Charge quantization of axion-dilaton black holes

We present axion-dilaton black-hole and multi-black-hole solutions of the low-energy string effective action. Under $\mathrm{SL}(2, R)$ electric-magnetic duality rotations only the "hair" (charges and asymptotic values of the fields) of our solutions is transformed. The functional form of the solutions is duality invariant. Axion-dilaton black holes with zero entropy and zero area of the horizon form a family of stable particlelike objects, which we call holons. We study the quantization of the charges of these objects and its compatibility with duality symmetry. In general the spectrum of black-hole solutions with quantized charges is not invariant under $\mathrm{SL}(2, R)$ but only under $\mathrm{SL}(2, Z)$ or one of its subgroups ${\Gamma }_l$. Because of their transformation properties, the asymptotic value of the axion-dilaton field of a black hole may be associated with the modular parameter $\tau$ of some complex torus and the integer numbers ($n, m$) that label its quantized electric and magnetic charges may be associated with winding numbers.