### N= 4 Dyons

Ashoke Sen
2009 Progress of Theoretical Physics Supplement
We review recent developments in our understanding of the spectrum of quarter BPS dyons in N = 4 supersymmetric string theories. It is a great honour for me to speak at this conference celebrating Professor Tohru Eguchi's 60th birthday conference. Professor Eguchi's work has influenced my own work in many ways. The famous Physics Reports article by Eguchi, Gilkey and Hanson 1) was my first introduction to the mathematical aspects of quantum field theories. In this talk I plan to describe some
more » ... to describe some recent progress in understanding the spectrum of quarter BPS dyons in N = 4 supersymmetric string theories. The plan of the lecture is as follows: 1. In the first part of the talk I shall describe some general properties of dyon partition function in a wide class of N = 4 supersymmetric string theories and convert them into specific conjectures for all N = 4 supersymmetric string theories. 2. In the second part of the talk I shall take up the particular example of heterotic string theory on T 6 and show how these general properties are realized in this theory. The study of dyo apectrum in N = 4 supersymmetric string theories was initiated in Ref. 2). Most of the material covered in this talk can be found in Refs. 3), 4) and references given in these papers. We begin by reviewing some general properties of N = 4 supersymmetric string theories. A generic N = 4 supersymmetric string theory in four dimensions has R U(1) gauge fields (R ≥ 6). 6 of these gauge fields may be identified as the graviphotons and the rest as part of (R − 6) matter multiplets. As a result a dyon carries (electric, magnetic) charges (Q, P ) where Q and P are R-dimensional vectors. We shall denote by d(Q, P ) the number of quarter BPS states with charge (Q, P ) weighted by (−1) F (2h) 6 /6!. Here F is the fermion number and h is the helicity carried by the state. d(Q, P ) can be shown to be a non-vanishing and protected index, in the sense that it does not change continuously as we vary the asymptotic moduli. However d(Q, P ) jumps across walls of marginal stability in the moduli space on which the dyon can decay into a pair of half-BPS states: (Q, P ) → (αQ+βP, γQ+δP )+(δQ−βP, −γQ+αP ) , αδ = βγ, α+δ = 1 . (1) Due to these jumps across the walls of marginal stability, d(Q, P ) depends not only on (Q, P ) but also on the domain in which the moduli lie. A convenient way of labelling the domains is as follows. Consider the ith wall bordering a domain on