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 -