Measure, integration and elements of harmonic analysis on generalized loop spaces

We present a new area law which is associated with the correlator of OPE blocks in higher dimensional conformal field theories (CFTs). The area law shows similar behaviour as black hole entropy or geometric entanglement entropy. It includes a leading term which is proportional to the area of the entanglement surface, and a logarithmic subleading term with degree q. We extract the UV cutoff independent coefficients and discuss various properties of the coefficients.

Higher dimensional self-similar spherical symmetric scalar field collapse and critical phenomena in black-hole formation

We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality of a residue field. One consequence of the main result is that it is provable in Zermelo–Fraenkel Set Theory with Choice (ZFC) that there is a Noetherian domain of cardinality ℵ1 with a finite residue field, but the statement “There is a Noetherian domain of cardinality ℵ2 with a finite residue field” is equivalent to the negation of the Continuum Hypothesis.

In this paper, we consider numerical and trigonometric series with a very general monotonicity condition. First, a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting. In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.

The complexity of linear problems in fields

Almost slender rings and compact rings

Let $K$ denote an $n$-dimensional local field. The aim of this expository paper is to survey the basic arithmetic theory of the $n$-dimensional local field $K$ together with its Milnor $K$-theory and Parshin topological $K$-theory; to review Kato's ramification theory for finite abelian extensions of the $n$-dimensional local field $K$, and to state the local abelian higher-dimensional $K$-theoretic generalization of local abelian class field theory of Hasse, which is developed by Kato and Parshin. The paper is geared toward non-abelian generalization of this theory.

Hilbert's 16th problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree , has been one of the most important driving forces for new developments in the qualitative theory of vector fields. Increasing the dimension, one cannot expect the existence of a finite upper bound for the number of limit cycles of, for instance, 3D polynomial vector fields of a given degree . Here, as an extension of such a problem in the 3D space, we investigate the number of isolated invariant tori in 3D polynomial vector fields. In this context, given a natural number , we denote by the upper bound for the number of isolated invariant tori of 3D polynomial vector fields of degree . Based on a recently developed averaging method for detecting invariant tori, our first main result provides a mechanism for constructing 3D differential vector fields with a number of normally hyperbolic invariant tori from a given planar differential vector field with hyperbolic limit cycles. The strength of our mechanism in studying the number lies in the fact that the constructed 3D differential vector field is polynomial provided that the given planar differential vector field is polynomial. Accordingly, our second main result establishes a lower bound for in terms of lower bounds for the number of hyperbolic limit cycles of planar polynomial vector fields of degree . Based on this last result, we apply a methodology due to Christopher and Lloyd to show that grows as fast as . Finally, the above‐mentioned problem is also formulated for higher dimensional polynomial vector fields.

We generalize, to any space-time dimension, the unitarity bounds of highest weight UIR's of the conformal groups with Lie algebras $so(2,d)$. We classify gauge theories invariant under $so(2,d)$, both integral and half-integral spins. A similar analysis is carried out for the algebras $so^*(2n)$. We study new unitary modules of the conformal algebra in $d>4$, that have no analogue for $d\leq 4$ as they cannot be obtained by "squaring" singletons. This may suggest the interpretation of higher dimensional non-trivial conformal field theories as theories of "tensionless" $p$-branes of which tensionless strings in $d=6$ are just particular examples.

It is shown that negative tension branes in higher dimensions may lead to an effective lower dimensional theory where the gauge-invariant vector fields associated with the fluctuations of the metric are always massless and localized on the brane. Explicit five-dimensional examples of this phenomenon are provided. Furthermore, it is shown that higher dimensional gauge fields can also be localized on these configurations with the zero mode separated from the massive tower by a gap.

Structures of Extended Quantum-Mechanical Supersymmetry in Higher Dimensional Dirac Field

We propose a new inflation scenario in flux compactification, where a zero mode scalar field of extra components of the higher dimensional gauge field is identified with an inflaton. The scalar field is a pseudo Nambu-Goldstone boson of spontaneously broken translational symmetry in compactified spaces. The inflaton potential is non-local and finite, which is protected against the higher dimensional non-derivative local operators by quantum gravity corrections thanks to the gauge symmetry in higher dimensions and the shift symmetry originated from the translation in extra spaces. We give an explicit inflation model in a six dimensional scalar QED, which is shown to be consistent with Planck 2018 data.

Characteristics of the solitary waves and rogue waves with interaction phenomena in a generalized (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml22" display="inline" overflow="scroll" altimg="si22.gif"><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>)-dimensional Kadomtsev–Petviashvili equation

The goal of this book is to present local class field theory from the cohomological point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions—primarily abelian—of “local” (i.e., complete for a discrete valuation) fields with finite residue field. For example, such fields are obtained by completing an algebraic number field; that is one of the aspects of “localisation”.

Let $F$ be a field, $\ell$ a prime and $D$ a central division $F$-algebra of $\ell$-power degree. By the Rost kernel of $D$ we mean the subgroup of $F^*$ consisting of elements $\lambda$ such that the cohomology class $(D)\cup (\lambda)\in H^3(F,\,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(2))$ vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by $i$-th powers of reduced norms from $D^{\otimes i},\,\forall i\ge 1$. Despite of known counterexamples, we prove some new cases of Suslin's conjecture. We assume $F$ is a henselian discrete valuation field with residue field $k$ of characteristic different from $\ell$. When $D$ has period $\ell$, we show that Suslin's conjecture holds if either $k$ is a $2$-local field or the cohomological $\ell$-dimension $\mathrm{cd}_{\ell}(k)$ of $k$ is $\le 2$. When the period is arbitrary, we prove the same result when $k$ itself is a henselian discrete valuation field with $\mathrm{cd}_{\ell}(k)\le 2$. In the case $\ell=\text{char}(k)$ an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.