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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.06042 |
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| _version_ | 1866909229890142208 |
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| author | Smilga, Andrei |
| author_facet | Smilga, Andrei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06042 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cartan monopoles Smilga, Andrei High Energy Physics - Theory 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. |
| title | Cartan monopoles |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2406.06042 |