Calibration group

In mathematical physics , a gauge group is the symmetry group ( gauge symmetry ) of a field theory . For non-commutative gauge groups, these are called Yang-Mills theories .

definition

Let be a Lie group with Lie algebra , and be a - principal bundle . The calibration group ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \pi :E\rightarrow M}$ ${\displaystyle G}$

${\displaystyle {\mathcal {G}}:=(C^{\infty }(E,G))^{G}}$

the - equivariant mappings of the total space in works through the space of connections ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {A}}\subset (\Omega ^{1}(E,{\mathfrak {g}}))^{G}}$

${\displaystyle g\omega :=g^{-1}\omega g+g^{-1}dg}$

for . ${\displaystyle g\in {\mathcal {G}},\omega \in {\mathcal {A}}}$

Gauge groups are infinite-dimensional Lie groups, their Lie algebra is the vector space of the differentiable mappings from to . ${\displaystyle M}$ ${\displaystyle {\mathfrak {g}}}$

See also

• Gauge theory

Web links

• Terence Tao : What is a gauge?