Terms Of Zeta Functions Research Articles

Anomalous Hall viscosity at the Weyl-semimetal–insulator transition

We show that 3D Lifshitz fermions arising as the critical theory at the Weyl semimetal/insulator transition naturally develop an anomalous Hall viscosity at finite temperature. We discuss how to couple the system to non-relativistic background sources for stress-tensor and momentum currents via a form of Newton-Cartan geometry with torsion and derive the Kubo formulas for the Hall viscosities. While the Lifshitz system that arises most naturally has scaling exponent $z=2$ we also generalize the theory for arbitrary Lifshitz scaling $z$ and show that, in the limit $z \to 0$, it may be given a Chern-Simons interpretation by dimensionally reducing along the anisotropic direction. The Hall viscosities are expressed in terms of zeta functions and their temperature dependence is dictated by the scaling exponent.

The Influence of Catechol-O-Methyltransferase (COMT) Val158Met Gene Polymorphism, Persistence, and Attentional Characteristics on Novelty Seeking

We provide a small C++ library with Mathematica and Python interfaces for computing thermal functions, defined which appear in finite-temperature quantum field theory and play a role in phase-transitions in the early Universe, including baryogenesis, electroweak symmetry breaking and the Higgs mechanism. The code is available from https://github.com/andrewfowlie/thermal_funcs.Program title: thermal_funcsProgram Files doi: http://dx.doi.org/10.17632/5nzrr2jjpd.1 Licensing provisions: BSD 3-ClauseProgramming language: C++, C interface to Mathematica and SWIG interface to PythonNature of problem: Thermal functions appear in finite-temperature quantum field theory and influence phase transitions in the early Universe. They have no closed-form solution. Studying phase transitions requires repeated evaluations of, inter alia, thermal functions. We provide several fast and accurate methods, and first and second derivatives.Solution method: We implement numerical quadrature, a Bessel function representation, asymptotic solutions expressed in terms of zeta functions and polylogarithms, and approximations and limits.

A fast C++ implementation of thermal functions

We provide a small C++ library with Mathematica and Python interfaces for computing thermal functions, defined JB/F(y2)≡ℜ∫0∞x2log1∓e−x2+y2dx,which appear in finite-temperature quantum field theory and play a role in phase-transitions in the early Universe, including baryogenesis, electroweak symmetry breaking and the Higgs mechanism. The code is available from https://github.com/andrewfowlie/thermal_funcs. Program summaryProgram title: thermal_funcsProgram Files doi: http://dx.doi.org/10.17632/5nzrr2jjpd.1Licensing provisions: BSD 3-ClauseProgramming language: C++, C interface to Mathematica and SWIG interface to PythonNature of problem: Thermal functions appear in finite-temperature quantum field theory and influence phase transitions in the early Universe. They have no closed-form solution. Studying phase transitions requires repeated evaluations of, inter alia, thermal functions. We provide several fast and accurate methods, and first and second derivatives.Solution method: We implement numerical quadrature, a Bessel function representation, asymptotic solutions expressed in terms of zeta functions and polylogarithms, and approximations and limits.

Integrals of logarithmic and hypergeometric functions

Abstract Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.

POLYLOGARITHMIC CONNECTIONS WITH EULER SUMS

Polylogarithmic functions are intrinsically connected with sums of harmonic numbers. In this paper we explore many relations and explicitly derive closed form representations of integrals of polylogarith- mic functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.

Summability for Nonunital Spectral Triples

This paper examines the issue of summability for spectral triples for the class of nonunital algebras introduced in (23). For the case of (p, ∞)-summability, we prove that the Dixmier trace can be used to define a (semifinite) trace on the algebra of the spectral triple. We show this trace is well-behaved, and provide a criteria for measurability of an operator in terms of zeta functions. We also show that all our hypotheses are satisfied by spectral triples arising from geodesically complete Riemannian manifolds. In addition, we indicate how the Local Index Theorem of Connes-Moscovici extends to our nonunital setting. AMS Classifications: 46L80, 46L89.

Congruences between derivatives of geometric L-functions

We prove a natural refinement of a theorem of Lichtenbaum describing the leading terms of Zeta functions of curves over finite fields in terms of Weil-etale cohomology. We then use this result to prove the validity of Chinburg’s Ω(3)-Conjecture for all abelian extensions of global function fields, to prove natural refinements and generalisations of the refined Stark conjectures formulated by, amongst others, Gross, Tate, Rubin and Popescu, to prove a variety of explicit restrictions on the Galois module structure of unit groups and divisor class groups and to describe explicitly the Fitting ideals of certain Weil-etale cohomology groups. In an Appendix coauthored with K.F. Lai and K.-S. Tan we also show that the main conjectures of geometric Iwasawa theory can be proved without using either crystalline cohomology or Drinfeld modules.

Triple integral identities and zeta functions

Some new identities are given for the representation of binomial sums. A master theorem is developed from which integral and closed form results, in terms of Zeta functions and harmonic numbers, are developed for sums of the type ?n?1 tn/n4(an+j/j)(bn+k/k)(cn+l/l).

String Scattering from Decaying Branes

We develop the general formalism of string scattering from decaying D-branes in bosonic string theory. In worldsheet perturbation theory, amplitudes can be written as a sum of correlators in a grand canonical ensemble of unitary random matrix models, with time setting the fugacity. An approach employed in the past for computing amplitudes in this theory involves an unjustified analytic continuation from special integer momenta. We give an alternative formulation which is well-defined for general momenta. We study the emission of closed strings from a decaying D-brane with initial conditions perturbed by the addition of an open string vertex operator. Using an integral formula due to Selberg, the relevant amplitude is expressed in closed form in terms of zeta functions. Perturbing the initial state can suppress or enhance the emission of high energy closed strings for extended branes, but enhances it for D0-branes. The closed string two point function is expressed as a sum of Toeplitz determinants of certain hypergeometric functions. A large N limit theorem due to Szego, and its extension due to Borodin and Okounkov, permits us to compute approximate results showing that previous naive analytic continuations amount to the large N approximation of the full result. We also give a free fermion formulation of scattering from decaying D-branes and describe the relation to a grand canonical ensemble for a 2d Coulomb gas.

A general BRST approach to string theories with zeta function regularizations

We propose a new general BRST approach to string and string-like theories that have a wider range of applicability than, e.g., the conventional conformal field theory method. The method involves a simple general regularization of all basic commutators, which makes all divergent sums expressible in terms of zeta functions from which finite values then may be extracted in a rigorous manner. The method is particularly useful in order to investigate possible state space representations to a given model. The method is applied to three string models: The ordinary bosonic string, the tensionless string, and the conformal tensionless string. We also investigate different state spaces for these models. The tensionless string models are treated in detail. Although we mostly rederive known results, they appear in a new fashion that deepens our understanding of these models. Furthermore, we believe that our treatment is more rigorous than most of the previous ones. In the case of the conformal tensionless string we find a new solution for d=4.