Questions tagged [gauge-theory]
Gauge theory in physics and mathematics refers to a field theory whose fields include principal bundles with connection.
78 questions with no upvoted or accepted answers
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Vortex equation on Riemann surface and a similar equation
Let's take a Riemann surface $(X,\omega)$ and a holomorphic line bundle $L$ on it with a hermitian metric $h$ on $L$. $g$ be a real valued smooth function on $X$ and we consider the following two sets ...
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Chern-Simons Functional for $S^1$-bundles over Riemann Surfaces
If $M$ is a closed, orientable 3-manifold $M$ and also happens to be an $S^1$ bundle over some Riemann surface $\Sigma$, then it seems like a natural Heegaard splitting of $M$ could come from ...
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Maxwell $U(1)$ gauge theory's electric and magnetic sources turned on simultaneously in the classical differential geometry
Question:
How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(...
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Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates
In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented.
Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
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Poincaré connection encode torsion and curvature
I'm trying to understand something that is written in Baez & Wise paper "Teleparallel Gravity as a Higher Gauge Theory". In section 4, they discuss Poincaré connection and, first of all, split ...
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Reference Request: A "Chevalley-Eilenberg"-style formulation of the $L_\infty$ algebra minimal model theorem?
The nicest definition of $L_\infty$-algebras ---which I will call a "Chevalley-Eilenberg" style definition after the obvious analogy with the Chevalley-Eilenberg differential of Lie algebras--- is the ...
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A Question about Hermitian Yang-Mills Equations
Let E be a holomorphic bundle over algebra surface X, let $H$ be a Hermitian metric of $E$, recall the Hermitian-Yang-mills equation is $\wedge F_H=\lambda.1$.
Let $H_t$ be Hermitian metrics over $E$ ...
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On nowhere zero self-dual 2-forms
Let $(X, g_x)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms over $X$. Then the Hodge-star decomposes $\Lambda^2$ into the space of self-dual ...
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Question in complex analysis arising from large $N$ gauge theory
This is a question in complex analysis that comes up in the treatment of large
$N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed ...
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"More stable" definitions of principal $G$-bundle
Let $G$ be a topological group. For any pointed topological space $X$, define $[X,G]$ to be the group whose underlying topological space is the space of pointed continuous maps from $X$ to $G$, with ...
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Question from Taubes' SW$\Rightarrow$ Gr
I am trying to understand Taubes' paper on SW$\Rightarrow$ Gr. I don't understand how either of the equations 2.16 or 2.17 appears, I would be happy to understand how the curvature term $F_a$ appears ...
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Spin(7)-instanton
Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the ...
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Does a gauge-invariant Caccioppoli inequality hold?
(I previously asked this question on Math.SE but got no responses after two weeks.)
Let $V \Subset U$ be domains in a Riemannian manifold $M$, and $W := U \setminus \overline V$. If $u: U \to \mathbb ...
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Is this a correct description of the BPS monopole of charge $1$?
I am reading the book "The Geometry and Dynamics of Magnetic Monopoles", by M.F. Atiyah and N.J. Hitchin, and I got to this part:
"... let $H$ be the Hopf line bundle over $S^2$ and ...
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Heat kernel coefficients for Laplacian in instanton background
The heat kernel coefficients $b_{2k}(x,y)$ of the covariant Laplacian in an $SU(2)$ instanton background (for simplicity let's say $q=1$ topological charge, so the 't Hooft solution) on $R^4$ is ...
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Instantons on the 4-sphere with respect to other Riemannian metrics
It is known that the moduli space of $\text{SU}(2)$ instantons of charge $1$ on $S^4$ is diffeomorphic to the five-ball $B^5$ if $S^4$ is endowed with the round metric.
Question: what does the moduli ...
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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}(...
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Uhlenbeck's compactness for abelian gauge
I am looking for a simpler proof (if possible) for the Uhlenbeck's compactness result (bound on connection up-to gauge from bound on curvature) on an open ball. I know that a proof exists using Hodge-...
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About irreducible connection
The irreducible connection is a connection whose holonomy group is just $G$ (let us just assume the base space $X$ is just connected). Otherwise, it is called reducible if the holonomy group $H_A$ ...
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Geometry of the complex Gauge group
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 $E$.
Is there a way to endow $\...
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Determine all possible magnetic monopole of gauge theories
In Wikipedia, it states about the magnetic monopole of the gauge theory is determined by the fact:
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It ...
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Fourier Lapalacian over periodic end
This is a technical question on Taubes' paper: Gauge theory over periodic end. on Page 378.
Recall that:
Let $Y$ be a closed manifold, with $b_1=1$, and $\tilde Y$ be the $\mathbb Z$-covering of $...
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Analog of Gauss-Bonnet formula for principal bundles over manifolds with boundary
The Gauss-Bonnet formula gives a topological invariant as an integral over a local density on the given manifold. In particular, when there is a boundary, GB formula has to be supplemented by a ...
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On the Hitchin fibration
I will refer to Simpson's "Higgs bundles and local systems".
Proposition 1.4:
When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
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Riemann curvature of $S^1$-principal bundle
Let $(M,g)$ be a Riemannian manifold and $\pi:P \to M$ be $S^1$- principal fiber bundle endowed with a connection $\Gamma$. For every $p\in P$ we have,
$$T_pP \simeq T_pV\oplus\Gamma_p$$
Where $V$ ...
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Obstruction to this gauge choice of the connection of a vector bundle
Let $M$ be a compact manifold with a nowhere-vanishing vector field $R$. Consider principal $G$-bundle $P$ over $M$, and $\mathcal{A}$ being the space of irreducible connections.
Let me denote a ...
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An interesting identity involving skew-Schur functions
Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9)
\begin{align*}
\prod_{k\geq1}(1+...
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Finding a unitary operator on L^{2}(\mathbb{R}^{2},dxdy)
I have a one parameter (r) family of self-adjoint representations of the universal enveloping algebra of some nilpotent Lie group on $L^{2}(\mathbb{R}^{2},dxdy)$ as follows:
$$\hat{X}^{r}=\hat{x}-i(r-...
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