New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)

An ABC Construction of Number Fields

We describe a general three step method for constructing number fields with Lie-type Galois groups and discriminants factoring into powers of specified primes. The first step involves extremal solutions of the matrix equation ABC = I. The second step involves extremal polynomial solutions of the equation A(x) + B(x) + C(x) = 0. The third step involves integer solutions of the generalized Fermat equation axp + byq + czr = 0. We concentrate here on details associated to the third step and give examples where the field discriminants have the form ±2a3b

Non-abelian ZZ-theory: Berends-Giele recursion for the α′\alpha'-expansion of disk integrals

We present a recursive method to calculate the $\alpha'$-expansion of disk integrals arising in tree-level scattering of open strings which resembles the approach of Berends and Giele to gluon amplitudes. Following an earlier interpretation of disk integrals as doubly partial amplitudes of an effective theory of scalars dubbed as $Z$-theory, we pinpoint the equation of motion of $Z$-theory from the Berends-Giele recursion for its tree amplitudes. A computer implementation of this method including explicit results for the recursion up to order $\alpha'^7$ is made available on the website http://repo.or.cz/BGap.gitComment: 58 pages, harvmac TeX, v2: cosmetic changes, published versio

Unitarity in three-dimensional flat space higher spin theories

We investigate generic flat-space higher spin theories in three dimensions and find a no-go result, given certain assumptions that we spell out. Namely, it is only possible to have at most two out of the following three properties: unitarity, flat space, non-trivial higher spin states. Interestingly, unitarity provides an (algebra-dependent) upper bound on the central charge, like c=42 for the Galilean $W_4^{(2-1-1)}$ algebra. We extend this no-go result to rule out unitary "multi-graviton" theories in flat space. We also provide an example circumventing the no-go result: Vasiliev-type flat space higher spin theory based on hs(1) can be unitary and simultaneously allow for non-trivial higher-spin states in the dual field theory.Comment: 34 pp, v2: added two paragraphs in section 5.3 + minor change

Classical and Stringy Properties of Black Holes

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Física Teórica. Fecha de Lectura: 14-09-2022This thesis is devoted to the study of dynamical and thermodynamical properties of black holes. It has two parts. Part I considers black holes in the context of the low energy effective actions of string theory. The first few higher-derivative corrections induced by finite-size effects in √ the string length ` s ∼ α0 , where α0 is the Regge slope parameter, are well understood for the heterotic superstring (HST). α0-corrected black hole solutions are available and computing their entropy is crucial given its relation to string microstates. However, the Iyer–Wald entropy formula gives a result that is not gauge invariant. This is due to the fact that the original computation assumes that all fields are tensors with no internal gauge freedom. In this thesis, Wald’s derivation is revisited using a formalism that accommodates gauge symmetry conveniently. The main result is a gauge-and Lorentz-invariant entropy formula that includes the first order corrections in α0 . It is also shown, in some particular theories, how magnetic-type terms can be included in the generic proofs of the laws of black hole thermodynamics, even though magnetic charges are not directly associated to gauge symmetry. Part II focuses on dynamical aspects of black holes in different contexts. Rotating black holes in higher-derivative theories are poorly understood due to the complexity of the equations of motion. The problem can be simplified by considering the near horizon geometry of an extremal, charged and rotating black hole. A non-perturbative solution of such a class is presented in a cubic theory called Einsteinian Cubic Gravity. It is the first example in which the entropy of a rotating black hole of higher-order gravity has been exactly computed. In the context of the AdS/CFT correspondence, NUT-charged AdS black holes describe equilibrium states of neutral fluids subject to non-trivial flows at the boundary. Physical transport properties, however, remain largely unexplored. The master equations governing gravitational fluctuations on a class of NUT-charged AdS black holes are derived in this thesis. These exhibit an intriguing relation to Landau quantisation. The gravitational quasinormal mode spectrum of a NUT-charged black hole is computed for the first time, and the spacetime appears to be robustly stable despite the existence of closed causal curves (“time machines”). There is an interesting class of quasi-hydrodynamic modes for which analytic dispersion relations are constructed as a definite holographic prediction for the dual fluid. The last chapter of this thesis deals with the tidal deformability of black holes. Tidal interactions, encoded linearly in the so-called tidal Love numbers, become significant in the last stages of the inspiral phase of a merger. In the case of vacuum, four-dimensional black holes, the tidal Love numbers are zero. The robustness of such a property is investigated by studying the static deformability of charged black holes. It is shown that tidal response coefficients keep on vanishing, in a very non-trivial way, from neutrality all the way down to extremality. This is true not only for gravity (spin-2), but also for spin-0 and spin-1 deformations. In higher dimensions, however, the tidal response is non-trivial and charging up the hole can excite new polarisation modes. One exception is the static response of spin-0 perturbations, which happens to vanish at extremality in any dimension. These results call for further investigation of the tidal deformability properties of black hole

Gauge theories on quantum spaces

We review the present status of gauge theories built on various quantum space-times described by noncommutative space-times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered.Comment: 141 pages. Review article. This is a preliminary versio