SL(2, Z) Multiplets of Type II Superstrings in D < 10

It has been shown recently that the toroidally compactified type IIB string effective action possesses an SL(2, R) invariance. Using this symmetry we construct an infinite family of macroscopic string-like solutions permuted by SL(2, Z) group for type II superstrings in 4 ≤ D < 10. These solutions, which formally look very similar to the corresponding solutions in D = 10, are characterized by two relatively prime integers corresponding to the ‘electric’ charges associated with the two antisymmetric tensor fields of the strings. Stability of these solutions is discussed briefly in the light of charge conservation and the tension gap equation. E-mail address: [email protected] It is well-known that the equations of motion of type IIB supergravity theory are invariant under an SL(2, R) group known as the supergravity duality group [1]. A discrete subgroup of this group has been conjectured to be the exact symmetry of the full quantum type IIB string theory [2]. As the string theory coupling constant transforms non-trivially under this SL(2, R) transformation [3,4,5], this symmetry is non-perturbative. So, in general it is not possible to prove this conjecture in the perturbative framework of string theory. A strong evidence in favor of this conjecture has been given by Schwarz [4] when he showed that certain BPS saturated macroscopic string-like solutions of type IIB string theory form an SL(2, Z) multiplet. These solutions, when characterized by two relatively prime integers corresponding to the charges associated with the two antisymmetric gauge fields (from NS-NS and R-R sectors), are stable and do not decay further into strings with lower charges [6]. The tensions as well as the charges associated with the strings have been shown to be given by SL(2, Z) covariant expressions. It has been shown recently that this SL(2, R) invariance of the type IIB theory survives the toroidal compactification [7,8]. In fact, this is not surprising since a symmetry in a higher dimensional theory should become a part of the bigger symmetry in the lower dimensional theory, although in this case, it requires quite non-trivial manipulation to prove the invariance. So, eventhough SL(2, R) remains a symmetry group of type II theory in D < 10, the whole symmetry group, known as the U-duality group [1,2] (E9, E8, E7, E6, SL(5), SO(5, 5), SL(3) × SL(2), SL(2) × SO(1, 1) for D = 2, 3, 4, 5, 6, 7, 8, 9 respectively) is much bigger and contains both the T-duality group [9] (O(10−D,10−D) in D-dimensions) and the S-duality group [10,11] (SL(2, R)) as the subgroup. In this paper, we will make use of the SL(2, R) invariance of type II theory in D-dimensions for 4 ≤ D < 10, to construct the SL(2, Z) multiplets of macroscopic string-like solutions. Since in ten dimensions an SL(2, Z) multiplet of string-like solutions in type IIB theory has already been shown to exist by Schwarz, it is not difficult to understand that such solutions should also exist in 4 ≤ D < 10 by straight dimensional reduction. For D ≤ 8 there must exist more string-like solutions which should form multiplets of bigger symmetry group of type II theory, but we will not construct those in this paper. As in the ten dimensional case, the solutions constructed here will be characterized by two relatively prime integers and will be shown to form a stable spectrum as they are prevented from decaying into other states by the charge conservation and the tension gap relation.