The gauge-theory tag has no usage guidance.

**24**

votes

**5**answers

3k views

### 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 ...

**24**

votes

**4**answers

2k views

### 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 ...

**17**

votes

**2**answers

2k 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 ...

**14**

votes

**2**answers

500 views

### 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 — corrections are ...

**13**

votes

**1**answer

775 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 ...

**12**

votes

**3**answers

2k views

### The “miracle” of Heegard Floer.

Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins ...

**12**

votes

**3**answers

642 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 ...

**11**

votes

**5**answers

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 ...

**9**

votes

**3**answers

661 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) = ...

**9**

votes

**1**answer

987 views

### Seiberg-Witten theory on 4-manifolds with boundary

What generalizations of Seiberg-Witten theory to 4-manifolds with boundary do exist?
I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to ...

**9**

votes

**3**answers

1k views

### Looking for reference on gauge fields as connections.

Can anyone give me references where I would see a detailed exposition of how to translate gauge field theory as known to physicists into the language of connections. I am looking for a detailed ...

**8**

votes

**2**answers

451 views

### What is the BRST-anti-BRST formalism?

What is the BRST-anti-BRST formalism?
Is the Sp(2) doublet the ghost, antighost pair?
Introductory accounts of this subject seem to be hard to find. I would appreciate a reference for someone who ...

**8**

votes

**0**answers

164 views

### Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$).
Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...

**8**

votes

**0**answers

804 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 ...

**8**

votes

**0**answers

249 views

### 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 ...

**7**

votes

**5**answers

797 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 ...

**7**

votes

**1**answer

1k views

### 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 ...

**7**

votes

**2**answers

332 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?

**7**

votes

**1**answer

750 views

### Casson's invariant and the trivial connection contribution to witten's 3-manifold invariant

This question might turn out not to make any sense, but here it is: Witten's (and Reshetikhin and Turaev's) 3-manifold invariant can be "defined" as an integral over the space of connections on the ...

**7**

votes

**0**answers

562 views

### Central Yang-Mills connections, and flat connections with prescribed holonomy

Let $X$ be $\Sigma^g$ which is the Riemann surface of genus $g$, and consider a trivial $G$-bundle over it.
1) In this $2$-d setting, the space of Yang-Mills central connections is the set of ...

**6**

votes

**3**answers

2k views

### Gauge theory construction of moduli of vector bundles

Given a smooth projective complex variety $X$, instead of using Mumford's GIT to construct the moduli of rank $n$ topologically trivial vector bundles, we can also take the gauge theory approach.
To ...

**6**

votes

**2**answers

217 views

### Flat connections on 3-manifold with boundary

Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}_{flat}(\Sigma)$ and $\overline {\mathcal ...

**6**

votes

**1**answer

351 views

### 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 Kähler structure on $G_{\mathbb{C}}$ which is $G$-invariant (maybe in a ...

**6**

votes

**2**answers

845 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 ...

**6**

votes

**3**answers

575 views

### Literature for gauge field theory on the lattice in geometrical formulation

I have found an article by Huebschmann, Rudolph and Schmidt: http://www.springerlink.com/content/b8v216v0m8h16264/ about "A Gauge Model for Quantum Mechanics on a Stratified Space" and I am very ...

**6**

votes

**3**answers

2k views

### 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 ...

**6**

votes

**1**answer

192 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 ...

**6**

votes

**1**answer

343 views

### Factors of automorphy from Chern connection

This question is inspired by a recent question about holomorphic bundles and factors of automorphy. Suppose $X$ is a compact, complex manifold whose universal cover $\widetilde{X}$ is Stein (the ...

**6**

votes

**0**answers

392 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 ...

**5**

votes

**1**answer

759 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 ...

**5**

votes

**1**answer

370 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 ...

**5**

votes

**1**answer

146 views

### Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map ...

**5**

votes

**1**answer

396 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 ...

**5**

votes

**2**answers

172 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 ...

**5**

votes

**1**answer

353 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 ...

**5**

votes

**1**answer

439 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
$$
g_{mn} = \left(
\begin{array}{cc}
g_{\mu\nu} & g_{\mu 5} \\
g_{5\nu} & g_{55} \\
\end{array}
\right)
$$
...

**5**

votes

**2**answers

542 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

**5**

votes

**1**answer

500 views

### Monopole classes on hyperbolic 3-manifolds

Let $M$ be a closed hyperbolic $3$-manifold, and $e \in H^2(M)$ an integral cohomology class which is the first Chern class of a $Spin^c$ structure on $M$. Suppose there is a solution to the monopole ...

**5**

votes

**0**answers

142 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 ...

**5**

votes

**0**answers

246 views

### 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 ...

**4**

votes

**2**answers

411 views

### 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 ...

**4**

votes

**2**answers

169 views

### What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?

Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.
Is it true that if $A$ is the connection 1-form of ...

**4**

votes

**1**answer

406 views

### How do you exponentiate a section of the adjoint bundle to get a gauge transformation?

Suppose $E$ is a vector bundle with structure group $G$ and let $P = F(E)$ be the frame bundle. Let $\mathfrak{g}_P$ denote the associated bundle to the adjoint representation of $G$ on its Lie ...

**4**

votes

**1**answer

266 views

### Uhlenbeck's theorem novelty

This link provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this ...

**4**

votes

**0**answers

150 views

### Is central extension of a group equivalent to a bundle with gauge field?

Let $\tilde G$ be a central extension of a group $G$ by $U(1)$.
One common elegant definition is that there should be a short exact sequence of groups: $0 \to U(1) \to \tilde G \to G \to 0$
...

**4**

votes

**0**answers

117 views

### Bochner-Weitzenbock formula for flat bundle Laplacian

Suppose $(M,g)$ is a compact Riemannian manifold and $(E, \nabla, \lambda, B)$ is the following data:
1) $E$ is a complex vector bundle over $M.$
2) $\nabla$ is a flat connection.
3) $B$ is a ...

**4**

votes

**0**answers

749 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, ...

**3**

votes

**2**answers

559 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) ...

**3**

votes

**2**answers

807 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 ...

**3**

votes

**1**answer

473 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 ...