₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction
We introduce families of four-regular graphs consisting of chains of hourglasses which are attached to a finite kernel. We prove a formula for the ₂ invariant of these hourglass chains, which only depends on the kernel. For different kernels, these hourglass chains typically give rise to different ₂...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211427 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction. Oliver Schnetz and Karen Yeats. SIGMA 17 (2021), 100, 26 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862592151376887808 |
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| author | Schnetz, Oliver Yeats, Karen |
| author_facet | Schnetz, Oliver Yeats, Karen |
| citation_txt | ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction. Oliver Schnetz and Karen Yeats. SIGMA 17 (2021), 100, 26 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce families of four-regular graphs consisting of chains of hourglasses which are attached to a finite kernel. We prove a formula for the ₂ invariant of these hourglass chains, which only depends on the kernel. For different kernels, these hourglass chains typically give rise to different ₂ invariants. An exhaustive search for the ₂ invariants of hourglass chains with kernels that have a maximum of ten vertices provides Calabi-Yau manifolds with point-counts which match the Fourier coefficients of modular forms whose weights and levels are [4,8], [4,16], [6,4], and [9,4]. Assuming the completion conjecture, we show that no modular form of weight 2 and level ≤ 1000 corresponds to the ₂ of such hourglass chains. This provides further evidence in favour of the conjecture that curves are absent in ₂ invariants of ⁴ quantum field theory.
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| first_indexed | 2026-03-13T21:15:07Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211427 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T21:15:07Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Schnetz, Oliver Yeats, Karen 2026-01-02T08:30:47Z 2021 ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction. Oliver Schnetz and Karen Yeats. SIGMA 17 (2021), 100, 26 pages 1815-0659 2020 Mathematics Subject Classification: 81T18 arXiv:2102.12383 https://nasplib.isofts.kiev.ua/handle/123456789/211427 https://doi.org/10.3842/SIGMA.2021.100 We introduce families of four-regular graphs consisting of chains of hourglasses which are attached to a finite kernel. We prove a formula for the ₂ invariant of these hourglass chains, which only depends on the kernel. For different kernels, these hourglass chains typically give rise to different ₂ invariants. An exhaustive search for the ₂ invariants of hourglass chains with kernels that have a maximum of ten vertices provides Calabi-Yau manifolds with point-counts which match the Fourier coefficients of modular forms whose weights and levels are [4,8], [4,16], [6,4], and [9,4]. Assuming the completion conjecture, we show that no modular form of weight 2 and level ≤ 1000 corresponds to the ₂ of such hourglass chains. This provides further evidence in favour of the conjecture that curves are absent in ₂ invariants of ⁴ quantum field theory. Both authors are deeply indebted to Dirk Kreimer for many years of encouragement and support. Oliver Schnetz is supported by a DFG grant SCHN 1240. Karen Yeats is supported by an NSERC Discovery grant and by the Canada Research Chairs program; during some of this work, she was visiting Germany as a Humboldt fellow. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction Article published earlier |
| spellingShingle | ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction Schnetz, Oliver Yeats, Karen |
| title | ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction |
| title_full | ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction |
| title_fullStr | ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction |
| title_full_unstemmed | ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction |
| title_short | ₂ Invariants of Hourglass Chains via Quadratic Denominator Reduction |
| title_sort | ₂ invariants of hourglass chains via quadratic denominator reduction |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211427 |
| work_keys_str_mv | AT schnetzoliver 2invariantsofhourglasschainsviaquadraticdenominatorreduction AT yeatskaren 2invariantsofhourglasschainsviaquadraticdenominatorreduction |