Curve counting theories via stable objects I. DT/PT correspondence

By Yukinobu Toda. 2009

162 citations recorded by the source corpus.

The Donaldson-Thomas invariant is a curve counting invariant on Calabi-Yau 3-folds via ideal sheaves. Another counting invariant via stable pairs is introduced by Pandharipande and Thomas, which counts pairs of curves and divisors on them. These two theories are conjecturally equivalent via generating functions, called DT/PT correspondence. In this paper, we show the Euler characteristic version of DT/PT correspondence, using the notion of weak stability conditions and the wall-crossing formula.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9d0f62bce1a6ce7432c1f4972e35cc33395cf998

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities (2011, unknown): “ll µ (6) ω,B-stableF with µω,B(F) = 0i. 12 AREND BAYER, EMANUELE MACR`I, AND YUKINOBU TODA In case B = 0, the above category contains the heart of the category of D0-D2-D6 bound states constructed in [Tod10]. We deﬁne a slope νω,B on Bω,B as follows. If E ∈ Bω,B satisﬁes ω2chB 1 (E) = 0, we set νω,B”
Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, methodology): “A G ] 2 9 Se p 20 09 The goal of the present paper is to show the transformation formula of DonaldsonThomas invariants on smooth projective Calabi-Yau 3-folds under birational transformations via categorical method.”
A theory of generalized Donaldson–Thomas invariants (2008, methodology): “We would like especially to thank Tom Bridgeland and Richard Thomas for lots of help with various parts of this project, and also to thank Kai Behrend, Jim Bryan, Daniel Fox, Spiro Karigiannis, Sheldon Katz, Bernhard Keller, Alastair King, Martijn Kool, Alan Lauder, Davesh Maulik, Sven Meinhardt, To”
The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds (2014, methodology): “We will use a variant of the notion of “weak stability” of [42], adapted to our situation: Definition 12.”
Gromov-Witten/pairs correspondence for the quintic 3-fold (2012, methodology): “For Calabi-Yau 3-folds, the formulation in terms of stable pairs [34] was proven to be equivalent in [4, 40].”
Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, methodology): “In Section 5, we give a proof of the rationality of the generating series of PT invariants in the framework of [37].”
Refined, Motivic, and Quantum (2009, background): “In the classical limit y → 1 (corresponding to q1 = q2 = q) all of these refinements reduce to the ordinary MacMahon function M(q), while in the “opposite” limit y → −1 they specialize to M(−q), which describes the contribution of the 0-dimensional subschemes to the D̂T-invariants of [24].”