The Segal-Bargmann Transform on Classical Matrix Lie Groups

We study the complex-time Segal-Bargmann transform $\mathbf{B}_{s,\tau}^{K_N}$ on a compact type Lie group $K_N$, where $K_N$ is one of the following classical matrix Lie groups: the special orthogonal group $\mathrm{SO}(N,\mathbb{R})$, the special unitary group $\mathrm{SU}(N)$, or the compact symplectic group $\mathrm{Sp}(N)$. Our work complements and extends the results of Driver, Hall, and Kemp on the Segal-Bargman transform for the unitary group $\mathrm{U}(N)$. We provide an effective method of computing the action of the Segal-Bargmann transform on \emph{trace polynomials}, which comprise a subspace of smooth functions on $K_N$ extending the polynomial functional calculus. Using these results, we show that as $N\to\infty$, the finite-dimensional transform $\mathbf{B}_{s,\tau}^{K_N}$ has a meaningful limit $\mathscr{G}_{s,\tau}^{(\beta)}$ (where $\beta$ is a parameter associated with $\mathrm{SO}(N,\mathbb{R})$, $\mathrm{SU}(N)$, or $\mathrm{Sp}(N)$), which can be identified as an operator on the space of complex Laurent polynomials

Holographic Butterfly Effect and Diffusion in Quantum Critical Region

We investigate the butterfly effect and charge diffusion near the quantum phase transition in holographic approach. We argue that their criticality is controlled by the holographic scaling geometry with deformations induced by a relevant operator at finite temperature. Specifically, in the quantum critical region controlled by a single fixed point, the butterfly velocity decreases when deviating from the critical point. While, in the non-critical region, the behavior of the butterfly velocity depends on the specific phase at low temperature. Moreover, in the holographic Berezinskii-Kosterlitz-Thouless transition, the universal behavior of the butterfly velocity is absent. Finally, the tendency of our holographic results matches with the numerical results of Bose-Hubbard model. A comparison between our result and that in the $O(N)$ nonlinear sigma model is also given.Comment: 41 pages, 7 figures, minor revisions, refs adde

Holographic Shear Viscosity in Hyperscaling Violating Theories without Translational Invariance

In this paper we investigate the ratio of shear viscosity to entropy density, $\eta/s$, in hyperscaling violating geometry with lattice structure. We show that the scaling relation with hyperscaling violation gives a strong constraint to the mass of graviton and usually leads to a power law of temperature, $\eta/s\sim T^\kappa$. We find the exponent $\kappa$ can be greater than two such that the new bound for viscosity raised in arXiv:1601.02757 is violated. Our above observation is testified by constructing specific solutions with UV completion in various holographic models. Finally, we compare the boundedness of $\kappa$ with the behavior of entanglement entropy and conjecture a relation between them.Comment: 38 pages, 8 figures: 1 appendix added, 2 figures added, 1 references adde

High-Energy Gamma-Rays from GRB X-ray Flares

The recent detection of X-ray flares during the afterglow phase of gamma-ray bursts (GRBs) suggests an inner-engine origin, at radii inside the forward shock. There must be inverse Compton (IC) emission arising from such flare photons scattered by forward shock afterglow electrons when they are passing through the forward shock. We find that this IC emission produces high energy gamma-ray flares, which may be detected by AGILE, GLAST and ground-based TeV telescopes. The anisotropic IC scattering between flare photons and forward shock electrons does not affect the total IC component intensity, but cause a time delay of the IC component peak relative to the flare peak. The anisotropic scattering effect may also weaken, to some extent, the suppression effect of the afterglow intensity induced by the enhanced electron cooling due to flare photons. We speculate that this IC component may already have been detected by EGRET from a very strong burst--GRB940217. Future observations by GLAST may help to distinguish whether X-ray flares originate from late central engine activity or from external shocks.Comment: 4 pages, Contributed talk presented at "The First GLAST Symposium", Feb.5-8 2007, Stanford Universit

On the magnetization of gamma-ray burst blast waves

The origin of magnetic fields that permeate the blast waves of gamma-ray bursts (GRBs) is a long-standing problem. The present paper argues that in four GRBs revealing extended emission at >100 MeV, with follow-up in the radio, optical and X-ray domains at later times, this magnetization can be described as the partial decay of the micro-turbulence that is generated in the shock precursor. Assuming that the bulk of the extended emission >100 MeV can be interpreted as synchrotron emission of shock accelerated electrons, we model the multi-wavelength light curves of GRB 090902B, GRB 090323, GRB 090328 and GRB 110731A, using a simplified then a full synchrotron calculation with power-law-decaying microturbulence \epsilon_B \propto t^{\alpha_t} (t denotes the time since injection through the shock, in the comoving blast frame). We find that these models point to a consistent value of the decay exponent -0.5 < \alpha_t < -0.4.Comment: 8 pages, 4 figures - discussion added, conclusions unchanged - version to appear in MNRA