SUPERFIELD FORMALISM OF TOPOLOGICAL YANG-MILLS THEORY

1989 ◽  
Vol 04 (27) ◽  
pp. 2625-2633 ◽  
Author(s):  
SHINICHI DEGUCHI
Keyword(s):  
Differential Forms ◽  
Yang Mills ◽  
Donaldson Invariants ◽  
Recursive Relations

Fields and their BRST transformations in the topological Yang-Mills theory are derived systematically using Bonora-Tonin’s superfield formalism. Anti-BRST transformations of the fields is also obtained. The differential forms which generate the Donaldson invariants and their recursive relations are natural consequences of the superfield formalism.

Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory

1982 ◽  
Vol 91 (3) ◽  
pp. 441-452 ◽  
Author(s):  
H. C. J. Sealey
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Differential Forms ◽  
Finite Energy ◽  
Energy Functional ◽  
Conformal Transformations ◽  
Second Variation ◽  
Yang Mills

In (5) it is shown that if m ≽ 3 then there is no non-constant harmonic map φ: ℝm → Sn with finite energy. The method of proof makes use of the fact that the energy functional is not invariant under conformal transformations. This fact has also allowed Xin(9), to show that any non-constant-harmonic map φ:Sm → (N, h), m ≽ 3, is not stable in the sense of having non-negative second variation.

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.

scholarly journals Twisted Donaldson invariants

2021 ◽  
pp. 1-54
Author(s):  
TSUYOSHI KATO ◽  
HIROFUMI SASAHIRA ◽  
HANG WANG
Keyword(s):  
Gauge Theory ◽  
Fundamental Group ◽  
Picard Group ◽  
Fundamental Groups ◽  
Smooth Structure ◽  
Yang Mills ◽  
Smooth Structures ◽  
New Variant ◽  
Donaldson Invariants

Abstract Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang–Mills gauge theory from twisting by the Picard group of a 4-manifold in the case when the fundamental group is free abelian. We then generalise it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic 4-manifolds.

Cartan's soldered spaces and conservation laws in physics

2015 ◽  
Vol 12 (09) ◽  
pp. 1550089 ◽  
Author(s):  
Joseph Kouneiher ◽  
Cécile Barbachoux
Keyword(s):  
Conservation Laws ◽  
Local Cohomology ◽  
Differential Forms ◽  
Einstein Equations ◽  
Vacuum Field ◽  
Order System ◽  
Mathematical Framework ◽  
Field Equations ◽  
Yang Mills ◽  
Integrability Conditions

In this paper, we will introduce a generalized soldering p-forms geometry, which can be the right framework to describe many new approaches and concepts in modern physics. Here we will treat some aspects of the theory of local cohomology in fields theory or more precisely the theory of soldering-form conservation laws in physics. We provide some illustrative applications, primarily taken from the Einstein equations of general theory of relativity and Yang–Mills theory. This theory can be considered to be a generalization of Noether's theory of conserved current to differential forms of any degree. An essential result of this, is that the conservation of the energy–momentum in general relativity, is linked to the fact that the vacuum field equations are equivalent to the integrability conditions of a first-order system of differential equations. We also apply the idea of the soldered space and the integrability conditions to the case of Yang–Mills theory. The mathematical framework, where these intuitive considerations would fit naturally, can be used to describe also the dynamics of changing manifolds.

scholarly journals Surface charges toolkit for gravity

2020 ◽  
Vol 29 (06) ◽  
pp. 2050040
Author(s):  
Ernesto Frodden ◽  
Diego Hidalgo
Keyword(s):  
Surface Charge ◽  
Gauge Theories ◽  
Differential Forms ◽  
Gravity Theory ◽  
Surface Charges ◽  
Local Surface ◽  
Yang Mills ◽  
Gravity Theories ◽  
Chern Simons ◽  
Matter Fields

These notes provide a detailed catalog of surface charge formulas for different classes of gravity theories. The present catalog reviews and extends the existing literature on the topic. Part of the focus is on reviewing the method to compute quasi-local surface charges for gauge theories in order to clarify conceptual issues and their range of applicability. Many surface charge formulas for gravity theories are expressed in metric, tetrads-connection, Chern–Simons connection, and even BF variables. For most of them, the language of differential forms is exploited and contrasted with the more popular metric components language. The gravity theory is coupled with matter fields as scalar, Maxwell, Skyrme, Yang–Mills, and spinors. Furthermore, three examples with ready-to-download notebook codes, show the method in full action. Several new results are highlighted through the notes.

scholarly journals High Energy Behavior in Maximally Supersymmetric Gauge Theories in Various Dimensions

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 104 ◽  
Author(s):  
Dmitry Kazakov ◽  
Leonid Bork ◽  
Arthur Borlakov ◽  
Denis Tolkachev ◽  
Dmitry Vlasenko
Keyword(s):  
Renormalization Constant ◽  
Finite Energy ◽  
High Energy ◽  
Four Dimensions ◽  
Yang Mills ◽  
High Energy Behavior ◽  
Planar Limit ◽  
Energy Behavior ◽  
Supersymmetric Gauge ◽  
Recursive Relations

Maximally supersymmetric field theories in various dimensions are believed to possess special properties due to extended supersymmetry. In four dimensions, they are free from UV divergences but are IR divergent on shell; in higher dimensions, on the contrary, they are IR finite but UV divergent. In what follows, we consider the four-point on-shell scattering amplitudes in D = 6 , 8 , 10 supersymmetric Yang–Mills theory in the planar limit within the spinor-helicity and on-shell supersymmetric formalism. We study the UV divergences and demonstrate how one can sum them over all orders of PT. Analyzing the R -operation, we obtain the recursive relations and derive differential equations that sum all leading, subleading, etc., divergences in all loops generalizing the standard RG formalism for the case of nonrenormalizable interactions. We then perform the renormalization procedure, which differs from the ordinary one in that the renormalization constant becomes the operator depending on kinematics. Solving the obtained RG equations for particular sets of diagrams analytically and for the general case numerically, we analyze their high energy behavior and find that, while each term of PT increases as a power of energy, the total sum behaves differently: in D = 6 two partial amplitudes decrease with energy and the third one increases exponentially, while in D = 8 and 10 the amplitudes possess an infinite number of periodic poles at finite energy.

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 Parity duality for the amplituhedron

2020 ◽  
Vol 156 (11) ◽  
pp. 2207-2262
Author(s):  
Pavel Galashin ◽  
Thomas Lam
Keyword(s):  
Scattering Amplitudes ◽  
Differential Forms ◽  
Twist Map ◽  
Yang Mills ◽  
Affine Permutations

The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

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