On the Quantum K-Theory of the Quintic
Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series (, , ) that satisfies a system of linear differential equations with respect to and -difference equations with respect to . With some mild assumptions on the vari...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211524 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On the Quantum K-Theory of the Quintic. Stavros Garoufalidis and Emanuel Scheidegger. SIGMA 18 (2022), 021, 20 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862589803945525248 |
|---|---|
| author | Garoufalidis, Stavros Scheidegger, Emanuel |
| author_facet | Garoufalidis, Stavros Scheidegger, Emanuel |
| citation_txt | On the Quantum K-Theory of the Quintic. Stavros Garoufalidis and Emanuel Scheidegger. SIGMA 18 (2022), 021, 20 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series (, , ) that satisfies a system of linear differential equations with respect to and -difference equations with respect to . With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small -function (, , 0), which, in the case of Fano manifolds, is a vector-valued -hypergeometric function. On the other hand, for the quintic 3-fold, we formulate an explicit conjecture for the small -function and its small linear -difference equation expressed linearly in terms of the Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants and the case of Fano manifolds, the coefficients of the small linear -difference equations are not Laurent polynomials, but rather analytic functions in two variables determined linearly by the Gopakumar-Vafa invariants of the quintic. Our conjecture for the small -function agrees with a proposal of Jockers-Mayr.
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| first_indexed | 2026-03-13T20:04:03Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211524 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T20:04:03Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Garoufalidis, Stavros Scheidegger, Emanuel 2026-01-05T12:24:47Z 2022 On the Quantum K-Theory of the Quintic. Stavros Garoufalidis and Emanuel Scheidegger. SIGMA 18 (2022), 021, 20 pages 1815-0659 2020 Mathematics Subject Classification: 14N35; 53D45; 39A13; 19E20 arXiv:2101.07490 https://nasplib.isofts.kiev.ua/handle/123456789/211524 https://doi.org/10.3842/SIGMA.2022.021 Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series (, , ) that satisfies a system of linear differential equations with respect to and -difference equations with respect to . With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small -function (, , 0), which, in the case of Fano manifolds, is a vector-valued -hypergeometric function. On the other hand, for the quintic 3-fold, we formulate an explicit conjecture for the small -function and its small linear -difference equation expressed linearly in terms of the Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants and the case of Fano manifolds, the coefficients of the small linear -difference equations are not Laurent polynomials, but rather analytic functions in two variables determined linearly by the Gopakumar-Vafa invariants of the quintic. Our conjecture for the small -function agrees with a proposal of Jockers-Mayr. The authors wish to thank the Max-Planck-Institute for Mathematics and the Bethe Center for Theoretical Physics in Bonn for inviting them to their workshop on Number Theoretic Methods in Quantum Physics in July 2019, where the first ideas were conceived. We also wish to thank Gaetan Borot, Alexander Givental, Todor Milanov, and Di Yang for useful conversations. E.S. wishes to thank the University of Melbourne for having him as a guest during 2020 and Southern University of Science and Technology for its hospitality in 2021. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Quantum K-Theory of the Quintic Article published earlier |
| spellingShingle | On the Quantum K-Theory of the Quintic Garoufalidis, Stavros Scheidegger, Emanuel |
| title | On the Quantum K-Theory of the Quintic |
| title_full | On the Quantum K-Theory of the Quintic |
| title_fullStr | On the Quantum K-Theory of the Quintic |
| title_full_unstemmed | On the Quantum K-Theory of the Quintic |
| title_short | On the Quantum K-Theory of the Quintic |
| title_sort | on the quantum k-theory of the quintic |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211524 |
| work_keys_str_mv | AT garoufalidisstavros onthequantumktheoryofthequintic AT scheideggeremanuel onthequantumktheoryofthequintic |