Higher rank K-theoretic Donaldson-Thomas Theory of points
We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$, in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$-fold ${{\mathbb {A}}}^3...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2021-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509421000049/type/journal_article |
| _version_ | 1811156192696729600 |
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| author | Nadir Fasola Sergej Monavari Andrea T. Ricolfi |
| author_facet | Nadir Fasola Sergej Monavari Andrea T. Ricolfi |
| author_sort | Nadir Fasola |
| collection | DOAJ |
| description | We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$, in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$-fold ${{\mathbb {A}}}^3$. We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$, that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$. Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$, where F is an equivariant exceptional locally free sheaf on a projective toric $3$-fold X. |
| first_indexed | 2024-04-10T04:47:24Z |
| format | Article |
| id | doaj.art-ae23d1d846bf4cda9632b4d6854a1d75 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:24Z |
| publishDate | 2021-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-ae23d1d846bf4cda9632b4d6854a1d752023-03-09T12:34:52ZengCambridge University PressForum of Mathematics, Sigma2050-50942021-01-01910.1017/fms.2021.4Higher rank K-theoretic Donaldson-Thomas Theory of pointsNadir Fasola0https://orcid.org/0000-0001-6222-3873Sergej Monavari1https://orcid.org/0000-0002-0181-8977Andrea T. Ricolfi2https://orcid.org/0000-0002-8172-2026SISSA Trieste, Via Bonomea 265, 34136 Trieste; E-mail:Mathematical Institute, Utrecht University, 3584 CD Utrecht; E-mail:SISSA Trieste, Via Bonomea 265, 34136 Trieste; E-mail:We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$, in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$-fold ${{\mathbb {A}}}^3$. We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$, that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$. Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$, where F is an equivariant exceptional locally free sheaf on a projective toric $3$-fold X.https://www.cambridge.org/core/product/identifier/S2050509421000049/type/journal_article14N3514C05 |
| spellingShingle | Nadir Fasola Sergej Monavari Andrea T. Ricolfi Higher rank K-theoretic Donaldson-Thomas Theory of points Forum of Mathematics, Sigma 14N35 14C05 |
| title | Higher rank K-theoretic Donaldson-Thomas Theory of points |
| title_full | Higher rank K-theoretic Donaldson-Thomas Theory of points |
| title_fullStr | Higher rank K-theoretic Donaldson-Thomas Theory of points |
| title_full_unstemmed | Higher rank K-theoretic Donaldson-Thomas Theory of points |
| title_short | Higher rank K-theoretic Donaldson-Thomas Theory of points |
| title_sort | higher rank k theoretic donaldson thomas theory of points |
| topic | 14N35 14C05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509421000049/type/journal_article |
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