scholarly journals Gauge-invariant theories and higher-degree forms

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
S. Salgado
Keyword(s):  
Lie Algebra ◽  
Covariant Derivative ◽  
Differential Algebra ◽  
Differential Forms ◽  
Mathematical Structure ◽  
Gauge Invariant ◽  
Standard Formula ◽  
Weil Theorem ◽  
Differential Algebras ◽  
Chern Simons

Abstract A free differential algebra is generalization of a Lie algebra in which the mathematical structure is extended by including of new Maurer-Cartan equations for higher-degree differential forms. In this article, we propose a generalization of the Chern-Weil theorem for free differential algebras containing only one p-form extension. This is achieved through a generalization of the covariant derivative, leading to an extension of the standard formula for Chern-Simons and transgression forms. We also study the possible existence of anomalies originated on this kind of structure. Some properties and particular cases are analyzed.

GRADED LIE ALGEBRA AND GENERALIZED CHERN-SIMONS ACTIONS IN ARBITRARY DIMENSIONS

1992 ◽  
Vol 07 (13) ◽  
pp. 1137-1147 ◽  
Author(s):  
NOBORU KAWAMOTO ◽  
YOSHIYUKI WATABIKI
Keyword(s):  
Lie Algebra ◽  
Essential Role ◽  
Three Dimensional ◽  
Gauge Invariant ◽  
Supersymmetric Version ◽  
Graded Lie Algebra ◽  
New Type ◽  
Arbitrary Dimensions ◽  
Chern Simons ◽  
Odd Dimensions

We obtain generalized Chern-Simons actions in arbitrary dimensions. A graded Lie algebra plays an essential role in extending into odd dimensions in contrast with the even-dimensional generalized Chern-Simons action which we have obtained previously. As a particular example we obtain three-dimensional generalized Chern-Simons action which includes the standard Chern-Simons action together with additional terms which are responsible for a new type of gauge symmetry as well as supersymmetry. As a particular supersymmetric version, we obtain an action whose partition function is Casson invariant. We also find series of gauge invariant physical observables.

scholarly journals Topological Gravity of Chern-Simons-Antoniadis-Savvidy in 2+1 Dimensions

2020 ◽  
Vol 9 (3) ◽  
pp. 65-71
Author(s):  
Suhaivi Hamdan ◽  
Defrianto Defrianto ◽  
Erwin Erwin ◽  
Saktioto Saktioto
Keyword(s):  
Field Strength ◽  
Poincaré Group ◽  
Gauge Invariant ◽  
Poincare Group ◽  
Lower Rank ◽  
Weil Theorem ◽  
Lagrangian Action ◽  
Strength Tensor ◽  
Topological Gravity ◽  
Chern Simons

Pada artikel ini akan ditunjukan analisa dari perluasan gauge invariant exact dan metric independent untuk menkontruksi lower-rank field-strength tensor. Hasil ini akan digunakan untuk mengkontruski ulang Chern-Simons-Antoniadis-Savvidy formasi (2n+1) pada dimensi genap dengan menggunakan pendekatan diferensial geometri. Selanjutnya akan dianalisa bentuk topological gravitasi 2-dimensi yang merupakan perluasan dari teorema Chern-Weil yang telah dikembangkan oleh Izurieta-Munoz-Salgado. Hasil dari penelitian ini memperlihatkan bahwa aksi Lagrangian yang sama seperti pada topological gravitasi Chern-Simons forms pada dimensi (2n+1) invariant terhadap Poincare group SO(D−1,1)  SO(D−1,2). This article determine and analyess of the extended gauge invariant exact and metric independent to construct the lower-rank field-strength tensor. These results used to construct Chern-Simons-Antoniadis-Savvidy (2n+1)-forms even dimensions using a differential geometry approach. This result analyzed 2-dimensional topological gravity forms that extended Chern-Weil theorem which has been developed by Izurieta-Munoz-Salgado. These results show similary topological gravity Lagrangian action of Chern-Simons forms (2n+1)-dimension invariant under Poincare group SO(D−1,1)  SO(D−1,2).Keywords: Gauge theory, field-strength tensor, Chern-Weill theorem, Chern-Simons-Antoniadis-Savvidy forms, topological gravity

scholarly journals Wronskian Envelope of a Lie Algebra

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Laurent Poinsot
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Free Lie Algebra ◽  
Sufficient Condition ◽  
Enveloping Algebra ◽  
Lie Bracket ◽  
Differential Algebras

The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.

scholarly journals GENERALIZED GAUGE THEORIES AND THE WEINBERG–SALAM MODEL WITH DIRAC–KÄHLER FERMIONS

2001 ◽  
Vol 16 (23) ◽  
pp. 3867-3895 ◽  
Author(s):  
NOBORU KAWAMOTO ◽  
HIROSHI UMETSU ◽  
TAKUYA TSUKIOKA
Keyword(s):  
Gauge Theory ◽  
Lie Algebra ◽  
Noncommutative Geometry ◽  
Gauge Theories ◽  
Differential Forms ◽  
Gauge Fields ◽  
Present Formulation ◽  
Yang Mills ◽  
Graded Lie Algebra ◽  
Chern Simons

We extend the previously proposed generalized gauge theory formulation of the Chern–Simons type and topological Yang–Mills type actions into Yang–Mills type actions. We formulate gauge fields and Dirac–Kähler matter fermions by all degrees of differential forms. The simplest version of the model which includes only zero and one-form gauge fields accommodated with the graded Lie algebra of SU (2|1) supergroup leads the Weinberg–Salam model. Thus the Weinberg–Salam model formulated by noncommutative geometry is a particular example of the present formulation.

scholarly journals DISCRETE AND CONTINUUM APPROACHES TO THREE-DIMENSIONAL QUANTUM GRAVITY

1991 ◽  
Vol 06 (39) ◽  
pp. 3591-3600 ◽  
Author(s):  
HIROSI OOGURI ◽  
NAOKI SASAKURA
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Analogue Model ◽  
Gauge Invariant ◽  
Invariant Functions ◽  
Dimensional Lattice ◽  
Chern Simons ◽  
Dimensional Surface

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

scholarly journals REMARKS ON A CHERN–SIMONS-LIKE COUPLING

2011 ◽  
Vol 26 (37) ◽  
pp. 2813-2821
Author(s):  
PATRICIO GAETE
Keyword(s):  
Gauge Theory ◽  
Vector Field ◽  
Static Potential ◽  
Gauge Invariant ◽  
Massive Vector ◽  
Hidden Sector ◽  
Dependent Variables ◽  
Qualitative Nature ◽  
Path Dependent ◽  
Chern Simons

We consider the static quantum potential for a gauge theory which includes a light massive vector field interacting with the familiar U (1) QED photon via a Chern–Simons-like coupling, by using the gauge-invariant, but path-dependent, variables formalism. An exactly screening phase is then obtained, which displays a marked departure of a qualitative nature from massive axionic electrodynamics. The above static potential profile is similar to that encountered in axionic electrodynamics consisting of a massless axion-like field, as well as to that encountered in the coupling between the familiar U (1) QED photon and a second massive gauge field living in the so-called U (1)h hidden-sector, inside a superconducting box.

scholarly journals THE UNIVERSAL LINK POLYNOMIAL

1992 ◽  
Vol 07 (05) ◽  
pp. 877-945 ◽  
Author(s):  
E. GUADAGNINI
Keyword(s):  
Field Theory ◽  
Closed Form ◽  
Lie Algebra ◽  
Wilson Line ◽  
Expectation Values ◽  
Simple Lie Algebra ◽  
Chern Simons

The solution of the non-Abelian SU (N) quantum Chern–Simons field theory defined in R3 is presented. It is shown how to compute the expectation values of the Wilson line operators, associated with oriented framed links, in closed form. The main properties of the universal link polynomial, defined by these expectation values, are derived in the case of a generic real simple Lie algebra. The resulting polynomials for some simple examples of links are reported.

Duality on the quantum space(3) with two parameters

Open Physics ◽  
2012 ◽  
Vol 10 (5) ◽  
Author(s):  
Muttalip Özavşar ◽  
Gürsel Yeşilot
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Differential Forms ◽  
Quantum Space ◽  
Dual Algebra ◽  
Commutation Relations ◽  
Dual Hopf Algebra ◽  
Invariant Differential ◽  
Leibniz Rules ◽  
Two Parameters

AbstractIn this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.

The algebra of differential forms on a full matric bialgebra

1993 ◽  
Vol 114 (1) ◽  
pp. 111-130 ◽  
Author(s):  
A. Sudbery
Keyword(s):  
Vector Space ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Differential Forms ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Algebra Of Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Necessary And Sufficient

AbstractWe construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

scholarly journals Explicit Connection Between Conformal Field Theory and 2+1 Chern–Simons Theory

1997 ◽  
Vol 12 (23) ◽  
pp. 1687-1697
Author(s):  
Daniel C. Cabra ◽  
Gerardo L. Rossini
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Wilson Loop ◽  
Simons Theory ◽  
Gauge Invariant ◽  
Conformal Field ◽  
Chern Simons ◽  
Chern Simons Theory

We give explicit field theoretical representations for the observables of (2+1)-dimensional Chern–Simons theory in terms of gauge-invariant composites of 2-D WZW fields. To test our identification we compute some basic Wilson loop correlators and re-obtain the known results.

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