BPS States, Weight Spaces and Vanishing Cycles

We review some simple group theoretical properties of BPS states, in relation with the singular homology of level surfaces. Primary focus is on classical and quantum N = 2 supersymmetric Yang-Mills theory, though the considerations can be applied to string theory as well. There has been dramatic recent progress in understanding non-perturbative properties of certain supersymmetric field and string theories. A common feature of such theories are singularities in the quantum moduli space, arising from certain BPS states becoming massless in these vacuum parameter regions. Much of the present discussion is centered at the properties of such states. Such states typically have various different representations, for example, a representation in terms of ordinary elementary fields, or, in a dual formulation, in terms of nonperturbative solitonic bound states. We will first discuss some generic properties of BPS states, with main emphasis on N=2 supersymmetric theories. We will then concretely specialize further below to classical and quantum Yang-Mills theory. The mass of a BPS state is directly given in terms of the central charge Z of the relevant underlying N=2 or N=4 supersymmetry algebra, m ≃ |Z| , Z ≡ ~q · ~a+ ~g · ~aD . (1) Here, ~q,~g are the electric and magnetic charges of the state in question, and ~a,~aD are the classical values of the Higgs field and “magnetic dual” Higgs field, respectively. An important insight is that Z can be represented as a period integral of some prime form λ on a suitable “level” surface X , ie.,