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Higher rank K-theoretic Donaldson-Thomas Theory of points

Part of: Projective and enumerative geometry Cycles and subschemes

Published online by Cambridge University Press:  02 March 2021

Nadir Fasola Open the ORCID record for Nadir Fasola [Opens in a new window] ,
Sergej Monavari Open the ORCID record for Sergej Monavari [Opens in a new window]  and
Andrea T. Ricolfi Open the ORCID record for Andrea T. Ricolfi [Opens in a new window]
Nadir Fasola
Affiliation:
SISSA Trieste, Via Bonomea 265, 34136Trieste; E-mail: nfasola@sissa.it
Sergej Monavari
Affiliation:
Mathematical Institute, Utrecht University, 3584 CDUtrecht; E-mail: s.monavari@uu.nl
Andrea T. Ricolfi
Affiliation:
SISSA Trieste, Via Bonomea 265, 34136Trieste; E-mail: aricolfi@sissa.it

Abstract

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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

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14C05: Parametrization (Chow and Hilbert schemes)
Type
Mathematical Physics
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e15
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Creative Commons
Creative Common License - CCCreative Common License - BY
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

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