RG flows with N=1 supersymmetry

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The dilaton effective action plays a key role for the recent proof of the a-theorem by Schwimmer and Komargodski. In the presence of other massless modes, one may ask if this proof is affected. In particular, in renormalization group (RG) flows with N=1 supersymmetry, there is a natural massless partner of the dilaton, namely an axion field. I will discuss RG flows, the a-theorem, and the form of the N=1 supersymmetric dilaton-axion effective action and its physics.