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21
votes
4answers
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 ...
20
votes
5answers
2k 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 ...
14
votes
2answers
941 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 ...
13
votes
2answers
442 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 ...
12
votes
3answers
589 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
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 ...
11
votes
3answers
1k 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 ...
9
votes
3answers
840 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 ...
7
votes
3answers
552 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) = ...
7
votes
1answer
852 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 ...
7
votes
2answers
252 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
1answer
648 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
0answers
257 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 ...
7
votes
0answers
207 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
0answers
493 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
1answer
538 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 ...
6
votes
3answers
1k 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
1answer
239 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
714 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
5answers
657 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 ...
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 ...
6
votes
3answers
486 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
1answer
165 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 ...
5
votes
1answer
313 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
1answer
273 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
1answer
395 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
1answer
468 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
0answers
176 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 ...
5
votes
0answers
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 ...
4
votes
1answer
461 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 ...
4
votes
1answer
288 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 ...
4
votes
2answers
483 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
4
votes
0answers
116 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 ...
3
votes
2answers
371 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
2answers
121 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 ...
3
votes
1answer
434 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 ...
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 ...
3
votes
2answers
730 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 ...
3
votes
1answer
159 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 $$ X^{+} = \beta\alpha' p^{+}\tau $$ $$ p^{+} = \frac{2\pi}{\beta} P^{\tau +} $$ ...
3
votes
2answers
616 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} ...
3
votes
1answer
295 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 ...
3
votes
0answers
152 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 $(X,g)$. ...
3
votes
0answers
285 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? ...
3
votes
0answers
249 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 ...
2
votes
1answer
183 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 ...
2
votes
1answer
179 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 ...
2
votes
1answer
402 views

gauge theory construction of vector bundles on singular varieties

This is sort of a follow-up to: Gauge theory construction of moduli of vector bundles If I have a complex compact algebraic curve with at worst nodal singularities, is there an analytic description ...
2
votes
1answer
341 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 ...
2
votes
2answers
277 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 ...
2
votes
0answers
182 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, ...