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### 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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### 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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### 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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### 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 ...

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### 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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### 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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### 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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### 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 ...

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492 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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### 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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### 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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### Gauge freedom in the tetrad

I'm reading the following paper about Petrov type D space times called "Type D vacuum metrics":
http://scitation.aip.org/content/aip/journal/jmp/10/7/10.1063/1.1664958
by Kinnersley. I have a ...

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242 views

### 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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76 views

### Cohomology of a flat principal connection

Let $M$ be a compact manifold, $G$ a compact Lie group, $P\to M$ a principal $G$-bundle and $A$ a flat principal connection on $P$. Then $(\Omega^\bullet(M;\operatorname{ad}P),d_A)$ forms a cochain ...

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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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105 views

### What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction of instantons. At the heart of this is the complex
$$
V ...

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206 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?