-normal Modes

We provide an algebraic way to calculate the quasi-normal modes of a black
hole, which possesses a hidden conformal symmetry. We construct an infinite
tower of quasi-normal modes from the highest-weight mode, in a simple and
elegant way. For the scalar, the hidden conformal symmetry manifest itself in
the fact that the scalar Laplacian could be rewritten in terms of the $SL(2,R)$
quadratic Casimir. For the vector and the tensor, the hidden conformal symmetry
acts on them through Lie derivatives. We show that for three-dimensional black
holes, with appropriate combination of the components the radial equations of
the vector and the tensor could be written in terms of the Lie-induced
quadratic Casimir. This allows the algebraic construction of the quasi-normal
modes feasible. Our results are in good agreement with the previous study.Comment: 23 pages; references added; typos corrected, more clarifications,
published versio

### Strong Subadditivity and Emergent Surface

In this paper, we introduce two bounds which we call the Upper Differential
Entropy and the Lower Differential Entropy for an infinite family of
intervals(strips) in quantum field theory. The two bounds are equal provided
that the theory is translational invariant and the entanglement entropy varies
smoothly with respect to the interval. When the theory has a holographic dual,
strong subadditivity of entanglement entropy indicates that there is always an
emergent surface whose gravitational entropy is exactly given by the bound.Comment: 18 pages, 8 figures, replace "residual entropy" to "differential
entropy

### R\'enyi Mutual Information for Free Scalar in Even Dimensions

We compute the R\'enyi mutual information of two disjoint spheres in free
massless scalar theory in even dimensions higher than two. The spherical twist
operator in a conformal field theory can be expanded into the sum of local
primary operators and their descendants. We analyze the primary operators in
the replicated scalar theory and find the ones of the fewest dimensions and
spins. We study the one-point function of these operators in the conical
geometry and obtain their expansion coefficients in the OPE of spherical twist
operators. We show that the R\'enyi mutual information can be expressed in
terms of the conformal partial waves. We compute explicitly the R\'enyi mutual
information up to order $z^d$, where $z$ is the cross ratio and $d$ is the
spacetime dimension.Comment: 29 pages; More discussion on the partition function of primary
operators, the contribution from spin-1 operator has been correcte

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