Saddling Timelike Liouville Theory
Abstract
The two-sphere partition function of timelike Liouville theory admits two complementary semi-classical descriptions whose relation presents a non-perturbative puzzle. A direct path-integral expansion around the round two-sphere saddle reproduces the perturbative small-β expansion, with β the Liouville coupling, of the analytically continued Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ) formula, while the latter contains an additional oscillatory factor suggestive of a second saddle contribution. We investigate the origin of this factor in the Liouville zero-mode sector using Picard-Lefschetz theory. Taking as input the Hankel-type contour motivated by the inverse-Gamma representation of the zero-mode integral, we determine its Lefschetz-thimble decomposition depending on β. For 0<β<1, it precisely reproduces the saddle structure encoded in the analytically continued DOZZ result. We identify the integration contour as the source of the difference between the zero-mode saddle descriptions. As β approaches $1$, the zero-mode saddle moves to the boundary of field space, while for β>1 the same contour is represented by a single contributing thimble. We further relate this Stokes reorganisation to recent probabilistic constructions of timelike Liouville theory.
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BibTeX
@misc{despontin2026saddling,
title = {{Saddling Timelike Liouville Theory}},
author = {Emilie Despontin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31438},
eprint = {2609.31438},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31438},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Saddling Timelike Liouville Theory
AU - Emilie Despontin
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31438
UR - https://arxiv.org/abs/2609.31438
AB - The two-sphere partition function of timelike Liouville theory admits two complementary semi-classical descriptions whose relation presents a non-perturbative puzzle. A direct path-integral expansion around the round two-sphere saddle reproduces the perturbative small-β expansion, with β the Liouville coupling, of the analytically continued Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ) formula, while the latter contains an additional oscillatory factor suggestive of a second saddle contribution. We investigate the origin of this factor in the Liouville zero-mode sector using Picard-Lefschetz theory. Taking as input the Hankel-type contour motivated by the inverse-Gamma representation of the zero-mode integral, we determine its Lefschetz-thimble decomposition depending on β. For 0<β<1, it precisely reproduces the saddle structure encoded in the analytically continued DOZZ result. We identify the integration contour as the source of the difference between the zero-mode saddle descriptions. As β approaches $1$, the zero-mode saddle moves to the boundary of field space, while for β>1 the same contour is represented by a single contributing thimble. We further relate this Stokes reorganisation to recent probabilistic constructions of timelike Liouville theory.
ER -