2016/11/20 by Luo, Wei, Zhu, Shengmao · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1611.06506
We study the open string integrality invariants (LMOV invariants) for toric Calabi-Yau 3-folds with Aganagic-Vafa brane (AV-brane). In this paper, we focus on the case of the resolved conifold with one out AV-brane in any integer framing τ, which is the large N duality of the Chern-Simons theory for a framed unknot with integer framing τ in S3. We compute the explicit formulas for the LMOV invariants in genus g=0 with any number of holes, and prove their integrality. For the higher genus LMOV invariants with one hole, they are reformulated into a generating function gm(q,a), and we prove that gm(q,a)∈ (q1/2-q-1/2)-2ℤ[(q1/2-q-1/2)2,a± 1/2] for any integer m≥ 1. As a by product, we compute the reduced open string partition function of ℂ3 with one AV-brane in framing τ. We find that, for τ≤ -1, this open string partition function is equivalent to the Hilbert-Poincaré series of the Cohomological Hall algebra of the |τ|-loop quiver. It gives an open string GW/DT correspondence.