The Schwarzian-Newton method for solving nonlinear equations, with applications

The Schwarzian-Newton method can be defined as the minimal method for solving nonlinear equations f(x) = 0 which is exact for any function f with constant Schwarzian derivative; exactness means that the method gives the exact root in one iteration for any starting value in a neighborhood of the root. This is a fourth order method which has Halley?s method as limit when the Schwarzian derivative tends to zero. We obtain conditions for the convergence of the SNM in an interval and show how this methodcan be applied for a reliable and fast solution of some problems, like the inversion of cumulative distribution functions (gamma and beta distributions) and the inversion of elliptic integrals.The author acknowledges financial support from Ministerio de Economía y Competitividad (project MTM2012-34787

Towards a bulk description of higher spin SYK

We consider on the bulk side extensions of the Sachdev--Ye--Kitaev (SYK) model to Yang--Mills and higher spins. To this end we study generalizations of the Jackiw--Teitelboim (JT) model in the BF formulation. Our main goal is to obtain generalizations of the Schwarzian action, which we achieve in two ways: by considering the on-shell action supplemented by suitable boundary terms compatible with all symmetries, and by applying the Lee--Wald--Zoupas formalism to analyze the symplectic structure of dilaton gravity. We conclude with a discussion of the entropy (including log-corrections from higher spins) and a holographic dictionary for the generalized SYK/JT correspondence.Comment: 42 pages; v2: Typos correcte

On the existence of nonoscillatory phase functions for second order differential equations in the high-frequency regime

We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate important implications of this fact. For example, that many special functions of great interest --- such as the Bessel functions $J_\nu$ and $Y_\nu$ --- can be evaluated accurately using a number of operations which is $O(1)$ in the order $\nu$. The present paper is devoted to the development of an analytical apparatus. Numerical aspects of this work will be reported at a later date

Menagerie of AdS2\boldsymbol{_2} boundary conditions

We consider different sets of AdS$_2$ boundary conditions for the Jackiw-Teitelboim model in the linear dilaton sector where the dilaton is allowed to fluctuate to leading order at the boundary of the Poincar\'e disk. The most general set of boundary condtions is easily motivated in the gauge theoretic formulation as a Poisson sigma model and has an $\mathfrak{sl}(2)$ current algebra as asymptotic symmetries. Consistency of the variational principle requires a novel boundary counterterm in the holographically renormalized action, namely a kinetic term for the dilaton. The on-shell action can be naturally reformulated as a Schwarzian boundary action. While there can be at most three canonical boundary charges on an equal-time slice, we consider all Fourier modes of these charges with respect to the Euclidean boundary time and study their associated algebras. Besides the (centerless) $\mathfrak{sl}(2)$ current algebra we find for stricter boundary conditions a Virasoro algebra, a warped conformal algebra and a $\mathfrak{u}(1)$ current algebra. In each of these cases we get one half of a corresponding symmetry algebra in three-dimensional Einstein gravity with negative cosmological constant and analogous boundary conditions. However, on-shell some of these algebras reduce to finite-dimensional ones, reminiscent of the on-shell breaking of conformal invariance in SYK. We conclude with a discussion of thermodynamical aspects, in particular the entropy and some Cardyology.Comment: 42 pp, 5 figs, v2: added ref

Non-commutative lattice modified Gel'fand-Dikii systems

We introduce integrable multicomponent non-commutative lattice systems, which can be considered as analogs of the modified Gel'fand-Dikii hierarchy. We present the corresponding systems of Lax pairs and we show directly multidimensional consistency of these Gel'fand-Dikii type equations. We demonstrate how the systems can be obtained as periodic reductions of the non-commutative lattice Kadomtsev-Petviashvilii hierarchy. The geometric description of the hierarchy in terms of Desargues maps helps to derive non-isospectral generalization of the non-commutative lattice modified Gel'fand-Dikii systems. We show also how arbitrary functions of single arguments appear naturally in our approach when making commutative reductions, which we illustrate on the non-isospectral non-autonomous versions of the lattice modified Korteweg-de Vries and Boussinesq systems.Comment: 12 pages, 1 figure; types corrected, conclusion section and new references added (v2