Title: Quiver Algebras from Counting BPS States

Abstract: The BPS states in a supersymmetric gauge theory are believed to form an algebra. For 4d N=1 theories arised from D-branes probing toric Calabi-Yau threefolds, the BPS algebras known as the quiver Yangians (and their toroidal and elliptic cousins) can be constructed from the crystal melting models used for BPS counting, where the crystals serve as the representations of the quiver Yangians. In this talk, I will discuss how the quiver BPS algebras can be recovered from the BPS partition functions using the JK residue formula. This would also allow us to see to what extent the quiver BPS algebras and the crystal representations can be extended to more general quivers. From the JK residue formula, there is another type of quiver algebras that can be constructed encoding the information of the BPS states. I will compare them with the quiver Yangians, and apply this different type of quiver algebras to 2d N=(0,2) theories such as those from toric Calabi-Yau fourfolds.