't Hooft's construction of free energy, electric and magnetic fluxes, and of the partition function with twisted boundary conditions, is extended to the case of N = 4 supersymmetric Yang-Mills theories based on arbitrary compact, simple Lie groups. The transformation of the fluxes and the free energy under S-duality is presented. We consider the partition function of N = 4 for a particular choice of boundary conditions, and compute exactly its leading infrared divergence. We verify that this partition function obeys the transformation laws required by S-duality. This provides independent evidence in favor of S-duality in N = 4 theories.
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We would like to thankC . Imbimboa nd S. Mukhi for helpfuld iscussionsW. e also thankS . Elitzur, E. RabinoviciA, . SchwimmerC, . Vafa and E. Witten.A .G. would like to thankt heD epartmenotf Physicso f NYU for its kind hospitalityL.. G. is supporteidn part by the Ministerod ell' Universite della RicercaS cientificae Tecnologicab,y INFN and by ECC, contractsS CI*-CT92-0789a nd CHRX-CT92-0035A. .G. is supporteidn part by BSF - American-IsraeBli -NationaSl cienceF oundationb,y the BRF - the Basic ResearchF oundationa ndby an Alon fellowshipM. .P. is supporteidn part by NSF under grantP HY-9318171A. .Z. is supportedin part by ECC, ProjectE RBCHGCTI93073, SCI-CT93-0340C, HRX-CT93-0340.