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6) where Pd−1 = e 0 sd−1 is the present value of the Planck length, if we take 0 to coincide with the present (constant) value of the dilaton. Although the use of the E-frame could simplify some calculations, and we shall see examples of this below, it should be kept in mind that the form of the corrections is no longer so simple. e. when it reaches a dilaton-dependent M. Gasperini, G. Veneziano / Physics Reports 373 (2003) 1 – 212 47 critical value. 5). For these reasons, although physical results are frame-independent, we shall always describe them with reference to the original S-frame metric in which the string length s is constant.
The ÿrst point concerns the ultraviolet cut-o . e. does not depend on the dilaton (at least in perturbation theory). 119) Obviously, higher-derivative corrections become relevant, in this frame, as soon as R ∼ independently of . 120) indicating that higher-derivative corrections become relevant at R˜ ∼ s−2 e , which is again the (dilaton-dependent) string scale expressed in Planck units. The second point concerns masses. Tree-level string masses are dilaton-independent in the S-frame, but depend on in the E-frame since the S-frame relation g p p = m2 becomes g˜ p p = m2 e in the E-frame.
Some of these ÿelds, the graviton, the dilaton, and the NS–NS two-form, are always present. g. gauge ÿelds and higher-order forms) may vary, depending on the particular superstring theory under study or, in an M-theory perspective, on the particular M-theory vacuum around which the generic solution is looked for. 4). The general solution of the ÿeld equations, in this regime, will consist of an arbitrary linear superposition of (long-wavelength) gravitational, dilatonic and axionic waves. From the point of view of target space (taken here, for the sake of illustration, to be (3 + 1)-dimensional), the generic solution depends upon four arbitrary functions of three coordinates [425] related to the metric, plus two more each for the dilaton and for the axion ÿeld associated to B .