Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

By M. Kontsevich, Y. Soibelman. 2008

894 citations recorded by the source corpus.

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).

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/90c3c7f264404ddf90365f70df4041b898f9ce69

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Citations in the corpus

Refined, Motivic, and Quantum (2009, methodology): “We claim that this is not an accident and the refined BPS invariants are actually the same as the motivic BPS invariants of Kontsevich and Soibelman, provided we identify L (resp. q) with the extra variable y2 that appears in the definition (1.4) of the refined BPS invariants5: Refined Motivic Quant”
Donaldson-Thomas theory and cluster algebras (2010, background): “ take a sequence k = (k) of length 1, then we have R(k),i= ( 0 i6= k, sk i= k. Hence we have AdT (k)[−1],±(x(k),i,±) = ( x(k),i,± i6= k, x(k),k,±(1+(yk,±) −1) i= k. (0.4) This recovers the results in [KS, pp143]. Composition ofclustertransformations Cluster algebras were introduced by Fomin and Zele”
Quantization of Integrable Systems and Four Dimensional Gauge Theories (2009, background): “the moment. Parallel developments: There are several recent developments which we feel are related to our story, including the obvious ones, such as the integrability in the AdS/CFT context, but also [43, 55, 58, 59, 60, 61]. It is tempting to conclude from the convergence of these topics that the p”
R-Twisting and 4d/2d Correspondences (2010, methodology): “This wall-crossing formula was discovered in the context of Donaldson-Thomas theory by Kontsevich-Soibelman in [3].”
Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, neutral): “These formulas will be deduced from the wallcrossing formula in the space of weak stability conditions on the derived category.”
Hilbert schemes and stable pairs: GIT and derived category wall crossings (2009, positive): “A quicker and apparently simpler proof was shown to us by Tom Bridgeland [4], but using the full force of Kontsevich and Soibelman’s theory [13] (which toric Calabi-Yau 3-folds using all of Joyce’s theory, and one using Kontsevich-Soibelman [18].”
Schur indices, BPS particles, and Argyres-Douglas theories (2015, background): “According to the wall-crossing formula of [16], crossing a wall of marginal stability the individual factors change but the operator O(q) is invariant.”