Article contents
Higher rank K-theoretic Donaldson-Thomas Theory of points
Published online by Cambridge University Press: 02 March 2021
Abstract
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.
As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.
MSC classification
- Type
- Mathematical Physics
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline9.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline10.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline11.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline12.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline13.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline14.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline15.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline16.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline17.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline18.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline19.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210302035453718-0013:S2050509421000049:S2050509421000049_inline20.png?pub-status=live)
- 15
- Cited by