Questions tagged [gauge-theory]

For questions about gauge theory in mathematical physics and differential geometry. Typical questions pertain to bundles, connections, spinors, and moduli spaces. Questions about the physics of gauge fields should be directed to physics.stackexchange.

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Relation between Poisson equation and Wilson lattice gauge invariance theory

I've recently started writing a library of numerical solvers for elliptic partial differential equations, with particular focus on the Poisson equation. If one considers typical Poisson equation in ...
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The map $C^\infty(M,U(A_F))\to C^\infty(M,U(A_F)/\mathfrak{H}(F))$ is an homomorphism.

I am reading "Noncommutative Geometry and Particle Physics" by van Suijlekom. I have problems to identify one map as homomorphism. Let $M\times F$ be an almost-commutative manifold. The ...
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The Grassmann connection is a connection

Let $\mathcal{A}$ be a *-algebra and $p\in M_N(\mathcal{A})$ an orthogonal projection. I need to show that $\nabla=p\circ d$ defines a connection on $\mathcal{E}=p\mathcal{A}^N$, where $d$ is acting ...
Schrödinger's cat's user avatar
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Intuition behind connection 1-forms and Ehresmann connections

I am learning about mathematical gauge theory and so far I have been able to develop an intuition behind all the objects I've read about such as principal bundles, associated bundles, and vertical/...
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Why does the adjoint representation appear in the definition of a connection 1-form?

When defining the a connection 1-form $A$ on a principal $G$-bundle we require that $$r^*_gA = Ad_{g^{-1}} \circ A, \quad \forall g \in G.$$ Here $r_g^*$ is the pullback of right multiplication by $g$,...
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Invariant differential forms?

In the first page of the paper "On the spaces of maps inducing isomorphic connections" by T.R. Ramadas, one can read that the automorphisms of a connection $\nabla$ on a principal $G$-bundle ...
Math learner's user avatar
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Trivial principal bundles and curvature.

Let $\mathcal{M}$ be a smooth manifold, $G$ a Lie group with Lie algebra $\mathfrak{g}$ and $\mathcal{P}\xrightarrow{\pi}\mathcal{M}$ a principal bundle. If $A\in\Omega^{1}(\mathcal{P},\mathfrak{g})$ ...
G. Blaickner's user avatar
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The set of all Yang-Mills connections is infinite dimensional

I am reading Jost's Riemannian Geometry and Geometric Analysis (7-Ed) and having a question about the Yang-Mills functional (page 182). Let $M$ be a compact manifold and $E$ be a metric bundle where ...
Justin Lien's user avatar
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Is gauge group infinite dimensional Lie group?

Let $P$ be a smooth principal bundle on a manifold $M$ with a structure Lie group $G$. Then we define the gauge group $\mathcal{G}$ by the automorphism group of $P$, that is, the group of ...
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How do gauge transformations reflect on the associated bundles?

The way we define an associated bundle to a principle bundle is by constructing the trivial bundle over the principle bundle with the desired fibre and then quotienting out by the structure group (...
Integral fan's user avatar
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Lagrangian field theories define a canonical $\mathbb R$-torsor/$\mathbb R$-bundle

Why is it possible to define an $\mathbb R$-torsor/$\mathbb R$-bundle out of the Lagrangian and to construct a projective limit? (I'm guessing this has to do with time evolution acting on the space of ...
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Residual gauge freedom and complete residual gauge fixing in lorenz gauge

I posted this question on physics stack exchange but nobody answers it so hope to get an answer on math stack exchange, What I understand after reading all answers from physics stack exchange related ...
Keshav shrestha's user avatar
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Cohomology of mixed degree differential forms

$\newcommand{\d}{\mathrm{d}}\newcommand{\H}{\mathrm{H}}$Consider the space, $\Omega^n(M)$, of differential $n$-forms on a smooth, torsion-free manifold, $M$, without boundary, of dimension $D$, and ...
ɪdɪət strəʊlə's user avatar
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When does the Cartan formalism have a gauge that makes the tetrad constant?

I hope I am using the terms correctly: by Cartan formalism I am thinking of a general (orthonormal) tetrad, defined with respect to a coordinate system $\mathbb{R}^n$, along with the spin connection. ...
Adam Herbst's user avatar
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Existence and uniqueness of a twisted one form which satisfies some inner product property

Let $P\rightarrow M$ be a principal bundle over a (pseudo)-Riemannian manifold $(M,g)$, with compact structure group $G$ ($G$ is assumed to be a real Lie group), and a connection one form $A$. Let $W$ ...
Chris's user avatar
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Deriving Contracted Bianchi Identity from Einstein-Hilbert Action

It is well known and often argued that the contracted Bianchi identity (and vanishing divergence of the stress-energy tensor) of General Relativity can be seen as a consequence of the theory's (and ...
Integral fan's user avatar
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Second Chern class $c_2$ of $SU(n)$ bundle over $S^4$ v.s. $\pi_3(SU(n))=\mathbb{Z}$

Let us compare the four statements: Consider the $SU(2)$ fiber over the $S^4$. Such that this $SU(2) = S^3$ fiber over $S^4$ gives a second Chern class $c_2$ on the $S^4$ with $c_2=1$. Consider the $...
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First Chern class $c_1$ of $U(1)$ bundle over $S^2$ v.s. $\pi_1(S^1)=\mathbb{Z}$

Let us compare the four statements: Consider the $U(1)$ fiber over the $S^2$. Such that this $S^1$ fiber over $S^2$ gives a first Chern class $c_1$ on the $S^2$ with $c_1=1$. Consider the $S^1$ ...
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Principal $PSU(N)$ bundles over $\mathbb R^3$

I am aware that principal $SU(N)$ bundles over 3-manifolds are trivial. Consider, however, a principal $PSU(N)$ bundle over $\mathbb R^3$. Is the topology of this bundle classified by $$\pi_2(SU(N)/U(...
dennis's user avatar
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Is there a sense in which two gauge connections are topologically equivalent?

We say that manifolds $M$ and $N$ are topologically equivalent iff there exists a homeomorphism between them. Is there are similar notion for gauge connections? I.e. is there a sense in which two ...
dennis's user avatar
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Examples of Physical Applications ofthe theory of Principal $G$-bundles

Let $(P,M,\pi)$ be a principal $G$-bundle, $\omega\in \mathbf \Omega^1(M,\mathfrak g)$ be a connection 1-form and $\Omega \in \mathbf \Omega^2(M,\mathfrak g)$ be its curvature 2-form, where $\mathfrak ...
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Sufficient condition for moduli space of slope-stable bundles to be non-empty

I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature. Let $X$ be a Kähler surface. Let $\mathscr{M}(...
holitinh's user avatar
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How do we vary the Einstein-Hilbert Action?

The Einstein-Hilbert actions is: $$ \begin{align} S=\int_M R(g)\text{dvol}_g=\int_M\sqrt{-g}Rd^4x \end{align} $$ I'm looking to vary the action in a coordinate free way without appealing to the ...
Chris's user avatar
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Curvature equation

The curvature equation of Seiberg-Witten equations is $F^+_A=\psi\otimes\psi^*-\frac{|\psi|^2}{2}Id$, where $F^+_A$ is selfdual part of curvature and $\psi$ is spinor. Could someone elaborate why ...
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Flat torsion-free connections

Can a manifold admit two different flat torsion-free connections? By different, I mean that there is not any diffeomorphism that makes the two equal. The question arises from me trying to explain to ...
Jack Fres's user avatar
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When is a principal bundle with group $G \times H$ the product of a principal bundle with group $G$ and one with group $H$?

In general, it is not true that a fiber bundle with a product fiber is the product of two fiber bundles with the factors as fibers (think of vector bundles). However, I read somewhere that every 2-...
Jack Fres's user avatar
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Boundary conditions for differential forms

I am trying to understand differential forms on manifolds with boundaries, and I am a bit confused with the boundary conditions. For the following, let $(M,g_M)$ be a smooth Riemannian manifold with ...
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Gauge Theory Gravity: transformation of vector potential $A'(x)$

In Gravity, Gauge Theories and Geometric Algebra by A. Lasenby, C. Doran and S. Gull develop a theory for gravity based on the use of a position-gauge field and a rotation-gauge field. The former is ...
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Gauge Transformation in E and M vs. U(1) theory over $\mathbb{R}^{1,3}$

In E and M we have that the fields exhibit gauge symmetry under the gauge transformations: $$V'=V-\frac{\partial \lambda}{\partial t}$$ $$M'=M+\nabla \lambda$$ for some $\lambda:\mathbb{R}^{1,3}\...
Chris's user avatar
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7 votes
1 answer
401 views

Mathematical Introduction to the Standard Model of Particle Physics

I am looking for textbooks, lecture notes, lecture videos on a rigorous introduction to the standard model of elementary particles. I'd prefer to not be referred to monographs for an introduction as ...
Song of Physics's user avatar
3 votes
1 answer
163 views

Transformation of Curvature 2-Form Under Global Bundle Autormphism

Let $F^A$ be a curvature two form on a principal bundle $P$ corresponding to the connection one form $A$. Let $f:P\rightarrow P$ be a global bundle automorphism; there then exists a map $\sigma_f:P\...
Chris's user avatar
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Deform a continuous mapping of $S^3$ into $G$ into a mapping into an $SU(2)$ subgroup of $G$.

I am reading The Uses of Instantons (1977) by Sidney Coleman (a physics text!) and he quotes a pretty strong and useful claim: For a general simple Lie group $G$, any continuous mapping of $S^3$ into ...
label's user avatar
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1 answer
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stabilizer of gauge group [closed]

Suppose $A$ is a flat connection on a fiber bundle $V$ over a manifold $M$, with fiber $G$. What is the stabilizer of the action of the gauge group on the space of all flat connections (i.e. $g(x)\...
math101's user avatar
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5 votes
1 answer
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Question on gauge fields "acting on different representations"

-First of all, in the end of this question, unfortunatelly, I'll kind of request the solution of an exercise. But, this isn't for any kind of classroom evaluation. It is just that I kind of grasp the ...
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Transformation of holonomy group under gauge transformation

I want to know how does holonomy group change when we apply gauge transformation. More explicitly, for given principal bundle $G\to P\to M$, if we have a holonomy group $G_A$ w.r.t. a connection $A$, ...
taiat's user avatar
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1 vote
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Why the "horizontal" in exterior covariant derivative

Let $P$ be a principal bundle, and $\omega$ a vector-valued 1-form on it. In standard textbooks, one define the exterior covariant derivative of $\omega$ to be $$D\omega(X_1, X_2) = d\omega(X_1^h, X_2^...
Alex's user avatar
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1 answer
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Curvature on an Associated Vector Bundle

I am in the midst of trying to solve problem 13 from chapter 5 of Hamilton's Mathematical Gauge theory text, and have come across some terms I am not sure what to do with. I will elaborate below. Let $...
Chris's user avatar
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1 vote
2 answers
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A determinant invariant with respect to a change of metric (covariant determinant?)

I use a multi-vector of G(2,R): $$ \mathbf{u} = a+x\sigma_x+y \sigma_y + b \sigma_x\sigma_y $$ its matrix representation is $$ \mathbf{u} \cong \pmatrix{a +x & y-b \\y+b & a-x} $$ and the ...
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Einstein field equation from covariant derivative

I am trying to derive the Einstein field equations from this approach: Let $\psi(t) = e^{-tM}$, where $M$ is a $4\times4$ matrix. Then $$ \frac{d}{dt}\psi(t) = -M\psi(t) $$ Let us now suppose that I ...
Anon21's user avatar
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2 votes
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Existence of symplectic vortices

Consider for a closed Riemann surface $(\Sigma,j_{\Sigma})$, a compact Lie group $G$ and $P$ a principal $G$ bundle over $\Sigma$ and a connection 1-form $A$ on the principal bundle $P$ and $u:P \to M$...
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Constant $SU(2)$ ASD connections on $\mathbb R^4$ are Flat

Let $A \in \mathfrak{su}(2) \otimes \mathbb R^4$, so $A$ is a collection of $4$ elements of $\mathfrak {su}(2), (A_0, \dots, A_3)$. We can consider the system of equations $$ [A_0, A_1] + [A_2, A_3] = ...
Holmes's user avatar
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5 votes
1 answer
194 views

Torsion Free Spin Connection

Ok I am not exactly sure how much of this common notation/terminology, and how much is unique to the book I'm reading, so bear with me for a moment here. First we have a vector bundle $E$ associated ...
Chris's user avatar
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4 votes
1 answer
192 views

Good reference for self study of Gauge theory [closed]

I am looking for the shortest way possible to study basic gauge theory. I am looking for some inspiring survey notes like this one: Christian Bär, Gauga Theory (rather than the great books by ...
user56980's user avatar
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2 answers
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Action of the Gauge Group on the Configuration Space of Seiberg-Witten Theory

In Seiberg-Witten theory, the group of gauge transformations is $Map(M,S^1)$. For a configuration $(A,\psi)$, where $A$ is a unitary connection on the determinant line bundle, $\psi$ is a spinor, and $...
viniciuscantocosta's user avatar
1 vote
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35 views

Topological equivalent vs Holomorphic equivalent of line bundle.

I know there are indeed some examples show that two line bundles over some projective manifold is topological equivalent but not holomorphic equivalent, but I find I give a "proof" shows ...
taiat's user avatar
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4 votes
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How to set up Chern-Simons theory, compute its "topological gauge shift" term, and compute its quantum path integrals with Wilson lines?

Suppose we want to consider Chern-Simons theory on an (odd-dimensional) compact boundaryless smooth manifold $X$ for a Lie group (the "structure/gauge group") $G$. Is it possible to do so ...
I.A.S. Tambe's user avatar
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1 vote
1 answer
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Gauge equivalence of Lie-valued forms on the base space of a principal bundle

Given a principal $G$-bundle $P\xrightarrow{\pi} M$: Assuming the bundle is globally trivial, we define two Lie$G$-valued 1-forms $A_1,A_2$ on $M$ to be gauge-equivalent if there is a principal bundle ...
I.A.S. Tambe's user avatar
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2 votes
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107 views

What is the "gauge field" on the base space in a gauge field theory?

Suppose we have a principal $G$-bundle $P\xrightarrow{\pi} M$, and we want to consider a classical gauge field theory (with a field Lagrangian) on $M$ for this bundle, in the physics sense. In the ...
I.A.S. Tambe's user avatar
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Moduli Space of Flat connection over Homology 3-Shpere

I'm trying to understand the space of flat connections over the trivial $SU(2)$ -bundle of a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer homology of it)...
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1 vote
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63 views

Standard metric on adjoint bundle

Consider we have a principal $G$-bundle $P$ over a closed manifold $V$. Denote $\mathfrak{g}_P$ by the associated bundle $P\times_G \mathfrak{g}$ where $G$ acts by adjoint action. Denote $\mathscr{G}$ ...
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