Motivic Donaldson–Thomas invariants of the conifold and the refined topological vertex

Abstract

We compute the motivic Donaldson-Thomas theory of the resolved conifold, in all chambers of the space of stability conditions of the corresponding quiver. The answer is a product formula whose terms depend on the position of the stability vector, generalizing known results for the corresponding numerical invariants. Our formulae imply in particular a motivic form of the DT/PT correspondence for the resolved conifold. The answer for the motivic PT series is in full agreement with the prediction of the refined topological vertex formalism.

Exploring 5d BPS Spectra with Exponential Networks

“…A standard physics description of generalized DT involves D6-D4-D2-D0 boundstates in type IIA string theory on X. The relation to BPS counting in M theory was explained in [3] (see also [4,5,[38][39][40][41][42]). On the other side, we will repackage these enumerative invariants as counting special Lagrangians in the mirror Calabi-Yau Y , a problem that arises naturally once we consider the insertion of defects in the 5d gauge theory, engineered by an M5 brane wrapped on a Lagrangian submanifold L ⊂ X.…”

“…Therefore both (more precisely, their KK modes) are expected to be captured by the 2d-4d wall-crossing described by the nonabelianization map. It would be especially interesting to carry out the computations in the conifold, to compare with [40,42]. Other interesting cases include local P 2 and elliptic fibrations over Hirzebrüch surfaces or del Pezzo surfaces.…”

Donaldson–Thomas transformations of moduli spaces of G-local systems

“…The following useful Lemma 1.8, due to [KS,Th.6] see also [MMNS,Lemma 1.12], expresses the series E q as a product, possibly infinite, of the "q-powers" of the quantum dilogarithm power series Ψ q pXq. Namely, given a formal Laurent series Ωpqq P Qppqqq, let us set Ψ q pXq˝Ω pqq " exp´ÿ ně1 p´1q n`1 Ωpq n q npq n´q´n q X n¯.…”

An Introduction to (Motivic) Donaldson-Thomas Theory

“…The case of zero potential has been intensively studied by M. Reineke in a series of papers [30,31,32]. Despite some computations of motivic or even numerical Donaldson-Thomas invariants for quivers with potential (see [2,6,4,27]), the true nature of Donaldson-Thomas invariants for quiver with potential remained mysterious for quite some time. A full understanding has been obtained recently and is the content of a series of papers [7,5,26,24].…”