Oscillator Calculus on Coadjoint Orbits and Index Theorems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908536244535296 |
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| author | Bykov, Dmitri Krivorol, Viacheslav Kuzovchikov, Andrew |
| author_facet | Bykov, Dmitri Krivorol, Viacheslav Kuzovchikov, Andrew |
| contents | We consider quantum mechanical systems of spin chain type, with finite-dimensional Hilbert spaces and $\mathcal{N}=2$ or $\mathcal{N}=4$ supersymmetry, described in $\mathcal{N}=2$ superspace in terms of nonlinear chiral multiplets. We prove that they are natural truncations of 1D sigma models, whose target spaces are $\mathsf{SU}(n)$ (co)adjoint orbits. As a first application, we compute the Witten indices of these finite-dimensional models showing that they reproduce the Dolbeault and de Rham indices of the target space. The problem of finding the exact spectra of generalized Laplace operators on such orbits is shown to be equivalent to the diagonalization of spin chain Hamiltonians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21024 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oscillator Calculus on Coadjoint Orbits and Index Theorems Bykov, Dmitri Krivorol, Viacheslav Kuzovchikov, Andrew High Energy Physics - Theory Mathematical Physics Differential Geometry Representation Theory We consider quantum mechanical systems of spin chain type, with finite-dimensional Hilbert spaces and $\mathcal{N}=2$ or $\mathcal{N}=4$ supersymmetry, described in $\mathcal{N}=2$ superspace in terms of nonlinear chiral multiplets. We prove that they are natural truncations of 1D sigma models, whose target spaces are $\mathsf{SU}(n)$ (co)adjoint orbits. As a first application, we compute the Witten indices of these finite-dimensional models showing that they reproduce the Dolbeault and de Rham indices of the target space. The problem of finding the exact spectra of generalized Laplace operators on such orbits is shown to be equivalent to the diagonalization of spin chain Hamiltonians. |
| title | Oscillator Calculus on Coadjoint Orbits and Index Theorems |
| topic | High Energy Physics - Theory Mathematical Physics Differential Geometry Representation Theory |
| url | https://arxiv.org/abs/2412.21024 |