Dirac-like Operators Research Articles

A generalized MIT Bag operator on spin manifolds in the non-relativistic limit

We consider Dirac-like operators with piecewise constant mass terms on spin manifolds, and we study the behaviour of their spectra when the mass parameters become large. In several asymptotic regimes, effective operators appear: the extrinsic Dirac operator and a generalized MIT Bag Dirac operator. This extends some results previously known for the Euclidean spaces to the case of general spin manifolds.

Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices

In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator $H_0 := D_S + V_0$ is the sum of a Dirac-like operator $D_S$ plus a periodic matrix valued potential $V_0$, and is assumed to have an open gap. The random potential $V_\omega$ is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family $H_\omega := H_0 + V_\omega$ is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.

Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices

In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator H_0 := D_S + V_0 is the sum of a Dirac-like operator D_S plus a periodic matrix-valued potential V_0 , and is assumed to have an open gap. The random potential V_\omega is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family H_\omega := H_0 +V_\omega is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.

Dirac-like operators on the Hilbert space of differential forms on manifolds with boundaries

The problem of self-adjoint extensions of Dirac-type operators in manifolds with boundaries is analyzed. The boundaries might be regular or non-regular. The latter situation includes point-like interactions, also called delta-like potentials, in manifolds of dimension higher than one. Self-adjoint boundary conditions for the case of dimension 2 are obtained explicitly.

A note on the semiclassical formulation of BPS states in four-dimensional N=2 theories

Vector spaces of (framed) BPS states of Lagrangian four-dimensional N=2 field theories can be defined in semiclassical chambers in terms of the $L^2$-cohomology of Dirac-like operators on monopole moduli spaces. This was spelled out previously for theories with only vectormultiplets, taking into account only a subset of the possible half-supersymmetric 't Hooft-Wilson line defects. This note completes the discussion by describing the modifications needed when including matter hypermultiplets together with arbitrary 't Hooft-Wilson line defects. Two applications of this extended discussion are given.

Supersymmetry of AdS and flat backgrounds in M-theory

We give a systematic description of all warped $AdS_n$ and ${\mathbb{R}}^{n-1,1}$ backgrounds of M-theory and identify the a priori number of supersymmetries that these backgrounds preserve. In particular, we show that $AdS_n$ backgrounds preserve $N= 2^{[{n\over2}]} k$ for $n\leq4$ and $N= 2^{[{n\over2}]+1} k$ for $4<n\leq 7$ supersymmetries while ${\mathbb{R}}^{n-1,1}$ backgrounds preserve $N= 2^{[{n\over2}]} k$ for $n\leq4$ and $N= 2^{[{n+1\over2}]} k$ for $4<n\leq7$, supersymmetries. Furthermore for $AdS_n$ backgrounds that satisfy the requirements for the maximum principle to hold, we show that the Killing spinors can be identified with the zero modes of Dirac-like operators on $M^{11-n}$ coupled to fluxes thus establishing a new class of Lichnerowicz type theorems. We also demonstrate that the Killing spinors of generic warped $AdS_n$ backgrounds do not factorize into products of Killing spinors on $AdS_n$ and Killing spinors on the transverse space.

Supersymmetry of AdS and flat IIB backgrounds

We present a systematic description of all warped $AdS_n\times_w M^{10-n}$ and ${\mathbb{R}}^{n-1,1}\times_w M^{10-n}$ IIB backgrounds and identify the a priori number of supersymmetries $N$ preserved by these solutions. In particular, we find that the $AdS_n$ backgrounds preserve $N=2^{[{n\over2}]} k$ for $n\leq 4$ and $N=2^{[{n\over2}]+1} k$ for $4<n\leq 6$ supersymmetries and for $k\in {\mathbb{N}}_{+}$ suitably restricted. In addition under some assumptions required for the applicability of the maximum principle, we demonstrate that the Killing spinors of $AdS_n$ backgrounds can be identified with the zero modes of Dirac-like operators on $M^{10-n}$ establishing a new class of Lichnerowicz type theorems. Furthermore, we adapt some of these results to ${\mathbb{R}}^{n-1,1}\times_w M^{10-n}$ backgrounds with fluxes by taking the AdS radius to infinity. We find that these backgrounds preserve $N=2^{[{n\over2}]} k$ for $2<n\leq 4$ and $N=2^{[{n+1\over2}]} k$ for $4<n\leq 7$ supersymmetries. We also demonstrate that the Killing spinors of $AdS_n\times_w M^{10-n}$ do not factorize into Killing spinors on $AdS_n$ and Killing spinors on $M^{10-n}$.

SMOOTH MODULI SPACES OF ASSOCIATIVE SUBMANIFOLDS

Let M a 7-dimensional manifold with holonomy in G2, and containing a compact associative 3-dimensional submanifold Y . It is known since McLean [9] that the problem of the associative deformations of Y is related to a Dirac-like operator on Y , hence is elliptic and of vanishing index. In this paper we adapt the Bochner method to give sufficient metric conditions which implies that Y is isolated. In the case where Y has boundary in a coassociative 4-dimensional submanifold X, it has been proved in [5] that the related deformation problem is still elliptic, and moreover its index is in general non vanishing. In this case, we give a vanishing theorem that mixes the ambient curvature and the geometry of the boundary, and which insures the smoothness of the moduli space of associative deformations of Y . Thanks, we are able to describe explicit smooth moduli spaces for some families of examples in R^7 and in Calabi-Yau extensions. In particular, we prove that the moduli space of associative deformations of product of a special lagrangian by S^1 in a Calabi-Yau manifold times S^1 is always smooth and of dimension b^1(L) + 1.

Applications of the potential algebras of the two-dimensional Dirac-like operators

Potential algebras can be used effectively in the analysis of the quantum systems. In the article, we focus on the systems described by a separable, 2×2 matrix Hamiltonian of the first order in derivatives. We find integrals of motion of the Hamiltonian that close centrally extended so(3), so(2,1) or oscillator algebra. The algebraic framework is used in construction of physically interesting solvable models described by the (2+1) dimensional Dirac equation. It is applied in description of open-cage fullerenes where the energies and wave functions of low-energy charge-carriers are computed. The potential algebras are also used in construction of shape-invariant, one-dimensional Dirac operators. We show that shape-invariance of the first-order operators is associated with the N=4 nonlinear supersymmetry which is represented by both local and nonlocal supercharges. The relation to the shape-invariant non-relativistic systems is discussed as well.

How to construct self/anti-self charge conjugate states for higher spins? Significance of the spin bases, mass dimension, and all that

We construct self/anti-self charge conjugate (Majorana-like) states for the (1/2, 0)⊕(0, 1/2) representation of the Lorentz group, and their analogs for higher spins within the quantum field theory. The problems of the basis rotations and that of the selection of phases in the Dirac-like and Majorana-like field operators are considered. The discrete symmetries properties (P, C, T) are studied. The corresponding dynamical equations are presented. In the (1/2, 0) ⊕ (0, 1/2) representation they obey the Diraclike equation with eight components, which has been first introduced by Markov. Thus, the Fock space for corresponding quantum fields is doubled (as shown by Ziino). The particular attention has been paid to the questions of chirality and helicity (two concepts which are frequently confused in the literature) for Dirac and Majorana states, and to the normalization (“the mass dimension”). We further review several experimental consequences which follow from the previous works of M. Kirchbach et al. on neutrinoless double beta decay, and G.J. Ni et al. on meson lifetimes. The results are generalized for spins 1, 3/2 and 2.

A Dirac Type Operator on the Non-Commutative Disk

We study an example of the index problem for a Dirac-like operator subject to Atiyah–Patodi–Singer boundary conditions on the noncommutative unit disk.

Bounded and unbounded Fredholm modules for quantum projective spaces

AbstractWe construct explicit generators of the K-theory and K-homology of the coordinate algebras of ‘functions’ on the quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Diraclike operators and spectral triples of any positive real dimension.

ON THE RIEMANN HYPOTHESIS, AREA QUANTIZATION, DIRAC OPERATORS, MODULARITY, AND RENORMALIZATION GROUP

Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert–Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line: sn = ½ + iρn. A detailed analysis of a one-dimensional Dirac-like operator with a potential V(x) is given that reproduces the spectrum of energy levels En = ρn, when the boundary conditions ΨE (x = -∞) = ± ΨE (x = +∞) are imposed. Such potential V(x) is derived implicitly from the relation [Formula: see text], where the functional form of [Formula: see text] is given by the full-fledged Riemann–von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the [Formula: see text] terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr–Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large [Formula: see text] has a one-to-one correspondence with the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the renormalization group program.

Local duality, chiral supersymmetry and fermion-like formulation of non-Abelian pure gauge fields

This work generalizes the fermion-like formulation of the Maxwell theory to the non-Abelian Yang–Mills theory without matter fields. This is a new representation of the Lie algebra valued electric and magnetic fields. The resulting equations of motion are invariant under the chiral transformation. In this formulation, duality is a kind of chirality. We may also define local duality transformations in terms of space-time dependent parameters. There is an N=1 supersymmetry for the Dirac-like operator in this representation.

Instantons on Noncommutative R4 and Projection Operators

I carefully study noncommutative version of ADHM construction of instantons, which was proposed by Nekrasov and Schwarz. Noncommutative ${\bf R}^4$ is described as algebra of operators acting in Fock space. In ADHM construction of instantons, one looks for zero-modes of Dirac-like operator. The feature peculiar to noncommutative case is that these zero-modes project out some states in Fock space. The mechanism of these projections is clarified when the gauge group is U(1). I also construct some zero-modes when the gauge group is U(N) and demonstrate that the projections also occur, and the mechanism is similar to the U(1) case. A physical interpretation of the projections in IIB matrix model is briefly discussed.

Applications in physics of the multiplicative anomaly formula involving some basic differential operators

In the framework leading to the multiplicative anomaly formula — which is here proven to be valid even in cases of known spectrum but non-compact manifold (very important in Physics) — zeta-function regularisation techniques are shown to be extremely efficient. Dirac-like operators and harmonic oscillators are investigated in detail, in any number of space dimensions. They yield a non-zero anomaly which, on the other hand, can always be expressed by means of a simple analytical formula. These results are used in several physical examples, where the determinant of a product of differential operators is not equal to the product of the corresponding functional determinants. The simplicity of the Hamiltonian operators chosen is aimed at showing that such a situation may be quite widespread in mathematical physics. However, the consequences of the existence of the determinant anomaly have often been overlooked.

Maslov index in Chern-Simons quantum mechanics.

We consider integrable Hamiltonian systems whose N conserved charges generate a U(1${)}^{\mathit{N}}$ symmetry group. We promote this global symmetry to a local one by adding appropriate gauge fields. The resulting action is supplemented by a (0+1)-dimensional Chern-Simons term. The model is used to study certain aspects of semiclassical quantization in ordinary quantum mechanics, in particular the origin of the Maslov index. It turns out that a consistent (semiclassical) quantization of the model requires the coefficients of the Chern-Simons term to be integer or half-integer. This parameter quantization implies the well-known semiclassical quantization rules for the action variables ${\mathit{J}}_{\mathit{i}}$, which coincide (by a kind of Gauss-law constraint) with the coefficients of the Chern-Simons term. Typically the gauged U(1${)}^{\mathit{N}}$ symmetry is free from local anomalies, but it can suffer from a global ${\mathit{Z}}_{2}$-type anomaly. Using a phase-space formalism, the (bosonic) fluctuations around a fixed classical trajectory are governed by a (0+1)-dimensional, Dirac-like operator coupled to a gauge field which assumes values in the Lie algebra of Sp(2N), the structure group of the tangent bundle over phase space. Because of ${\mathrm{\ensuremath{\Pi}}}_{1}$(Sp(2N))=Z, the space of gauge transformations decomposes into topologically inequivalent sectors, thus giving rise to the possibility of global anomalies. The ${\mathit{Z}}_{2}$ anomaly manifests itself in a nonzero Maslov index ${\mathrm{\ensuremath{\mu}}}_{\mathit{i}}$ in the semiclassical quantization prescription ${\mathit{J}}_{\mathit{i}}$=${\mathit{n}}_{\mathit{i}}$+1/4${\mathrm{\ensuremath{\mu}}}_{\mathit{i}}$, ${\mathit{n}}_{\mathit{i}}$\ensuremath{\in}Z. In the Hamiltonian formulation the anomaly is controlled by the Atiyah-Patodi-Singer index theorem, and in the Lagrangian formulation the Morse index theorem plays an analogous role. One finds that the Maslov indices ${\mathrm{\ensuremath{\mu}}}_{\mathit{i}}$ are related to the winding numbers for ${\mathrm{\ensuremath{\Pi}}}_{1}$(Sp(2N)) of a set of one-parameter families of symplectic matrices constructed out of the Jacobi fields around the classical trajectories.

Supersymmetric chiral effective action and nonabelian anomalies

The definition of the one-loop supersymmetric effective action for chiral superfields coupled to a nonabelian gauge superfield is carefully discussed. The variation of the odd parity part is not always integrable after regularisation, if anomalies are present. When this is compensated by a local additional term, so as to obtain an integral variation, a supersymmetric expression for the nonabelian anomaly, satisfying the Wess-Zumino consistency conditions, is simply obtained. The nonabelian anomaly is shown to be reproduced by relating it to the index for a Dirac-like operator defined on a two-dimensional disc in the space of gauge superfields. The integrated odd parity part of the effective action is further identified formally with the spectral asymmetry of a related Dirac-like operator involving one additional boson coordinate. The results are also expressed in terms of superforms which allow a connection to be made to general topological discussions of the anomaly. This depends on an analysis of the cohomology of superforms over flat superspace.