Cartan monopoles

TL;DR

This paper generalizes Hamiltonians for chiral supersymmetric gauge theories at small volume, describing Cartan monopoles as lattice monopole systems with quantized parameters, for various gauge groups.

Contribution

It introduces explicit Hamiltonian constructions for Cartan monopoles in certain gauge groups, extending monopole concepts to supersymmetric gauge theories.

Findings

Explicit Hamiltonians for SU(3), SU(4), and SU(5)

Description of Cartan monopoles as lattice monopole systems

Quantization of the generalized magnetic charge

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 $SU(N \geq 3), \ Spin(4n+2)$ with $n \geq 1$ and $E_6$. We give explicit expressions for these Hamiltonians for $SU(3)$, $SU(4) \simeq Spin(6)$ 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 {\it 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.