We prove the transformation formula of Donaldson-Thomas (DT) invariants counting two dimensional torsion sheaves on Calabi-Yau 3-folds under flops. The error term is described by the Dedekind eta function and the Jacobi theta function, and our result gives evidence of a 3-fold version of Vafa-Witten's S-duality conjecture. As an application, we prove a blow-up formula of DT type invariants on the total spaces of canonical line bundles on smooth projective surfaces. It gives an analogue of the similar blow-up formula in the original S-duality conjecture by Yoshioka, Li-Qin and G "ottsche.
A flop formula for DT type curve counting invariants was obtained in the papers @cite_20 , @cite_21 , @cite_0 , @cite_3 . Among them, the papers @cite_0 , @cite_3 (also see @cite_36 ) use similar Hall algebra methods, but we need to work with the relevant abelian category @math which did not appear in the above papers. On the other hand, there are few mathematical literatures in which DT invariants of the form @math are studied. @cite_4 , the modularity of these invariants is discussed for nodal K3 fibrations using degeneration formula. @cite_40 , @cite_38 , some relationship between the invariants @math and DT type curve counting invariants are studied. @cite_5 , the invariant @math on local @math is studied for small @math . In physics literatures, a few of the D4 brane counting which corresponds to the invariants of the form @math are computed @cite_35 , @cite_32 . Also the flop formula of D4D2D0 bound states on the resolved conifold is studied in @cite_14 , @cite_16 using Kontsevich-Soibelman's wall-crossing formula @cite_23 . The result of Theorem is interpreted as a mathematical justification and a generalization of the arguments in the physics articles @cite_14 , @cite_16 .
@cite_38 @cite_35 @cite_14 @cite_4 @cite_36 @cite_21 @cite_32 @cite_3 @cite_0 @cite_40 @cite_23 @cite_5 @cite_16 @cite_20
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We describe a correspondence between the Donaldson–Thomas invariants enumerating D0–D6 bound states on a Calabi–Yau 3-fold and certain Gromov–Witten invariants counting rational curves in a family of blowups of weighted projective planes. This is a variation on a correspondence found by Gross–Pandharipande, with D0–D6 bound states replacing representations of generalised Kronecker quivers. We build on a small part of the theories developed by Joyce–Song and Kontsevich–Soibelman for wall-crossing formulae and by Gross–Pandharipande–Siebert for factorisations in the tropical vertex group. Along the way we write down an explicit formula for the BPS state counts which arise up to rank 3 and prove their integrality. We also compare with previous “noncommutative DT invariants” computations in the physics literature. We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ("number of BPS states with given charge" in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit "as the motive of affine line approaches to 1" we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum). We prove a comparison formula for the Donaldson-Thomas curve-counting invariants of two smooth and projective Calabi-Yau threefolds related by a flop. By results of Bridgeland any two such varieties are derived equivalent. Furthermore there exist pairs of categories of perverse coherent sheaves on both sides which are swapped by this equivalence. Using the theory developed by Joyce we construct the motivic Hall algebras of these categories. These algebras provide a bridge relating the invariants on both sides of the flop. We study the wall-crossing phenomena of D4-D2-D0 bound states with two units of D4-branechargeontheresolvedconifold. We identify the walls of marginal stability and evaluate the discrete changes of the BPS indices by using the Kontsevich-Soibelman wall-crossing formula. In particular, we find that the field theories on D4-branes in two large radius limits are properly connected by the wall-crossings involving the flop transition of the conifold. We also find that in one of the large radius limits there are stable bound states of two D4-D2-D0 fragments.