# 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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### 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 ...
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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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### Non existence of a preferred Horizontal subspace on a bundle. Why not ? (Basics)

If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=T^2$ (2-torus) and $B=S^1$. Can I choose to put my finger on the identity element of the group over a point ...
1 vote
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### calculate the dimension of moduli space

I'm learning gauge theoretic topics about 4-manifolds, and I get stuck when I try to calculate the dimension of ASD moduli space. For an oriented closed 4-manifold $M$, we first fix a riemannian ...
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1 vote
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### local model of gauge theoretic moduli space

I'm a beginner of gauge theory and I find that most materials state the theorem(without proof) that: Given a principal bundle $P$, for a connection $A$ over $P$, a isotopy group $G_A$ of $A$ consist ...
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### Principal connections on $P$ and covariant derivatives on associated vector bundle $E=P\times_\rho V$

I would like to have a concrete proof or reference to the following fact: Let $P\rightarrow M$ be a principal $G$-bundle over an $n$-dimensional manifold $M$, and let $E=P\times_\rho V$ be an ...
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### What does it mean for a gauge field to have no curvature?

The electromagnetic gauge field is $A + d\theta$, where $\theta \colon \mathbb{R}^n \to \mathbb{R}$ comes from a gauge function, $e^{i\theta(\vec x)}$. Let's set $A=0$. The curvature form is $0$ ...
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### Moduli space of connection on line bundle

I want to show that for a $U(1)$ bundle $P$ over a connected smooth 4-manifold $X$, the moduli space of Yang-Mills connection over $P$ is the torus $H^1(X,\mathbb{R})/H^1(X,\mathbb{Z})$. Now I reduce ...
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### stabilizer of connection on SU(2) bundle

Suppose $P$ a principal bundle over connected manifold $B$ with correspondent Lie group $G=SU(2)$, and $A$ a connection on $P$. We say a map $\sigma \in Aut(P)$ a stabilizer of $A$ if $\sigma^*A=A$....
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### differentiate gauge transformation

Suppose $A_0$ a connection on principal bundle $P$, pick $\xi$ an elements in $\Omega^1(adP)$ and $e^{t\xi} \in \Omega^0(AdP)$ a one-parameter group generated by $\xi$. （edited：Suppose $G$ the ...
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### Understanding gauge transformations and relation to Lie Groups?

I am starting to study gauge theory. I have a background in basic group theory, multivariable calculus, and the idea of symmetry in relation to group theory. I am trying to understand why the ...
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1 vote
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### On the definition of the transformation of the connection $1-$form.

Introduction I'm studying principal fiber bundles to deal with the gauge sector of standard model lagrangian. Now, references $[1]$ and $[2]$, gives us a definition on how the connection $1$-form ...
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### Gauging the GL(4,R) group

I am told from numerous sources that general relativity can be understood as the product of gauging the GL(4,R) group. However, no one seems able to provide me with a source, or a derivation. It is ...
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### Gauge theory and well-posedness of a PDE

I am trying to understand the relationship between gauge theories (for example General Relativity) and the well-posedness of the underlying theory. In General Relativity, it is known that you must be ...
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