Lie algebras and BPS states

Question

1. I would like to know what is a charge lattice and how it is related to the root lattice.
2. Is there a relationship between mutation of quiver and the Weyl group?
3. How do BPS states correspond to the weight in a given representation?

This post imported from StackExchange Physics at 2014-04-13 14:46 (UCT), posted by SE-user user30656

Comments

Will a moderator or the OP correct all the typos in the posed question? I can't seem to do it.

Answer 1

Let $Q=(Q_0,Q_1)$ be a quiver with vertices $Q_0$ and arrows $Q_1$, A representation of a quiver, associates a vector space to each vertex. Let $\mathbb{V}=\oplus_{j\in Q_0} V_j$. Define the dimension vector as: $\mathbf{n}=(n_j)$, where $j\in Q_0$ and $n_j=\text{dim}(V_j)$ is the dimension of the $j$-the vector space. 

In some physical examples, it turns out that BPS states might be realised as the representations of a quiver with the dimension vector being mapped on to the charge vector of the BPS state. Below are three some references where quivers are used to describe half-BPS states in $\mathcal{N}=2$ quantum field theory and string theory.

1. BPS Quivers and Spectra of Complete N=2 Quantum Field Theories
2. The spectrum of BPS branes on a noncompact Calabi-Yau (the appendix has a nice introduction to quivers)
3. D-branes, Exceptional Sheaves and Quivers on Calabi-Yau manifolds: From Mukai to McKay

4. Generalized quiver mutations and single-centered indices 

As  discussed in reference 1, there are two kinds of walls across which the quiver description gets modified. As one crosses a wall, the quiver changes and in some (all?) cases, it can be understood as a mutation of a quiver. Additional data in the form of  the central charge, a complex valued function, $Z(\mathbf{n})$ determines the walls. Mutations also provide a realization of Seiberg duality (see ref 4 above as well), In quivers with loops, there might be an associated superpotential (see references above for the definition of a superpotential). 

A connection to  Lie algebra follows from Gabriel's theorem and its generalizations which relates quivers to finite Lie algebras. One suspects that more general Lie algebras might be associated to all quivers but there is no such general theorem that I am aware of. However, see this paper: On the algebras of BPS states.