# 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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### Differential of a connection form of a Principal Bundle

In Hamilton's "Mathematical Gauge Theory", he defines the curvature as follows, $$F(X, Y) = dA(\pi^H(X), \pi^H(Y))$$ However, he hasn't defined what $d$ is for a vector-valued one form is. ...
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### Fourier transform of differential form

I'm trying to make sense of how to take the Fourier transform(FT) of the Yang-Mills gauge field $A=A_\mu dx^\mu \in \Omega^1(M)$, where $M$ is $\mathbb{R}^4$, let's say. A shortcut to this problem ...
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### Geometry of electrodynamics

In this question, I'd like to go over the physics - math dictionary occurring in the geometric structure (Principal bundle/spin bundles etc.) of Maxwell electrodynamics and the Dirac field. Consider ...
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### Vanishing of connection matrices for flat principal $G$-bundle

Background Recall that for a real vector bundle, there is a well known integrability theorem. Theorem. Suppose there is a vector bundle $E$ with fiber $\mathbb R^n$. If $A$ is a flat connection on $E$,...
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### Equating two definitions of principal fiber bundles

I am following these lectures on principal fiber bundles. Here, a principal fiber bundle is defined as a fiber bundle of which total space $P$ has a right free action of some Lie group and which is ...
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### 3-connection on nontrivial 3-manifold

I'm studying Chern-Simons theory on topological nontrivial 3-manifold (I come from a physics background, so I'm new to some mathematical concepts). If the first homology group $H_1(M)$ is nontrivial ...
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### Understanding the Geometry Spawned from Quotient Spaces $GL^+(4,R)/SO(3,1)$, $GL^+(4,R)/Spin(3,1)$, and $GL^+(4,R)/Spin^c(3,1)$

I'm working on a theoretical framework where I explore different quotient spaces formed with GL$^+$(4,R) and various groups. Specifically, I'm interested in the types of geometry that arise from the ...
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### Formal definition of a $U(1)$ connection

Let $\pi:P \rightarrow M$ be a $U(1)$ principal bundle. I often see people refer to a "$U(1)$ connection" but I cannot find a formal definition of this term. The closest I got was this ...
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### Differential form taking values in a vector bundle?

Let $\pi: P \rightarrow M$ be a $G$-principal bundle. Define $\Omega^k_{\text{hor}}(P, \mathfrak{g})^{\text{Ad}}$ to be the set of $k$-forms $\omega$ taking values in the Lie algebra $\mathfrak{g}$ ...
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### Sum of two gauge dependents being gauge independent

I don't know much about Gauge theory but from some bits and pieces I know about it, the following problem seems to be related to it. Consider the classical Lagrangian $$\mathcal{L}=T-U$$ Where $T$ is ...
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### Invariant connection on principal bundle

Suppose I have a principal bundle, and some group G acting on the principal bundle. Is it always possible to find a G-invariant connection on the principle bundle? If G is compact, then I can imagine ...
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### Does a gauge-invariant Caccioppoli inequality hold?

(I suspect that this question has an elementary resolution. But perhaps it would be more appropriate on MathOverflow, and if so I would not be opposed to migrating it there.) Let $V \Subset U$ be ...
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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 ...
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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 $A\in\Omega^{1}(\mathcal{P},\mathfrak{g})$ ...
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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 ...
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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 (...
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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 ...
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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 ...
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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 $W$ ...
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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 ...
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