Higher-dimensional Theories Research Articles

Computational higher-dimensional type theory

Formal constructive type theory has proved to be an effective language for mechanized proof. By avoiding non-constructive principles, such as the law of the excluded middle, type theory admits sharper proofs and broader interpretations of results. From a computer science perspective, interest in type theory arises from its applications to programming languages. Standard constructive type theories used in mechanization admit computational interpretations based on meta-mathematical normalization theorems. These proofs are notoriously brittle; any change to the theory potentially invalidates its computational meaning. As a case in point, Voevodsky's univalence axiom raises questions about the computational meaning of proofs. We consider the question: Can higher-dimensional type theory be construed as a programming language? We answer this question affirmatively by providing a direct, deterministic operational interpretation for a representative higher-dimensional dependent type theory with higher inductive types and an instance of univalence. Rather than being a formal type theory defined by rules, it is instead a computational type theory in the sense of Martin-Löf's meaning explanations and of the NuPRL semantics. The definition of the type theory starts with programs; types are specifications of program behavior. The main result is a canonicity theorem stating that closed programs of boolean type evaluate to true or false.

Superconformal Algebraic Approach to Hadron Structure

Fundamental aspects of nonperturbative QCD dynamics which are not obvious from its classical Lagrangian, such as the emergence of a mass scale and confinement, the existence of a zero mass bound state, the appearance of universal Regge trajectories and the breaking of chiral symmetry are incorporated from the onset in an effective theory based on superconformal quantum mechanics and its embedding in a higher dimensional gravitational theory. In addition, superconformal quantum mechanics gives remarkable connections between the light meson and nucleon spectra. This new approach to hadron physics is also suitable to describe nonperturbative QCD observables based on structure functions, such as GPDs, which are not amenable to a first-principle computation. The formalism is also successful in the description of form factors, the nonperturbative behavior of the strong coupling and diffractive processes. We also discuss in this article how the framework can be extended rather successfully to the heavy-light hadron sector.

On the embedding of Weyl manifolds

We discuss the possibility of extending different versions of the Campbell-Magaard theorem, which have already been established in the context of semi-Riemannian geometry, to the context of Weyl’s geometry. We show that some of the known results can be naturally extended to the new geometric scenario, although new difficulties arise. In pursuit of solving the embedding problem, we have obtained some no-go theorems. We also highlight some of the difficulties that appear in the embedding problem, which are typical of the Weylian character of the geometry. The establishing of these new results may be viewed as part of a program that highlights the possible significance of embedding theorems of increasing degrees of generality in the context of modern higher-dimensional space-time theories.

Towards Generic Cosmological Evolution Without Singularity and Fundamental Symmetry of Space-time

Dynamical systems methods are used to investigate cosmological model with non-minimally coupled scalar field. Existence of an asymptotically unstable de Sitter state distinguishes values of the non-minimal coupling constant parameter $\frac{3}{16}\le\xi<\frac{1}{4}$, which correspond to conformal coupling in higher dimensional theories of gravity.

Remarks on the effects of the Gribov copies on the infrared behavior of higher dimensional Yang-Mills theory

In this paper, we discuss non-perturbative infrared features of Yang-Mills theory in Euclidean space-time dimensions greater than four in the Landau gauge and within the Refined Gribov-Zwanziger framework, which enables us to take into account the existence of gauge copies by restricting the domain of integration in the path integral to the Gribov region. Evidences for a decoupling/massive solution for the gluon propagator in higher dimensions are provided. This behavior is strengthened the bigger the dimension is. Further, we show that, by a dimensional reduction of the Refined Gribov-Zwanziger action from five to four dimensions, a non-perturbative coupling between the inverse of the Faddeev-Popov operator and the scalar field corresponding to the fifth component of the gauge field naturally arises, being in agreement with the recently proposed mechanism \cite{Capri:2014bsa} to generalize the Refined Gribov-Zwanziger construction to the matter sector.

Jost–Lehmann–Dyson representation in higher dimensional field theories

Abstract The Jost–Lehmann–Dyson representation is derived for massive scalar field theories in higher spacetime dimensions, D > 4 , for the four point scattering amplitude. The representation is very crucial to investigate the analyticity properties of the amplitude. The axiomatic approach of Lehmann–Symanzik–Zimmermann is adopted to show the existence of such a representation. Consequently, a host of interesting results will follow from derivation of JLD representation such as proof of analyticity properties and asymptotic behavior of the amplitude.

Monopole correlation functions and holographic phases of matter in 2+1 dimensions

The strong coupling dynamics of a 2+1 dimensional U(1) gauge theory coupled to charged matter is holographically modeled via a top-down construction with intersecting D3- and D5-branes. We explore the resulting phase diagram at finite temperature and charge density using correlation functions of monopole operators, dual to magnetically charged particles in the higher-dimensional bulk theory, as a diagnostic.

Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories.

The low-energy spectra of many body systems on a torus, of finite size L, are well understood in magnetically ordered and gapped topological phases. However, the spectra at quantum critical points separating such phases are largely unexplored for (2+1)D systems. Using a combination of analytical and numerical techniques, we accurately calculate and analyze the low-energy torus spectrum at an Ising critical point which provides a universal fingerprint of the underlying quantum field theory, with the energy levels given by universal numbers times 1/L. We highlight the implications of a neighboring topological phase on the spectrum by studying the Ising* transition (i.e. the transition between a Z_{2} topological phase and a trivial paramagnet), in the example of the toric code in a longitudinal field, and advocate a phenomenological picture that provides qualitative insight into the operator content of the critical field theory.

Conformal frames in cosmology

From higher dimensional theories, e.g. string theory, one expects the presence of nonminimally coupled scalar fields. We review the notion of conformal frames in cosmology and emphasize their physical equivalence, which holds at least at a classical level. Furthermore, if there is a field, or fields, which dominates the universe, as it is often the case in cosmology, we can use such notion of frames to treat our system, matter and gravity, as two different sectors. On one hand, the gravity sector which describes the dynamics of the geometry and on the other hand, the matter sector which has such geometry as a playground. We use this interpretation to build a model where the fact that a curvaton couples to a particular frame metric could leave an imprint in the cosmic microwave background (CMB).

Chow group of 0-cycles with modulus and higher-dimensional class field theory

One of the main results of this paper is a proof of the rank one case of an existence conjecture on lisse l-adic sheaves on a smooth variety over a finite field due to Deligne and Drinfeld. The problem is translated into the language of higher dimensional class field theory over finite fields, which describes the abelian fundamental group by Chow groups of zero cycles with moduli. A key ingredient is the construction of a cycle theoretic avatar of refined Artin conductor in ramification theory originally studied by Kazuya Kato.

Fundaments of quaternionic Clifford analysis II: splitting of equations

Quaternionic Clifford analysis is a recent new branch of Clifford analysis, a higher dimensional function theory which refines harmonic analysis and generalizes to higher dimension the theory of holomorphic functions in the complex plane. So-called quaternionic monogenic functions satisfy a system of first-order linear differential equations expressed in terms of four interrelated Dirac operators. The conceptual significance of quaternionic Clifford analysis is unraveled by showing that quaternionic monogenicity can be characterized by means of generalized gradients in the sense of Stein and Weiss. At the same time, connections between quaternionic monogenic functions and other branches of Clifford analysis, viz Hermitian monogenic and standard or Euclidean monogenic functions are established as well.

Quantum revivals in conformal field theories in higher dimensions

We investigate the behavior of the return amplitude following a quantum quench in a conformal field theory (CFT) on a compact spatial manifold of dimension and linear size , from a state of extensive energy with short-range correlations. After an initial gaussian decay reaches a plateau value related to the density of available states at the initial energy. However for this value is attained from below after a single oscillation. For a holographic CFT the plateau persists up to times at least , where is the dimensionless Stefan–Boltzmann constant. On the other hand for a free field theory on manifolds with high symmetry there are typically revivals at times . In particular, on a sphere of circumference , there is an action of the modular group on implying structure near all rational values of t/L, similar to what happens for rational CFTs in .

A functional programmer's guide to homotopy type theory

Dependent type theories are functional programming languages with types rich enough to do computer-checked mathematics and software verification. Homotopy type theory is a recent area of work that connects dependent type theory to the mathematical disciplines of homotopy theory and higher-dimensional category theory. From a programming point of view, these connections have revealed that all types in dependent type theory support a certain generic program that had not previously been exploited. Specifically, each type can be equipped with computationally relevant witnesses of equality of elements of that type, and all types support a generic program that transports elements along these equalities. One mechanism for equipping types with non-trivial witnesses of equality is Voevodsky’s univalence axiom, which implies that equality of types themselves is witnessed by type isomorphism. Another is higher inductive types, an extended datatype schema that allows identifications between different datatype constructors. While these new mechanisms were originally formulated as axiomatic extensions of type theory, recent work has investigated their computational meaning, leading to the development of new programming languages that better support them. In this talk, I will illustrate what univalence and higher inductive types mean in programming terms. I will also discuss how studying some related semantic settings can reveal additional structure on types; for example, moving from groupoids (categories where all maps are invertible) to general categories yields an account of coercions instead of equalities. Overall, I hope to convey some of the beauty and richness of these connections between disciplines, which we are just beginning to understand.

The Spectroscopy and Form Factors of Nucleon Resonances from Superconformal Quantum Mechanics and Holographic QCD

The superconformal algebraic approach to hadronic physics is used to construct a semiclassical effective theory for nucleons which incorporates essential nonperturbative dynamical features, such as the emergence of a confining scale and the Regge resonance spectrum. Relativistic bound-state equations for nucleons follow from the extension of superconformal quantum mechanics to the light front and its holographic embedding in a higher dimensional gravity theory. Superconformal algebra has been used elsewhere to describe the connections between the light mesons and baryons, but in the present context it relates the fermion positive and negative chirality states and uniquely determines the confinement potential of nucleons. The holographic mapping of multi-quark bound states also leads to a light-front cluster decomposition of form factors for an arbitrary number of constituents. The remarkable analytical structure which follows incorporates the correct scaling behavior at high photon virtualities and also vector dominance at low energies.

Fischer Decomposition and Cauchy–Kovalevskaya Extension in Fractional Clifford Analysis: The Riemann–Liouville Case

AbstractIn this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann–Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy–Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on ℝd. Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form xPl and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.

The IR obstruction to UV completion for Dante's Inferno model with higher-dimensional gauge theory origin

We continue our investigation of large field inflation models obtained from higher-dimensional gauge theories, initiated in our previous study [1]. We focus on Dante's Inferno model which was the most preferred model in our previous analysis. We point out the relevance of the IR obstruction to UV completion, which constrains the form of the potential of the massive vector field, under the current observational upper bound on the tensor to scalar ratio. We also show that in simple examples of the potential arising from DBI action of a D5-brane and that of an NS5-brane that the inflation takes place in the field range which is within the convergence radius of the Taylor expansion. This is in contrast to the well known examples of axion monodromy inflation where inflaton takes place outside the convergence radius of the Taylor expansion. This difference arises from the very essence of Dante's Inferno model that the effective inflaton potential is stretched in the inflaton field direction compared with the potential for the original field.

Conformal field theory out of equilibrium: a review

We provide a pedagogical review of the main ideas and results in non-equilibrium conformal field theory and connected subjects. These concern the understanding of quantum transport and its statistics at and near critical points. Starting with phenomenological considerations, we explain the general framework, illustrated by the example of the Heisenberg quantum chain. We then introduce the main concepts underlying conformal field theory (CFT), the emergence of critical ballistic transport, and the CFT scattering construction of non-equilibrium steady states. Using this we review the theory for energy transport in homogeneous one-dimensional critical systems, including the complete description of its large deviations and the resulting (extended) fluctuation relations. We generalize some of these ideas to one-dimensional critical charge transport and to the presence of defects, as well as beyond one-dimensional criticality. We describe non-equilibrium transport in free-particle models, where connections are made with generalized Gibbs ensembles, and in higher-dimensional and non-integrable quantum field theories, where the use of the powerful hydrodynamic ideas for non-equilibrium steady states is explained. We finish with a list of open questions. The review does not assume any advanced prior knowledge of conformal field theory, large-deviation theory or hydrodynamics.

Photo-production of a 750 GeV di-photon resonance mediated by Kaluza-Klein leptons in the loop

We consider the phenomenology of a 750 GeV resonance X which can be produced at the LHC by only photon fusion and subsequently decay into di-photons. We propose that the spin-zero state X is coupled to a heavy lepton that lives in the bulk of a higher-dimensional theory and interacts only with the photons of the Standard Model. We compute the di-photon rate in these models with two and more compact extra dimensions and demonstrate that they allow for a compelling explanation of the di-photon excess recently observed by the ATLAS and CMS collaborations. The central role in our approach is played by the summation over the Kaluza-Klein modes of the new leptons, thus providing a significant enhancement of the X → γγ loops for the production and decay subprocesses. It is expected that the jet activity accompanying these purely electromagnetic (at the partonic level) processes is numerically suppressed by factors such as $$ {\alpha}_{\mathrm{em}}^2{\mathcal{C}}_{q\overline{q}}/{\mathcal{C}}_{\gamma \gamma}\sim 1{0}^{-3} $$ .

MHz gravitational waves from short-term anisotropic inflation

We reveal the universality of short-term anisotropic inflation. As a demonstration, we study inflation with an exponential type gauge kinetic function which is ubiquitous in models obtained by dimensional reduction from higher dimensional fundamental theory. It turns out that an anisotropic inflation universally takes place in the later stage of conventional inflation. Remarkably, we find that primordial gravitational waves with a peak amplitude around 10−26∼ 10−27 are copiously produced in high-frequency bands 10 MHz∼100 MHz. If we could detect such gravitational waves in future, we would be able to probe higher dimensional fundamental theory.

SUSY’s Ladder: reframing sequestering at Large Volume

Theories with approximate no-scale structure, such as the Large Volume Scenario, have a distinctive hierarchy of multiple mass scales in between TeV gaugino masses and the Planck scale, which we call SUSY's Ladder. This is a particular realization of Split Supersymmetry in which the same small parameter suppresses gaugino masses relative to scalar soft masses, scalar soft masses relative to the gravitino mass, and the UV cutoff or string scale relative to the Planck scale. This scenario has many phenomenologically interesting properties, and can avoid dangers including the gravitino problem, flavor problems, and the moduli-induced LSP problem that plague other supersymmetric theories. We study SUSY's Ladder using a superspace formalism that makes the mysterious cancelations in previous computations manifest. This opens the possibility of a consistent effective field theory understanding of the phenomenology of these scenarios, based on power-counting in the small ratio of string to Planck scales. We also show that four-dimensional theories with approximate no-scale structure enforced by a single volume modulus arise only from two special higher-dimensional theories: five-dimensional supergravity and ten-dimensional type IIB supergravity. This gives a phenomenological argument in favor of ten dimensional ultraviolet physics which is different from standard arguments based on the consistency of superstring theory.