We construct the half-supersymmetric instanton solutions that are electric-magnetically dual to the recently discussed half-supersymmetric Q7-branes. We call these instantons `Q-instantons. Whereas the D-instanton is most conveniently described using the RR axion chi and the dilaton phi, the Q-instanton is most conveniently described using a different set of fields chi and T, where chi is an axionic scalar. The real part of the Q-instanton on-shell action is a function of T and the imaginary part is linear in chi. Discrete shifts of the axion chi correspond to PSL(2,Z) transformations that are of finite order. These are e.g. pure S-duality transformations relating weak and strongly coupled regimes. We argue that near each orbifold point of the quantum axion-dilaton moduli space PSL(2,Z)PSL(2,R)/SO(2) the higher order R^4 terms in the string effective action contain contributions from an infinite sum of single multiply-charged instantons with the Q-instantons corresponding to the orbifold points tau=i,rho where tau is the complex axion-dilaton field.
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