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We compute the partition function of four-dimensional abelian gauge theory ona general four-torus T4 with flat metric using Dirac quantization. In additionto an SL(4, Z) symmetry, it possesses SL(2,Z) symmetry that is electromagneticS-duality. We show explicitly how this SL(2, Z) S-duality of the 4d abeliangauge theory has its origin in symmetries of the 6d (2,0) tensor theory, bycomputing the partition function of a single fivebrane compactified on T2 x T4,which has SL(2,Z) x SL(4,Z) symmetry. If we identify the couplings of theabelian gauge theory \tau = {\theta\over 2\pi} + i{4\pi\over e^2} with thecomplex modulus of the T2 torus, \tau = \beta^2 + i {R_1\over R_2}, then in thesmall T2 limit, the partition function of the fivebrane tensor field can befactorized, and contains the partition function of the 4d gauge theory. In thisway the SL(2,Z) symmetry of the 6d tensor partition function is identified withthe S-duality symmetry of the 4d gauge partition function. Each partitionfunction is the product of zero mode and oscillator contributions, where theSL(2,Z) acts suitably. For the 4d gauge theory, which has a Lagrangian, thisproduct redistributes when using path integral quantization.

Electric-magnetic Duality of Abelian Gauge Theory on the Four-torus, from the Fivebrane on T2 x T4, via their Partition Functions