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

In this work we extend the first part of the previous paper [F4] to higher dimensional local fields. We introduce a nontrivial translation invariant measure on the additive group of higher dimensional local fields, and then develop elements of integration and harmonic analysis. We also discuss its relation with several other measure theories. For an n-dimensional local field F a translation invariant measure μ is defined on a certain ring A of measurable sets and takes values in R ((X1)) . . . ((Xn−1)) (which itself is an n-dimensional local field whose last residue field is the archimedean field R). The algebra A in the case of higher dimensional fields with finite last residue field is the algebra generated by characteristic functions of shifts of fractional ideals, i.e. a + bO with a, b ∈ F and O the ring of integers of F with respect to the n-dimensional structure. The measure is countably additive in a refined sense (see sections 7 and 8). Elements of integration theory are introduced in sections 9–13. The additive group of a higher dimensional field has a certain topology on it with respect to which it is not locally compact for n > 1. This topology plays a key role in higher class field theory [F1–F3]. The additive group of F is self dual, which together with the measure and integration leads to analogs of many classical results in Fourier analysis (section 14). In particular, for functions in certain space we introduce their transform

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

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.

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 $1+1$ dimensional Dirac field in a finite volume is quantized canonically by using two different methods in this paper. First, we take the boundary conditions (BCs) as primary Dirac constraints, and prove that those BCs as well as the intrinsic constraints are entangled and they form the second class constraints. The quantization is performed canonically by using Dirac's procedure. And then, we study this model in the reduced phase space. It shows that the Poisson brackets in the reduced phase space are equivalent to the Dirac brackets in the original phase space.

The physical property of pseudo spin of electrons in graphene is investigated. In contrast to a recent description [Phys. Rev. Lett. 106 (2011) 116803], we show that pseudo spin in graphene is not completely a real angular momentum. The pseudo spin only in the direction perpendicular to the graphene sheet is real angular momentum while the pseudo spin parallel to the graphene plane is still not real angular momentum. Interestingly, it is also shown that the Newtonian-like force and pseudo spin torque of massive Dirac electrons in graphene under strain field mimic gravitomagnetic force and gravitomagnetic spin torque, respectively. This is due to the equivalence of pseudo spin and velocity operators of (2+1)-dimensional massive electrons in graphene, different from that in real (3+1)-dimensional Dirac fields. This work reveals the new physical property of graphene as a pseudo gravitomagnetic material.

Donaldson-Thomas theory on a Calabi-Yau can be described in terms of a certain six-dimensional cohomological gauge theory. We introduce a certain class of defects in this gauge theory which generalize surface defects in four dimensions. These defects are associated with divisors and are defined by prescribing certain boundary conditions for the gauge fields. We discuss generalized instanton moduli spaces when the theory is defined with a defect and propose a generalization of Donaldson-Thomas invariants. These invariants arise by studying torsion free coherent sheaves on Calabi-Yau varieties with a certain parabolic structure along a divisor, determined by the defect. We discuss the case of the affine space as a concrete example. In this case the moduli space of parabolic sheaves admits an alternative description in terms of the representation theory of a certain quiver. The latter can be used to compute the invariants explicitly via equivariant localization. We also briefly discuss extensions of our work to other higher dimensional field theories.

Studying quantum field theories numerically on the continuum and a new look at perturbation theory

In this work, we develop a new technique for the numerical study of quantum field theory. The procedure, borrowed from non-relativistic quantum mechanics, is that of finding the eigenvalues of a finite Hamiltonian matrix. The matrix is created by evaluating the matrix elements of the Hamiltonian operator on a finite basis of states. The eigenvalues and eigenvectors of the finite dimensional matrix become an accurate approximation to those of the physical system as the finite basis of states is extended to become more complete. We study a model of scalars coupled to fermions in 0+1 dimensions as a simple field theory to consider in the course of developing the technique. We find in the course of studying this model a change of basis which diagonalizes the Hamiltonian in the large coupling limit. The importance of this transformation is that it can be generalized to higher dimensional field theories involving a trilinear coupling between a Bose and a Fermi field. Having developed the numerical and analytical techniques, we consider a Fermi field coupled to a Bose field in 1+1 dimensions with the Yukawa coupling λΨΨΨ. We extend the large coupling limit basis of the 0+1 dimensional model to this case using a Bogoliubov transformation on the fermions. Although we do not use this basis in the numerical work due to its complexity, it provides a handle on the behavior of the system in the large coupling limit. In this model we consider the effects of renormalization and the generation of bound states.

Construction of higher dimensional field theory with dynamical boundary conditions

We study the polarization tensor of a Dirac field in $(3+1)$ dimensions confined to a half space -- a problem motivated by applications to the condensed matter physics, and to Topological Insulators in particular. Although the Pauli-Villars regularization scheme has a number of advantages, like explicit gauge invariance and decoupling of heavy modes, it is not applicable on manifolds with boundaries. Here, we modify this scheme by giving an axial mass to the regulators and to the physical field. We compute the renormalized polarization tensor in coordinate representation. We discuss then the induced Chern-Simons type action on the boundary and compare it to the effective action of a $(2+1)$ dimensional Dirac fermion.