Questions tagged [gauge-theory]
Gauge theory in physics and mathematics refers to a field theory whose fields include principal bundles with connection.
14 questions from the last 365 days
4
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1
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177
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Exact forms, gauge transformations, and the Hodge decomposition in non-abelian Gauge theory
I am trying to understand how the Hodge decomposition is affected by gauge transformations in non-abelian in gauge theory (eg $\mathrm{SU}(N)$). In particular, I am searching for a way to generalise ...
- 41
5
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0
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185
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Nullity of a self-dual connection
I consider Yang-Mills theory in the critical dimension $4$ on a $SU(2)$-bundle with positive Chern-Class. It is well known that self-dual connections ($*F=F$) are minimizers of the Yang-Mills ...
- 914
1
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0
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81
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"More stable" definitions of principal $G$-bundle
Let $G$ be a topological group. For any pointed topological space $X$, define $[X,G]$ to be the group whose underlying topological space is the space of pointed continuous maps from $X$ to $G$, with ...
- 863
4
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1
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121
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Riemann surface invariants from vortex equation
The vortex equations are often pitched as a toy model of the Seiberg-Witten equations. While the SW equations are frequently referenced in the context of providing geometric invariants on the base ...
- 447
4
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0
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78
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Higher-dimensional analogue of the relation between stable Higgs bundles and constant curvature metrics
In Hitchin's famous paper[1] on the self-dual Yang-Mills equations, he discussed the relation between the stable Higgs bundles and the Teichmüller space for a compact Riemann surface. Namely, through ...
- 353
4
votes
1
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139
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The smoothness of solutions to the Hitchin self-dual equations within a stable orbit after Sobolev completion
First, let me introduce the background of the problem: While studying chapter 4 of the Hitchin's paper "THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE", I encountered an issue. Hitchin ...
- 41
2
votes
0
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74
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Is it likely that gradient flow trajectories of a $G$-invariant function pass through degenerate points?
This question may be posed somewhat vaguely, but I'm interested to actually get an idea of what to expect, so I try to not target it at a specific result.
Assume that $G$ is a compact Lie group, ...
- 139
0
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1
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143
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Gauge invariance issues of YM theories in 2D Euclidean space
In order to be clear, I will write down every component explicitly. Also, I assume Euclidean metric in this post, so that spacetime indices are written as $i,j$ rather than $\mu, \nu$.
Following Wiki, ...
- 3,477
1
vote
2
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157
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Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold
I've already asked this question in: https://math.stackexchange.com/questions/4899825/question-about-the-index-of-two-elliptic-operators-over-a-4-dimensional-riemanni, and I've been suggested to ask ...
- 335
3
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0
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147
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Generalizing the Narasimhan–Seshadri theorem
There is a theorem that (stable, topologically trivial) holomorphic $G$-bundles are in one-to-one correspondence to flat $K$-bundles (with the appropriate corresponding condition), where $K$ is the ...
- 563
0
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0
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117
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An interesting identity involving skew-Schur functions
Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9)
\begin{align*}
\prod_{k\geq1}(1+...
- 43.2k
1
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0
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97
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Question from Taubes' SW$\Rightarrow$ Gr
I am trying to understand Taubes' paper on SW$\Rightarrow$ Gr. I don't understand how either of the equations 2.16 or 2.17 appears, I would be happy to understand how the curvature term $F_a$ appears ...
- 954
4
votes
1
answer
238
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Taubes' SW$\Rightarrow$ Gr
I am reading Taubes' paper on SW$\Rightarrow$ Gr and lost in some analysis, can anyone help me to see how to get equation 2.19 from equation 2.18? Is this some version of Kato for the Laplacian?
- 954
1
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0
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72
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Spin(7)-instanton
Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the ...
- 954