The d = 6, N = 1 heterotic string does not ’live’ in six dimensions

We discuss why the N6 = 1 heterotic string has to be viewed as something similar to a “non-compact orbifold”. Only the perturbative spectrum is forced to satisfy the constraints imposed by the vanishing of six-dimensional anomalies. These do not apply to the states of the non-perturbative spectrum, such as those appearing when small instantons shrink to zero size. Research supported by the “Marie Curie” fellowship HPMF-CT-1999-00396. Extended version of a work presented at ”Modern Trends in String Theory”, Lisbon, 13–17 July 2001. e-mail: [email protected] The massless spectrum of the heterotic string in ten dimensions is determined by the constraint imposed by the vanishing of the Green–Schwarz anomaly, that requires the presence, together with theN10 = 1 supergravity multiplet, of 496 gauge bosons; the only two solutions for the gauge group G are G = E8 × E8 and G = SO(32). The (massless) spectrum of the string in lower dimensions is obtained by “dimensional reduction”. Namely, the degrees of freedom remain the same, but they are differently interpreted, in terms of lower dimensional fields, arranged into multiplets of the appropriate Nd supersymmetry in d dimensions. In less than ten dimensions, the Green-Schwarz anomaly constraint doesn’t apply anymore, and the gauge and matter spectrum can be varied, by introducing Wilson lines. In six dimensions, it is also possible to reduce supersymmetry from the N6 = 2, as derived by toroidal compactification of the N10 = 1, to N6 = 1, by compactifying the four coordinates on a curved space. As is known, this space is unique, the K3 surface. Although not uniquely determined, the massless spectrum is nevertheless highly constrained: it must satisfy a constraint derived by requiring the cancellation of the N6 = 1, TrR 4 anomaly. This constraint reads [1]: NH − NV + 29NT = 273 . (0.1) The N6 = 2 heterotic string has been conjectured to be dual to the type IIA string compactified on the K3 surface [2]. When both the theories are further compactified on a two-torus, T , this translates into a duality between N4 = 4 theories, that maps a modulus of the gravity multiplet into a modulus of the vector manifold [3]. The N6 = 1 theory on the other hand does not possess a type IIA dual. However, when this theory is toroidally compactified on a two-torus, it is expected to be dual to the type IIA string compactified on a K3 fibration. The heterotic dilaton–axion field maps then to the volume form of the fibration. It may seem a bit strange that, while the N4 = 4 duality exists also in six dimensions, this duality of the N4 = 2 theory appears only in four dimensions. ¿From a technical point of view, this is related to the fact that it is not possible to construct an N6 = 1 type II theory. But it also means that, if string-string duality has to be taken seriously, the operation of compactification of the N6 = 1 heterotic theory on T 2 is not so an innocuous one: something very special must happen at the level of the “string theory”, namely the theory conjectured to be the basic one, underlying all the specific manifestations, whether heterotic or type II, or type I string constructions. ¿From that point of view, the two-torus must not be a “flat” space. In order to understand what is going on, we start by considering in detail the heterotic/type IIA duality map in the N4 = 2 theory. The correspondence of the volume form of the base of the type IIA, K3 fibration, with the heterotic dilaton can be observed by looking at the effective theory, built on the massless states of both the constructions. The duality map is therefore “perturbative/non-perturbative”, and requires for consistency that also the modulus parameterizing the coupling of the type IIA string, belonging to the hypermultiplets, has a heterotic counterpart. On the heterotic side, the hypermultiplets correspond to