Constructive Quantum Field Theory on Curved Surfaces and Related Topics
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913964742410240 |
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| author | Lin, Jiasheng |
| author_facet | Lin, Jiasheng |
| contents | This is the Ph.D. thesis of the author. In this thesis, we construct the $ P(ϕ)_2 $ Quantum Field Theory (QFT) model on curved surfaces and show that it satisfies Segal's axioms (arXiv:2403.12804). An important ingredient in this construction is the use of a local regularization procedure to define the interaction as a random variable with respect to the Gaussian Free Field (GFF). We provide a counterexample demonstrating that spectral truncation regularization violates locality (arXiv:2312.15511). We then explain how Segal's formalism can be extended to the gluing of surfaces with slits, which offers a geometric interpretation of the entanglement entropy. Using this interpretation, we exploit the Polyakov anomaly formula in Conformal Field Theory (CFT) and apply a simple renormalization procedure to define a quantity corresponding to entanglement entropy within this geometric interpretation. We then show that this quantity behaves like a CFT correlation function. This allows us to rigorously derive an entropy calculation of Cardy and Calabrese (arXiv:2501.19014). Finally, Segal's formalism is also related to the asymptotics of zeta determinants on surfaces of large genus where the genus tends to infinity (arXiv:2505.01586). We provide a geometric proof--independent of Segal's axioms--of the corresponding result using heat kernels, in addition to another proof based on Segal's axioms. Both proofs are presented in the thesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21655 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructive Quantum Field Theory on Curved Surfaces and Related Topics Lin, Jiasheng Quantum Physics High Energy Physics - Theory Mathematical Physics Differential Geometry Probability This is the Ph.D. thesis of the author. In this thesis, we construct the $ P(ϕ)_2 $ Quantum Field Theory (QFT) model on curved surfaces and show that it satisfies Segal's axioms (arXiv:2403.12804). An important ingredient in this construction is the use of a local regularization procedure to define the interaction as a random variable with respect to the Gaussian Free Field (GFF). We provide a counterexample demonstrating that spectral truncation regularization violates locality (arXiv:2312.15511). We then explain how Segal's formalism can be extended to the gluing of surfaces with slits, which offers a geometric interpretation of the entanglement entropy. Using this interpretation, we exploit the Polyakov anomaly formula in Conformal Field Theory (CFT) and apply a simple renormalization procedure to define a quantity corresponding to entanglement entropy within this geometric interpretation. We then show that this quantity behaves like a CFT correlation function. This allows us to rigorously derive an entropy calculation of Cardy and Calabrese (arXiv:2501.19014). Finally, Segal's formalism is also related to the asymptotics of zeta determinants on surfaces of large genus where the genus tends to infinity (arXiv:2505.01586). We provide a geometric proof--independent of Segal's axioms--of the corresponding result using heat kernels, in addition to another proof based on Segal's axioms. Both proofs are presented in the thesis. |
| title | Constructive Quantum Field Theory on Curved Surfaces and Related Topics |
| topic | Quantum Physics High Energy Physics - Theory Mathematical Physics Differential Geometry Probability |
| url | https://arxiv.org/abs/2507.21655 |