A Physical Approach to Polya's Conjecture

The similarity between the Polya's conjecture and the Bonomol'nyi bound remind us to consider a physical approach to Polya's conjecture. We conjecture a duality between the waves and the soliton solutions on the surface. We consider the special case in the disc.Comment: 6 pages, no figure

The Entropy of BTZ Black Hole from Loop Quantum Gravity

In this paper, we calculated the entropy of the BTZ black hole in the framework of loop quantum gravity. We got the result that the horizon degrees of freedom can be described by the 2D SO(1,1) punctured BF theory. Finally we got the area law for the entropy of BTZ black hole.Comment: 12 pag

Holograph in noncommutative geometry: Part 1

In this paper, we consider the holograph principle emergent from noncommutative geometry, based on the spectral action principle. We show that under some appropriate conditions, the gravity theory on a manifold with boundary could be equivalent to a gauge theory $SU(N)$ on the boundary. Then an expression for $N$ with the geometrical quantities of the manifold is given. Based on this result, we find that the volume of the manifold and the boundary have some discrete structure. Applying the result to the black hole, we get that the radium of the Schwarzschild black hole is quantized. We also find an explanation why the extremal RN-black hole has zero temperature but with finite entropy.Comment: 9 pages, no figure

Compact Chiral Boson Fields on the Horizon of BTZ Black Hole

In the previous work, it was shown that the degrees of freedom on the horizon of BTZ black hole can be described by two chiral massless scalar fields with opposite chirality. In this paper, we continuous this research. It is found that the scalar field is actually a compact boson field on a circle. The compactness results in the quantization of the black hole radius. Then we quantize the two boson fields and get two abelian Kac-Moody algebras. From the boson field, one can construct the full $W_{1+\infty}$ algebra which was used to classify the BTZ black holes.Comment: 8 pages, comments are welcom

W-hairs of black holes in three dimensional spacetime

In the previous paper (arXiv:1804.09438) we found that the near horizon symmetry algebra of black holes is a subalgebra of the $W_{1+\infty}$ symmetry algebra of quantum Hall fluid in three dimensional spacetime. In this paper, we give a slightly different representation of the former algebra from the latter one. Similar to the horizon fluff proposal, based on the $W_{1+\infty}$ algebra, we count the number of the microstates of the BTZ black holes and obtain the Bekenstein-Hawking entropy.Comment: 7 pages, comments are welcome

Black hole as topological insulator (I): the BTZ black hole case

Black holes are extraordinary massive objects which can be described classically by general relativity, and topological insulators are new orders of matter that could be use to built a topological quantum computer. They seem to be different objects, but in this paper, we claim that the black hole can be considered as kind of topological insulator. For BTZ black hole in three dimensional $AdS_3$ spacetime we give two evidences to support this claim: the first evidence comes from the black hole "membrane paradigm", which says that the horizon of black hole behaves like an electrical conductor. On the other hand, the vacuum can be considered as an insulator. The second evidence comes from the fact that the horizon of BTZ black hole can support two chiral massless scalar field with opposite chirality. Those are two key properties of 2D topological insulator. We also consider the coupling with the electromagnetic field to show that the boundary modes can conduct the electricity. For higher dimensional black hole the first evidence is still valid. So we conjecture that the higher dimensional black hole can also be considered as higher dimensional topological insulators. This conjecture will have far-reaching influences on our understanding of quantum black hole and the nature of gravity.Comment: 7 pages, title changed, adding a section to discuss the coupling with electromagnetic fiel

Central charges for Kerr and Kerr-AdS black holes in diverse dimensions

In the previous work we give a microscopic explanation of the entropy for the BTZ black hole and four-dimensional Kerr black hole based on the massless scalar field theory on the horizon. An essential input is the central charges of those black holes. In this paper, we calculate the central charges for Kerr black holes and Kerr-AdS black holes in diverse dimensions by rewriting the entropy formula in a suggesting way. Then we also give the statistical explanation for the entropy of those black holes based on the scalar field on the horizon which similar to 4D kerr black hole.Comment: 6 pages, comments are welcomed;v2, make connection with the thermodynamic potential;v3, bring the interacting scalar field theory which has a special form of Hamiltonian

Black hole as topological insulator (II): the boundary modes

In the previous paper Ref.[1], it was claimed that the black hole can be considered as a kind of topological insulator. For BTZ black hole in three dimensional $AdS_3$ spacetime two evidences were given to support this claim: the first evidence comes from the black hole "membrane paradigm", and the second evidence comes from the fact that the horizon of BTZ black hole can support two chiral massless scalar field with opposite chirality. Those are two key properties of 2D topological insulator. For higher dimensional black hole the first evidence is still valid but the second fails. In this paper, starting from the boundary BF theory, which can be used to describe the boundary degrees of freedom of black hole in arbitrary dimension, we shown that the isolated horizon of $3+1-$D black hole can support massless scalar field and vector field. Those two fields can be used to construct a massless Dirac field through the $2+1-$dimensional bosonization, which also appears on the boundary of $3+1-$D topological insulators.Comment: 6 pages, adding canonical mass dimension analysi

Sharp Hardy-Littlewood-Sobolev inequality on the upper half space

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $\lambda=n-\alpha$ (that is for the case of $\alpha>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.Comment: This is a detailed version. A short version has been submitte

Reversed Hardy-Littewood-Sobolev inequality

The classical sharp Hardy-Littlewood-Sobolev inequality states that, for $1<p, t<\infty$ and $0<\lambda=n-\alpha <n$ with $1/p +1 /t+ \lambda /n=2$, there is a best constant $N(n,\lambda,p)>0$, such that $$|\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(x)|x-y|^{-\lambda} g(y) dx dy|\le N(n,\lambda,p)||f||_{L^p(\mathbb{R}^n)}||g||_{L^t(\mathbb{R}^n)}$$ holds for all $f\in L^p(\mathbb{R}^n), g\in L^t(\mathbb{R}^n).$ The sharp form is due to Lieb, who proved the existence of the extremal functions to the inequality with sharp constant, and computed the best constant in the case of $p=t$ (or one of them is 2). Except that the case for $p\in ((n-1)/n, n/\alpha)$ (thus $\alpha$ may be greater than $n$) was considered by Stein and Weiss in 1960, there is no other result for $\alpha>n$. In this paper, we prove that the reversed Hardy-Littlewood-Sobolev inequality for $0<p, t<1$, $\lambda<0$ holds for all nonnegative $f\in L^p(\mathbb{R}^n), g\in L^t(\mathbb{R}^n).$ For $p=t$, the existence of extremal functions is proved, all extremal functions are classified via the method of moving sphere, and the best constant is computed.Comment: Comment on recent development on the application of the reversed HLS inequality is added (in introduction section), new references ([7] and [22]) are added. Some typoes are correcte