On the Existence of States Saturating the Bogomol’nyi Bound in N=4 Supersymmetry

We give an argument showing that in N=4 supersymmetric gauge theories there exists at least one bound state saturating the Bogomol’nyi bound with electric charge p and magnetic charge q, for each p and q relatively prime, and we comment on the uniqueness of such state. This result is a necessary condition for the existence of an exact S-duality in N=4 supersymmetric theories. On leave of absence from I.N.F.N., sez. di Pisa, Pisa, Italy. E-mail: [email protected] A mounting body of evidence [1, 2, 3] supports the conjecture that N=4 rigid supersymmetric theories possess an SL(2, Z) duality. This SL(2, Z) contains as a subgroup the strong-weak duality of Montonen and Olive [4, 5, 6]. Let us confine our attention to an N=4 theory with gauge group SU(2) spontaneously broken to U(1), for simplicity. In this case the duality acts by fractional transformations on the complex number S = θ/2π + i4π/g, which combines together the theta angle θ and the gauge coupling constant g S → aS + b cS + d , a, b, c, d ∈ Z, ad− bc = 1. (1) The meaning of duality is that theories whose coupling constants are related by a transformation of type (1) are physically equivalent. In particular, the stable physical states of the two theories are the same, up to an eventual relabeling of them. Among the physical states of the theory there is a special subset: those that saturate the Bogomol’nyi bound [7]. These states belong to short multiplets of N=4 and thus their mass is non renormalized [8]. By denoting with p, q ∈ Z the electric and magnetic charge of one such state, its mass is given by the following formula [8, 2] M(p, q) = 4πV 2 ImS (p + 2ReSpq + |S|q). (2) Here V is a constant independent of p, q and S. Not all assignments of p and q need give a stable state. Indeed, a (p, q) state may decay in a pair [(p1, q1), (p2, q2)] p1 + p2 = p, q1 + q2 = q, whenever M(p, q) ≥ M(p1, q1) +M(p2, q2). (3) Since eq. (2) defines a positive-definite scalar product (ImS > 0), the triangular inequality holds: M(p, q) ≤ M(p1, q1) +M(p2, q2), with the equality attained when p1/q1 = p2/q2 = p/q. The charges are integers, thus this is possible only when p and q have a common divisor p = Nm, q = Nn, N, n,m ∈ Z. (4) This argument, due to Sen [2, 3], does not guarantees that states obeying the above equations actually decay, but it does shows that all Bogomol’nyi states with p, q relatively prime are stable. Moreover, Bogomol’nyi states are transformed by S-duality. If this duality exists, eq. (2) must be invariant under the SL(2, Z) transformation (1), together with a relabeling of p and q, which turns out to be p → ap− bq, q → −cp+ dq. (5) If S-duality holds, eq. (5) maps the multiplet with p = 1, q = 0 (the electrically charged vector multiplet of the SU(2) theory, corresponding to a broken generator of the gauge group) into states with p = a, q = −c. These states must also be stable, by consistency. This property holds because the constraints ad− bc = 1, a, b, c, d ∈ Z imply that p and q are relatively prime. Stable states with p = 0, 1 and q = 0, 1 are well known: they are, respectively, the elementary charged states of the SU(2) gauge theory, the BPS monopole, and the dyon. A crucial test of S-duality is to verify whether all states with p, q relatively prime do indeed exist with the right