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Virasoro Conformal Blocks and modular functor from Liouville CFT

  • Guillaume Baverez
  • Colin Guillarmou
  • Antti Kupiainen
  • Rémi Rhodes
  • Yuxiao Xie
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Abstract

In this article, we give a global construction of the Virasoro conformal blocks on Teichmüller space of Riemann surfaces of genus g with m marked points, when the central charge is c>25. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which encodes their conformal invariance. For the sphere with $3$ points, the space of conformal blocks is one-dimensional. For a given marked Riemann surface and a marked pair of pants decomposition, it is in general an infinite dimensional Hilbert space, isomorphic to L2(R+3g−3+m) with respect to some measure involving the DOZZ structure constants. We show that it carries a projective unitary representation of the mapping class group. More generally, there are unitary isomorphisms for each Moore-Seiberg move, changing a marked pair of pants decomposition into another one.

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Cite

BibTeXRIS
BibTeX
@misc{baverez2026virasoro,
  title = {{Virasoro Conformal Blocks and modular functor from Liouville CFT}},
  author = {Guillaume Baverez and Colin Guillarmou and Antti Kupiainen and Rémi Rhodes and Yuxiao Xie},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03540},
  eprint = {2610.03540},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03540},
}
RIS
TY  - UNPB
TI  - Virasoro Conformal Blocks and modular functor from Liouville CFT
AU  - Guillaume Baverez
AU  - Colin Guillarmou
AU  - Antti Kupiainen
AU  - Rémi Rhodes
AU  - Yuxiao Xie
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03540
UR  - https://arxiv.org/abs/2610.03540
AB  - In this article, we give a global construction of the Virasoro conformal blocks on Teichmüller space of Riemann surfaces of genus g with m marked points, when the central charge is c>25. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which encodes their conformal invariance. For the sphere with $3$ points, the space of conformal blocks is one-dimensional. For a given marked Riemann surface and a marked pair of pants decomposition, it is in general an infinite dimensional Hilbert space, isomorphic to L^2(R_+^3g−3+m) with respect to some measure involving the DOZZ structure constants. We show that it carries a projective unitary representation of the mapping class group. More generally, there are unitary isomorphisms for each Moore-Seiberg move, changing a marked pair of pants decomposition into another one.
ER  -