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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. ...
Jeff's user avatar
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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 ...
Richard122's user avatar
2 votes
1 answer
51 views

Yang-Mills action is invariant under conformal change of the metric

Let $(M,g)$ be a pseudo-Riemannian 4-manifold with a principal bundle $P\to M$. Prove that the Yang–Mills action $S_{YM}[A]$ is invariant under a conformal change of the metric $g$: $\quad g'=\mathrm{...
Siyuan Yin's user avatar
1 vote
0 answers
53 views

Gauge Theory : Mathematics + Physics

I'm interested in learning mathematical gauge theory, particularly its applications in physics, focusing on (topological) quantum field theory with an emphasis on condensed matter. I'm looking for ...
user82261's user avatar
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1 vote
0 answers
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What is the topology of a Clifford algebra?

I'm reading Hamilton's Mathematical Gauge Theory and I'm currently studying the Pin and Spin groups. Hamilton defines them as specific subsets of a Clifford algebra, and it is understood that the ...
Níckolas Alves's user avatar
2 votes
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69 views

Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold

Let $X$ be a compact Riemannian 4-manifold, $P$ a principal $G$-bundle over $X$, and $\mathfrak{g}$ be its adjoint bundle. Let $\omega$ be a self-dual conneciton on $P$ (i.e. its curvature $\Omega \in ...
user302934's user avatar
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1 vote
1 answer
42 views

Two $SO(2)$ connections on a smooth $SO(2)$-vector bundle

Let $L\to M$ be a smooth $SO(2)$-vector bundle. We can identify the Lie algebra $\mathfrak{so}(2)$ with $i\Bbb R$. Suppose $\nabla, \nabla'$ are two $SO(2)$-connections on $L$ such that $\nabla=\nabla'...
blancket's user avatar
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When does the map from Fundamental Group to Holonomy Group Injective?

We know that over a rank $n$ vector bundle $E$ (with base space being $M$, a manifold), if the connection $\nabla$ is flat, then the parallel transport along loop $\gamma:I\to M$ for $E$ will only ...
BoyanLiu's user avatar
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33 views

Definition Gauge Field / Connection on principal bundle

Reading Division Algebras and Supersymmetry I by John C. Baez and John Huerta, one passage confused me a bit: A connection $A$ on a principal $G$ bundle over $M$. Since the bundle is trivial, we ...
anonymous250's user avatar
2 votes
1 answer
50 views

An action on the space of connections on a $SO(3)$-vector bundle

Let $M$ be a Riemannian manifold and $E$ a smooth $SO(3)$-vector bundle over $M$ with fiberwise metric. Suppose there is an action of a finite cyclic group $\Bbb Z_n$ on $M$ by isometries, and there ...
blancket's user avatar
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Two connections on a smooth $SO(3)$-vector bundle

Let $E\to M$ be a smooth oriented real vector bundle of rank 3, so that its structure group can be reduced to $SO(3)$. Suppose $E\to M$ is given a Riemannian metric and a connection $\nabla:\Omega^0(E)...
blancket's user avatar
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4 votes
1 answer
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Self duality of a connection is invariant under a gauge transformation

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $P\to M$ be a (smooth) principal $G$-bundle over an oriented Riemannian smooth 4-manifold $(M,g)$. Let $E=P\times_{\text{Ad}}\mathfrak{g}...
blancket's user avatar
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1 vote
1 answer
45 views

Covariant derivative induced by pullback connection under an automorphism

Let $G$ be a Lie group and $\pi:P\to M$ a smooth principal $G$-bundle. Let $\omega$ be a connection on $P$; it is a $\mathfrak{g}$-valued 1-form on $P$ where $\mathfrak{g}$ is the Lie algebra of $G$. (...
blancket's user avatar
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29 views

How does one strictly define "the number of components that can be fixed by the choice of gauge"?

It is common for books on physics to contain something along these lines: "We have $X$ parameters and $Y$ conditions, hence, the number of free parameters is $X-Y$" or "we have $X$ ...
Daigaku no Baku's user avatar
2 votes
1 answer
40 views

Do (ineffective/locally effective) Cartan geometries have a unique maximal atlas?

In Sharpe's book on Cartan geometry, he mentions in kind of an offhand way that Cartan atlases extend to unique maximal Cartan atlases, but I'm feeling skeptical. For a Klein pair $(G,H)$ with Lie ...
subrosar's user avatar
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Mathematical references for gauge theory in condensed matter physics

I am currently trying to go through some literature on the classification of symmetry protected topological phases. One shortcoming in my background is that I am unable to understand the physics-...
user82261's user avatar
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5 votes
1 answer
155 views

What is the correct sign for the four-vector potential gauge transform; $A_\mu\to A_\mu\pm\partial_\mu\lambda$ and where does this gauge originate? [closed]

I have three questions regarding the following extract(s), I have marked red the parts for which I do not understand for later reference. The convention followed for the Minkowski metric in these ...
Sirius Black's user avatar
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37 views

$\mathbb Z_N$ Gauge Theory

I am currently trying to go through some literature on symmetry protected topological phases and gauge theories defined on lattices. I am looking for a mathematically precise reference that discusses $...
user82261's user avatar
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4 votes
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157 views

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 ...
Integral fan's user avatar
1 vote
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108 views

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 $E$,...
Mohith Nagaraju's user avatar
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1 answer
106 views

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 ...
Lourenco Entrudo's user avatar
1 vote
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65 views

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 ...
polology's user avatar
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0 answers
35 views

Torsion spin-c structures on $S^1\times S^2$

I have been reading the paper https://arxiv.org/pdf/1902.04050.pdf by Zemke and at some point we have the following: My question is, how can one make sense of torsion $\mathrm{Spin}^c$ structures on $...
horned-sphere's user avatar
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39 views

Formulating $Spin^c(3,1)$ Connection and Curvature on a $GL+(4,R)/Spin^c(3,1)$ Structured Manifold

I am exploring a geometric framework where the usual metric tensor role (as in $GL^+(4,\mathbb{R})/\text{SO}(3,1) $) is replaced by a structure defined by the quotient $ GL^+(4,\mathbb{R})/\text{Spin}^...
Anon21's user avatar
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3 votes
1 answer
119 views

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 ...
Anon21's user avatar
  • 2,589
0 votes
1 answer
89 views

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 ...
CBBAM's user avatar
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2 votes
1 answer
139 views

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 $\mathfrak{g}$ ...
CBBAM's user avatar
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1 vote
1 answer
130 views

Confusion about parallel transport on principal bundles

Let $G$ be a matrix Lie group with Lie algebra $\mathfrak{g} = T_e G$. Let $P \to X$ be a $G$-principal bundle with a connection 1-form $A \in \Omega^1(P,\mathfrak{g})$. Suppose we have a loop $\gamma\...
KuSi's user avatar
  • 181
1 vote
0 answers
39 views

Union of spin-c structures

Let's say we have two 4-manifolds with boundary $M_1$ and $M_2$, with spin-$\mathbb{C}$ structures $s_1$ and $s_2$. Let's say also there is an orientation reversing diffeomorphism $f:\partial M_1 \to \...
horned-sphere's user avatar
0 votes
0 answers
24 views

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 ...
GedankenExperimentalist's user avatar
2 votes
1 answer
129 views

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 ...
jaws93's user avatar
  • 21
1 vote
0 answers
41 views

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 ...
Aidan Backus's user avatar
0 votes
0 answers
47 views

Is every connection form $\omega$ the extremal of some functional $S[\omega]$?

Is every connection form $\omega$ the extremal of some functional $S[\omega]$ ? Context: Palatini action $S_{Pal}$ of General Relativity is (assuming Cosmological constant $\Lambda=0)$: $$ S_{Pal}[e,\...
Powder's user avatar
  • 931
2 votes
0 answers
32 views

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 ...
Akhaim's user avatar
  • 51
1 vote
1 answer
38 views

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 ...
Schrödinger's cat's user avatar
2 votes
0 answers
40 views

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
7 votes
1 answer
697 views

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/...
CBBAM's user avatar
  • 6,255
2 votes
2 answers
240 views

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 $g$,...
CBBAM's user avatar
  • 6,255
0 votes
0 answers
99 views

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
4 votes
1 answer
101 views

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
1 vote
0 answers
148 views

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
0 votes
1 answer
166 views

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 ...
s.h's user avatar
  • 476
1 vote
0 answers
184 views

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
1 vote
0 answers
66 views

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 ...
MrHolmes's user avatar
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1 vote
1 answer
120 views

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
2 votes
0 answers
66 views

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
1 vote
1 answer
68 views

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
  • 3,431
1 vote
1 answer
177 views

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
0 votes
0 answers
237 views

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 $...
wonderich's user avatar
  • 5,969
1 vote
1 answer
419 views

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 $S^1$ ...
wonderich's user avatar
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