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# Communications in Number Theory and Physics

## Volume 3 (2009)

### Number 1

### Borcherds–Kac–Moody symmetry of $\CN=4$ dyons

Pages: 59 – 110

DOI: http://dx.doi.org/10.4310/CNTP.2009.v3.n1.a2

#### Authors

#### Abstract

We consider compactifications of heterotic string theory tofour dimensions on CHL (Chaudhuri-Hockney-Lykken) orbifolds of the type $T^6/\mathbb{Z}_N$ with $\CN=4$ supersymmetry. The exactpartition functions of the quarter-BPS (Bogomol'nyi-Prasad-Sommerfeld) dyons in thesemodels are given in terms of genus-two Siegel modularforms. Only the $N=1,2,3$ models satisfy a certainfiniteness condition, and in these cases one can identify aBorcherds–Kac–Moody superalgebra underlying the symmetrystructure of the dyon spectrum. We identify the real roots,and find that the corresponding Cartan matrices exhaust aknown classification. We show that the Siegel modular formsatisfies the Weyl denominator identity of the algebra,which enables the determination of all root multiplicities.Furthermore, the Weyl group determines the structure ofwall-crossings and the attractor flows of the theory. For$N> 4$, no such interpretation appears to be possible.