Complex Chern-Simons Theory with $k=8\mathbb{N}$ and An Improved Spinfoam Model with Cosmological Constant
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| Format: | Preprint |
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2025
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| _version_ | 1866909630149427200 |
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| author | Han, Muxin Pan, Qiaoyin |
| author_facet | Han, Muxin Pan, Qiaoyin |
| contents | This paper presents an improvement to the four-dimensional spinfoam model with cosmological constant ($Λ$-SF model) in loop quantum gravity. The original $Λ$-SF model, defined via ${\rm SL}(2,\mathbb{C})$ Chern-Simons theory on graph-complement 3-manifolds, produces finite amplitudes and reproduces curved 4-simplex geometries in the semi-classical limit. However, extending the model to general simplicial complexes necessitated ad hoc, non-universal phase factors in face amplitudes, complicating systematic constructions. We resolve this issue by redefining the vertex amplitude using a novel set of phase space coordinates that eliminate the extraneous phase factor, yielding a universally defined face amplitude. Key results include: (1) The vertex amplitude is rigorously shown to be well-defined for Chern-Simons levels $k \in 8\mathbb{N}$, compatible with semi-classical analysis ($k \to \infty$). (2) The symplectic structure of the Chern-Simons phase space is modified to accommodate ${\rm SL}(2,\mathbb{C})$ holonomies, relaxing quantization constraints to $\mathrm{Sp}(2r,\mathbb{Z}/4)$. (3) Edge amplitudes are simplified using constraints aligned with colored tensor models, enabling systematic gluing of 4-simplices into complexes dual to colored graphs. (4) Stationary phase analysis confirms consistency of critical points with prior work, recovering Regge geometries with curvature determined by $Λ$. These advancements streamline the spinfoam amplitude definition, facilitating future studies of colored group field theories and continuum limits of quantum gravity. The results establish a robust framework for 4D quantum gravity with non-zero $Λ$, free of previous ambiguities in face amplitudes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complex Chern-Simons Theory with $k=8\mathbb{N}$ and An Improved Spinfoam Model with Cosmological Constant Han, Muxin Pan, Qiaoyin General Relativity and Quantum Cosmology This paper presents an improvement to the four-dimensional spinfoam model with cosmological constant ($Λ$-SF model) in loop quantum gravity. The original $Λ$-SF model, defined via ${\rm SL}(2,\mathbb{C})$ Chern-Simons theory on graph-complement 3-manifolds, produces finite amplitudes and reproduces curved 4-simplex geometries in the semi-classical limit. However, extending the model to general simplicial complexes necessitated ad hoc, non-universal phase factors in face amplitudes, complicating systematic constructions. We resolve this issue by redefining the vertex amplitude using a novel set of phase space coordinates that eliminate the extraneous phase factor, yielding a universally defined face amplitude. Key results include: (1) The vertex amplitude is rigorously shown to be well-defined for Chern-Simons levels $k \in 8\mathbb{N}$, compatible with semi-classical analysis ($k \to \infty$). (2) The symplectic structure of the Chern-Simons phase space is modified to accommodate ${\rm SL}(2,\mathbb{C})$ holonomies, relaxing quantization constraints to $\mathrm{Sp}(2r,\mathbb{Z}/4)$. (3) Edge amplitudes are simplified using constraints aligned with colored tensor models, enabling systematic gluing of 4-simplices into complexes dual to colored graphs. (4) Stationary phase analysis confirms consistency of critical points with prior work, recovering Regge geometries with curvature determined by $Λ$. These advancements streamline the spinfoam amplitude definition, facilitating future studies of colored group field theories and continuum limits of quantum gravity. The results establish a robust framework for 4D quantum gravity with non-zero $Λ$, free of previous ambiguities in face amplitudes. |
| title | Complex Chern-Simons Theory with $k=8\mathbb{N}$ and An Improved Spinfoam Model with Cosmological Constant |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2504.06427 |