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
![\ mathfrak {g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
![\ pi: E \ rightarrow M](https://wikimedia.org/api/rest_v1/media/math/render/svg/a92a4f1d68248e1603d5b177d2cf3d321c86ee2d)
![{\ mathcal {G}}: = (C ^ \ infty (E, G)) ^ G](https://wikimedia.org/api/rest_v1/media/math/render/svg/dccadd9a960cb20d742acfd39a2d465f5e02cd6f)
the - equivariant mappings of the total space in
works through
the space of connections![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ mathcal {A}} \ subset (\ Omega ^ 1 (E, \ mathfrak {g})) ^ G](https://wikimedia.org/api/rest_v1/media/math/render/svg/38353eb938f4425825960acbc5cb6d434cd6df14)
![g \ omega: = g ^ {- 1} \ omega g + g ^ {- 1} dg](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd99c649b0a1bb10d6d29a35808a3bb8a80fe1f1)
for .
![g \ in {\ mathcal {G}}, \ omega \ in {\ mathcal {A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65144ff8997c5905e93d9b85f99fa47ee796514f)
Gauge groups are infinite-dimensional Lie groups, their Lie algebra is the vector space of the differentiable mappings from to .
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![\ mathfrak {g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
See also
Web links