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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.
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- Mathematical Physics
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- 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.
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- © The Author(s), 2021. Published by Cambridge University Press
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