A Strong Coupling Test of S-duality

By studying the partition function of N = 4 topologically twisted supersymmetric Yang-Mills on four-manifolds, we make an exact strong coupling test of the Montonen-Olive strong-weak duality conjecture. Unexpected and exciting links are found with two-dimensional rational conformal field theory. One of the most remarkable known quantum field theories in four dimensions is the N = 4 supersymmetric Yang-Mills theory. This theory has the largest possible number of supersymmetries for a four-dimensional theory without gravity. It is believed to be exactly finite and conformally invariant. A long-standing conjecture asserts that this theory has a symmetry exchanging strong and weak coupling and exchanging electric and magnetic fields. This conjecture originated with work of Montonen and Olive, who [1] proposed a symmetry with the above properties and also exchanging the gauge group G with the dual group G (whose weight lattice is the dual of that of G). It was soon realized that this duality was more likely to hold supersymmetrically [2] and in fact the N = 4 theory was seen to be the most likely candidate [3] since only in that case the elementary electrons and monopoles have the same quantum numbers. (It has recently been argued that an analog of Montonen-Olive duality does hold for a certain N = 2 theory with matter hypermultiplets [4].) While Montonen-Olive duality was originally proposed as a Z 2 symmetry involving the coupling constant only, the N = 4 theory has one more parameter that should be included, namely the θ angle. As was originally recognized in lattice models [5,6] and string theory [7,8], when the θ angle is included, it is natural to combine it with the gauge coupling constant g in a complex parameter τ = θ 2π + 4πi g 2. Then the Z 2 originally proposed by Olive and Montonen can be extended to a full SL(2, Z) symmetry acting on τ in the familiar fashion τ → aτ + b cτ + d ; (1.2) here a, b, c, and d are integers with ad − bc = 1, so that the matrix a b c d (1.3) has determinant 1. Indeed, SL(2, Z) is generated by the transformations S = 0 1 −1 0 (1.4) 1 and T = 1 1 0 1. (1.5) Invariance under T is the assertion that physics is periodic in θ with period 2π, and S is equivalent at θ = 0 …