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.
271
questions
2
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0
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29
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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 ...
- 51
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0
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23
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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 ...
2
votes
0
answers
26
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 ...
6
votes
1
answer
237
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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/...
- 4,287
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votes
1
answer
85
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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 $g$,...
- 4,287
0
votes
0
answers
70
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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 ...
- 11
4
votes
1
answer
69
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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 $A\in\Omega^{1}(\mathcal{P},\mathfrak{g})$ ...
- 629
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0
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67
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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 ...
- 597
0
votes
1
answer
81
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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 ...
- 454
1
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0
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73
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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 (...
- 696
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0
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61
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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 ...
- 455
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27
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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 ...
2
votes
0
answers
61
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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 ...
- 263
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When does the Cartan formalism have a gauge that makes the tetrad constant?
I hope I am using the terms correctly: by Cartan formalism I am thinking of a general (orthonormal) tetrad, defined with respect to a coordinate system $\mathbb{R}^n$, along with the spin connection. ...
- 209
1
vote
1
answer
56
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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 $W$ ...
- 2,073
1
vote
1
answer
102
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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 ...
- 696
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0
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106
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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 $...
- 5,821
1
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1
answer
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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 $S^1$ ...
- 5,821
0
votes
1
answer
72
views
Principal $PSU(N)$ bundles over $\mathbb R^3$
I am aware that principal $SU(N)$ bundles over 3-manifolds are trivial.
Consider, however, a principal $PSU(N)$ bundle over $\mathbb R^3$. Is the topology of this bundle classified by $$\pi_2(SU(N)/U(...
- 473
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0
answers
41
views
Is there a sense in which two gauge connections are topologically equivalent?
We say that manifolds $M$ and $N$ are topologically equivalent iff there exists a homeomorphism between them. Is there are similar notion for gauge connections? I.e. is there a sense in which two ...
- 473
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0
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53
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Examples of Physical Applications ofthe theory of Principal $G$-bundles
Let $(P,M,\pi)$ be a principal $G$-bundle, $\omega\in \mathbf \Omega^1(M,\mathfrak g)$ be a connection 1-form and $\Omega \in \mathbf \Omega^2(M,\mathfrak g)$ be its curvature 2-form, where $\mathfrak ...
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0
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Sufficient condition for moduli space of slope-stable bundles to be non-empty
I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature.
Let $X$ be a Kähler surface. Let $\mathscr{M}(...
- 11
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0
answers
98
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How do we vary the Einstein-Hilbert Action?
The Einstein-Hilbert actions is:
$$
\begin{align}
S=\int_M R(g)\text{dvol}_g=\int_M\sqrt{-g}Rd^4x
\end{align}
$$
I'm looking to vary the action in a coordinate free way without appealing to the ...
- 2,073
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vote
0
answers
23
views
Curvature equation
The curvature equation of Seiberg-Witten equations is
$F^+_A=\psi\otimes\psi^*-\frac{|\psi|^2}{2}Id$,
where $F^+_A$ is selfdual part of curvature and $\psi$ is spinor.
Could someone elaborate why ...
- 233
1
vote
1
answer
148
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Flat torsion-free connections
Can a manifold admit two different flat torsion-free connections? By different, I mean that there is not any diffeomorphism that makes the two equal. The question arises from me trying to explain to ...
- 333
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votes
1
answer
114
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When is a principal bundle with group $G \times H$ the product of a principal bundle with group $G$ and one with group $H$?
In general, it is not true that a fiber bundle with a product fiber is the product of two fiber bundles with the factors as fibers (think of vector bundles). However, I read somewhere that every 2-...
- 333
6
votes
1
answer
248
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Boundary conditions for differential forms
I am trying to understand differential forms on manifolds with boundaries, and I am a bit confused with the boundary conditions.
For the following, let $(M,g_M)$ be a smooth Riemannian manifold with ...
- 183
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votes
1
answer
100
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Gauge Theory Gravity: transformation of vector potential $A'(x)$
In Gravity, Gauge Theories and Geometric Algebra by A. Lasenby, C. Doran and S. Gull develop a theory for gravity based on the use of a position-gauge field and a rotation-gauge field. The former is ...
- 63
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0
answers
27
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Gauge Transformation in E and M vs. U(1) theory over $\mathbb{R}^{1,3}$
In E and M we have that the fields exhibit gauge symmetry under the gauge transformations:
$$V'=V-\frac{\partial \lambda}{\partial t}$$
$$M'=M+\nabla \lambda$$
for some $\lambda:\mathbb{R}^{1,3}\...
- 2,073
7
votes
1
answer
401
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Mathematical Introduction to the Standard Model of Particle Physics
I am looking for textbooks, lecture notes, lecture videos on a rigorous introduction to the standard model of elementary particles. I'd prefer to not be referred to monographs for an introduction as ...
- 253
3
votes
1
answer
163
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Transformation of Curvature 2-Form Under Global Bundle Autormphism
Let $F^A$ be a curvature two form on a principal bundle $P$ corresponding to the connection one form $A$. Let $f:P\rightarrow P$ be a global bundle automorphism; there then exists a map $\sigma_f:P\...
- 2,073
3
votes
0
answers
68
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Deform a continuous mapping of $S^3$ into $G$ into a mapping into an $SU(2)$ subgroup of $G$.
I am reading The Uses of Instantons (1977) by Sidney Coleman (a physics text!) and he quotes a pretty strong and useful claim:
For a general simple Lie group $G$, any continuous mapping of $S^3$ into ...
- 322
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1
answer
54
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stabilizer of gauge group [closed]
Suppose $A$ is a flat connection on a fiber bundle $V$ over a manifold $M$, with fiber $G$. What is the stabilizer of the action of the gauge group on the space of all flat connections (i.e. $g(x)\...
- 21
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votes
1
answer
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Question on gauge fields "acting on different representations"
-First of all, in the end of this question, unfortunatelly, I'll kind of request the solution of an exercise. But, this isn't for any kind of classroom evaluation. It is just that I kind of grasp the ...
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1
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60
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Transformation of holonomy group under gauge transformation
I want to know how does holonomy group change when we apply gauge transformation.
More explicitly, for given principal bundle $G\to P\to M$, if we have a holonomy group $G_A$ w.r.t. a connection $A$, ...
- 1,107
1
vote
2
answers
153
views
Why the "horizontal" in exterior covariant derivative
Let $P$ be a principal bundle, and $\omega$ a vector-valued 1-form on it. In standard textbooks, one define the exterior covariant derivative of $\omega$ to be
$$D\omega(X_1, X_2) = d\omega(X_1^h, X_2^...
- 245
5
votes
1
answer
182
views
Curvature on an Associated Vector Bundle
I am in the midst of trying to solve problem 13 from chapter 5 of Hamilton's Mathematical Gauge theory text, and have come across some terms I am not sure what to do with. I will elaborate below.
Let $...
- 2,073
1
vote
2
answers
55
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A determinant invariant with respect to a change of metric (covariant determinant?)
I use a multi-vector of G(2,R):
$$
\mathbf{u} = a+x\sigma_x+y \sigma_y + b \sigma_x\sigma_y
$$
its matrix representation is
$$
\mathbf{u} \cong \pmatrix{a +x & y-b \\y+b & a-x}
$$
and the ...
- 2,455
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votes
0
answers
61
views
Einstein field equation from covariant derivative
I am trying to derive the Einstein field equations from this approach:
Let $\psi(t) = e^{-tM}$, where $M$ is a $4\times4$ matrix.
Then
$$
\frac{d}{dt}\psi(t) = -M\psi(t)
$$
Let us now suppose that I ...
- 2,455
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votes
0
answers
63
views
Existence of symplectic vortices
Consider for a closed Riemann surface $(\Sigma,j_{\Sigma})$, a compact Lie group $G$ and $P$ a principal $G$ bundle over $\Sigma$ and a connection 1-form $A$ on the principal bundle $P$ and $u:P \to M$...
- 3,935
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1
answer
71
views
Constant $SU(2)$ ASD connections on $\mathbb R^4$ are Flat
Let $A \in \mathfrak{su}(2) \otimes \mathbb R^4$, so $A$ is a collection of $4$ elements of $\mathfrak {su}(2), (A_0, \dots, A_3)$. We can consider the system of equations
$$
[A_0, A_1] + [A_2, A_3] = ...
- 821
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votes
1
answer
194
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Torsion Free Spin Connection
Ok I am not exactly sure how much of this common notation/terminology, and how much is unique to the book I'm reading, so bear with me for a moment here. First we have a vector bundle $E$ associated ...
- 2,073
4
votes
1
answer
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Good reference for self study of Gauge theory [closed]
I am looking for the shortest way possible to study basic gauge theory. I am looking for some inspiring survey notes like this one: Christian Bär, Gauga Theory (rather than the great books by ...
- 325
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votes
2
answers
95
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Action of the Gauge Group on the Configuration Space of Seiberg-Witten Theory
In Seiberg-Witten theory, the group of gauge transformations is $Map(M,S^1)$.
For a configuration $(A,\psi)$, where $A$ is a unitary connection on the determinant line bundle, $\psi$ is a spinor, and $...
1
vote
0
answers
35
views
Topological equivalent vs Holomorphic equivalent of line bundle.
I know there are indeed some examples show that two line bundles over some projective manifold is topological equivalent but not holomorphic equivalent, but I find I give a "proof" shows ...
- 1,107
4
votes
0
answers
117
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How to set up Chern-Simons theory, compute its "topological gauge shift" term, and compute its quantum path integrals with Wilson lines?
Suppose we want to consider Chern-Simons theory on an (odd-dimensional) compact boundaryless smooth manifold $X$ for a Lie group (the "structure/gauge group") $G$.
Is it possible to do so ...
- 2,391
1
vote
1
answer
82
views
Gauge equivalence of Lie-valued forms on the base space of a principal bundle
Given a principal $G$-bundle $P\xrightarrow{\pi} M$:
Assuming the bundle is globally trivial, we define two Lie$G$-valued 1-forms $A_1,A_2$ on $M$ to be gauge-equivalent
if there is a principal bundle ...
- 2,391
2
votes
0
answers
107
views
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 ...
- 2,391
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votes
0
answers
61
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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)...
user388493
1
vote
0
answers
63
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Standard metric on adjoint bundle
Consider we have a principal $G$-bundle $P$ over a closed manifold $V$. Denote $\mathfrak{g}_P$ by the associated bundle $P\times_G \mathfrak{g}$ where $G$ acts by adjoint action. Denote $\mathscr{G}$ ...
user388493