Connections on principal bundles, gauge groups, Yang-Mills connections, (anti-)self-dual connections, stability of vector bundles, Donaldson invariants, the Seiberg-Witten equations and invariants, the Bogomolnyi (monopole) equation, Chern-Simons invariant, Donaldson-Thomas theory, relations to ...

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Effect of gauge transformation on connection 1-form of a principal connection

Let $(P,\pi,M,G)$ be a principal fibration, $A$ a principal connection on $P$ (i.e. $\forall p \in P, T_pP = A_p \oplus V_p$), $\omega$ the connection 1-form of $A$, $f$ a gauge transformation of $P$, ...
2
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0answers
35 views

Basic question: Curvature transforms under Complexified Gauge Transformation

Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ is a self adjoint complexified gauge ...
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1answer
52 views

Quick question: Extension of vector bundles on a compact Riemann surface

Given the following short exact sequence of holomorphic vector bundles on a compact Riemann surface: $0\rightarrow M\rightarrow E \rightarrow N\rightarrow 0$ Fix a hermitian metric on $E$ and $n=...
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0answers
36 views

Invariant connection form

Let $\pi : E \to \mathbb{R}^3$ define a vector bundle with a connection form $\nabla$ on $\mathbb{R}^3$. The text that i'm am reading then goes on to say that $\nabla$ is $\mathbb{R}$-invariant in ...
3
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2answers
57 views

Elementary question: local computation of curvature on principal bundle

Let $G$ be a Lie group and $S=[0,1]^2$. Let $\omega$ be a connection $1$-form on the trivial principal $G-$bundle $P=S\times G$ over $S$. Let $(x_1,x_2)$ be coordinates on the base $S$. We can choose ...
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0answers
44 views

Quick question: Curvature form of a connection on the trivial bundle

Let $L=\mathbb{R}^2\times U(1)$ be the trivial $U(1)$-bundle over $\mathbb{R}^2$. Define a connection $\nabla=d+A$ where $A=fdx+gdy$ is an $\mathbb{R}$ valued $1$-form on $L$. That is, $\nabla$ gives ...
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10 views

local sections transformation formula

I am trying to prove a formula which states the relations between two local sections in a principal bundle: Let $P(M,G)$ be a principal bundle let $\{ U_\alpha \}_{\alpha \in I}$ be an open cover for ...
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1answer
28 views

Apply connections to a gauge transformation?

I'm reading Donaldson's book, Floer homology groups in Yang-Mills theory. On page 82, he considers a trivial bundle $P$ over a $4$-manifold $X$ with tubular ends which is equipped with a connection $...
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14 views

Local gauge transformation law on a principal bundle

I am referring to the answer by Henry to a related old question. Since it has been a long time I post it up as a new question instead of appending to the old one as a comment. The local gauge ...
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1answer
24 views

Choice of a horizontal tangent space of a principal bundle

Let $\pi:P\to M$ be a principal bundle with group $G=\pi^{-1}(p)$, and let $u\in P$ and $p=\pi(u)$. As I understand it, the choice of the vertical tangent space $V_uP=\mathrm{ker}(\pi_*)$ is natural, ...
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40 views

A question about Levi-Civita connection and curvature over 3 manifold

Give a 3-manifold M and Riemannian metric $g$, denote $A$ as the Levi-Civita connection on 3-manifold M corresponds to the metric $g$. Denote the curvature of $A$ as $F_A$, choose three bases ${e_1,...
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1answer
70 views

The Moduli Space of Gauge Theory

Physicists often speak of "Connections modulo gauge transformations" as the natural configuration space of a gauge field. In this sense, the fundamental object of study in gauge theory is the space of ...
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19 views

Number of SU(2) that can be embedded in SU(n)

Consider the Lie algebra su(3). Its generators $\lambda_i$ span 3-1 = 2 different Cartan subalgebra, which can be used to form two sets of ladder operators for each generator $H_i$ of the Cartan ...
6
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1answer
83 views

Poincaré duality for currents and non-closed forms

In page 8 of Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology by Szabo, the author claims that Poincaré duality holds for non-closed forms as long as the other form (...
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1answer
27 views

Are there multiple non-isomorphic principal $G$-bundles on Euclidean space? [duplicate]

I'm pretty sure the answer is out there, see this MathOverflow question, but that is unfortunately way over my head :). I'm interested in the case that $G$ is a Lie group (e.g. $U(1)$), but I don't ...
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51 views

What is the mathematical understanding behind what physicists call a gauge fixing?

I'm learning fiber bundle from my poor physicist point of view. I understand that a gauge transformation (physicist language) corresponds to the transformation of the connections built from an ...
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2answers
136 views

How do I derive this formula from gauge theory?

This is Exercise 3.4.14 in R. W. Sharpe's Differential Geometry. Suppose $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and $H$ is a Lie subgroup of $G$. Let $\theta$ be a $\mathfrak{g}$-...
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47 views

Can I split this integral to a sum over three contours?

I have the following integral $$ Z = \frac{1}{2\pi i} \int dx \, \frac{1}{(x-a_1)(x-a_2)(x-a_3)}\times \frac{1}{(x+\epsilon - a_1)(x + \epsilon - a_2)(x+ \epsilon - a_3)} $$ and this integral has ...
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2answers
60 views

Gauge transformations for line bundle where the manifold is simply connected.

Im trying to understand the significance of the manifold being simply connected for the following (or any really) case to do with basic yang mills theory. We are considering a U(1) line bundle, L, ...
3
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0answers
55 views

Sobolev Multiplication theorem for Fibre bundles

Let $X$ be a compact, oriented, four dimensional Riemannian manifold and $Q\longrightarrow X$ be a principal $G$-bundle over $X$ for a smooth, compact Lie group $G$. Let $M$ be a manifold admitting a ...
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43 views

Proof of $d^\ast A =0$ where $D=d+A$ is Yang Mill connection

Recall $$ F =( dA_{ij} + A_{il}\wedge A_{lj} )\mu_i \otimes \mu_j^\ast $$ Hence if rank of $E$ is $2$, then $$ F= dA $$ since $A$ is skewsymmetric. If $D$ is Yang Mill connection then $ D^\ast F=0$. ...
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97 views

Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot $ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
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1answer
16 views

Difficulty proving gauge invariance on an SU(N)-valued potential

Say we have a four-dimensional spherically symmetric $\mathfrak{su}(N)$ gauge potential in standard Schwarzschild co-ordinates which can be written \begin{equation} \mathcal{A}=Adt+Bdr+\frac{1}{2}(C-...
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2answers
66 views

About gauge transformation

If $E$ is a vector bundle with a bundle metric, so we have ${\rm Aut}\ (E)$ whose fiber at $x\in M$ is the group of orthogonal transformation in $E_x$. Then gauge transformation is a section of ${\rm ...
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1answer
51 views

Curvature of $K$-invariant connection (principal bundles)

Here is a proposition from Kobayashi & Nomizu's Foundations of Differential Geometry. I don't understand how they obtain the final line of the proof. They write: \begin{align} 2\Omega(\tilde{...
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59 views

Rigorously what is this integral?

I've been studying some gauge theories approach to problems in mechanics in order to get a better understanding of the ideas from gauge theories and to see some applications of fibre bundle theory. ...
0
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1answer
13 views

Compute variation left action subgroup

I consider a Lie group $G$, with a group element $g$ parametrised in some manner with parameter $\theta_i$, $i=1,\cdots, \dim G$. Suppose that $K\subset G$. I want to compute the variation of an group ...
1
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1answer
33 views

'Large' closed subgroup

I am working through a paper in the field of differential geometry (Yang-Mills theory) and the author writes: 'We assume the Riemannian manifold $(M,h)$ admits a large closed subgroup $K$ of the ...
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2answers
208 views

Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can take $...
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53 views

Bianchi Identity - Gauge Theory

I am reading through some lecture notes (found here) and following a proof of the Bianchi identity in the context of principal bundles. That is, $h^*\Omega = 0$, where $\Omega$ is the curvature 2-...
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1answer
58 views

Gauge transformation on a principal bundle

I am reading through lecture notes found here and on pg 11 they define a map $\overline{\phi}_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow G$ by $\overline{\phi}_{\alpha}(p)=g_{\alpha}(\Phi(p))g_{\alpha}(...
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1answer
33 views

Adjoint representation for matrix groups (Gauge theory)

This is a question in regards to an identity in Gauge theory. Let $\omega$ be the connection one form on a principal bundle $\pi:P\rightarrow M$ and let $A_{\alpha}:=s_{\alpha}^*\omega$ be the gauge ...
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31 views

Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
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1answer
51 views

Gauge theory on a trivial bundle

I am learning gauge theory, so I tried to understand what happens in the case of a trivial principal bundle. However I have some problems understanding how a connection looks like in that case. Here's ...
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1answer
277 views

Intuition for Exotic $\mathbb R^4$'s

Today one of my professors told me that $\mathbb R^4$ admits uncountably many non-diffeomorphic differential structures. When I asked him whether there's an intuitive reason to expect a result like ...
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43 views

Contraction of second exterior covariant derivative with metric

Let $G \hookrightarrow P \to M$ be a principal $G$ bundle, $P \times_\rho V$ be a vector bundle associated to representation $\rho$ of $G$ on $V$. If $\omega$ is a connection $1$-form on $P$ then we ...
2
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89 views

How to calculate the Maurer-Cartan form in the adjoint representation?

While I am reading a paper, I come across a difficulty. Here, we have a Lie group and we know its Lie algebra defined as $[G_a,G_b]=f_{ab}^{\phantom{ab}c}G_c$ with $G_a\in\mathfrak g$. Under the ...
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0answers
88 views

Lecture notes on holomorphic Yang-Mills theory

Some time ago I've found these lecture notes on the gauge theory. In particular, in these lecture notes the author introduces and studies the Yang-Mills equations in the case of real bundles and ...
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96 views

Gauge covariant derivative on principal bundle over $\mathbb R^d$

I try to understand the physical definition of covariant derivative in gauge theories in terms of the exterior covariant derivative of vector-valued forms defined as the horizontal projection wrt a ...
11
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2answers
318 views

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial x^i}F_{ij}+[A_i,F_{ij}]=0 $$ from ...
4
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1answer
400 views

Principle G bundles v.s. Flat G connection

What is the difference between Principle G bundles v.s. Flat G connection? I heard that for a discrete group $G$ (in physics, or a finite group $G$ in math), the principle G bundles is the same as ...
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0answers
40 views

Differentiable structure on the gauge group?

In this paper I have come across a formulation involving differentiation in the gauge group of a principal bundle which I do not understand (found at the very top of p. 369). Let $P\rightarrow M$ be ...
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0answers
43 views

Equivalence between pullback connections of smoothly homotopic maps

Let $f,g:M\rightarrow N$ be smooth maps between smooth manifolds such that there exist a smooth homotopy $H:M\times [0,1]\rightarrow N$ between them. If we have a principal bundle $P\rightarrow N$, we ...
3
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1answer
152 views

Connections in non-Riemannian geometry

In case of Riemannian geometry the connection $\Gamma^i_{jk}$ as is derived from the derivatives of the metric tensor $g_{ij}$ is ought to be symmetric wrt to its lower two indices. But in the case of ...
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97 views

The magnetic monopole and the Hopf bundle

Consider the vector field $\textbf{B} = \frac{1}{\rho^2}\textbf{e}_\rho$ on $\mathbb{R}^3 - \{0\}$ where $(\rho, \theta, \phi)$ are the usual spherical coordinates and $\textbf{e}_\rho = \frac{\...
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29 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T [0,\beta]...
4
votes
1answer
3k views

Trace of tensor product vs Tensor contraction

I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality: $$ \text{Tr}(T_1\otimes T_2)=\text{Tr}(T_1)\text{Tr}(T_2)\;\...
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120 views

The trace of a wedge product of matrices

I'm trying understand a computation on Besse's book (p. 371). I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form relative to the direct sum ...
4
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65 views

Natural Action of Killing vector fields on space of connections

I try to understand some mathematical aspects of supersymmetric Yang Mills theory following the book "Quantum fields and strings - a course for mathematicians". In this context the following question ...
3
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2answers
404 views

Principal bundle automorphism generating global gauge transformations

Consider a principal $G$-bundle $P$ with connection form $\omega$. An automorphism $f$ of $P$ is by definition a (smooth) $G$-equivariant map: $f(p \cdot g) =f(p) \cdot g$ for all $p\in P$ and $g\in G$...