Cartan monopoles

Abstract

The effective Hamiltonians for chiral supersymmetric gauge theories at small spatial volume are generalizations of the Hamiltonians describing the motion of a scalar or a spinor particle in a field of Dirac monopoles (we are dealing in fact with a certain lattice of monopoles supplemented with a periodic singular potential). The gauge fields in such Hamiltonians belong to the Cartan subalgebras of the corresponding gauge algebras. Such a construction exists for all groups admitting complex representations, i.e., for SU(N≥3), Spin(4n+2) with n≥1 and E6. We give explicit expressions for these Hamiltonians for SU(3), SU(4)≃Spin(6) and for SU(5). The simplified version of such a Hamiltonian, deprived of fermion terms, of the extra scalar potential and when only one node of the lattice is taken into consideration, describe a 3r-dimensional motion (r being the rank of the group) in the field what we call a “Cartan monopole.” As is the case for the ordinary monopole, the Lagrangian of this system enjoys gauge symmetry, rotational symmetry, and the parameter, generalizing the notion of magnetic charge for Cartan monopoles, is quantized. Published by the American Physical Society 2024