# Questions tagged [gauge-theory]

For questions about gauge theory in mathematical/theoretical physics, which a field theory whose fields include principal bundles with connection.

141 questions
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### Self dual solution of Yang-Mills equation on the Nahm case

I'm reading this notes, and I have a question about the construction on page 48. Let $M=S^{1}\times \mathbb{R}^{3}$ and $E\rightarrow M$ a vector bundle over $M$. If $A$ is the $1$-form matrix ...
0answers
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### Navier-Stokes smoothness problem and Gauge Theory

Recently, I came across this paper where the author describes an analogy between electrodynamics and fluid dynamics. He develops a one-to-one correspondence between the equations of electrodynamics ...
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### Adjoint of the Coupled Covariant Derivative on Spinors

I want to understand the proof for a $C^0$ bound of solutions to the Seiberg-Witten equations. Among other places, it can be found in Kronheimer, Mrowka: "The genus of embedded surfaces in the ...
2answers
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### Going from one notation to another in Yang-Mills

In many books on Yang-Mills theories, written from a physicist's point of view, the curvature tensor is written as: $$F_{\mu \nu}^a = \partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^bA_\nu^c$$...
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### Computing curvature of the quotient of the tautological connection

I am trying to understand a certain passage in the book "Geometry of Four-Manifolds" by Donaldson and Kronheimer (specifically, a computation in section 5.2.3). I am confused on the proof of ...
1answer
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### Reference request for gauge theory in low dimensional topology

I've been studying 3 and 4 manifold topology and it seems to me that lots of very powerful invariants come from a mysterious place called "gauge theory". When I peer into this place, I am confronted ...
1answer
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### What constitutes a gauge theory? Help me understand electromagnetism as the prototype of all gauge theories

In The Geometry of Physics and Knots, Sir Michael Atiyah says that electromagnetism is the prototype of all gauge theories: The prototype of all gauge theories is electromagnetism. From the ...
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### Prerequisites to Yang-Mills Theory

Does anyone know what prerequisite is necessary to understand Yang-Mills Theory? And Some of the most important books necessary of the field in Math and Physic about the subject starting from the ...
1answer
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### Literature suggestion for understanding Gauge theory from the perspective of a Mathematician.

Can anyone please suggest some good literature or references for understanding Gauge theory from the perspective of a mathematician (from the point of view of differential geometry)? Being a ...
1answer
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### Gauge transformations when base space is contractible

Let $\pi: P \rightarrow M$ be a principal $G$ bundle. Then the gauge group is defined as the automorphisms of the bundle lifting the identity on $M$. Suppose that $M$ is contractible, then the bundle ...
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### Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $-2\pi i \Omega$...
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### Number of automorphisms of some principal $G$-bundles.

Let $M$ be a manifold and let $G$ be a finite group. Let $P \to M$ be a principal $G$-bundle. I have some basic questions regarding the automorphism group of $P$ as a principal $G$-bundle over $M.$. ...
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### Use of GIT for moduli problems

Was there an actual use of GIT technics (moment map, Kempf-Ness theorem) in the different proofs of the Kobayashi-Hitchin correspondence (by Uhlenbeck-Yau and Donaldson)? Or this was rather a ...
1answer
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### Isotropy group of connection is isomorphic to centraliser of holonomy group

I am asking for a proof of Lemma (4.2.8) of Donaldson, Kronheimer: The Geometry of Four-Manifolds. Let $P \rightarrow X$ be a principal bundle with structure group $G$. Denote by $\mathcal{G}$ the ...
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1answer
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### Symmetric tensor of Lie algebra of $su(N)$

I am interested in knowing the exact form of the anti-commutation of two generators of $su(N)$ lie algebra. Let us denote $T^a$ to be the generator of $su(N)$ lie algebra in the defining ...
1answer
118 views

### Symmetric tensor of Lie algebra of $su(N)$

I am interested in knowing the exact form of the anti-commutation of two generators of $su(N)$ lie algebra. Let us denote $T^a$ to be the generator of $su(N)$ lie algebra in the defining ...
0answers
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### A new gauge field [Cech 1-cocycle] acts a symmetry by multiplying itself to transition functions

For a gauge theory of a gauge group $G$ (better here: a simply-connected and compact $G$; but if you wish, you can comment more general cases), we view the spacetime as the base manifold and the ...
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1answer
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### Elliptic Bootstrapping for Gauge Transformations

I'm reading John Morgan's book "Seiberg-Witten equations and applications to the topology of smooth manifolds". I'm stuck in the proof of Lemma 4.5.3, which says the following. Suppose $(a_n,\psi_n)$ ...
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### Doubt in proof of Uhlenbeck's theorem in Donaldson & Kronheimer

Theorem 2.3.8 in 'The Geometry of Four-Manifolds' by Donaldson and Kronheimer reads: There is a constant $\epsilon>0$ such that if $A$ is a ASD (anti self dual) connection on the trivial bundle ...
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63 views

### Is the determinant of the holonomy gauge invariant / significant?

I am currently reading the book Baez & Muniain - "Gauge Fields, Knots, and Gravity". Chapter 2 of part II defines the holonomy on a bundle $E\to M$ with a gauge group $G$ and connection $D$. They ...
1answer
108 views

### Vector-valued forms inside the first jet bundle

On page 433 of "Self-duality in four-dimensional Riemannian geometry" by Atiyah, Hitchin and Singer, it is written that $p^*(E \otimes \Lambda^1) \subset p^*J_1(E)$, where $\Lambda^1 \to X$ is the ...
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### Dimension of the Marsden-Weinstein reduction of a coadjoint orbit in the dual of the Lie algebra of the gauge group (Atiyah-Bott context)

Let $P\to \Sigma$ be a $\mathrm{SU}(2)$-principal bundle over a smooth orientable closed genus $g$ real surface $\Sigma$. Let $\mathcal{A}$ be the space of connexions over $P$ and let $\mathcal{G}$ be ...
2answers
280 views

### Yang–Mills theory and mass gap

I am interested in widening my knowledge into the formal aspects of Yang–Mills theory. In particular, I would like to study the current mathematical and physical research literature about this ...
1answer
177 views

### Different definitions of irreducible $\mathrm{SU}(2)$ connections

Let $Y$ be Poincaré's integral homology 3-sphere. Let $\pi:P\to Y$ be a (necessarily trivial) $\mathrm{SU}(2)$-principal bundle over $Y$. Fix $x_0\in Y$ and $p_0\in P_x:=\pi^{-1}(x_0)$. The ...
3answers
174 views