O ( d , d ) invariance at two and three loops

We show that in a two-dimensional σ-model whose fields only depend on one target space coordinate , the O(d, d) invariance of the conformal invariance conditions observed at one loop is preserved at two loops (in the general case with torsion) and at three loops (in the case without torsion). Duality invariance has proved to be an immensely fruitful concept in string theory. A particular aspect of duality which has been valuable in string cosmology (for a recent comprehensive review of this subject, with an extensive list of references, see Ref. [1]) is the concept of O(d, d) invariance. This is displayed in the case where the fields depend only on time in an arbitrary number of dimensions (time and d spatial dimensions). It was discovered some time ago[2] that the lowest order string effective action (in the simplest case with only the metric, dilaton and antisymmetric tensor field) then exhibits continuous, global O(d, d) invariance. (This is also reminiscent of the O(d, d) invariance previously observed in the context of string compactifications[3].) The O(d, d) invariance of the string action was later found to persist in the presence of matter or gauge fields[4]. Now the conformal invariance conditions for the fields are related to the field equations of the string action, and so, at least at lowest order, the O(d, d) invariance can be used to transform between different conformal backgrounds. Duality invariance in general can be understood as a consequence of an isometry in an underlying theory[5]; and the O(d, d) invariance can be viewed as the result of gauging d abelian isometries[6]. In Refs. [2] and [7] it was argued that the O(d, d) invariance should be maintained to all orders. In Ref. [8] the two-loop string action was considered in the context of fields depending only on time, and it was shown that after a suitable field redefinition it could be written in an explicitly O(d, d) invariant form. However, it was pointed out in Refs. [9] (in the context of T duality) that the invariance of the action does not manifestly guarantee that the conformal invariance conditions transform appropriately. Therefore it seems to us that it is necessary to check explicitly the transformation properties of the conformal invariance conditions in order to complete the proof that O(d, d) invariance is preserved. At one loop this has been done in Ref. [2]. The main purpose of …