Derived categories of small toric Calabi-Yau 3-folds and counting invariants

Source: arXiv


We first construct a derived equivalence between a small crepant resolution
of an affine toric Calabi-Yau 3-fold and a certain quiver with a
superpotential. Under this derived equivalence we establish a wall-crossing
formula for the generating function of the counting invariants of perverse
coherent systems. As an application we provide certain equations on
Donaldson-Thomas, Pandeharipande-Thomas and Szendroi's invariants. Finally, we
show that moduli spaces associated with a quiver given by successive mutations
are realized as the moduli spaces associated the original quiver by changing
the stability conditions.

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    • "This is called wall-crossing phenomena, which make it valuable and interesting to study counting the BPS states. The BPS counting problem and the wall-crossing phenomena are important also in many other research areas, such as black hole microstates [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Donaldson- Thomas invariants [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25], topological strings and instanton counting [26] [27] [28] [29] [30] [31] [32], M-theory viewpoint [33] [34], exact counting of N = 4 dyons [35] [36] [37] [38] [39], supersymmetric gauge theories [40] [41] [42] [43] and many others [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]. "
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    ABSTRACT: We study the spectrum of BPS D5-D3-F1 states in type IIB theory, which are proposed to be dual to D4-D2-D0 states on the resolved conifold in type IIA theory. We evaluate the BPS partition functions for all values of the moduli parameter in the type IIB side, and find them completely agree with the results in the type IIA side which was obtained by using Kontsevich-Soibelman's wall-crossing formula. Our result is a quite strong evidence for string dualities on the conifold.
    Preview · Article · Jul 2011 · Journal of High Energy Physics
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    • "Interestingly enough, only for symmetric brane tilings there are known explicit formulas for the classical Donaldson-Thomas invariants (also called generalized Donaldson-Thomas invariants [19]), see e.g. [13] [27] [28] [33] [36] [37]. The author expects that the motivic Donaldson-Thomas invariants can also be computed in all these cases. "
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    ABSTRACT: We prove wall-crossing formulas for the motivic invariants of the moduli spaces of framed objects in the ind-constructible abelian categories. Developed techniques are applied in the case of the motivic Donaldson-Thomas invariants of quivers with potentials. Another application is a new proof of the formula for the motivic invariants of smooth models of quiver moduli spaces.
    Preview · Article · Apr 2011 · The Quarterly Journal of Mathematics
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    • "One class of such systems, which we also analyze in this paper, involves string theory on toric Calabi-Yau manifolds (more precisely, those which contain no compact four-cycles, as we will explain in what follows). This class has been studied both from physical [6] [7] [8] [9] [10] and mathematical [11] [12] [13] [14] [15] points of view. Physically it concerns the counting of bound states of D0 and D2 branes, wrapping cycles of such toric manifolds, to a single D6-brane. "
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    ABSTRACT: We consider wall-crossing phenomena associated to the counting of D2-branes attached to D4-branes wrapping lagrangian cycles in Calabi-Yau manifolds, both from M-theory and matrix model perspective. Firstly, from M-theory viewpoint, we review that open BPS generating functions in various chambers are given by a restriction of the modulus square of the open topological string partition functions. Secondly, we show that these BPS generating functions can be identified with integrands of matrix models, which naturally arise in the free fermion formulation of corresponding crystal models. A parameter specifying a choice of an open BPS chamber has a natural, geometric interpretation in the crystal model. These results extend previously known relations between open topological string amplitudes and matrix models to include chamber dependence. Keywords:Matrix Models–Topological Strings–M-Theory
    Preview · Article · Mar 2011 · Journal of High Energy Physics
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