Exact calculations beyond charge neutrality in timelike Liouville field theory
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912885249146880 |
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| author | Chatterjee, Sourav |
| author_facet | Chatterjee, Sourav |
| contents | Timelike Liouville field theory (also known as imaginary Liouville theory or imaginary Gaussian multiplicative chaos) is expected to describe two-dimensional quantum gravity in a positive-curvature regime, but its path integral is not a probability measure and rigorous exact computations are currently available only in the charge-neutral (integer screening) case. In this paper we show that at the special coupling $b=1/\sqrt{2}$, the Coulomb-gas expansion of the timelike path integral becomes explicitly computable beyond charge neutrality. The reason is that the $n$-fold integrals generated by the interaction acquire a Vandermonde/determinantal structure at $b=1/\sqrt{2}$, which allows exact evaluation in terms of classical special functions. We derive Mellin-Barnes type representations (involving the Barnes $G$-function and, in a three-point case, Gauss hypergeometric functions) for the zero- and one-point functions, for an antipodal two-point function, and for a three-point function with a resonant insertion $α_2=b$. We then address the subtle zero-mode integration: after a Gaussian regularization we obtain an explicit renormalized partition function $C(1/\sqrt{2},μ)=e(4π\sqrt2 μ)^{-1}$, identify distributional limits in the physically relevant regime $α_j=\frac{1}{2}Q+\mathrm{i} P_j$, and compare with the Hankel-contour prescription recently proposed in the physics literature. These results provide the first rigorously controlled family of exact calculations in timelike Liouville theory outside charge neutrality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19097 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact calculations beyond charge neutrality in timelike Liouville field theory Chatterjee, Sourav Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Probability 81T40, 17B69, 81T20 Timelike Liouville field theory (also known as imaginary Liouville theory or imaginary Gaussian multiplicative chaos) is expected to describe two-dimensional quantum gravity in a positive-curvature regime, but its path integral is not a probability measure and rigorous exact computations are currently available only in the charge-neutral (integer screening) case. In this paper we show that at the special coupling $b=1/\sqrt{2}$, the Coulomb-gas expansion of the timelike path integral becomes explicitly computable beyond charge neutrality. The reason is that the $n$-fold integrals generated by the interaction acquire a Vandermonde/determinantal structure at $b=1/\sqrt{2}$, which allows exact evaluation in terms of classical special functions. We derive Mellin-Barnes type representations (involving the Barnes $G$-function and, in a three-point case, Gauss hypergeometric functions) for the zero- and one-point functions, for an antipodal two-point function, and for a three-point function with a resonant insertion $α_2=b$. We then address the subtle zero-mode integration: after a Gaussian regularization we obtain an explicit renormalized partition function $C(1/\sqrt{2},μ)=e(4π\sqrt2 μ)^{-1}$, identify distributional limits in the physically relevant regime $α_j=\frac{1}{2}Q+\mathrm{i} P_j$, and compare with the Hankel-contour prescription recently proposed in the physics literature. These results provide the first rigorously controlled family of exact calculations in timelike Liouville theory outside charge neutrality. |
| title | Exact calculations beyond charge neutrality in timelike Liouville field theory |
| topic | Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Probability 81T40, 17B69, 81T20 |
| url | https://arxiv.org/abs/2601.19097 |