Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index
We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the An and BCn root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in terms of the lens elliptic gamma function, a generalisation of the e...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209451 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index / A. P. Kels, M. Yamazaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 41 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862741416763981824 |
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| author | Kels, A.P. Yamazaki, M. |
| author_facet | Kels, A.P. Yamazaki, M. |
| citation_txt | Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index / A. P. Kels, M. Yamazaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 41 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the An and BCn root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in terms of the lens elliptic gamma function, a generalisation of the elliptic gamma function that depends on an additional integer variable, as well as a complex variable and two elliptic nomes. As an application of our results, we prove an equality between S¹×S³/Zr supersymmetric indices, for a pair of four-dimensional N=1 supersymmetric gauge theories related by Seiberg duality, with gauge groups SU(n+1) and Sp(2n). This provides one of the most elaborate checks of the Seiberg duality known to date. As another application of the An integral, we prove a star-star relation for a two-dimensional integrable lattice model of statistical mechanics, previously given by the second author.
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| first_indexed | 2025-12-07T20:19:59Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209451 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T20:19:59Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kels, A.P. Yamazaki, M. 2025-11-21T19:04:03Z 2018 Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index / A. P. Kels, M. Yamazaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 41 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C67; 33E20; 81T60; 81T13; 82B23; 16T25 arXiv: 1704.03159 https://nasplib.isofts.kiev.ua/handle/123456789/209451 https://doi.org/10.3842/SIGMA.2018.013 We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the An and BCn root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in terms of the lens elliptic gamma function, a generalisation of the elliptic gamma function that depends on an additional integer variable, as well as a complex variable and two elliptic nomes. As an application of our results, we prove an equality between S¹×S³/Zr supersymmetric indices, for a pair of four-dimensional N=1 supersymmetric gauge theories related by Seiberg duality, with gauge groups SU(n+1) and Sp(2n). This provides one of the most elaborate checks of the Seiberg duality known to date. As another application of the An integral, we prove a star-star relation for a two-dimensional integrable lattice model of statistical mechanics, previously given by the second author. The main results in Theorem 3.1, and Theorem 4.1, were presented in March 2017 at the workshop “Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics”, at the Erwin Schr¨odinger Institute, in Vienna, and APK thanks the participants and organisers, particularly V.P. Spiridonov, for their comments. We also thank the anonymous referees for helpful comments, which led us to use a change of variables to write the An transformation in a way where the right hand side of (3.5) is manifestly periodic in both the complex and integer variables. Particularly, the periodicity is necessary to allow the (mod 2r) and (mod r), in the balancing condition (3.1). MY would like to thank Harvard University for its hospitality, where part of this work was performed. APK is an overseas researcher under the Postdoctoral Fellowship of Japan Society for the Promotion of Science (JSPS). MY is supported by the WPI program (MEXT, Japan), by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, by JSPS KAKENHI Grant No. 15K17634, and by JSPS-NRF research fund. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index Article published earlier |
| spellingShingle | Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index Kels, A.P. Yamazaki, M. |
| title | Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index |
| title_full | Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index |
| title_fullStr | Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index |
| title_full_unstemmed | Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index |
| title_short | Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index |
| title_sort | elliptic hypergeometric sum/integral transformations and supersymmetric lens index |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209451 |
| work_keys_str_mv | AT kelsap elliptichypergeometricsumintegraltransformationsandsupersymmetriclensindex AT yamazakim elliptichypergeometricsumintegraltransformationsandsupersymmetriclensindex |