M5-branes and Wilson Surfaces

We investigate the M5-brane description of the Wilson surface operators in six-dimensional (2,0) superconformal field theory from AdS/CFT correspondence. We consider the Wilson surface operators in high-dimensional representation, whose description could be M5-brane string soliton solutions in $AdS_7\times S^4$ background. We construct such string soliton solutions from the covariant M5-brane equations of motion and discuss their properties. The supersymmetry analysis shows that these solutions are half-BPS. We also discuss the subtle issue on the boundary terms.Comment: 30 pages, Latex; little revision;Typos corrected, references added, JHEP published versio

Hausdorff dimension of certain sets arising in continued fraction expansions

AbstractThis paper is concerned with the fractional dimensions of some sets of points with their partial quotients obeying some restrictions in their continued fraction expansions. The Hausdorff dimension of the following set, which shares a dichotomy law according to Borel–Bernstein's theorem, is completely determinedE(ϕ)={x∈[0,1):an(x)⩾ϕ(n), i.o. n}. Our result significantly strengthens the results of Good and Lúczak

A 7.0\% Determination of the Baryon Fraction in the Intergalactic Medium from Localized Fast Radio Bursts

The dispersion measure (DM)--redshift relation of fast radio bursts (FRBs) has been proposed as a potential new tool for probing intergalactic medium (IGM) and for studying cosmology. However, the poor knowledge of the baryon fraction in the IGM ($f_{\mathrm{IGM}}$) and its degeneracy with cosmological parameters impose restrictions on the cosmological applications of FRBs. Furthermore, DMs contributed by the IGM ($\mathrm{DM_{IGM}}$) and host galaxy ($\mathrm{DM_{host}}$), the important cosmological quantities, cannot be exactly extracted from observations, which would bring uncontrolled systematic uncertainties in FRB cosmology. In this work, we use seventeen localized FRBs to constrain $f_{\mathrm{IGM}}$ and possibly its redshift evolution. Other cosmological probes such as Type Ia supernovae, baryon acoustic oscillations, and cosmic microwave background radiation are combined to break parameter degeneracy. Taking into account the probability distributions of $\mathrm{DM_{IGM}}$ and $\mathrm{DM_{host}}$ derived from the the IllustrisTNG simulation, we obtain a robust measurement of $f_{\mathrm{IGM}}=0.857\pm0.060$, representing a precision of 7.0\%. We find that there is no strong evidence for the redshift dependence of $f_{\mathrm{IGM}}$ at the current observational data level. The rapid progress in localizing FRBs will significantly improve the constraints on $f_{\mathrm{IGM}}$.Comment: 9 pages, 3 figures, 2 table

On the fast Khintchine spectrum in continued fractions

For $x\in [0,1)$, let $x=[a_1(x), a_2(x),...]$ be its continued fraction expansion with partial quotients ${a_n(x), n\ge 1}$. Let $\psi : \mathbb{N} \rightarrow \mathbb{N}$ be a function with $\psi(n)/n\to \infty$ as $n\to \infty$. In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set E(\psi):=\Big{x\in [0,1): \lim_{n\to\infty}\frac{1}{\psi(n)}\sum_{j=1}^n\log a_j(x)=1\Big} is completely determined without any extra condition on $\psi$.Comment: 10 page