Flat context

In mathematics , flat relationships in geometry and gauge theory are important.

definition

Be a Lie group and a - Principal bundle . ${\displaystyle G}$ ${\displaystyle \pi :E\rightarrow M}$ ${\displaystyle G}$

A flat connection is a connection whose curvature form vanishes . ${\displaystyle \omega \in \Omega ^{1}(E,{\mathfrak {g}})}$ ${\displaystyle \Omega :=d\omega +{\frac {1}{2}}\left[\omega ,\omega \right]=0}$

From Ambrose-Singer's theorem it follows that a -principal bundle with a flat connection is a flat bundle of form ${\displaystyle G}$

${\displaystyle E_{\rho }:={\widetilde {M}}\times G/\sim }$

with (abhängende from the flat connection) for a representation is. is the name of the holonomy representation of the flat context. ${\displaystyle (\gamma x,g)\sim (x,\rho (\gamma )g)}$ ${\displaystyle \rho \colon \pi _{1}M\to G}$ ${\displaystyle \rho }$

Module space with flat connections

The space of all relationships of a given principal bundle is with the topology. The subspace of flat connections is denoted by. The gauge group acts on through , it forms from within itself. ${\displaystyle {\mathcal {A}}:=\Omega ^{1}(M,{\mathfrak {g}})}$ ${\displaystyle C^{\infty }}$ ${\displaystyle {\mathcal {A}}_{F}}$ ${\displaystyle {\mathcal {G}}=C^{\infty }(M,G)}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle g\omega =g^{-1}\omega g+g^{-1}dg}$ ${\displaystyle {\mathcal {A}}_{F}}$

If the bundle is (topologically) trivializable, the holonomy representation mediates a bijection between

${\displaystyle {\mathcal {A}}_{F}/{\mathcal {G}}}$

and a connected component of the representation variety

${\displaystyle Hom(\pi _{1}M,G)/conjugation}$.

The module space of flat connections is

${\displaystyle {\mathcal {M}}={\mathcal {A}}_{F}/{\mathcal {G}}}$.

Its tangent space in a flat connection is ${\displaystyle A\in {\mathcal {M}}}$

${\displaystyle T_{A}{\mathcal {M}}=H^{1}(M,d_{A})}$

With

${\displaystyle d_{A}a=da+\left[A,a\right]}$

for . ${\displaystyle A\in {\mathcal {M}},a\in \Omega ^{*}(M,{\mathfrak {g}})}$

The set of Narasimhan-Seshadri identifies the module room flat correlations above a compact Riemann surface with a complex manifold , namely the multiplicity of stable vector bundle over . ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$

swell

1. ↑ Narasimhan, Seshadri: Stable and Unitary Vector Bundles on a Compact Riemann Surface
2. ↑ Donaldson: A new proof of a theorem of Narasimhan and Seshadri ( Memento of the original from February 1, 2017 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  

Web links

• Flat connections on oriented 2-manifolds