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Journal articles on the topic 'Hodge-Dirac operator'

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Author: Grafiati

Published: 10 December 2022

Last updated: 29 January 2023

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1

Sushch, Volodymyr. "2D Discrete Hodge–Dirac Operator on the Torus." Symmetry 14, no. 8 (July 28, 2022): 1556. http://dx.doi.org/10.3390/sym14081556.

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We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups.

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2

Cosmo, Fabio di, and Alessandro Zampini. "Dirac operators on the S3 and S2 spheres." International Journal of Geometric Methods in Modern Physics 14, no. 08 (May 11, 2017): 1740005. http://dx.doi.org/10.1142/s0219887817400059.

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We describe both the Hodge–de Rham and the spin manifold Dirac operator on the spheres [Formula: see text] and [Formula: see text], following the formalism introduced by Kähler, and exhibit a complete spectral resolution for them in terms of suitably globally defined eigenspinors.

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3

Di Cosmo, Fabio, Giuseppe Marmo, Juan Manuel Pérez-Pardo, and Alessandro Zampini. "A Hodge–de Rham Dirac operator on the quantum SU(2)." International Journal of Geometric Methods in Modern Physics 15, no. 02 (January 24, 2018): 1850030. http://dx.doi.org/10.1142/s0219887818500305.

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We describe how it is possible to define a Hodge–de Rham Dirac operator associated to a suitable Cartan-Killing metric form upon the exterior algebra over the quantum spheres [Formula: see text] equipped with a three-dimensional left covariant calculus.

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4

Leopardi, Paul, and Ari Stern. "The Abstract Hodge--Dirac Operator and Its Stable Discretization." SIAM Journal on Numerical Analysis 54, no. 6 (January 2016): 3258–79. http://dx.doi.org/10.1137/15m1047684.

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5

MALIK, R. P. "ONE-FORM ABELIAN GAUGE THEORY AS THE HODGE THEORY." International Journal of Modern Physics A 22, no. 21 (August 20, 2007): 3521–62. http://dx.doi.org/10.1142/s0217751x07037135.

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We demonstrate that the two (1+1)-dimensional (2D) free 1-form Abelian gauge theory provides an interesting field theoretical model for the Hodge theory. The physical symmetries of the theory correspond to all the basic mathematical ingredients that are required in the definition of the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, constitute an algebra that is reminiscent of the algebra obeyed by the de Rham cohomological operators. The topological features of the above theory are discussed in terms of the BRST and co-BRST operators. The super-de Rham cohomological operators are exploited in the derivation of the nilpotent (anti-)BRST, (anti-)co-BRST symmetry transformations and the equations of motion for all the fields of the theory, within the framework of the superfield formulation. The derivation of the equations of motion, by exploiting the super-Laplacian operator, is a completely new result in the framework of the superfield approach to BRST formalism. In an Appendix, the interacting 2D Abelian gauge theory (where there is a coupling between the U(1) gauge field and the Dirac fields) is also shown to provide a tractable field theoretical model for the Hodge theory.

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6

RODRIGUES, WALDYR A., and QUINTINO A. G. SOUZA. "AN AMBIGUOUS STATEMENT CALLED THE "TETRAD POSTULATE" AND THE CORRECT FIELD EQUATIONS SATISFIED BY THE TETRAD FIELDS." International Journal of Modern Physics D 14, no. 12 (December 2005): 2095–150. http://dx.doi.org/10.1142/s0218271805008157.

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The names tetrad, tetrads, cotetrads have been used with many different meanings in the physics literature, not all of them equivalent from the mathematical point of view. In this paper, we introduce unambiguous definitions for each of those terms, and show how the old miscellanea made many authors introduce in their formalism an ambiguous statement called the "tetrad postulate," which has been the source of much misunderstanding, as we show explicitly by examining examples found in the literature. Since formulating Einstein's field equations intrinsically in terms of cotetrad fields θa, a = 0, 1, 2, 3 is a worthy enterprise, we derive the equation of motion of each θausing modern mathematical tools (the Clifford bundle formalism and the theory of the square of the Dirac operator). Indeed, we identify (giving all details and theorems) from the square of the Dirac operator some noticeable mathematical objects, namely, the Ricci, Einstein, covariant D'Alembertian and the Hodge Laplacian operators, which permit us to show that each θasatisfies a well-defined wave equation. Also, we present for completeness a detailed derivation of the cotetrad wave equations from a variational principle. We compare the cotetrad wave equation satisfied by each θawith some others appearing in the literature, and which are unfortunately in error.

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7

Abreu-Blaya, Ricardo, Juan Bory-Reyes, and Michael Shapiro. "The Cauchy Transform for the Hodge/De Rham System and Some of its Properties." gmj 14, no. 1 (March 2007): 1–20. http://dx.doi.org/10.1515/gmj.2007.1.

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Abstract We study the analogue of the Cauchy transform for the theory of solutions of the Hodge/de Rham system in the case of a rectifiable surface of integration which additionally satisfies an Ahlfors/David regularity condition and we prove the Cauchy integral formula, the Plemelj/Privalov theorem and the Sokhotski/Plemelj theorem for it, as well as the necessary and sufficient condition for the possibility to extend a given 𝑘-form from such a surface to a harmonic 𝑘-form in the domain. A formula for the square of the singular Cauchy transform is given. The proofs of all these facts are based on a close relation between algebra-valued null-solutions of the Dirac operator in the Euclidean space and hyperholomorphic functions of Clifford analysis.

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8

van Neerven, Jan, and Rik Versendaal. "$$L^p$$-Analysis of the Hodge–Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds." Journal of Geometric Analysis 28, no. 4 (November 4, 2017): 3109–38. http://dx.doi.org/10.1007/s12220-017-9947-4.

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9

LUSANNA, LUCA. "CLASSICAL YANG-MILLS THEORY WITH FERMIONS II: DIRAC’S OBSERVABLES." International Journal of Modern Physics A 10, no. 26 (October 20, 1995): 3675–757. http://dx.doi.org/10.1142/s0217751x95001753.

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For pure Yang-Mills theory on Minkowski space-time, formulated in functional spaces where the covariant divergence is an elliptic operator without zero modes, and for a trivial principal bundle over the fixed time Euclidean space with a compact, semisimple, connected and simply connected structure Lie group, a Green function for the covariant divergence has been found. It allows one to solve the first class constraints associated with Gauss’ laws and to identify a connection-dependent coordinatization of the trivial principal bundle. In a neighborhood of the global identity section, by using canonical coordinates of the first kind on the fibers, one has a symplectic implementation of the Lie algebra of the small gauge transformations generated by Gauss’ laws and one can make a generalized Hodge decomposition of the gauge potential one-forms based on the BRST operator. This decomposition singles out a pure gauge background connection (the BRST ghost as Maurer-Cartan one-form on the group of gauge transformations) and a transverse gauge-covariant magnetic gauge potential. After an analogous decomposition of the electric field strength into the transverse and the longitudinal part, Dirac’s observables associated with the transverse electric and magnetic components are identified as their restriction to the global identity section of the trivial principal bundle. The longitudinal part of the electric field can be re-expressed in terms of these electric and magnetic transverse parts and of the constraints without Gribov ambiguity. The physical Lagrangian, Hamiltonian, non-Abelian and topological charges have been obtained in terms of transverse Dirac’s observables, also in the presence of fermion fields, after a symplectic decoupling of the gauge degrees of freedom; one has an explicit realization of the abstract “Riemannian metric” on the orbit space. Both the Lagrangian and the Hamiltonian are nonlocal and nonpolynomial; like in the Coulomb gauge they are not Lorentz-invariant, but the invariance can be enforced on them if one introduces Wigner covariance of the observables by analyzing the various kinds of Poincare orbits of the system and by reformulating the theory on suitable spacelike hypersurfaces, following Dirac. By extending to classical relativistic field theory the problems associated with the Lorentz noncovariance of the canonical (presymplectic) center of mass for extended relativistic systems, in the sector of the field theory with P2>0 and W2≠0 one identifies a classical invariant intrinsic unit of length, determined by the Poincare Casimirs, whose quantum counterpart is the ultraviolet cutoff looked for by Dirac and Yukawa: it is the Compton wavelength of the field configuration (in an irreducible Poincare representation) multiplied by the value of its spin.

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10

Cerejeiras, P., and J. Cnops. "Hodge—dirac operators for hyperbolic spaces." Complex Variables, Theory and Application: An International Journal 41, no. 3 (May 2000): 267–78. http://dx.doi.org/10.1080/17476930008815254.

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11

Duse, Erik. "Second order elliptic equations and Hodge-Dirac operators." Journal of Functional Analysis 281, no. 12 (December 2021): 109267. http://dx.doi.org/10.1016/j.jfa.2021.109267.

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12

Duse, Erik. "Second order elliptic equations and Hodge-Dirac operators." Journal of Functional Analysis 281, no. 12 (December 2021): 109267. http://dx.doi.org/10.1016/j.jfa.2021.109267.

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13

McIntosh, Alan, and Sylvie Monniaux. "Hodge–Dirac, Hodge-Laplacian and Hodge–Stokes operators in $L^p$ spaces on Lipschitz domains." Revista Matemática Iberoamericana 34, no. 4 (December 17, 2018): 1711–53. http://dx.doi.org/10.4171/rmi/1041.

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14

MALIK, R. P. "DUAL BRST SYMMETRY FOR QED." Modern Physics Letters A 16, no. 08 (March 14, 2001): 477–88. http://dx.doi.org/10.1142/s0217732301003668.

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We show the existence of a co(dual)-BRST symmetry for the usual BRST invariant Lagrangian density of an Abelian gauge theory in two dimensions of space–time where a U(1) gauge field is coupled to the Noether conserved current (constructed by the Dirac fields). Under this new symmetry, it is the gauge-fixing term that remains invariant and the symmetry transformations on the Dirac fields are analogous to the chiral transformations. This interacting theory is shown to provide a tractable field theoretical model for the Hodge theory. The Hodge dual operation is shown to correspond to a discrete symmetry in the theory and the extended BRST algebra for the generators of the underlying symmetries turns out to be reminiscent of the algebra obeyed by the de Rham cohomology operators of differential geometry.

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15

Hytönen, Tuomas, Alan McIntosh, and Pierre Portal. "Holomorphic functional calculus of Hodge-Dirac operators in L p." Journal of Evolution Equations 11, no. 1 (September 24, 2010): 71–105. http://dx.doi.org/10.1007/s00028-010-0082-y.

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16

Yuying, Qiao, Swanhild Bernstein, Sirkka-Liisa, and John Ryan. "Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space." Journal d'Analyse Mathématique 98, no. 1 (December 2006): 43–64. http://dx.doi.org/10.1007/bf02790269.

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17

MALIK, R. P. "NEW LOCAL SYMMETRY FOR QED IN TWO DIMENSIONS." Modern Physics Letters A 15, no. 34 (November 10, 2000): 2079–85. http://dx.doi.org/10.1142/s0217732300002681.

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A new local, covariant and nilpotent symmetry is shown to exist for the interacting BRST invariant U(1) gauge theory in two dimensions of space–time. Under this new symmetry, it is the gauge-fixing term that remains invariant and the corresponding transformations on the Dirac fields turn out to be the analogue of chiral transformations. The extended BRST algebra is derived for the generators of all the underlying symmetries, present in the theory. This algebra turns out to be the analogue of the algebra obeyed by the de Rham cohomology operators of differential geometry. Possible interpretations and implications of this symmetry are pointed out in the context of BRST cohomology and Hodge decomposition theorem.

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18

Frey, Dorothee, Alan McIntosh, and Pierre Portal. "Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in L P." Journal d'Analyse Mathématique 134, no. 2 (February 2018): 399–453. http://dx.doi.org/10.1007/s11854-018-0013-3.

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