scholarly journals Note on the bundle geometry of field space, variational connections, the dressing field method, & presymplectic structures of gauge theories over bounded regions

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
J. François ◽  
N. Parrini ◽  
N. Boulanger
Keyword(s):  
Gauge Theories ◽  
Field Method ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Field Independent ◽  
Field Dependent ◽  
Presymplectic Structure ◽  
The One ◽  
Phase Space Methods ◽  
Special Case

Abstract In this note, we consider how the bundle geometry of field space interplays with the covariant phase space methods so as to allow to write results of some generality on the presymplectic structure of invariant gauge theories coupled to matter. We obtain in particular the generic form of Noether charges associated with field-independent and field-dependent gauge parameters, as well as their Poisson bracket. We also provide the general field-dependent gauge transformations of the presymplectic potential and 2-form, which clearly highlights the problem posed by boundaries in generic situations. We then conduct a comparative analysis of two strategies recently considered to evade the boundary problem and associate a modified symplectic structure to a gauge theory over a bounded region: namely the use of edge modes on the one hand, and of variational connections on the other. To do so, we first try to give the clearest geometric account of both, showing in particular that edge modes are a special case of a differential geometric tool of gauge symmetry reduction known as the “dressing field method”. Applications to Yang-Mills theory and General Relativity reproduce or generalise several results of the recent literature.

scholarly journals Bundle geometry of the connection space, covariant Hamiltonian formalism, the problem of boundaries in gauge theories, and the dressing field method

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
J. François
Keyword(s):  
Gauge Theories ◽  
Hamiltonian Formalism ◽  
Field Method ◽  
Simons Theory ◽  
Geometric Framework ◽  
Gauge Transformations ◽  
Connection Space ◽  
Presymplectic Structure ◽  
Chern Simons ◽  
Invariant Gauge

Abstract We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satisfying two restricting hypothesis. In particular, we derive the general field-dependent gauge transformations of the presymplectic potential and presymplectic 2-form in both cases. We point-out that a generalisation of the standard bundle geometry, called twisted geometry, arises naturally in the study of non-invariant gauge theories (e.g. non-Abelian Chern-Simons theory). These results prove that the well-known problem of associating a symplectic structure to a gauge theory over bounded regions is a generic feature of both classes. The edge modes strategy, recently introduced to address this issue, has been actively developed in various contexts by several authors. We draw attention to the dressing field method as the geometric framework underpinning, or rather encompassing, this strategy. The geometric insight afforded by the method both clarifies it and clearly delineates its potential shortcomings as well as its conditions of success. Applying our general framework to various examples allows to straightforwardly recover several results of the recent literature on edge modes and on the presymplectic structure of general relativity.

CONFORMALLY COVARIANT DRESSING TRANSFORMATIONS FOR SUPER SELF-DUALITY EQUATIONS IN D=4

1989 ◽  
Vol 04 (10) ◽  
pp. 971-982
Author(s):  
J. AVAN
Keyword(s):  
Linear System ◽  
Gauge Transformations ◽  
Four Dimensions ◽  
Yang Mills ◽  
Self Duality ◽  
Loop Algebras ◽  
Field Dependent

A set of conformally covariant dressing transformations is constructed for the supersym-metric N=3 self-duality equations in four dimensions, using the associated covariant linear system. They form a closed, 5+6-index algebra, up to field-dependent gauge transformations, containing the previously known loop algebras as a particular subset. This construction generalizes the formerly built set of conformally covariant DT for ordinary self-dual Yang-Mills.

scholarly journals On the component structure of one-loop effective actions in 6D, 𝒩 = (1,0) and 𝒩 = (1,1) supersymmetric gauge theories

2019 ◽  
Vol 35 (09) ◽  
pp. 2050060 ◽  
Author(s):  
I. L. Buchbinder ◽  
A. S. Budekhina ◽  
B. S. Merzlikin
Keyword(s):  
Effective Potential ◽  
Effective Action ◽  
Gauge Theories ◽  
Scalar Fields ◽  
Low Energy ◽  
Supersymmetric Gauge Theories ◽  
Component Structure ◽  
Yang Mills ◽  
Supersymmetric Gauge ◽  
The One

We study the six-dimensional [Formula: see text] and [Formula: see text] supersymmetric Yang–Mills (SYM) theories in the component formulation. The one-loop divergencies of effective action are calculated. The leading one-loop low-energy contributions to bosonic sector of effective action are found. It is explicitly demonstrated that the contributions to effective potential for the constant background scalar fields are absent in the [Formula: see text] SYM theory.

Quantum gauge theories

2021 ◽  
pp. 223-268
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Gauge Theories ◽  
Brst Symmetry ◽  
Background Field ◽  
Field Method ◽  
Extensive Discussion ◽  
Functional Integrals ◽  
Yang Mills ◽  
Gauge Dependence ◽  
Background Field Method ◽  
Popov Method

This chapter, which is the last chapter in Part I, is devoted to an extensive discussion of quantum gauge theories, which is based on functional integrals and Lagrangian quantization. After introducing the notion of a Yang-Mills gauge theory, the Faddeev-Popov method (also known as the DeWitt-Faddeev-Popov procedure) is explained. Starting from this point, the BRST symmetry is formulated, and the corresponding Ward identities (called Slavnov-Taylor identities in some cases) established. More specialized subjects, such as the gauge dependence of effective action and the background field method, are dealt with in detail. In addition, Yang-Mills theory is analyzed as a primary example of general theorems concerning the renormalization of gauge theories.

GEOMETRIC REGULARIZATION AND GAUGE INVARIANCE IN CHERN–SIMONS THEORIES

1992 ◽  
Vol 07 (02) ◽  
pp. 235-256 ◽  
Author(s):  
MANUEL ASOREY ◽  
FERNANDO FALCETO
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Gauge Condition ◽  
Simons Theory ◽  
Coupling Constant ◽  
Gauge Transformations ◽  
Yang Mills ◽  
The One ◽  
Chern Simons ◽  
Three Dimensional Space

Some perturbative aspects of Chern–Simons theories are analyzed in a geometric-regularization framework. In particular, we show that the independence from the gauge condition of the regularized theory, which insures its global meaning, does impose a new constraint on the parameters of the regularization. The condition turns out to be the one that arises in pure or topologically massive Yang–Mills theories in three-dimensional space–times. One-loop calculations show the existence of nonvanishing finite renormalizations of gauge fields and coupling constant which preserve the topological meaning of Chern–Simons theory. The existence of a (finite) gauge-field renormalization at one-loop level is compensated by the renormalization of gauge transformations in such a way that the one-loop effective action remains gauge-invariant with respect to renormalized gauge transformations. The independence of both renormalizations from the space–time volume indicates the topological nature of the theory.

scholarly journals General Yang–Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical frameworkorFrom Bianchi identities to twisted Courant algebroids

2014 ◽  
Vol 12 (01) ◽  
pp. 1550009 ◽  
Author(s):  
Melchior Grützmann ◽  
Thomas Strobl
Keyword(s):  
Gauge Theories ◽  
Differential Forms ◽  
Coupled System ◽  
Scalar Fields ◽  
Gauge Fields ◽  
Courant Algebroids ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Derived Bracket ◽  
Form Degree

Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into what is called a Q-structure or, equivalently, an L∞-algebroid. This has many technical and conceptual advantages: complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines and is given by what is called the derived bracket construction. This paper aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form degree p, we pay particular attention to p = 2, i.e. 1- and 2-form gauge fields coupled nonlinearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g. one as a particular kind of a vector-bundle twisted Courant algebroid.

TWISTOR-LIKE APPROACH TO SUPERSYMMETRIC GAUGE THEORIES

1994 ◽  
Vol 09 (32) ◽  
pp. 5635-5649
Author(s):  
HIROYUKI YAMASHITA
Keyword(s):  
Infinite Series ◽  
Gauge Theories ◽  
Supersymmetric Gauge Theories ◽  
Yang Mills ◽  
Constraint Equations ◽  
Abelian Case ◽  
Supersymmetric Gauge ◽  
Special Case

We consider the constraint conditions on the supersymmetric Yang-Mills theories in D=6, N=1, which are gauge- and super-covariant. These constraint conditions have been introduced to remove superfluous fields. We present a method to tell how and to what degree the constraint restricts the theory in the D=6, N=1 Abelian case by analogy with the twistor method for self-dual equations. The constraint is transformed into an infinite series of constraint equations. We find that a previously known theory in D=6, N=1 corresponds to a special case which is chosen so that the series is finite.

scholarly journals Super heat kernel and one-loop divergence of super Yang–Mills theory in conformal supergravity

2019 ◽  
Vol 2019 (10) ◽  
Author(s):  
Ka-Hei Leung
Keyword(s):  
Heat Kernel ◽  
Iterative Scheme ◽  
Kernel Method ◽  
Background Field ◽  
Field Method ◽  
Conformal Supergravity ◽  
Quantum Theories ◽  
Yang Mills ◽  
Background Field Method ◽  
The One

Abstract We consider super Yang–Mills (SYM) theory in $N=1$ conformal supergravity. Using the background field method and the Faddeev–Popov procedure, the quantized action of the theory is presented. Its one-loop effective action is studied using the heat kernel method. We shall develop a non-iterative scheme, generalizing the non-supersymmetric case, to obtain the super heat kernel coefficients. In particular, the first three coefficients, which govern the one-loop divergence, will be calculated. We shall also demonstrate how to schematically derive the higher-order coefficients. The method presented here can be readily applied to various quantum theories. We shall, as an application, derive the full one-loop divergence of SYM in conformal supergravity.

scholarly journals Harmonic Superspace Approach to the Effective Action in Six-Dimensional Supersymmetric Gauge Theories

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 68 ◽  
Author(s):  
Ioseph Buchbinder ◽  
Evgeny Ivanov ◽  
Boris Merzlikin ◽  
Konstantin Stepanyantz
Keyword(s):  
Effective Action ◽  
Gauge Theories ◽  
Proper Time ◽  
Background Field ◽  
Field Method ◽  
Quantum Corrections ◽  
Supersymmetric Gauge Theories ◽  
Loop Approximation ◽  
Supersymmetric Gauge ◽  
The One

We review the recent progress in studying the quantum structure of 6 D , N = ( 1 , 0 ) , and N = ( 1 , 1 ) supersymmetric gauge theories formulated through unconstrained harmonic superfields. The harmonic superfield approach allows one to carry out the quantization and calculations of the quantum corrections in a manifestly N = ( 1 , 0 ) supersymmetric way. The quantum effective action is constructed with the help of the background field method that secures the manifest gauge invariance of the results. Although the theories under consideration are not renormalizable, the extended supersymmetry essentially improves the ultraviolet behavior of the lowest-order loops. The N = ( 1 , 1 ) supersymmetric Yang–Mills theory turns out to be finite in the one-loop approximation in the minimal gauge. Furthermore, some two-loop divergences are shown to be absent in this theory. Analysis of the divergences is performed both in terms of harmonic supergraphs and by the manifestly gauge covariant superfield proper-time method. The finite one-loop leading low-energy effective action is calculated and analyzed. Furthermore, in the Abelian case, we discuss the gauge dependence of the quantum corrections and present its precise form for the one-loop divergent part of the effective action.

TOPOLOGICAL QUANTUM FIELD THEORIES AND GAUGE INVARIANCE IN STOCHASTIC QUANTIZATION

1991 ◽  
Vol 06 (16) ◽  
pp. 2793-2803 ◽  
Author(s):  
Laurent Baulieu
Keyword(s):  
Gauge Theories ◽  
Three Dimensional ◽  
Langevin Equations ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Topological Quantum ◽  
Brst Charge ◽  
Deep Relationship

The Langevin equations describing the quantization of gauge theories have a geometrical structure. We show that stochastically quantized gauge theories are governed by a single differential operator. The latter combines supersymmetry and ordinary gauge transformations. Quantum field theory can be defined on the basis of a Hamiltonian of the type [Formula: see text], where Q has has deep relationship with the conserved BRST charge of a topological gauge theory, and [Formula: see text] is its adjoint. We display the examples of Yang-Mills theory and of 2D gravity. Interesting applications are for first order actions, in particular for the theories defined by the three dimensional Chern Simons action as well as the “two dimensional” ∫M2TrϕF.

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