Lectures on BPS states and spectral networks

Abstract

These are notes for a lecture series on BPS states and spectral networks, delivered at Park City Mathematics Institute, July 2019. The first part is a general review of the notions of BPS state and BPS index. The second part discusses the specific case of BPS states in N = (2, 2) supersymmetric field theories in two dimensions, and introduces the notion of spectral network as a way of computing the BPS indices in that context. The last part discusses the more general case of 2d and 4d BPS indices associated to surface defects in four-dimensional field theories of class S. 1. Lecture 1: What is a BPS state? BPS states appear very frequently in geometric applications of quantum field theory. The aim of this lecture is to explain rather generally what a BPS state is and some of their basic properties. In one sentence: we’ll study a representation H = H0 ⊕H1 of a certain super Lie algebra A = A0 ⊕A1, and the BPS states are the ones in irreps annihilated by nontrivial subspaces of A1. 1.1. Quantum mechanics A time-independent quantum system involves the following data: • A Hilbert space H, • A formally self-adjoint operator H : H→ H. We think of iH as generating the abelian Lie algebra (1.1.1) Lie(ISO(0, 1)) = Lie(Isom(R0,1)) ' R. Eigenvectors ofH are called bound states; each bound state thus spans a 1-dimensional irreducible representation of ISO(0, 1). The fact that H is formally self-adjoint implies that this representation is unitary. The eigenvalues of H are called bound state energies. Example 1.1.2 (Particle on the line). In your first course on quantum mechanics you study the “particle on the line.” For this example you need to fix a function V : R→ R. Then there is a time-independent quantum system T1[R,V] with: 2010 Mathematics Subject Classification. Primary ????; Secondary ????