Volume complexity for Janus AdS3 geometries

We investigate the complexity=volume proposal in the case of Janus AdS3 geometries, both at zero and finite temperature. The leading contribution coming from the Janus interface is a logarithmic divergence, whose coefficient is a function of the dilaton excursion. In the presence of the defect, complexity is no longer topological and becomes temperature-dependent. We also study the time evolution of the extremal volume for the time-dependent Janus BTZ black hole. This background is not dual to an interface but to a pair of entangled CFTs with different values of the couplings. At late times, when the equilibrium is restored, the couplings of the CFTs do not influence the complexity rate. On the contrary, the complexity rate for the out-of-equilibrium system is always smaller compared to the pure BTZ black hole background.

Divergent reflections around the photon sphere of a black hole

From any location outside the event horizon of a black hole there are an infinite number of trajectories for light to an observer. Each of these paths differ in the number of orbits revolved around the black hole and in their proximity to the last photon orbit. With simple numerical and a perturbed analytical solution to the null-geodesic equation of the Schwarzschild black hole we will reaffirm how each additional orbit is a factor $$e^{2 \pi }$$ e 2 π closer to the black hole’s optical edge. Consequently, the surface of the black hole and any background light will be mirrored infinitely in exponentially thinner slices around the last photon orbit. Furthermore, the introduced formalism proves how the entire trajectories of light in the strong field limit is prescribed by a diverging and a converging exponential. Lastly, the existence of the exponential family is generalized to the equatorial plane of the Kerr black hole with the exponentials dependence on spin derived. Thereby, proving that the distance between subsequent images increases and decreases for respectively retrograde and prograde images. In the limit of an extremely rotating Kerr black hole no logarithmic divergence exists for prograde trajectories.

A calculation of the Weyl anomaly for 6D conformal higher spins

In this work we continue the study of the one-loop partition function for higher derivative conformal higher spin (CHS) fields in six dimensions and its holographic counterpart given by massless higher spin Fronsdal fields in seven dimensions.In going beyond the conformal class of the boundary round 6-sphere, we start by considering a Ricci-flat, but not conformally flat, boundary and the corresponding Poincaré-Einstein space-filling metric. Here we are able to match the UV logarithmic divergence of the boundary with the IR logarithmic divergence of the bulk, very much like in the known 4D/5D setting, under the assumptions of factorization of the higher derivative CHS kinetic operator and WKB-exactness of the heat kernel of the dual bulk field. A key technical ingredient in this construction is the determination of the fourth heat kernel coefficient b6 for Lichnerowicz Laplacians on both 6D and 7D Einstein manifolds. These results allow to obtain, in addition to the already known type-A Weyl anomaly, two of the three independent type-B anomaly coefficients in terms of the third, say c3 for instance.In order to gain access to c3, and thus determine the four central charges independently, we further consider a generic non Ricci-flat Einstein boundary. However, in this case we find a mismatch between boundary and bulk computations for spins higher than two. We close by discussing the nature of this discrepancy and perspectives for a possible amendment.

Logarithmic Divergence Measure for Fuzzy Matrix and Application

This paper introduces a new divergence measure for a fuzzy matrix with proof of its validity. In addition, the properties are proved for the new fuzzy divergence measure. A method to solve decision making problem is developed by using the proposed fuzzy divergence measure. Finally, the application of this fuzzy divergence measure to decision making is shown using real-life example

Robust anomalous metallic states and vestiges of self-duality in two-dimensional granular In-InOx composites

Many experiments investigating magnetic-field tuned superconductor-insulator transition (H-SIT) often exhibit low-temperature resistance saturation, which is interpreted as an anomalous metallic phase emerging from a ‘failed superconductor’, thus challenging conventional theory. Here we study a random granular array of indium islands grown on a gateable layer of indium-oxide. By tuning the intergrain couplings, we reveal a wide range of magnetic fields where resistance saturation is observed, under conditions of careful electromagnetic filtering and within a wide range of linear response. Exposure to external broadband noise or microwave radiation is shown to strengthen the tendency of superconductivity, where at low field a global superconducting phase is restored. Increasing magnetic field unveils an ‘avoided H-SIT’ that exhibits granularity-induced logarithmic divergence of the resistance/conductance above/below that transition, pointing to possible vestiges of the original emergent duality observed in a true H-SIT. We conclude that anomalous metallic phase is intimately associated with inherent inhomogeneities, exhibiting robust behavior at attainable temperatures for strongly granular two-dimensional systems.

Non-singular vortices with positive mass in 2+1-dimensional Einstein gravity with AdS3 and Minkowski background

In previous work, black hole vortex solutions in Einstein gravity with AdS3 background were found where the scalar matter profile had a singularity at the origin r = 0. In this paper, we find numerically static vortex solutions where the scalar and gauge fields have a non-singular profile under Einstein gravity in an AdS3 background. Vortices with different winding numbers n, VEV v and cosmological constant Λ are obtained. These vortices have positive mass and are not BTZ black holes as they have no event horizon. The mass is determined in two ways: by subtracting the numerical values of two separate asymptotic metrics and via an integral that is purely over the matter fields. The mass of the vortex increases as the cosmological constant becomes more negative and this coincides with the core of the vortex becoming smaller (compressed). We then consider the vortex with gravity in asymptotically flat spacetime for different values of the coupling α = 1/(16πG). At the origin, the spacetime has its highest curvature and there is no singularity. It transitions to an asymptotic conical spacetime with angular deficit that increases significantly as α decreases. For comparison, we also consider the vortex without gravity in flat spacetime. For this case, one cannot obtain the mass by the first method (subtracting two metrics) but remarkably, via a limiting procedure, one can obtain an integral mass formula. In the absence of gauge fields, there is a well-known logarithmic divergence in the energy of the vortex. With gravity, we present this divergence in a new light. We show that the metric acquires a logarithmic term which is the 2 + 1 dimensional realization of the Newtonian gravitational potential when General Relativity is supplemented with a scalar field. This opens up novel possibilities which we discuss in the conclusion.

Counterterms, Kounterterms, and the variational problem in AdS gravity

We show that the Kounterterms for pure AdS gravity in arbitrary even dimensions coincide with the boundary counterterms obtained through holographic renormalization if and only if the boundary Weyl tensor vanishes. In particular, the Kounterterms lead to a well posed variational problem for generic asymptotically locally AdS manifolds only in four dimensions. We determine the exact form of the counterterms for conformally flat boundaries and demonstrate that, in even dimensions, the Kounterterms take exactly the same form. This agreement can be understood as a consequence of Anderson’s theorem for the renormalized volume of conformally compact Einstein 4-manifolds and its higher dimensional generalizations by Albin and Chang, Qing and Yang. For odd dimensional asymptotically locally AdS manifolds with a conformally flat boundary, the Kounterterms coincide with the boundary counterterms except for the logarithmic divergence associated with the holographic conformal anomaly, and finite local terms.

Two-dimensional Brinkman flows and their relation to analogous Stokes flows

2D Stokes flows often exhibit the Stokes paradox: logarithmic growth of the fluid velocity in the far field. Analogous Brinkman flows are governed by the same equations apart from an additional term involving a parameter $\alpha$. Although these equations reduce to those for Stokes flow when $\alpha =0$, we show that the Brinkman solutions do not approach the corresponding Stokes solutions as $\alpha \to 0$; instead, logarithmic divergence with $\alpha$ is found. We also show that Brinkman flows do not exhibit a Stokes-like paradox. These results are given in detail for two specific problems, namely flow past a rigid circular cylinder and flow past a thin rigid strip.

RG-inspired machine learning for lattice field theory

Machine learning has been a fast growing field of research in several areas dealing with large datasets. We report recent attempts to use renormalization group (RG) ideas in the context of machine learning. We examine coarse graining procedures for perceptron models designed to identify the digits of the MNIST data. We discuss the correspondence between principal components analysis (PCA) and RG flows across the transition for worm configurations of the 2D Ising model. Preliminary results regarding the logarithmic divergence of the leading PCA eigenvalue were presented at the conference. More generally, we discuss the relationship between PCA and observables in Monte Carlo simulations and the possibility of reducing the number of learning parameters in supervised learning based on RG inspired hierarchical ansatzes.