Modular Exercises for Four-Point Blocks - I
The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In this work, we prove that sphere four-point chiral blocks of r...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2025 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2025
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/212878 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Modular Exercises for Four-Point Blocks - I. Miranda C.N. Cheng, Terry Gannon and Guglielmo Lockhart. SIGMA 21 (2025), 013, 61 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862739922014699520 |
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| author | Cheng, Miranda C.N. Gannon, Terry Lockhart, Guglielmo |
| author_facet | Cheng, Miranda C.N. Gannon, Terry Lockhart, Guglielmo |
| citation_txt | Modular Exercises for Four-Point Blocks - I. Miranda C.N. Cheng, Terry Gannon and Guglielmo Lockhart. SIGMA 21 (2025), 013, 61 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In this work, we prove that sphere four-point chiral blocks of rational VOAs are vector-valued modular forms for the groups Γ(2), Γ₀(2), or SL₂(ℤ). Moreover, we prove that the four-point correlators, combining the holomorphic and anti-holomorphic chiral blocks, are modular invariant. In particular, in this language, the crossing symmetries are simply modular. This gives the possibility of exploiting the available techniques and knowledge about modular forms to determine or constrain the physically interesting quantities, such as chiral blocks and fusion coefficients, which we illustrate with a few examples. We also highlight the existence of a sphere-torus correspondence equating the sphere quantities of certain theories ₛ with the torus quantities of another family of theories ₜ. A companion paper will delve into more examples and explore this sphere-torus duality.
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| first_indexed | 2026-03-21T18:31:12Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212878 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:31:12Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cheng, Miranda C.N. Gannon, Terry Lockhart, Guglielmo 2026-02-13T13:49:43Z 2025 Modular Exercises for Four-Point Blocks - I. Miranda C.N. Cheng, Terry Gannon and Guglielmo Lockhart. SIGMA 21 (2025), 013, 61 pages 1815-0659 2020 Mathematics Subject Classification: 81T40; 17B69; 11F03 arXiv:2002.11125 https://nasplib.isofts.kiev.ua/handle/123456789/212878 https://doi.org/10.3842/SIGMA.2025.013 The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In this work, we prove that sphere four-point chiral blocks of rational VOAs are vector-valued modular forms for the groups Γ(2), Γ₀(2), or SL₂(ℤ). Moreover, we prove that the four-point correlators, combining the holomorphic and anti-holomorphic chiral blocks, are modular invariant. In particular, in this language, the crossing symmetries are simply modular. This gives the possibility of exploiting the available techniques and knowledge about modular forms to determine or constrain the physically interesting quantities, such as chiral blocks and fusion coefficients, which we illustrate with a few examples. We also highlight the existence of a sphere-torus correspondence equating the sphere quantities of certain theories ₛ with the torus quantities of another family of theories ₜ. A companion paper will delve into more examples and explore this sphere-torus duality. We thank Vassilis Anagiannis, Christopher Beem, Francesca Ferrari, Alex Maloney, Greg Moore, and Gim Seng Ng for useful conversations. We are also grateful to the anonymous referees for their helpful suggestions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curiegrantagreement No 708045. The work of M.C. and G.L. is supported by an ERC starting grant H2020 #640159. The work of M.C. has also received support from an NWO Vidi grant (number 016. Vidi. 189.182). The work of T.G. is supported by an NSERC Discovery grant. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Modular Exercises for Four-Point Blocks - I Article published earlier |
| spellingShingle | Modular Exercises for Four-Point Blocks - I Cheng, Miranda C.N. Gannon, Terry Lockhart, Guglielmo |
| title | Modular Exercises for Four-Point Blocks - I |
| title_full | Modular Exercises for Four-Point Blocks - I |
| title_fullStr | Modular Exercises for Four-Point Blocks - I |
| title_full_unstemmed | Modular Exercises for Four-Point Blocks - I |
| title_short | Modular Exercises for Four-Point Blocks - I |
| title_sort | modular exercises for four-point blocks - i |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212878 |
| work_keys_str_mv | AT chengmirandacn modularexercisesforfourpointblocksi AT gannonterry modularexercisesforfourpointblocksi AT lockhartguglielmo modularexercisesforfourpointblocksi |