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A note on Zamolodchikov's recursion relation for the torus conformal block and its light limit

  • Armen Poghosyan
  • Hasmik Poghosyan
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Abstract

In this paper, we review the Zamolodchikov-like recursion relations for torus one-point conformal blocks in both Liouville and A2 Toda theories. Starting from these relations, we derive the corresponding recursion relations in the light asymptotic limit. In Liouville theory, the light-limit recursion reproduces the known expression for the one-point light conformal block in terms of the Gauss hypergeometric function. For A2 Toda theory, our recursion relation provides a new, efficient method for explicit calculations, and we have verified that it is in full agreement with previously known results in the literature. Last but not least, we present closed-form expressions for the coefficients of the insertion-point expansion for both the generic and light A2 Toda one-point blocks.

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Cite

BibTeXRIS
BibTeX
@misc{poghosyan2026note,
  title = {{A note on Zamolodchikov's recursion relation for the torus conformal block and its light limit}},
  author = {Armen Poghosyan and Hasmik Poghosyan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.09177},
  eprint = {2607.09177},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.09177},
}
RIS
TY  - UNPB
TI  - A note on Zamolodchikov's recursion relation for the torus conformal block and its light limit
AU  - Armen Poghosyan
AU  - Hasmik Poghosyan
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2607.09177
UR  - https://arxiv.org/abs/2607.09177
AB  - In this paper, we review the Zamolodchikov-like recursion relations for torus one-point conformal blocks in both Liouville and A_2 Toda theories. Starting from these relations, we derive the corresponding recursion relations in the light asymptotic limit. In Liouville theory, the light-limit recursion reproduces the known expression for the one-point light conformal block in terms of the Gauss hypergeometric function. For A_2 Toda theory, our recursion relation provides a new, efficient method for explicit calculations, and we have verified that it is in full agreement with previously known results in the literature. Last but not least, we present closed-form expressions for the coefficients of the insertion-point expansion for both the generic and light A_2 Toda one-point blocks.
ER  -