TY - JOUR

T1 - Higher rank K-theoretic Donaldson-Thomas Theory of points

AU - Fasola, Nadir

AU - Monavari, Sergej

AU - Ricolfi, Andrea T.

N1 - Publisher Copyright:
© The Author(s), 2021. Published by Cambridge University Press.

PY - 2021/3/2

Y1 - 2021/3/2

N2 - We exploit the critical structure on the Quot scheme QuotA3(O⊕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 A3. 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 r > 1, that the invariants do not depend on the equivariant parameters of the framing torus (*)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 A3 in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

AB - We exploit the critical structure on the Quot scheme QuotA3(O⊕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 A3. 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 r > 1, that the invariants do not depend on the equivariant parameters of the framing torus (*)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 A3 in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

KW - 14C05

KW - 14N35

KW - 2020 Mathematics subject classification

UR - http://www.scopus.com/inward/record.url?scp=85102006671&partnerID=8YFLogxK

U2 - 10.1017/fms.2021.4

DO - 10.1017/fms.2021.4

M3 - Article

AN - SCOPUS:85102006671

SN - 2050-5094

VL - 9

SP - 1

EP - 51

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e15

ER -