Curated collection

Gauge groups and global forms

Compact groups, central quotients, and covering maps relevant to gauge theory. Group data constrain a theory; matter and line-operator choices belong to a physical setup.

Read the conventions and classification

31 released records · classifications and individual objects are labeled separately.

Gauge groups and global forms
Object	Kind	Essential data	Coverage
$\mathrm{SU}(2)$ Special unitary group	Group Lie theory	dimℝ 3	executable 7 linked workbenchs
$\mathrm{SO}(3)$ Rotation group in three dimensions	Group Lie theory	dimℝ 3	executable 2 linked workbenchs
$\mathrm{SU}(3)$ Special unitary group	Group Lie theory	dimℝ 8	executable 8 linked workbenchs
$\mathrm{PSU}(3)$ Projective special unitary group	Group Lie theory	dimℝ 8	executable 1 linked workbench
$\mathrm{U}(1)$ Circle group	Group Lie theory	dimℝ 1	reference Sourced record
$\mathrm{Spin}(4)$ Euclidean spin group in four dimensions	Group Lie theory	dimℝ 6	reference Sourced record
$\mathrm{SU}(4)$ Special unitary group in four dimensions	Group Lie theory	dimℝ 15	executable 4 linked workbenchs
$\mathrm{PSU}(4)$ Adjoint compact group of type A₃	Group Lie theory	dimℝ 15	reference Sourced record
$\mathrm{SU}(5)$ Special unitary group in five dimensions	Group Lie theory	dimℝ 24	executable 4 linked workbenchs
$\mathrm{U}(2)$ Unitary group in two dimensions	Group Lie theory	dimℝ 4	reference Sourced record
$\mathrm{SO}(4)$ Rotation group in four dimensions	Group Lie theory	dimℝ 6	reference Sourced record
$\mathrm{SO}(5)$ Rotation group in five dimensions	Group Lie theory	dimℝ 10	reference Sourced record
$\mathrm{SO}(6)$ Rotation group in six dimensions	Group Lie theory	dimℝ 15	reference Sourced record
$\mathrm{Sp}(2)$ Compact symplectic group of quaternionic rank two	Group Lie theory	dimℝ 10	executable 1 linked workbench
$\mathrm{Sp}(3)$ Compact symplectic group of quaternionic rank three	Group Lie theory	dimℝ 21	reference Sourced record
$\mathrm{Spin}(8)$ Simply connected compact spin group in eight dimensions	Group Lie theory	dimℝ 28	executable 1 linked workbench
$\mathrm{SO}(8)$ Rotation group in eight dimensions	Group Lie theory	dimℝ 28	reference Sourced record
$\mathrm{PSO}(8)$ Adjoint compact group of type D₄	Group Lie theory	dimℝ 28	reference Sourced record
$\mathrm{Spin}(10)$ Simply connected compact spin group in ten dimensions	Group Lie theory	dimℝ 45	executable 1 linked workbench
$\mathrm{SO}(10)$ Rotation group in ten dimensions	Group Lie theory	dimℝ 45	reference Sourced record
$\mathrm{G}_{2}\;\text{compact}$ Compact group of octonion automorphisms	Group Lie theory	dimℝ 14	executable 1 linked workbench
$\mathrm{F}_{4}\;\text{compact}$ Compact exceptional group of type F₄	Group Lie theory	dimℝ 52	executable 1 linked workbench
$\mathrm{E}_{6}\;\text{compact, simply connected}$ Simply connected compact exceptional group of type E₆	Group Lie theory	dimℝ 78	executable 1 linked workbench
$\mathrm{E}_{7}\;\text{compact, simply connected}$ Simply connected compact exceptional group of type E₇	Group Lie theory	dimℝ 133	executable 1 linked workbench
$\mathrm{E}_{8}\;\text{compact}$ Compact exceptional group of type E₈	Group Lie theory	dimℝ 248	executable 1 linked workbench
$\mathrm{E}_{6}\;\text{compact, adjoint}$ Adjoint compact exceptional group of type E₆	Group Lie theory	dimℝ 78	reference Sourced record
$\mathrm{E}_{7}\;\text{compact, adjoint}$ Adjoint compact exceptional group of type E₇	Group Lie theory	dimℝ 133	reference Sourced record
$SU(3)\times SU(2)\times U(1)$ Standard Model compact covering product	Group Lie theory	dimℝ 12	executable 3 linked workbenchs
$G_{\mathrm{SM}}^{(2)}$ Standard Model global group with diagonal Z2 quotient	Group Lie theory	dimℝ 12	executable 3 linked workbenchs
$G_{\mathrm{SM}}^{(3)}$ Standard Model global group with diagonal Z3 quotient	Group Lie theory	dimℝ 12	executable 3 linked workbenchs
$G_{\mathrm{SM}}^{(6)}$ Standard Model global group with diagonal Z6 quotient	Group Lie theory	dimℝ 12	executable 3 linked workbenchs