Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866913868102500352 |
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| author | Sueishi, Naohisa Kamata, Syo Misumi, Tatsuhiro Ünsal, Mithat |
| author_facet | Sueishi, Naohisa Kamata, Syo Misumi, Tatsuhiro Ünsal, Mithat |
| contents | We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with $N $ minima on $S^{1}$. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of $N$-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with 't Hooft anomaly for even $N$ and global inconsistency for odd $N$. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_06586 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$ Sueishi, Naohisa Kamata, Syo Misumi, Tatsuhiro Ünsal, Mithat Quantum Physics High Energy Physics - Theory Mathematical Physics We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with $N $ minima on $S^{1}$. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of $N$-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with 't Hooft anomaly for even $N$ and global inconsistency for odd $N$. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence. |
| title | Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$ |
| topic | Quantum Physics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2103.06586 |