Branes, Quivers and BPS Algebras

Abstract

These lecture notes cover a brief introduction into some of the algebro-geometric techniques used in the construction of BPS algebras. The first section introduces the derived category of coherent sheaves as a useful model of branes in toric Calabi-Yau three-folds. This model allows a rather simple derivation of quiver quantum mechanics describing low-energy dynamics of various brane systems. Vacua of such quantum mechanics can be identified with the critical equivariant cohomology of the moduli space of quiver representations. These are often counted by various crystal configurations. Using correspondences in algebraic geometry, one can construct rich families of affine-Yangian representations. We conclude with an exploration of different algebraic structures naturally appearing in our story. The material was covered in a 4-lecture mini-course within the Second PIMS Summer School on Algebraic Geometry in High-Energy Physics. The text contains some new ideas, examples and remarks that are going to be covered in detail in a joint work with Dylan Butson. Contents 1. Physical motivation 2. Quivers from branes 2.1. Derived category of coherent sheaves as a brane category 2.2. Morphisms in the brane category 2.3. Framed quivers 2.4. Higgs field 2.5. Potential 2.6. Examples of framed quivers with potential 3. Supersymmetric vacua 3.1. Quiver quantum mechanics 3.2. Twisted masses and equivariance 3.3. Critical equivariant cohomology 3.4. Example of equivariant cohomology 3.5. D2 and 1d partitions 3.6. D4 and 2d partitions 3.7. D6 and 3d partitions 4. Modules from correspondences 4.1. Correspondences 4.2. D2-brane and the Weyl algebra 4.3. D4-brane and the p gl 1 Kac-Moody algebra 4.4. D6-brane and the MacMahon module 5. Cherednik algebras and W-algebras 5.1. Affine Yangian of gl 1 and its shifts 5.2. Coproduct

BPS Hall algebra of scattering Hall states

“…The notion of the BPS algebra goes back to [3], and has been developed intensely since (see [4,5] and references therein for recent reviews). We will further nail down our interest mostly to a specific family of BPS algebras known as quiver Yangians [6,7] and their generalizations (see e.g.…”

“…The reader is encouraged to review an expository summary in [9,59,60]. If one considers a brane of a lower dimension wrapping a divisor inside a brane covering the whole CY 3 , for example, a C 2 -cycle inside C 3 [5,[61][62][63] one could cut out of the 3d crystal a 2d slice of one level points in, say, the R-charge direction. In this case eigen values of σ-fields acquire expectation values corresponding to flavor weights of path operators in the complex σ-plane.…”

Crystal melting, BPS quivers and plethystics

“…As we have an infinite number of BPS degeneracies with some structures therein, it is natural to expect a BPS algebra acting on the BPS states [37]. In the C 3 case which has been extensively studied in literature, the affine Yangian of gl 1 , Y gl 1 , acts on the plane partition and it enumerates the BPS states [38][39][40][41][42]. In particular, the BPS partition function is the character for the vacuum module of Y gl 1 .…”

Crystal Melting, BPS Quivers and Plethystics

“…Quiver Yangians As we have an infinite number of BPS degeneracies with some structures therein, it is natural to expect a BPS algebra acting on the BPS states [37]. In the C 3 case which has been extensively studied in literature, the affine Yangian of gl 1 , Y gl 1 , acts on the plane partition and it enumerates the BPS states [38][39][40][41][42]. In particular, the BPS partition function is the character for the vacuum module of Y gl 1 .…”