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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}\\ \...
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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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 ...
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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 $...
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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 ...
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$S^3$ 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. It’s well known that $S^3$ can be ...
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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-...
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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]...
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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 ...
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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 $E$. ...
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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(\...
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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 ...
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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 ...
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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 ...
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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.)...
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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 ...
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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.
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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 $S_+$ ...
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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+...
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Analogue of Temperley-Lieb algebra of higher rank

In Piunikhin’s 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 ...
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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?
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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 ...
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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,\...
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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{...
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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. ...
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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,...
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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 = ...
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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{\...
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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 ...
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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 $X$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...