Super Liouville theory on the torus: bootstrap and the super Virasoro minimal string
Abstract
Motivated by applications to low-dimensional worldsheet string constructions, we analyze 𝒩 = 1 super Liouville theory on the torus. Specifically, we re-examine the recursive representations of the torus one-point super Virasoro blocks that have been constructed in the literature and correct several of their ingredients. In the process we identify a new block in the odd spin structure, associated with a privileged superdescendant whose torus one-point function transforms like that of a Virasoro primary. We then demonstrate that the structure constants for both spacelike and timelike super Liouville theory satisfy modular crossing. Furthermore, we use a bootstrap argument to construct modular crossing kernels for all four spin structure sectors, valid for central charges c ∈ ℂ∖(−∞,3/2], and verify that they correctly implement the crossing transformations of the blocks. As a concrete application of the technology, we analyze the super Virasoro minimal string. There are four such theories, 0A± and 0B±, which are distinguished by the GSO projection together with the choice of worldsheet supercharge. The resulting NS-sector sphere three-point and torus one-point amplitudes are shown to agree with topological recursion on the spectral curves of the dual matrix integrals, which were recently derived from 3d supergravity. Notably, the one-point function of the privileged superdescendant plays an essential role in the contribution of the odd spin structure to the torus amplitude via a careful treatment of the picture changing operator. We also identify an exceptional zero-momentum Ramond insertion in the 0A− theory that we interpret geometrically as a Ramond puncture, and match its mixed sphere three-point amplitude to the corresponding intersection theory prediction.
Related
- Resurgence in the Virasoro minimal string and 3d gravity
- Semiclassical Liouville Theory on the Real Projective Plane: A Complex Interpretation of the Bootstrap
- Instanton-induced closed-string amplitudes in minimal superstring theory at subleading order
- The Virasoro Bootstrap: from the c∼1 desert to coupled Potts
- virasoro_blocks: Software for the 2d bootstrap
BibTeX
@misc{collier2026super,
title = {{Super Liouville theory on the torus: bootstrap and the super Virasoro minimal string}},
author = {Scott Collier and Beatrix Mühlmann and Mukund Rangamani and Ioannis Tsiares and Jianming Zheng},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01909},
eprint = {2610.01909},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05785898v1},
}
RIS
TY - UNPB
TI - Super Liouville theory on the torus: bootstrap and the super Virasoro minimal string
AU - Scott Collier
AU - Beatrix Mühlmann
AU - Mukund Rangamani
AU - Ioannis Tsiares
AU - Jianming Zheng
PY - 2026
DA - 2026/10/01
PB - HAL
DO - 10.48550/arxiv.2610.01909
UR - https://hal.science/hal-05785898v1
AB - Motivated by applications to low-dimensional worldsheet string constructions, we analyze 𝒩 = 1 super Liouville theory on the torus. Specifically, we re-examine the recursive representations of the torus one-point super Virasoro blocks that have been constructed in the literature and correct several of their ingredients. In the process we identify a new block in the odd spin structure, associated with a privileged superdescendant whose torus one-point function transforms like that of a Virasoro primary. We then demonstrate that the structure constants for both spacelike and timelike super Liouville theory satisfy modular crossing. Furthermore, we use a bootstrap argument to construct modular crossing kernels for all four spin structure sectors, valid for central charges c ∈ ℂ∖(−∞,3/2], and verify that they correctly implement the crossing transformations of the blocks. As a concrete application of the technology, we analyze the super Virasoro minimal string. There are four such theories, 0A^± and 0B^±, which are distinguished by the GSO projection together with the choice of worldsheet supercharge. The resulting NS-sector sphere three-point and torus one-point amplitudes are shown to agree with topological recursion on the spectral curves of the dual matrix integrals, which were recently derived from 3d supergravity. Notably, the one-point function of the privileged superdescendant plays an essential role in the contribution of the odd spin structure to the torus amplitude via a careful treatment of the picture changing operator. We also identify an exceptional zero-momentum Ramond insertion in the 0A^− theory that we interpret geometrically as a Ramond puncture, and match its mixed sphere three-point amplitude to the corresponding intersection theory prediction.
ER -