Heterotic-Type II String Duality and the H-Monopole Problem

Since T-duality has been proved only perturbatively and most of the heterotic states map into solitonic, non-perturbative, type II states, the 6-dimensional string-string duality between the heterotic string and the type II string is not sufficient to prove the S-duality of the former, in terms of the known T-duality of the latter. We nevertheless show in detail that the perturbative T-duality, together with the heterotictype II duality, does imply the existence of heterotic H-monopoles, with the correct multiplicity and multiplet structure. This construction is valid at a generic point in the moduli space of heterotic toroidal compactifications. CERN-TH/95-217 August 1995 and Theory Division, CERN, CH-1211 Geneva 23, Switzerland e-mail [email protected] e-mail [email protected] Laboratoire Propre du CNRS UPR A.0014 e-mail [email protected] The standard approach to string theory is intrinsically perturbative: one is given a recipe whereby one computes, say, the g-loop contribution to an S-matrix element in terms of a (super)conformal 2-d field theory on a genus-g Riemann surface. Naturally, any technique shedding light on the non-perturbative dynamics of strings is of the utmost importance. One such technique is based on the conjecture that some strongly interacting string models can be rewritten in terms of other, weakly interacting, “dual,” string models. One of the better understood among string dualities is that between the heterotic string, compactified to 6 dimensions on a 4-torus T4, and the type IIA superstring, compactified on K3 [1, 2]. Evidence supporting this conjecture has been given in refs. [1, 2, 3, 4]. If this 6-dimensional heterotic-type II duality holds, it implies various results. One of the most important is that, upon further compactification of both the heterotic and type II strings to 4 dimensions on a two-torus T2, there exists a “duality of dualities” [5] (see also [6]) between the two strings. This property consists in the following: both the heterotic string, compactified on T4 × T2, and the type II string, compactified on K3 × T2, have N = 4, d = 4 supersymmetry. They are both invariant under a discrete group of target space dualities (see [7] for a review on this matter). This group contains the direct product SL(2, Z)⊗SL(2, Z). The first SL(2, Z) acts on the complex structure of the torus T2, and it is called “U-duality.” The second, called “T-duality,” acts by fractional linear transformations on the complex field T = B56 + i √ G, where √ G is the volume of the two-torus and B56 comes from the dimensional reduction on T2 of the universal antisymmetric tensor of strings. Both theories are also conjectured to be invariant under a coupling-constant duality, the “S-duality,” which also forms an SL(2, Z) group. As shown in [5, 2], under heterotic-type II duality, the Tand S-dualities are interchanged. Thus, one may be tempted to conclude that S-duality follows automatically from the 6-dimensional heterotic-type II duality, since T-duality is a well-established, perturbative symmetry of strings . This statement is not correct as it stands: perturbative T-duality is not sufficient to prove S-duality in the dual string. One obvious reason is that, for instance, the type II perturbative spectrum contains no state charged under the vectors coming from the Ramond-Ramond sector. These vectors are mapped by the heterotic-type II duality into gauge fields in the Cartan subalgebra of the heterotic gauge group (E8 ⊗ E8, for instance). Conversely, heterotic states charged under the gauge group must be mapped by heterotic-type II duality into solitonic (nonperturbative) states of the type II string. This means that in order to prove, say, that the heterotic string compactified on T4 × T2 is S-dual, one needs to prove that the type II string is T-dual non-perturbatively. Thus one has to find the action of T-duality on the non-perturbative, solitonic spectrum of this string etc. This task is obviously as complicated as a direct proof of S-duality for the heterotic string. On the other hand, perturbative T-duality of the type II string can still be of use in trying to prove S-duality of the heterotic string: one may discover perturbative states of the type II string, transforming among themselves under T-duality, which map under heterotic-type II duality into perturbative, as well as non-perturbative states of the heterotic string, transforming among themselves under S-duality. The purpose of this paper is to study in detail this scenario, where the perturbative Tduality of one string gives non-perturbative information about S-duality on the dual string. In i.e. a symmetry holding order by order in the string loop expansion.