scholarly journals DIRAC OPERATORS ON LIE ALGEBROIDS

2021 ◽  
pp. 983
Author(s):  
Arezo Tarviji ◽  
Morteza Mirmohammad Rezaei
Keyword(s):  
Dirac Operator ◽  
Spin Structure ◽  
Dirac Operators ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Eigenvalue Bounds ◽  
Base Manifold ◽  
Spinor Bundle ◽  
Complex Spin ◽  
The One

We compare the Dirac operator on transitive Riemannian Lie algebroid equipped by spin or complex spin structure with the one defined on to its base manifold‎. Consequently we derive upper eigenvalue bounds of Dirac operator on base manifold of spin Lie algebroid twisted with the spinor bundle of kernel bundle‎.

scholarly journals Transitive double Lie algebroids via core diagrams

2021 ◽  
Vol 13 (3) ◽  
pp. 403
Author(s):  
Madeleine Jotz Lean ◽  
Kirill C. H. Mackenzie
Keyword(s):  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Base Manifold ◽  
Lie Groupoid ◽  
The Core ◽  
Fixed Base ◽  
The One

<p style='text-indent:20px;'>The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called <i>transitive</i>. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.</p><p style='text-indent:20px;'>The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.</p><p style='text-indent:20px;'>Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.</p>

scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

scholarly journals Higher spectral flow and an entire bivariant JLO cocycle

2012 ◽  
Vol 11 (1) ◽  
pp. 183-232 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
Alan L. Carey
Keyword(s):  
Dirac Operator ◽  
Chern Character ◽  
Dirac Operators ◽  
Cyclic Cohomology ◽  
Closed Manifold ◽  
Spectral Flow ◽  
Smooth Functions ◽  
Base Manifold ◽  
K Theory ◽  
Higher Spectral Flow

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

scholarly journals Integrable lifts for transitive Lie algebroids

2018 ◽  
Vol 29 (09) ◽  
pp. 1850062 ◽  
Author(s):  
Iakovos Androulidakis ◽  
Paolo Antonini
Keyword(s):  
Extra Dimensions ◽  
Lie Algebroids ◽  
The Other ◽  
Lie Algebroid ◽  
Base Manifold ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Other Hand ◽  
De Rham ◽  
Simply Connected

Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid is the quotient of a finite-dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an “Almeida–Molino” integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a “de Rham” integrable lift for any given transitive Abelian Lie algebroid.

scholarly journals Twistor spaces and the adiabatic limits of Dirac operators

2001 ◽  
Vol 164 ◽  
pp. 53-73 ◽  
Author(s):  
Masayoshi Nagase
Keyword(s):  
Dirac Operator ◽  
Spin Structure ◽  
Twistor Space ◽  
Dirac Operators ◽  
Twistor Spaces ◽  
Adiabatic Limits ◽  
Twisted Dirac Operator

We show that a (Spinq-style) twistor space admits a canonical Spin structure. The adiabatic limits of η-invariants of the associated Dirac operator and of an intrinsically twisted Dirac operator are then investigated.

scholarly journals TRANSVERSAL KILLING AND TWISTOR SPINORS ASSOCIATED TO THE BASIC DIRAC OPERATORS

2013 ◽  
Vol 25 (08) ◽  
pp. 1330011 ◽  
Author(s):  
ADRIAN MIHAI IONESCU ◽  
VLADIMIR SLESAR ◽  
MIHAI VISINESCU ◽  
GABRIEL EDUARD VÎLCU
Keyword(s):  
Riemannian Manifold ◽  
Dirac Operator ◽  
Mean Curvature ◽  
Spin Structure ◽  
Natural Extension ◽  
Dirac Operators ◽  
Harmonic Mean ◽  
Standard Definition ◽  
Killing Spinors ◽  
Harmonic Spinors

We study the interplay between the basic Dirac operator and the transversal Killing and twistor spinors. In order to obtain results for the general Riemannian foliations with bundle-like metric, we consider transversal Killing spinors that appear as natural extension of the harmonic spinors associated with the basic Dirac operator. In the case of foliations with basic-harmonic mean curvature, it turns out that this type of spinors coincide with the standard definition. We obtain the corresponding version of classical results on closed Riemannian manifold with spin structure, and extending some previous results.

scholarly journals ON EIGENVALUE ESTIMATES FOR THE SUBMANIFOLD DIRAC OPERATOR

2002 ◽  
Vol 13 (05) ◽  
pp. 533-548 ◽  
Author(s):  
NICOLAS GINOUX ◽  
BERTRAND MOREL
Keyword(s):  
Dirac Operator ◽  
Lower Bounds ◽  
Dirac Operators ◽  
Extrinsic Curvature ◽  
Eigenvalue Estimates ◽  
Killing Spinors ◽  
Spinor Fields ◽  
Twisted Dirac Operators

We give lower bounds for the eigenvalues of the submanifold Dirac operator in terms of intrinsic and extrinsic curvature expressions. We also show that the limiting cases give rise to a class of spinor fields generalizing that of Killing spinors. We conclude by translating these results in terms of intrinsic twisted Dirac operators.

scholarly journals On local integration of Lie brackets

2020 ◽  
Vol 2020 (760) ◽  
pp. 267-293 ◽  
Author(s):  
Alejandro Cabrera ◽  
Ioan Mărcuţ ◽  
María Amelia Salazar
Keyword(s):  
Vector Field ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Theory ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Local Integration ◽  
Lie Brackets ◽  
Complete Account

AbstractWe give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional proof of the fact that the category of germs of local Lie groupoids is equivalent to that of Lie algebroids.

Structure and Magnetism of REPtMg (RE = Pr,Nd, Sm)

2002 ◽  
Vol 57 (5) ◽  
pp. 488-494 ◽  
Author(s):  
Rainer Kraft ◽  
Gunter Kotzyba ◽  
Rolf-Dieter Hoffmann ◽  
Rainer Pöttgen
Keyword(s):  
Spin Structure ◽  
Three Dimensional ◽  
Point Of View ◽  
Single Crystal Diffraction ◽  
Magnetic Susceptibility Data ◽  
X Ray ◽  
Susceptibility Data ◽  
Complex Spin ◽  
Ferromagnetic Ordering ◽  
Ordered State

New magnesium based intermetallic compounds PrPtMg, NdPtMg and SmPtMg were synthesized from the elements by reaction in sealed tantalum tubes in a high-frequency furnace. The three compounds were investigated by X-ray powder and single crystal diffraction: ZrNiAl type, space group P6̄2m, a = 752.34(8), c = 412.66(4) pm, wR2 = 0.0668, 341 F2 values, 14 variables for PrPtMg, a = 748.80(8), c = 411.52(4) pm, wR2 = 0.0521, 196 F2 values, 14 variables for NdPtMg and a = 743.90(5), c = 409.80(3) pm, wR2 = 0.0489, 248 F2 values, 12 variables for SmPtMg. From a geometrical point of view these structures are composed of two types of platinum centered trigonal prisms, i. e. [Pt1Mg3RE6] and [Pt2Mg6RE3]. These prisms are condensed via common edges and faces. Together the platinum and magnesium atoms build three-dimensional [PtMg] networks in which the rare earth atoms are located in distorted pentagonal channels. Magnetic susceptibility data of PrPtMg show Curie-Weiss behaviour with an experimentalmagnetic moment of 3.59(2) μB and a paramagnetic Curie temperature of 7.5(5) K. Ferromagnetic ordering is detected at TC = 8.0(5) K with a magnetic moment of 1.75(5) μB/Pr at 4.5 K and 5 T. SmPtMg orders ferromagnetically below 52(1) K with a presumably complex spin structure in the ordered state.

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