The gauge-theory tag has no wiki summary.

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### Gauge-theoretic formulation of Maxwell equations [duplicate]

Does any one know how to write the Maxwell equations as an equation on a principal $U(1)$-bundle?
In Freed & Uhlenbeck's Instantons and Four manifolds, the authors claim that the Maxwell ...

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126 views

### Differentiable structure on the Gauge group of a principal bundle?

I am currently reading this paper in which we have a map $g:I\rightarrow Gauge(P)$ for some principal bundle $P$ which is differentiated. I am looking for a reference or explanation what the "most ...

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290 views

### Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?

So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle ...

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226 views

### Existence of diagonalizing coordinates for the metric tensor

Solving for metrics that are Einstein, i.e that satisfy $R_{\mu \nu} = \Lambda g_{\mu \nu}$ is highly non-trivial as soon as $g_{\alpha \beta}$ is allowed to have off-diagonal components. However, ...

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**1**answer

191 views

### What is the meaning of Yang-Mills action evaluated on Levi-Civita connection?

On a Riemannian manifold $M$ with riemann curvature tensor $R_{\mu\nu\rho\sigma}$ written as (endomorphism valued) curvature two-tensor of the Levi-Civita connection $R=R_{\mu\nu}dx^\mu\wedge ...

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555 views

### How can gauge theory techniques be useful to study when topological manifolds can be triangulated?

I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be ...

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303 views

### Large vs Small Gauge transformations and Physical theories

I can't decipher the difference between large and small gauge transformations especially in its applications in physics.If perhaps one can engineer a simple physical theory that has such a ...

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272 views

### Curvature as infinitesimal holonomy

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy ...

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### Vector bundle connection over complex manifold vs. over underlying real manifold

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$ of rank $r$. Denote by $(X^{\mathbb{R}},g^{\mathbb{R}})$ the underlying Riemannian manifold of $(X,g)$.
...

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290 views

### On Corlette's paper Flat G-bundles with canonical metrics

I am now reading Kevin Corlette's paper:
Flat G-bundles with canonical metrics, JDG, 1988
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214442469
I ...

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257 views

### What is the definition of picture changing operation?

What is the definition of picture changing operation?
What is a standard reference where it is defined - not just used?

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291 views

### A gentle introduction to CFT [closed]

1) Which is the definition of a conformal field theory?
2) Which are the physical prerequisites one would need to start studying conformal field theories?
(i.e Does one need to know supersymmetry? ...

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124 views

### 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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162 views

### Does fixing the reparameterization invariance of the string action correspond to some kind of orbifolding?

Does fixing the reparameterization invariance of the string action, for example by choosing the light-cone gauge
$$
X^{+} = \beta\alpha' p^{+}\tau
$$
$$
p^{+} = \frac{2\pi}{\beta} P^{\tau +}
$$
...

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**1**answer

402 views

### Why does closed string theory have only one dilaton field instead of $22$? [closed]

Looking at $5D$ Kaluza-Klein theory, the Kaluza-Klein metric is given by
$$
g_{mn} = \left(
\begin{array}{cc}
g_{\mu\nu} & g_{\mu 5} \\
g_{5\nu} & g_{55} \\
\end{array}
\right)
$$
...

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155 views

### Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...

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256 views

### Representation variety vs. space of flat connections

The holonomy provides a bijection from
the space of flat G-connections (modulo gauge equivalence) on a trivial G-bundle over M
to
a connected component of the representation variety ...

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592 views

### Representing SU(3) with 3 ropes in 3 dimensions

The short question is: how exactly is SU(3) realized with ropes?
The long question: There is this idea that deformations of a configuration of three infinitely long, flexible ropes that cross each ...

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295 views

### Non simply connected HyperKähler 4-manifolds without ALE metrics

In a 1989 paper Peter Kronheimer showed that each simply connected HyperKÃ¤hler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?

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384 views

### How many principal bundles are there over a given base?

I'm currently considering principal bundles and classifying spaces in the context of gauge theory and a crucial question came to my mind:
Is there a way to say how many (isomorphism classes of) ...

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167 views

### Conjecture on homotopy groups of moduli space of self-dual connections

In the paper Stability in Yang-Mills Theories (1983), Taubes puts a (topological) bound on the Hessian of the YM-action on $S^4$. He consequently conjectured:
"The inclusion ...

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567 views

### The classifying space of a gauge group

Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by
$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...

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306 views

### On Dimension of Instanton Moduli Space

I am reading Charles Nash's book on differential topology and QFT. In particular, I have question on the part calculating dimension of instanton moduli space. The question split into conceptual part ...

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993 views

### How to understand Chern-Simons action

Hi all. The question I have should be a rather simple one, but I just can't think it through.
So the Chern-Simons action reads
\begin{equation}
S = \int_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A ...

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195 views

### Torsion-free $G$-Structures

I have the following question. Let $G \subset SO(n)$ be a Lie Group and $M$ be a smooth manifold of dimension $n$. Furthermore let $P$ be a $G$-structure on $M$ i.e. $P$ is a principal subbundle of ...

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118 views

### Transversality for Chern-Simons functional on a rational homology sphere

When Floer defined instanton Floer homology for an integer homology sphere. He chose a finite set of loops in the manifold and consider the holonomy along these loops as a perturbation on Chern-Simons ...

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315 views

### Qustions on R.Bryant's papaer “Calibrated embeddings in the special Lagrangian and coassociative cases”

I am reading the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by R.Bryant (here the link: http://arxiv.org/abs/math/9912246) and there are certain things that are ...

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### Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds?

My question begins with a caveat: I sometimes spend time with topologists, but do not consider myself to be one. In particular, my apologies for any errors in what I say below â€” corrections are ...

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### Wedge Product of Lie Algebra Valued One-Form

I've been reading about the formal structure of gauge theories and am a little confused by the notation. Could someone clarify this for me?
Suppose that $A$ is a Lie algebra valued 1-form ...

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### Isometric embedding of a compact Lie Group in $M(n,\mathbb{C})$

Greetings,
Let $G$ be a compact Lie group with a bi-invariant inner product $h$ on it. Can one embedd $G$ in $M(n,\mathbb{C})$ isometrically for some $n \in \mathbb{N}$. By isometrically I mean that ...

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### Kähler form on complex Lie group

Hallo,
Let $G$ be a semi-simple, compact Lie Group. Consider its complexification $G_{\mathbb{C}}$. Does there exist a KÃ¤hler structure on $G_{\mathbb{C}}$ which is $G$-invariant (maybe in a ...

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477 views

### Relation of SW and Donaldson Invariant

My question is:
I am request for the reference that Is there any relationship between the Seiberg-Witten Invariant and Donaldson's Invariant? Or the relationship between Seiberg-Witten Moduli Space ...

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### Seiberg-Witten curve for product SU(2)^N gauge theories

In equation 2.10 of this article, the author gives the Seiberg-Witten curve for a $U(N)$ gauge theory with $L<2N$ massive flavours with masses given by $m_i$ as:
$y^{2}=\left\langle ...

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718 views

### Floer homology and Invariants for Einstein Field Equations?

Motivation: There have been the instanton (anti-self dual connection) solutions to the Yang-Mills equation $d_A^\ast F_A=0$ which extremize the YM energy $\int_M|F_A|^2$, leading to the Donaldson ...

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441 views

### The number of simply connected 4-dimension manifold

For a simply connected four-dimension manifold, we know the Freedmen's work.
My question is: For every integer N, Is the number of simply connected 4-manifolds which the second betti number is ...

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### What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...

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682 views

### Symplectic structures from Lagrangians?

In Witten's paper 'Quantization of Chern-Simons Gauge Theory with Complex Gauge Group,' he makes the statement (p. 35) that the symplectic form on the moduli space of flat $G_\mathbb{C}$ connections ...

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292 views

### How to compute the Monopole Floer Homology for Surface $\times S^1$ ?

We know that Monopole Floer homology of a 3-manifold $M$ depends on a spin-c structure. My question is that if $M$ is $F\times S^1$ ($F$ is a surface of genus larger than 1) then how can we compute ...

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1k views

### Yang-Mills and Chern-Simons functionals as Morse functions

Can the Yang-Mills or Chern-Simons action functionals be considered as [possibly perfect] Morse functions? I assume we would be in an equivariant scenario due to considering the configuration spaces ...

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### Variation of the Chern connection according to the variation of hermitian metric

Whats is the relation between the Chern connections of tow Hermitian metrics in a holomorphic vector bundle?

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320 views

### Reference request: Seminal papers in gauge-theoretic mathematics [closed]

Following on from previous question I was also searching for seminal papers in gauge theory.
Would be greatly appreciative of references to such.

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488 views

### Reference request: Introductions to current mathematics derived from / related to gauge theories

I was searching for introductions to current mathematics related to gauge theories.
Can someone suggest some good references?
E.g.
Topics in Physical Mathematics by K. Marathe

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373 views

### Has anyone seen this Hitchin-like system?

Let $(M,g)$ be a riemannian manifold and let $P\to M$ be a principal $G$-bundle with connection $A$. Let $\alpha \in \Omega^1(M;\mathrm{ad}P)$ be a one-form on $M$ with values in the adjoint bundle ...

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### Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of a vector bundle over P?

I am an analyst struggling through some geometry used in physics.
Some background: For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection ...

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### K.Uhlenbeck's preprint “A priori estimates for Yang-Mills fields”

Does anyone have a copy of the unpublished preprint of Karen Uhlenbeck A priori estimates for Yang-Mills fields from around 1986?
It appears to have circulated for some time, and it is quoted in ...

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751 views

### Why must a reducible flat SU(2)-connection over a homology sphere be trivial?

Let $M$ be a homology sphere. Suppose $P=M\times SU(2)$ is the trivial $SU(2)$ principal bundle. Let $R$ be all reducible connections on $P$. Here $A$ in $R$ is reducible if the gauge transformation ...

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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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### What is the precise statement of the correspondence between stable Higgs bundles on a Riemann surface, solutions to Hitchin's self-duality equations on the Riemann surface, and representations of the fundamental group of the Riemann surface?

I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and ...

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### Is symplectic reduction interesting from a physical point of view?

Do you think that symplectic reduction (Marsden Weinstein reduction) is interesting from a physical point of view? If so, why? Does it give you some new physical insights?
There are some possible ...

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319 views

### module of sections of the horizontal bundle

Some times ago I posted this question here. There I carelessly assumed that if you have a set of sections of a vector bundle which span every fiber pointwise, they also generate the module of smooth ...