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Questions tagged [gauge-theory]

A gauge theory has internal degrees of freedom that do not affect the foretold physical outcomes of the theory. The theory has a Lie group of *continuous symmetries* of these internal degrees of freedom, *i.e.* the predicted physics under any transformation in this group on the degrees of freedom. Examples include the $U(1)$-symmetric quantum electrodynamics and other Yang-Mills theories wherein non-Abelian groups replace the $U(1)$ gauge group of QED.

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Is the space of gauge invariant states a Hilbert space?

In a theory with a gauge symmetry, as I understand it, the gauge symmetry is not a symmetry of a system, but rather its a redundancy in description. The procedure goes like this start with some ...
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Twisted Chern-Simons, and Twisted Wess-Zumino Term

I am asking this question about Chern-Simons theory from the paper "Quantum Field Theory and Jones Polynomial" by Edward Witten. Let $M$ be a closed three dimensional manifold, and $P\rightarrow M$ ...
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What is meant by the coupling term $g_{\mu\nu}T^{\mu\nu}$ in Supergravity?

In the "Cambridge Lectures on Supersymmetry and Extra Dimensions" of F.Quevedo it is written on page 59 ($T^{\mu\nu}$ stands for the energy-momentum tensor): The metric $g_{\mu\nu}$ as gauge field ...
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Why not regard all large gauge transformations as genuine ones?

A large gauge transformation is a gauge transformation that is not connected to the identity. When quantizing a gauge theory, we must take configurations related by ordinary gauge transformations to ...
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Do higher homotopy groups play any role in gauge theory?

As is more-or-less well-known, the magnetic monopoles of a gauge theory are classified by the first homotopy group of the gauge group, $\pi_1(G)$ (cf. Lubkin (1963)). The second homotopy group is ...
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Updating link variables in lattice $SU(N)$ gauge theory

I'm currently writing a basic program in python to simulate a 1 + 1 dimensional yang mills gauge theory with symmetry group $SU(2)$. On the lattice you work with link variables, which are $SU(N)$ ...
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What justifies compactifying space and spacetime, in the context of instantons?

When studying Yang-Mills instantons, there are two instances where one compactifies a space. When classifying vacuum states, one demands $A_\mu(\mathbf{x})$ to become a constant as $\mathbf{x} \to \...
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Faddeev-Popov Determinant of Chern-Simons Theory

I am asking this question in order to figure out the expression of the Faddeev-Popov determinant given by Edward Witten is his paper "Quantum Field Theory and Jones Polynomial". Starting from the ...
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Can anomalies exist without gauge fields?

In Schwartz's QFT book, it is stated that anomalies cannot exist in a theory without gauge fields. This is because anomalies always give equations like $$\partial_\mu j^\mu \sim F \tilde{F}$$ where ...
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Is gauge invariance the same as gauge symmetry?

I am studying Gauge theory using Wikipedia and need a little clarification on the difference between gauge symmetry and gauge invariance. For sure it has probably already been addressed on this site, ...
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U(1) Dirac string moved to the SU(2) or SO(3) gauge theory

Dirac string describes the string connecting the U(1) magnetic monopole to the U(1) anti-magnetic monopole in the U(1) gauge theory. Since U(1) is a subgroup of SU(2) and SO(3), we may embed the U(1) ...
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Decomposition of the gauge group of a sigma model

I'm following the Chapter 5 (specifically, section 5.4) of Quigg's book Gauge Theories of the Strong, Weak and Electromagnetic Interactions and am confused with the following: Studying a sigma model ...
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2D anomaly-free condition for a gauge theory

Take a $SU(2)$ gauge theory in 2d spacetime, say there are $n_1$ left-handed Weyl fermion in spin-1 written as $$ 1_L, $$ and $n_0$ left-handed Weyl fermion in spin-0 written as $$ 0_L . $$ and $n_{1/...
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Why is the interpolation between two connections related via a gauge transformation still a connection?

I am studying the theory of anomalies in gauge field. Let $A$ be a gauge field (or a connection for mathematicians). Let $A_{U}$ be an equivalent gauge related via a local gauge transformation $$A_{...
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Penrose paragraph on Bundle Cross-section?

I am reading "Road to Reality" by Rogen Penrose. In chapter 15, Fibre and Gauge Connection ,while going through Clifford Bundle, he says: ...Of course, this in itself does not tell us why the ...
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Massless states on a compact manifold

I would like to understand the following statement found on page 6 second paragraph of 0305098. The authors discuss the quantization of the level $k$ of the WZW model for compact (and non-compact) ...
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Is the gauge transform field in electromagnetism a Lagrange multiplier?

In a draft answer to another question about gauge transformations, I played around with demonstrating the action of a gauge transformation on the Lagrangian density. Beginning with the classical ...
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1answer
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What makes an operator “baryonic”?

I am trying to work my way through arXiv:1712.00020, but there is a statement I don't quite get. On §2.6 (p. 21), the author claims that the simplest baryonic operator in the gauge theory $\mathrm{SU}(...
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Hamilton or Hamilton-Jacobi formalism with Hamiltonian equal to zero

I have the Lagrange function: $$L=\sqrt{\frac{\dot{x}^2+\dot{y}^2}{-y}}.\tag{1}$$ The energy is then: $$H=\dot{x}\frac{\partial L}{\partial \dot{x}}+\dot{y}\frac{\partial L}{\partial \dot{y}}-L=0.\...
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Topological Charge in 2 dimensional U(1) Gauge Theory

The topological charge in 2 dimensions for $U(1)$ gauge fields is defined by $Q \propto \int d^2x ~\epsilon_{\mu\nu}F_{\mu\nu}$ with the field strength tensor $F_{\mu\nu}=\partial_\mu A_\nu - \...
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Breaking a gauge group through a monopole$.$

I am trying to understand the paper arXiv:1712.08639 and, in particular, the discussion in §5. In this section, the authors take a gauge theory with group $\mathrm{SO}(N)$, and they add a "unit of ...
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226 views

The Hilbert space of Chern-Simons on a torus, part one$.$

There is a key result in 2+1 dimensional Chern-Simons theory, which was first discussed in ref.1.: the Hilbert space of the theory, when quantised on $T^2\times\mathbb R$, is isomorphic to $$ \frac{\...
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QCD from chirally segregated, gauged $SU(3)_L \times SU(3)_R$?

There are already theory papers out there in which color $SU(3)_C$ is actually the diagonal subgroup of multiple $SU(3)$ factors. But due to a comment by @zooby, a new twist on this idea occurred to ...
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Super-gauge transformation in two dimensional $\mathcal{N}= (0,2)$ superspace

I'm trying to couple matter to $\mathcal{N}=(0,2)$ SYM in 2d using superfield formalism. There are some paper (this on Sec. 6, or this on Sec. 3 [whose notation will be used here]) that construct what ...
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Relationship between TQFTs and gauge theories

The motivation for this question comes from observing that the toric code model has the properties of a TQFT (robust ground state degeneracy) and of a $\mathbb Z_2$ gauge theory (local spin flips don'...
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Gauge anomaly from conformal dimension?

According to ref.1, the Chern-Simons theory $\mathrm{SU}(N)_k$ has a $\mathbb Z_N$ one-form symmetry with anomaly $$ \eta=\exp\left[-2\pi i \frac{k}{N}\right]\tag{4.12} $$ which, apparently, can be ...
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Does a SUSY Chern-Simons term prevent the dualising of the gauge potential to a scalar?

In 3D $\mathcal{N}=2$ supersymmetric field theory with abelian gauge fields, the gauge field $A_{\mu}$ is often dualised to a real scalar $\gamma$. Does a Chern-Simons term prevent this dual ...
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Flux attachment to dynamical gauge field

It is a question about flux attachment. When I attach fluxes to dynamical gauge fields, something weird happened: an extra Hall conductivity term. We start from the action \begin{equation} \mathcal{L}...
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1answer
249 views

Is general relativity resulted from diffeomorphism invariance?

Any action expressed as the integral of a 4-form in 4-dimensional spacetime is diffeomorphism invariant. For example the following 4-form topological (Pontryagin) action $$ S = \int F\wedge F $$ is ...
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In 3D ${\cal N}=2$ SUSY, the linear multiplet contains a global current. How is this related to the gauge field?

I am reading the paper on 3d $\mathcal{N}=2$ supersymmetry by O. Aharony et al. (https://arxiv.org/abs/hep-th/9703110) and I am a bit confused about linear multiplets in section 2.3. A linear ...
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Is it possible to use the Faddeev-Popov trick for discrete gauge symmetries?

I was thinking of this previous question of mine, where I was trying to implement a path-integral over the half-line: $$ Z=\int_{\varphi\ge0}\ \mathrm e^{iS[\varphi]}\mathrm d\varphi\tag1 $$ It seems ...
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Gauge symmetry and symmetry breaking?

In quantum field theory we say that gauge symmetry is a redundancy, and also, in Xiao-Gang Wen's book, it reads that gauge symmetry is not a symmetry, so it can never be broken. And the Higgs ...
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Motivation for Non-Abelian Gauge Invariance

I have a very similar question to the one asked below: Why are non-Abelian gauge theories Lorentz invariant quantum mechanically? In particular, the setup to my question is essentially the same: ...
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Comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons

I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of: Chern class (1st, 2nd), and ...
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Transformation of fields in non-abelian gauge theories

Let us consider a gauge group, e.g. $SU(N)$. One usually says that a fermionic field $\psi$ belongs to the fundamental representation of the group. As far as I understand, the fundamental ...
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Going from full non-Abelian gauge transformation to its infinitesimal version in component notation

Let $A_\mu^a(x)$ be a non-Abelian gauge field, with $\mathrm{SU}(N)$ generators $T_a$. We can write the field as a Lie-algebra-valued object $$ \mathbf{A}_\mu \equiv A_\mu^a T_a.$$ The full local ...
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2D ${\cal N}=(2,2)$ Super Yang-Mills with Superspace

I'm reading this famous paper by Witten. There is the expression of field strength for the abelian vector multiplet (eq. (2.16)): $$\Sigma = \frac{1}{\sqrt{2}}\bar{D}_+D_- V\;.\tag{2.16}$$ I'm ...
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Why is $SU(3)$ and not $U(3)$ the correct gauge symmetry? [duplicate]

If quarks come in three colours $r$, $g$ and $b$ than (neglecting all other quantum numbers and spacial freedom for now) a state of a quark would be a vector in $\mathbb{C}^3$. If we are now looking ...
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Question about the Faddeev-Popov method for gauge-fixing

The correct way to gauge-fix in a path integral is to insert the Faddeev-Popov determinant, and add a delta-functional constraint. The final action contains three contributions: a Yang-Mills (I'm ...
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Physical manifold with a natural linear connection on them

Of course in many situation a manifold raised from a physical situation (like spacetime or configuration manifold and so on) are really much more richer than an abstract manifold. for example phase ...
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Anyon condensation – what's the precise definition?

Say I have an anyonic system modelled as a Chern-Simons system with group $G$. If the centre of $G$ is non-trivial, one may also study the system described by $G/\Gamma$, where $\Gamma$ is a discrete ...
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1answer
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Showing dual field strength tensor term in Lagrangian is a total derivative

I am trying to prove the identity: $$\text{Tr}\left\{\star F_{\mu\nu}F^{\mu\nu}\right\}=\partial^{\mu}K_{\mu} \tag{1}$$ where $K_{\mu}$ is given by: $$K_\mu=\epsilon_{\mu\nu\rho\sigma}\text{Tr}\...
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1answer
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Non-abelian gauge theories are non-linear

To preface this, I know very little about Standard model physics and nonabelian gauge theory, so please correct me if my understanding is incorrect. I was reading about the Standard Model, and I ...
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Are these the only gauge-invariant functions of $A_\mu$?

I know off course that $F_{\mu\nu}$ is a gauge invariant function of $A_\mu$ in the abelian case. Also we have $\epsilon^{\alpha\beta\mu\nu} F_{\alpha\beta}F_{\mu\nu}$ in that case. Are there any ...
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Conformal vs. scale invariance of ${\cal N} = 4$ Supersymmetric Yang-Mills theory

I will quote the following from the Wikipedia article on Supersymmetry Nonrenormalization theorems. "In ${\cal N} = 4$ super Yang–Mills the $\beta$-function is zero for all couplings, meaning that ...
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1answer
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What is really this Gauge-Gravity duality all about?

I have no background in string theory but have a reasonable exposure to quantum field theory including the quantization of gauge theories. In simple terms what is this Gauge-Gravity duality all about ...
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Lie algebra valued potential vector [closed]

Maybe it is a simple question but I have some difficulty to understand the explicit matrix form of this usual relation: $$A_\mu=A^a_\mu \tau_a$$ where $A^a_\mu $ is the Lie algebra valued potential ...
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One-Particle Self-Energy (Yang-Mills)

I've been asking about an interesting question when i was calculating the self energy of a Gauge boson in Yang-Mills theory. I think that the correct way to think this problem is: the incident ...
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3answers
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What is the Lorenz condition for potentials?

I was just going through the "Electromagnetic Waves" chapter in the Classical Field Theory book by Landau. Here he mentions that they impose an auxiliary condition and it is known as the Lorenz ...
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Why is the Standard Model gauge group a simple product?

The Standard Model gauge group is always given as $\text{SU}(3)\times\text{SU}(2)\times\text{U}(1)$ (possibly quotiented by some subgroup that acts trivially on all konwn fields). However, at the ...