Abstract
We introduce a class of new algebras, the shifted quiver Yangians, as the BPS algebras for type IIA string theory on general toric Calabi-Yau three-folds. We construct representations of the shifted quiver Yangian from general subcrystals of the canonical crystal. We derive our results via equivariant localization for supersymmetric quiver quantum mechanics for various framed quivers, where the framings are determined by the shape of the subcrystals.Our results unify many known BPS state counting problems, including open BPS counting, non-compact D4-branes, and wall crossing phenomena, simply as different representations of the shifted quiver Yangians. Furthermore, most of our representations seem to be new, and this suggests the existence of a zoo of BPS state counting problems yet to be studied in detail.
Highlights
It has been a fascinating problem in supersymmetric gauge theories and string theory to identify the BPS algebras [1] underlying the BPS (Bogomol’nyi-Prasad-Sommerfield) states and their enumerative invariants
Instead of the particular crystal from [2], we consider a general subset of the crystal, and construct a representation of the quiver Yangian acting on its molten crystals
The quiver Yangian Y(Q, W ) is the BPS algebra for type IIA string theory compactified on an arbitrary non-compact toric Calabi-Yau three-fold, in the so-called non-commutative DT chamber and without the open BPS states involved [2, 3]
Summary
It has been a fascinating problem in supersymmetric gauge theories and string theory to identify the BPS algebras [1] underlying the BPS (Bogomol’nyi-Prasad-Sommerfield) states and their enumerative invariants. Instead of the particular crystal from [2] (called the canonical crystal in this paper), we consider a general subset of the crystal, and construct a representation of the quiver Yangian acting on its molten crystals Such a general discussion requires us to extend the definition of the quiver Yangian to the shifted quiver Yangian, which is a new algebra we define in this paper. Our discussion incorporates many of the known phenomena in the studies of BPS states in the literature, such as the wall crossing phenomena and the inclusion of various non-compact D-branes (some of which gives rise to the so-called “open/closed BPS state counting”) All these different-looking BPS counting problems are unified as different representations of the shifted quiver Yangian. The main players of this story, together with the sections they appear in, are summarized in figure 1
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