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Questions tagged [gauge-theory]

For questions about gauge theory in mathematical/theoretical physics, which a field theory whose fields include principal bundles with connection.

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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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24 views

Navier-Stokes smoothness problem and Gauge Theory

Recently, I came across this paper where the author describes an analogy between electrodynamics and fluid dynamics. He develops a one-to-one correspondence between the equations of electrodynamics ...
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24 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 ...
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2answers
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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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78 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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1answer
109 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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55 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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1answer
79 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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1answer
21 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 ...
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19 views

Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $-2\pi i \Omega$...
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Number of automorphisms of some principal $G$-bundles.

Let $M$ be a manifold and let $G$ be a finite group. Let $P \to M$ be a principal $G$-bundle. I have some basic questions regarding the automorphism group of $P$ as a principal $G$-bundle over $M.$. ...
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Use of GIT for moduli problems

Was there an actual use of GIT technics (moment map, Kempf-Ness theorem) in the different proofs of the Kobayashi-Hitchin correspondence (by Uhlenbeck-Yau and Donaldson)? Or this was rather a ...
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1answer
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Isotropy group of connection is isomorphic to centraliser of holonomy group

I am asking for a proof of Lemma (4.2.8) of Donaldson, Kronheimer: The Geometry of Four-Manifolds. Let $P \rightarrow X$ be a principal bundle with structure group $G$. Denote by $\mathcal{G}$ the ...
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51 views

Can I Find ALL Connection 1-Forms from the Gauge Group?

I would like to know if its possible that given a principal fiber bundle $(P, \pi, M, G)$, the gauge group $GA(P)$ of all gauge transformations $f:P\to P$, and a single connection $1$-form $\omega_0 \...
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1answer
34 views

The degrees of freedom in 3+1 General Relativity

I would like to understand what are the degrees of freedom in GR. I have read a few previous posts already, but none of them really help me. Below, I will try to write down the entangled web of ...
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1answer
49 views

references for physical gauge theory and spinors.

Does anyone know of any good references for physical examples of gauge theory (as a mathematically precise theory of connection on principal bundles). Simple examples will do (e.g the $U(1)$ ...
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Deformation theory of holomorphic vector bundles in Donaldson-Kronheimer

There is the Proposition 6.4.3 in Donaldson-Kronheimer as follows: Proposition (6.4.3) (i) There is a holomorphic map $\psi$ from a neighborhood of $0$ in $H^1(\operatorname{End} \mathscr{E})...
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1answer
46 views

Symmetric tensor of Lie algebra of $su(N)$

I am interested in knowing the exact form of the anti-commutation of two generators of $su(N)$ lie algebra. Let us denote $T^a$ to be the generator of $su(N)$ lie algebra in the defining ...
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1answer
118 views

Symmetric tensor of Lie algebra of $su(N)$

I am interested in knowing the exact form of the anti-commutation of two generators of $su(N)$ lie algebra. Let us denote $T^a$ to be the generator of $su(N)$ lie algebra in the defining ...
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40 views

A new gauge field [Cech 1-cocycle] acts a symmetry by multiplying itself to transition functions

For a gauge theory of a gauge group $G$ (better here: a simply-connected and compact $G$; but if you wish, you can comment more general cases), we view the spacetime as the base manifold and the ...
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Why is the interpolation between two connections related via a gauge transformation still a connection?

I am studying the theory of anomalies in gauge field. Let $A$ be a gauge field (or a connection for mathematicians). Let $A_{U}$ be an equivalent gauge related via a local gauge transformation $$A_{...
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Is the comparator $U(y,x)$ in Gauge Theory the same as a holonomy?

For Gauge theories you have a comparator that transforms as $$U(y,x) = e^{i\alpha(y)}U(y,x) e^{-i\alpha(x)}$$ Is this the same thing as the holonomy?
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364 views

What rigorous mathematical theorems has Edward Witten discovered?

I read that Ed Witten's 1990 Fields Medal was somewhat controversial among mathematicians, because even though no one questioned his deep conceptual understanding of important new mathematical ideas, ...
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Embedding of gauge map

Let $X$ be a compact $n$-manifold, and $p\geq n/2$. Q How to show that there is a real polynomial $P_{m,p}(x)$ of degree $m+1$ with trivial zero-degree term, such that for any $u:X\to S^1$, we have $...
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57 views

Effect of gauge transformations on connections. Passage in the proof

Given a gauge transformation $u$ for a vector bundle $E\to M$, and a connection $\nabla$, we define: $$u^*\nabla = u\nabla(u^{-1}s)$$ I want to prove the well-known identity $$u^*\nabla = \nabla -(d_{...
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1answer
144 views

Elliptic Bootstrapping for Gauge Transformations

I'm reading John Morgan's book "Seiberg-Witten equations and applications to the topology of smooth manifolds". I'm stuck in the proof of Lemma 4.5.3, which says the following. Suppose $(a_n,\psi_n)$ ...
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45 views

Doubt in proof of Uhlenbeck's theorem in Donaldson & Kronheimer

Theorem 2.3.8 in 'The Geometry of Four-Manifolds' by Donaldson and Kronheimer reads: There is a constant $\epsilon>0$ such that if $A$ is a ASD (anti self dual) connection on the trivial bundle ...
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63 views

Is the determinant of the holonomy gauge invariant / significant?

I am currently reading the book Baez & Muniain - "Gauge Fields, Knots, and Gravity". Chapter 2 of part II defines the holonomy on a bundle $E\to M$ with a gauge group $G$ and connection $D$. They ...
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1answer
108 views

Vector-valued forms inside the first jet bundle

On page 433 of "Self-duality in four-dimensional Riemannian geometry" by Atiyah, Hitchin and Singer, it is written that $p^*(E \otimes \Lambda^1) \subset p^*J_1(E)$, where $\Lambda^1 \to X$ is the ...
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Dimension of the space of harmonic $\mathrm{Ad}P$-valued differential 1-forms on a surface

Let $P\to \Sigma$ be a $\mathrm{SU}(2)$-principal bundle over a smooth oriented closed genus $g\geq 2$ real Riemannian surface $\Sigma$. Let $\mathcal{A}$ be the space of connexions over $P$. Let $\...
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88 views

Connections on Bundles with Trivial Determinant

I want to let $E$ be a rank $n$ complex vector bundle on a smooth manifold $X$. A connection $\nabla$ gives us a way of differentiating local or global sections of $E$. We define it to be a $\mathbb{...
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Dimension of the Marsden-Weinstein reduction of a coadjoint orbit in the dual of the Lie algebra of the gauge group (Atiyah-Bott context)

Let $P\to \Sigma$ be a $\mathrm{SU}(2)$-principal bundle over a smooth orientable closed genus $g$ real surface $\Sigma$. Let $\mathcal{A}$ be the space of connexions over $P$ and let $\mathcal{G}$ be ...
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280 views

Yang–Mills theory and mass gap

I am interested in widening my knowledge into the formal aspects of Yang–Mills theory. In particular, I would like to study the current mathematical and physical research literature about this ...
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1answer
177 views

Different definitions of irreducible $\mathrm{SU}(2)$ connections

Let $Y$ be Poincaré's integral homology 3-sphere. Let $\pi:P\to Y$ be a (necessarily trivial) $\mathrm{SU}(2)$-principal bundle over $Y$. Fix $x_0\in Y$ and $p_0\in P_x:=\pi^{-1}(x_0)$. The ...
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174 views

Holonomy group is closed or not?

Let $P\to M$ be a $\mathrm{SU}(2)$-principal bundle over a closed connected manifold $M$. Let $A$ be a flat connection on $P$, fix $x\in M$ and let $H:=\mathrm{Hol}_{x,A}(\pi_1(M,x))<\mathrm{Aut}(...
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Holonomy group and irreducible $\mathrm{SU}(2)$-connections

Let $P\to M$ be a $\mathrm{SU}(2)$-principal bundle over a closed connected manifold $M$ and let $A$ be a flat connexion form on $P$. Fixing $x\in M$ we have a homomorphism $$ \mathrm{Hol}_x(A) : \...
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Smooth Principal Bundle from continuous transition functions?

Setting: Let $M$ be a smooth manifold and $\{U_\alpha\}_{\alpha \in \mathcal{I}}$ a locally finite covering of open subsets. Furthermore let $G$ be a smooth Lie group. Now assume we are given a family ...
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Structures on Vector Bundles with Reduced Structure Group

I am interested in considering vector bundles $E$ of rank $n$ over a base $X$ with fiber $V \cong \mathbb{C}^{n}$ such that the structure group $G \subseteq \text{GL}_{n}(\mathbb{C})$ is possibly a ...
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55 views

Yang-Mills energy gap in dimension 2

Let $(\Sigma,g)$ be a smooth genus $k$ closed oriented surface endowed with a Riemannian metric $g$. Let $P_\Sigma$ be a (necessarily) trivial $\mathrm{SU}(2)$-principal bundle over $\Sigma$. Let $\...
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Codimension of infinite space

Let $B$ be a finite dimensional manifold, $H$ be a complete Hilbert space and $V$ a finite dimensional subspace of $H$. We set $\Gamma:=\Gamma(B;H)$, i.e. the smooth maps $B\to H$ and $W:=\{s\in \...
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1answer
168 views

Gauge Curvature by example of the curvature of the graphs of $y=f(x), z=F(x,y)$ and $w=H(x,y,z)$.

I am trying to understand Guage Curvature - or perhaps, more precisely, the Gaussian curvature of a Riemannian manifold in gauge coordinates. The best way I can think is by example: finding the ...
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1answer
59 views

Connections and Local Trivialisation

Suppose we have a principal $G$-bundle $\pi:P\rightarrow M$ with a connection $\omega:TP\rightarrow \mathfrak{g}$, where $\mathfrak{g}$ is the lie algebra of $G$. Now suppose I have an open set $U$ in ...
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Gauge transformation as Lie Group [closed]

How can I prove that the family of Gauge transformations $$ A_\mu \mapsto A'_\mu = A_\mu +\partial_\mu \phi$$ where $\phi (x)$ is an arbitrary scalar function, is a Lie Group?
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1answer
77 views

Chern-Simons form on a 5d manifold

In a physics paper entitled "Five-Dimensional Supersymmetric Gauge Theories and Degenerations of Calabi-Yau Spaces" by Intriligator, Morrison and Seiberg, the authors write an equation $$2\pi \frac{...
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Gauge fixing function in the quantization of gauge field theory

A gauge fixing function in the process of quantization of a gauge field is the function $f: M \rightarrow \mathfrak{g}$ , where $M$ is the space of fields and $\mathfrak{g}$ is the Lie algebra ...
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1answer
197 views

Homotopy groups of integer homology 3-spheres

Let $Y^3$ be a integer homology 3-sphere, i.e. $H_k(Y;\mathbb{Z}) = H_k(S^3;\mathbb{Z}), \forall k$, i.e. $$ H_0(Y;\mathbb{Z}) = H_3(Y;\mathbb{Z}) = \mathbb{Z} \quad \text{and} \quad H_1(Y;\mathbb{Z}) ...
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Abelian instantons for generic metric

Let $(X,g)$ be a Riemannian 4-manifold and let $G \to X$ be a principal $G$ bundle. The ASD equations $F^+=-F$ are satisfied for $F=dA$ the curvature two-form of the $G$-equivariant connection $A$. ...
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193 views

Clifford action of exterior derivative of self-dual 2-form

I am confused about a claim on page 30 of the lectures notes of Hutchings and Taubes on Seiberg-Witten theory (https://math.berkeley.edu/~hutching/pub/tn.pdf). Background: Let $M$ is a Riemannian 4-...
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1answer
189 views

Trivial connections on trivial bundles

In Cliff Taubes' book "Differential geometry", he explains how to define the trivial connection on the product principal bundle $U \times G,$ where $G$ is a Lie group. Namely, the trivial connection $...