Yang–Mills theory is a QFT, a *gauge theory* normally symmetric under a compact non-Abelian Lie group relying on (originally massless) gauge vector fields. YM theories describe the strong and electroweak interactions of elementary particle physics, the Standard Model.

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Non-abelian current commutators

There many articles, in which non-abelian current commutators are computed. The general result is that quantum corrections lead to additional term in commutator $$[J^a_\mu (x), J^b_\nu (y)] \delta ...
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67 views

Why is Seiberg duality called an electromagnetic duality?

An electromagnetic duality is a duality that maps electric to magnetic degrees of freedom of two distinct theories. Apart from source-less Maxwell electrodynamics, other theories require magnetic ...
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131 views

Completeness Relations of Polarization Vectors in QCD

What are the completeness relations of the polarization vectors of (external) particles in QCD amplitude calculation? (I assume the polarization vectors depend on the gauge and even so still have some ...
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What S means in S-duality?

As I know, there are many dualities related to S-duality. For example, Montonen-Olive duality, Seiberg duality. and so on. so, I wonder that what "S" means in the term "S-duality". If this is a stupid ...
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Where is the BRST symmetry?

When quantizing YM we start from the gauge fixed path integral (to remove redundancy of integrating over Gauge symmetric configurations) $$\int \mathcal{D}A \delta(G(A)) \text{det} \Delta_{FP}e^{i\int ...
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Classical Yang Mills vacuum

What is the vacuum of classical Yang Mills theory $$\mathcal{L} = - \frac14 F^{a \mu \nu} F^a_{\mu \nu}~?$$ Is it simply $A^a_\mu=0$ for all its components?
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Question from Terning's book

In Chapter 7 of Terning's book (Modern Supersymmetry), the first example considered is that of an SO(3) gauge theory, a complex scalar in the triplet representation of SO(3) and a potential term: ...
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36 views

Equality of renormalized coupling constant

I want to show, that the renormalized coupling constants of a SU(N) Yang-Mills field with fermions included, are all equal. In the most textbooks it is written, that this could been shown by the ...
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53 views

Infinitesimal gauge invariance of Yang--Mills Lagrangian

Under an infinitesimal gauge transformation $g(x) = 1 - i\alpha{}_i(x)T{}^i$, where $[T{}^a, T{}^b] = if{}^{ab}{}_c T{}^c$, I want to know what happens to the Lagrangian $\mathcal{L} = ...
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57 views

Unphysical degrees of freedom in the Yang Mills Lagrangian [duplicate]

Im taking my first course in QFT and has stumbled upon something that I do not understand. Given the Yang Mills lagrangian $$\mathcal{L} = -\frac{1}{4}F^{a}_{\mu \nu}F^{a\mu \nu}$$ with $F^{\mu ...
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69 views

Is the standard model a quantized gauge theory?

I have studied some quantum field theory and gauge theory but I am definitely not an expert. I am aware that in quantizing electrodynamics one has to fix a gauge. I have read that for general gauge ...
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Gauge invariance in classical electrodynamics

I think that I don't fully understand concept of gauge invariance. Suppose we have a Lagrangian for classical ED which is: $$\mathcal{L} = -\frac{1}{4} (F_{\mu \nu})^2 - j^{\mu}A_{\mu}.$$ First part ...
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On the action of superconformal generators in maximally supersymmetric Yang-Mills

Consider maximally supersymmetric Yang-Mills theory in 3+1 dimensions. This theory has 32 supercharges: 16 ordinary ones, conventionally labeled $Q$; and 16 superconformal ones, conventionally ...
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Doesn't modelling using Lie Groups assume spacetime is continuous?

Lie groups are used to some behaviors of quantum mechanics, as well as forming a basis for Kaluza-Klein, Yang-Mills, and String theory. But Lie groups are defined as involving a differentiable ...
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409 views

$SU(2)$ Yang-Mills EOM

I'm having trouble with some indices on my yang-mills lagrangian. I have a gauge group $SU(2)$ and a field strength tensor $$ ...
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$\mathcal{N} = 4$ Super-Yang Mills propagators

In $\mathcal{N} = 4$ Super-Yang mills there are only massless particles. If one wishes to obtain a heavy quark one can see the SYM theory as a stack of (N+1)-branes in AdS$_5 \times$S$^5$ where one ...
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Correlation functions in a zero dimensional QFT?

I would like to ask about correlation functions in a 0-dimensional matrix model QFT. What information do these correlators give? I know only of correlators between two different spatial positions.
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Why is quantising gravity so difficult? [duplicate]

Since gravity is so similar with the Yang-Mills theory, the Christoffel connection is the gauge potential, the Riemann curvature is the field strength, then why is quantising gravity so difficult when ...
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38 views

Coupling constant in the Yang-Mills action

Intuitively, gauge coupling defines the strength of interactions between fields. But how to interpret the coupling $1/g^2$ in front of the kinetic term of Yang-Mills theories, ...
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227 views

Gauge fields and strings: Loop equations

I am trying to derive Eq. (7.25) (p. 117) of Polyakov's book: $$ \delta \Psi (C) ~=~ \int_{0}^{2\pi} {\rm P} \left(F_{\mu\nu}(x(s)) \exp \oint_C A_\mu dx^\mu \right)\dot{x}_\nu \delta x_\mu(x) \, ...
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Number of components of a One-Form Superfield

A Supersymmetric Yang Mills theory has 8 bosonic and 8 fermionic components. Since SUSY YM Theory is described via the Vector Superfield, $$V=C(x)+i\theta\chi(x)-i\bar{\theta}\bar{\chi}(x) \dots.$$ ...
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Yang-Mills theories, confinement and chiral symmetry breaking

I was thinking about hadrons in general Yang-Mills theories and I have some doubts that I'd like to discuss with you. Suppose that we have a Yang-Mills theory that, like QCD, tend to bind quarks ...
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58 views

Integrals of Chern class, $c_i$ in YM theories

I am a bit confused with the definition of the 1st (and 2nd by extension) Chern class in YM theories. I understand that in general $c_i \in H^{2i}(M,\mathbb{Z})$ where $M$ is a smooth manifold. Then, ...
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Gravity as a gauge theory

Currently, (classical) gravity (General Relativity) is NOT a gauge theory (at least in the sense of a Yang-Mills theory). Why should "classical" gravity be some (non-trivial or "special" or ...
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Gauge transformations in gravity [duplicate]

The Maxwell equations are invariant under the transformation $$A_{\mu} \rightarrow A_{\mu} - \dfrac{1}{e}\partial_{\mu}\alpha(x)$$ where $\alpha(x)$ is a phase transformation varying from point to ...
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Good Books on Gauge Theory [duplicate]

Possible Duplicate: Comprehensive book on group theory for physicists? I'm having a hard time trying to get my head around the fundamentals of gauge theory. I've taken classes in QFT and ...
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237 views

Invariance of supersymmetric Yang-Mills theory under supersymmetry

I was following Brink, Scherk and Schwartz, "Supersymmetric Yang-Mills theories". The variation of the Lagrangian w.r.t a supersymmetry transformation can be reduced to $$ \delta L = -igf_{a b c} ...
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474 views

What is Motivic mathematics and how is it used in physics?

In a few videos I've seen where he discusses the new approach to calculating the super Yang Mills scattering amplitudes, Nima Arkani-Hamed sometimes alludes to the use of Motivic methods as being ...
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95 views

Reason for not using Higgs mechanism to solve mass gap problem

I am curious as to precisely why one can't introduce masses for gluons in Yang-Mills by a Higgs-type mechanism as in electroweak theory. Is it because then one would end up with an unwanted massless ...
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47 views

Nahm's theorem and related with SYM ;other theories

To study $S$-duality in more detail, i tried to read, Electric-magnetic duality and the Geometric Langlands Program. In section 2.1, there is a comment about 10d SYM. Excerpt from the paper above, ...
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Unitarity of the S-matrix and Feynman Diagrams

There are several questions on the unitarity of the S matrix, but unfortunately non of them answers directly the following question. The S matrix is unitary and that can be proven by the fact that ...
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Why is the Yang–Mills existence and mass gap problem so fundamental?

Ref: Yang–Mills existence and mass gap Can anyone please explain why the Yang–Mills Existence and Mass Gap problem is so important / fundamental to contemporary mathematics (and, presumably, ...
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What exactly is the “diagonal embedding” in the supersymmetric topological twist?

Consider $\mathcal{N}=2$ pure SYM theory. If we want to put the theory in a 4-manifold we take its topological twist. The global symmetry group $$G= SU(2)_{+} \times SU(2)_{-} \times SU(2)_I \times ...
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How can I understand instantons as sheaves?

In specific, instantons are considered or interpeted as torsion free coherent sheaves. Why is that the case? Is there a nice way to understand this relation and of course also understand how the two ...
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Gauge theory for mathematicians?

I'm looking for a textbook or set of lecture notes on gauge theory for mathematicians that assumes only minimal background in physics. I'd prefer a text that uses more sophisticated mathematical ...
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SU(2) critical point and volume dependence

I am doing multi-dimensional plots of $\beta_j$ for SU(2) for infinite volume to understand the flow behavior and I was wondering, before I go too much further, if anyone knew off the top of their ...
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Yang and Mills' (and others') justification for local gauge invariance

In most physics textbooks, local gauge invariance is simply postulated---you start with a global symmetry, e.g. the global phase, then allow it to depend on the spacetime point, make the necessary ...
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The BPS mass formula for 3d N=4

Is the mass formula (or equivalently the central charge formula) known for BPS particles in 3d $\mathcal{N}=4$ susy gauge theories? In particular, what is its dependence on the Fayet-Iliopoulos ...
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287 views

Einstein-Yang-Mills Connections

I am playing around with coupling a classical $SU(2)$ Yang-Mills theory to Einstein's equations. Assuming spherical symmetry, the $SU(2)$ connection can be written \begin{equation} A = ...
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What are the global symmetries of $\mathcal{N}=2$ SYM before twisting and after twisting?

I am confused with the global symmetries of $\mathcal{N}=2$ SYM. On one hand I know that the theory has a $U(2)_R = SU(2)_R \times U(1)_R$ symmetry. Now, there exists also a $U(1)_B$ global symmetry. ...
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Vector potential in gauge transformation

While applying Gauge transformation, $\psi\prime = U \psi$ , where $ U= e^{i q \lambda(x)}$ , transformation law for "Vector Potential" comes out to be : $$ A_{\mu}\prime= ...
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Are correlators constructed out of Wilson loops singular in pure Yang-Mills?

If I have some gauge invariant function of two Wilson loops (such as $\left<\text{Tr}W_1 \text{Tr}W_2\right>$) does the expectation value diverge when the loops coincide the same way ...
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Is it true that particles propagate on geodesics in Yang-Mills theory?

I mean free particles. Sorry for the inaccurate wording, I'm new in this field of physics.
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How are quadruple gluon vertices related to $SU(2)$ and $SU(3)$?

I once read that the non-commutativity of the Lie Groups $SU(2)$ and $SU(3)$ is the reason that the weak and strong interactions have Feynman diagrams with quadruple vertices, where four gauge bosons ...
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How does the Higgs field relate to the Yang-Mills fields and gauge theories in general?

I asked this in astronomy How does the Higgs field relate to the Yang-Mills fields and gauge theories in general? but they suggested I ask it here. It is very confusing. Is there an easy ...
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Transformation Law for Covariant Derivative in $SU(2)$ Yang-Mills

In page 488 of Peskin and Schroeder, it is stated (emphasis mine): It is not difficult to check using (15.27) and (15.21) that, even for finite transformations, the covariant derivative has the ...
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Why is QCD hard to solve if I know the beta functions?

Why is it still hard to solve QCD if we know the beta functions of the coupling? Aren't only the loops causing problems? And am I not able to write every possible interaction exact at tree-level with ...
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Why is $U(1)$ special when defining global charges?

For gauge groups like $SU(2)$ and $SU(3)$ etc. we know that observable states such as mesons or baryons must be charge neutral. However, for a $U(1)$ gauge group we can have charged initial states in ...
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Intro to Super Yang Mills theory

I'm looking to start learning Super Yang Mills theory. Currently I have studied Peskin and Schroeder up to the Renormalization Group, but don't know supersymmetry yet. I know some Conformal Field ...