On an Example of Quiver Donaldson–Thomas/Relative Gromov–Witten Correspondence

Abstract

We explain and generalize a recent example of quiver Donaldson–Thomas/relative Gromov–Witten correspondence due to Reineke–Weist by showing how to reduce it to the Gromov–Witten/Kronecker correspondence by a degeneration and blow-up. We also refine the result by working with all genera on the Gromov–Witten side and with refined Donaldson–Thomas invariants on the quiver side.

Stable maps to Looijenga pairs

“…Three of these topics have been of immediate relevance for this paper: the topological vertex formalism was one of the main tools for the proof of Theorem 1.5; the integrality of Klemm-Pandharipande invariants in Theorem 1.6 then follows effortlessly as a corollary via the strips-quivers correspondence; and finally, the higher genus open BPS statement of Theorem 1.7 is immediately suggested by recognising Ω d (Y (D)) as an LMOV partition function [87,94]. The relation to BPS invariants echoes very similar 4 statements relating log GW theory to DT and LMOV invariants in [13,17], and in particular it partly demystifies the interpretation of log GW partition functions as related to some putative open curve counting theory on a Calabi-Yau 3-fold in [17, §9] by realising the open BPS count in terms of actual, explicit special Lagrangians in a toric Calabi-Yau threefold. We defer the analysis of other strands of implications of the log/open correspondence to future work.…”

“…The following result is a new example of correspondence between Gromov-Witten invariants and quiver DT-invariants. Earlier examples can be found in [13,17,41,42,66,67,86,[127][128][129]140].…”

Witten-Dijkgraaf-Verlinde-Verlinde equation and its application to relative Gromov-Witten theory

“…We remark that in [Bou18], Reineke and Weist's result on the correspondence between relative GW-invariants and quiver Donaldson-Thomas invariants is extended to more general cases. But it is still unclear whether our recursive formula can be derived via such generalized correspondence.…”

Refined floor diagrams from higher genera and lambda classes

“…This trick to exchange tangency conditions and blow-ups has been used since the early days of Gromov-Witten theory, see for example [19]. For another examples of application of this technique closely related to the present paper, we refer to [6,7,21].…”

“…One can probably obtain a direct proof of Theorem 7.1 using a degeneration to the normal cone of the non-horizontal toric divisors of X . One should apply to this degeneration an argument in log Gromov-Witten theory, as in [6][7][8], and use Lemma 3.3. Given such direct proof, one could reverse the logic and derive Theorem 5.12 from [8], but this would go against the spirit of the present paper, which was to remain in the realm of relative Gromov-Witten theory and to not use any logarithmic technology.…”