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.
180
questions
0
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0answers
24 views
Definitions of the quadratic Casimir of $SU(N)$
Particle physics, especially QCD, deals a lot with $SU(N)$ and therefore also with $\mathfrak{su}(N)$. In the QCD literature it is normal to define the (quadratic) Casimir element in a representation ...
0
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0answers
39 views
What is the covariant derivative of a general linear transformation?
Is it possible to obtain general relativity as a gauge theory from the general linear group?
The starting point is:
$$
M'=GM
$$
where $M',M,G$ are elements of $GL(4,\mathbb{R})$.
I believe the second ...
4
votes
0answers
56 views
What books would you recommend to understand gauge theories?
I am a math student who got interested in the topics above also I want to learn about the Dirac matter the spinors the Einsteins GR and the Yang- Mills Maxwell Anderson -Higgs theories and models and ...
7
votes
1answer
49 views
Compactness of the moduli space of representations of fundamental group
Let $M$ be a compact manifold and $G$ a compact Lie group. I am trying to deduce that the moduli space of flat connections on $M$ is compact, and for that I have shown that there is a correspondence ...
0
votes
1answer
40 views
Gauge transformation (about mathematical definition of wikipedia, other sources)
I've had some exposure to vector bundles, but as good as none to principal bundles. I'm trying to get some orientation in the subjects concerning vector bundles and gauge theory.
On Wiki, there's a ...
1
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0answers
27 views
Metric tensor of a space as an induced metric from embedding in complex space, and dilations thereof.
Apologies, I know the title's a mouth-full.
We can consider the standard unit three-sphere $S^{3}$ as being embedded in $\mathbb{C}^{2}$ with coordinates:
$$\Phi=\begin{array}{c}
\phi^{1}+i\phi^{2}\\
\...
1
vote
0answers
35 views
What kind of mathematical object is the quantized Yang-Mills field?
I have a question regarding the mathematical description of Yang-Mills theories. In classical Yang-Mills theory, the Yang-Mills field (or field strength in physical language) is given by the curvature ...
5
votes
1answer
96 views
What is a Gauge symmetry, intuitively (string theory)?
I'm writing an essay for a popular (but mathematically mature) audience on the history of mathematical physics, wherein I have a section devoted to string theory. Unfortunately, neither I (nor my ...
2
votes
1answer
45 views
Computing the curvature associated to “quaternionic” connection
I am attempting to verify that for a matrix $A=\begin{pmatrix}\alpha &\beta\\\gamma & \delta\end{pmatrix}\in\text{Sp}(2)$ where $\text{Sp}(2)$ is the compact sympletic group and the matrix ...
1
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0answers
169 views
How graduate students get to work in homological mirror symmetry
My question is probably an odd one here but I would very much like to work in Homological Mirror Symmetry. An example of a course I'd like to be able to take and understand is https://faculty.math....
3
votes
3answers
186 views
Why is the divergence of curl expected to be zero?
I was reading the proof of Helmholtz decomposition theorem where I found the relation between the rotational and the irrotational fields are not symmetric. And by that I mean if the divergence of the ...
1
vote
0answers
25 views
If a field in a gauge theory is covariant is that the same as the covariant derivatives of the field are 0?
If a field in a gauge theory is covariant is that the same as the covariant derivatives of the field are 0? I'd like a formal answer, coordinate free. E.g. the covariant derivative can be written as
d ...
1
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0answers
43 views
Is torsion-free affine connection not unique, but there is only one that is compatible with the metric?
On page 79 of David Bleecker's book Gauge Theory and Variational Principles, the author states a theorem:
6.2.5 Theorem. There is a unique connection $\theta$ (called the Levi-Civita connection) on $...
1
vote
0answers
36 views
Product of flat differential characters
Let $A,A'$ be two differential characters of degrees $k, l \in \mathbb N$ on a smooth manifold $X$. Their curvatures $F \in \Omega^{k}$, $F' \in \Omega^{l}(X)$ are differential forms with integral ...
1
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0answers
24 views
$S^3$ hyperspherical harmonics in terms of spherical harmonics
In a problem I might want to work one, I would find myself having to work explicitly with hyperspherical harmonics on a three-sphere and their tensor decomposition. Itās well known that $S^3$ can be ...
0
votes
1answer
67 views
Electromagnetism & the Gauge Theory
A Gauge Theory obtains from Maxwell's equations from a slight generalization of the target space and geometry: Consider matrix-valued objects instead of scalar-valued objects along with a scalar-...
0
votes
1answer
70 views
Winding number of discontinuous map
Is the winding number of a map $f: S^k \to S^k$ that is discontinous even defined? What happens if one plugs in a discontinuous function $U$ inside the winding number formula? Is it an integer?
$$ W[U]...
2
votes
0answers
27 views
When can a principle $G$-bundle over the boundary be extended into the bulk?
Let $M$ be an $m$-dimensional smooth manifold with boundary $\partial M$, and let $G$ be a compact Lie group. When can a principal $G$-bundle with base manifold $\partial M$ be extended to a principal ...
0
votes
0answers
70 views
Geometry of the complex Gauge group
This is a pretty naive question:
Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$. ...
0
votes
0answers
9 views
Upper estimation of radial distance via Hausdorff Distance
I found this inequality (without proof) in a paper and I . If $A$ and $B$ are convex compact and contain the ball $B(0,r)$, we have the following inequality:
$$\sup\limits_{\theta \in (0,2\pi]} |u_A(\...
0
votes
0answers
9 views
Generalized Discrete Fourier Transform and su(J) x su(K) gauge Transformations.
The discrete Fourier transform is used in physics in situations where there is a finite Abelian symmetry group. The classes of an Abelian group G of size N each have a single group element so the ...
0
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0answers
26 views
Canonical connection on $\mathcal{A}\times X$
Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
7
votes
1answer
75 views
Nontrivial bundle in physics
A student of mine asked me why we are studying bundles in theoretical physics, and why it isn't enough to look at product spaces.
I gave the example of the tangent bundle of the sphere, and of the ...
0
votes
0answers
16 views
Induced Projective map of a vector $X$
I've been going through connections on fibre bundles in Nakahara's Topology, Geometry and Physics from 2003 and I wondered if someone could answer this question for me (from page 395, exercise 10.1 a.)...
0
votes
1answer
37 views
Yang-Mills Field Strength Compatibility Function
I've been going through Nakahara's Topology, Geometry and Physics from 2003 and I'm struggling to fully understand this derivation from page 410:
Say there exists two different fields, $\mathcal{F}_i ...
1
vote
0answers
25 views
What are SIMPLE examples of Noether identities?
The only specific examples I know of Noether identities are for Einstein related theories and sigma models. These are all rather complicated for expository purposes.
9
votes
0answers
89 views
Curvature of canonical connection on 4-manifold with self-dual harmonic 2 form
Let $X$ be an compact oriented Riemannian 4-manifold with $b_2^+\geq1$. Let $\omega$ be a self-dual harmonic two form vanishing transversely. On $X\setminus \omega^{-1}(0)$, the spinor bundle $S_+$ ...
1
vote
0answers
25 views
Geometric interpretation of a multivector of $Cl_{3,1}(\mathbb{R})$
A general multivector of $Cl_{3,1}(\mathbb{R})$ is
$$
\begin{align}
\mathbf{u}:=&a+b\gamma_0\gamma_1\gamma_2\gamma_3\\
&+t \gamma_0+x\gamma_1+y\gamma_2+z\gamma_3\\
&+E_x\gamma_0\gamma_1+...
0
votes
1answer
26 views
Analogue of Temperley-Lieb algebra of higher rank
In Piunikhinās Turaev-Viro and Kauffman invariants for 3-manifolds coincide, the exact relation between Kauffman algebra and the representation of the quantum group $U_q(sl_2)$ is shown.
This seems ...
2
votes
1answer
64 views
Tri-vector for Nambu bracket
The Poisson bracket can be given via a bi-vector.
Is there a tri-vector for the Nambu 3-bracket? and higher?
1
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0answers
49 views
Obstruction theory on gauge transformations
Let $X$ be an $n$-dimensional smooth manifold with an SO(3)-bundle $P$. Then we have the gauge group $\mathscr{G}(P) = \Gamma(P\times_{\text{Ad}}\text{SO}(3))$. The claim (in Froyshov's "Equivariant ...
1
vote
2answers
136 views
Flat connection over a simply connectied manifold
I am looking for a proof of the following result.
Let $p:P\to M$ be a $G$-principal bundle over a connected and simply connected manifold.
Suppose there is a flat connection $A\in\Omega^1(P,\...
1
vote
1answer
51 views
Solutions of the Yang-Mills-Higgs equations in the plane
Is there any reference describing the analytical properties of the critical points of the functional
$$
E(\phi,A)=\int_{\mathbb{R}^2}\frac{1}{2}|(\partial_j-iA_j)\phi|^2+\frac{1}{4}F_{jk}F_{jk}+\frac{...
1
vote
2answers
85 views
When does the gauge potential vanish?
Question: Given a $U(1)$-principal bundle over the circle group $S^1$, when does the gauge potential vanish?
Attempt: Firstly, I believe that the principal bundle $P_M = U(1) \times S^1$ is trivial. ...
1
vote
0answers
37 views
About integer spin representations and the degree counting
I am following this statement / answer given in https://physics.stackexchange.com/a/230266/42982:
About degrees of freedom (dof):
"There are two spinor representation of SO(3,1) (1/2,0) and (0,...
3
votes
2answers
136 views
On the definition of a connection 1-form
Let $P\to M$ be a principal $G$-bundle. I'm reading Mathematical Gauge Theory by Mark Hamilton and he defines a connection $1$-form as $A\in \Omega^1(P,\mathfrak{g})$ satisfying
(i) $({\rm R}_g)^*A = ...
2
votes
1answer
45 views
Asymptotic map on Seiberg-Witten moduli space on manifold with cylindrical end
I am reading Liviu I. Nicolaescu's book "Notes on Seiberg-Witten Theory". I am confused that in Corollary 4.3.22(b) (and Remark 4.4.2), it appears that the boundary map $\partial_\infty:\widehat{\...
2
votes
1answer
52 views
Multiplication subscript
I'm teaching myself some differential geometry in the hope to understand gauge theory properly. In the definition of the the pullback bundle I came across a strange notation that I've never seen ...
1
vote
0answers
37 views
Does $SW_X\not\neq 0 \implies \hat{A}(X)\neq 0$?
Let $X^4$ be a closed spin 4-manifold with $b_2^+\geq 2$ and let
$$
SW_X:\mathrm{Spin}^c(X)\to \mathbb{Z}
$$
be the Seiberg-Witten map. If either $\hat{A}(X)\neq 0$ or $SW_X\not \equiv 0$, then $X$ ...
1
vote
0answers
17 views
A question on choosing the gauge $A_1 = 0$
I am reading a little about magnetic monopoles, with gauge group $SU(2)$. In the book "The Geometry and Dynamics of Magnetic Monopoles", Atiyah and Hitchin consider a pair $(A,\phi)$ where $A$ is a ...
0
votes
2answers
126 views
E-valued differential forms as sections of the tensor product bundle of E with differential forms
Let $E$ be a vector bundle over $M$. Wikipedia says an $E$-valued differential form over $M$ can be defined as a bundle morphism $TM\otimes...\otimes TM \rightarrow E$ that is skew-symmetric.
I am ...
1
vote
0answers
33 views
Self dual solution of Yang-Mills equation on the Nahm case
I'm reading this notes, and I have a question about the construction on page 48.
Let $M=S^{1}\times \mathbb{R}^{3}$ and $E\rightarrow M$ a vector bundle over $M$. If $A$ is the $1$-form matrix ...
4
votes
1answer
75 views
Adjoint of the Coupled Covariant Derivative on Spinors
I want to understand the proof for a $C^0$ bound of solutions to the Seiberg-Witten equations.
Among other places, it can be found in Kronheimer, Mrowka: "The genus of embedded surfaces in the ...
1
vote
2answers
76 views
Going from one notation to another in Yang-Mills
In many books on Yang-Mills theories, written from a physicist's point of view, the curvature tensor is written as:
$$F_{\mu \nu}^a = \partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^bA_\nu^c$$...
9
votes
0answers
182 views
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 ...
7
votes
1answer
284 views
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 ...
5
votes
1answer
217 views
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 ...
2
votes
0answers
266 views
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 ...
5
votes
1answer
228 views
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 ...
0
votes
1answer
41 views
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 ...