Integral Transforms for Finite Gauge Theory
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866908782611660800 |
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| author | Van Dyke, Jackson |
| author_facet | Van Dyke, Jackson |
| contents | This paper shows that quantization of $π$-finite spaces, as a functor out of a higher category of spans, is equivariant in two ways: Symmetries of a given polarization/Lagrangian always induce coherent symmetries of the quantization. On the other hand, symmetries of the entire phase space a priori only induce projective symmetries, with an invertible once-categorified theory, the anomaly theory, encoding the projectivity.
We give projective symmetries of three-dimensional finite gauge theories a concrete description via a twice-categorified analogue of Blattner-Kostant-Sternberg kernels and the associated integral transforms, such as the Fourier transform. This establishes an analogy between certain instances of the $π$-finite quantization procedure considered herein and the geometric quantization of a symplectic vector space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00117 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Integral Transforms for Finite Gauge Theory Van Dyke, Jackson Quantum Algebra Other Condensed Matter Mathematical Physics This paper shows that quantization of $π$-finite spaces, as a functor out of a higher category of spans, is equivariant in two ways: Symmetries of a given polarization/Lagrangian always induce coherent symmetries of the quantization. On the other hand, symmetries of the entire phase space a priori only induce projective symmetries, with an invertible once-categorified theory, the anomaly theory, encoding the projectivity. We give projective symmetries of three-dimensional finite gauge theories a concrete description via a twice-categorified analogue of Blattner-Kostant-Sternberg kernels and the associated integral transforms, such as the Fourier transform. This establishes an analogy between certain instances of the $π$-finite quantization procedure considered herein and the geometric quantization of a symplectic vector space. |
| title | Integral Transforms for Finite Gauge Theory |
| topic | Quantum Algebra Other Condensed Matter Mathematical Physics |
| url | https://arxiv.org/abs/2312.00117 |