2
votes
1answer
154 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 ...
5
votes
0answers
215 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 ...
3
votes
0answers
143 views

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 $(X,g)$. ...
2
votes
0answers
103 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 ...
1
vote
0answers
118 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 ...
3
votes
1answer
151 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 $$ X^{+} = \beta\alpha' p^{+}\tau $$ $$ p^{+} = \frac{2\pi}{\beta} P^{\tau +} $$ ...
0
votes
1answer
147 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 ...
3
votes
0answers
239 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 ...
0
votes
2answers
274 views

Non simply connected HyperKähler 4-manifolds without ALE metrics

In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?
3
votes
1answer
268 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 ...
9
votes
2answers
673 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 ...
1
vote
1answer
181 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 ...
4
votes
0answers
111 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
1answer
303 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 ...
1
vote
1answer
364 views

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 ...
2
votes
1answer
177 views

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 ...
6
votes
1answer
232 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
2answers
695 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 ...
7
votes
0answers
198 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 ...
11
votes
5answers
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 ...
0
votes
0answers
190 views

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?
5
votes
0answers
368 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 ...
3
votes
3answers
1k 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
1answer
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 ...
1
vote
2answers
304 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 ...
2
votes
1answer
328 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 ...
6
votes
3answers
472 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 ...
1
vote
1answer
291 views

Reference for some elementary facts about principal bundles

Let $(P,\pi,B,G)$ be a principal bundle with total space $P$, base $B$, projection $\pi$ and structure group $G$. Now I am searching for a good reference (with proofs) for the following facts: 1) ...
7
votes
0answers
478 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 ...
1
vote
3answers
518 views

Can all G-connections on a Riemann surface X be induced by maps from X to G

There is the invariant Maurer–Cartan 1-form on a compact semi-simple Lie group G. So if we pull it back using a map from X to G then we get a G-connection on X. The question is, can all G-connections ...
7
votes
1answer
824 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 ...
3
votes
2answers
610 views

Deriving symmetries of a Gauge theory

Hello, I don't know if this is a good place for exposing my problem but I'll try... I have a gauge theory with action: $S=\int\;dt L=\int d^4 x \;\epsilon^{\mu\nu\rho\sigma} B_{\mu\nu\;IJ} ...
9
votes
3answers
809 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 ...