Generalized holonomy in String-corrected Spacetimes

Abstract

Abstract. The quartic-curvature corrections derived from string theory have a specific impact on the geometry of target-space manifolds of special holonomy. In the cases of Calabi-Yau manifolds, D = 7 manifolds of G2 holonomy and D = 8 manifolds of Spin 7 holonomy, string theory α corrections conspire to preserve the unbroken supersymmetry of these backgrounds despite the fact that the α corrections cause the Riemannian holonomy to lose its special character. We show how this supersymmetry preservation is expressed in the language of generalized holonomy for the Killing spinor operator. IntroductionThe effective action for the massless modes of a given string theory consists at the lowest order of the action for the corresponding supergravity theory, which is then modified by an infinite sequence of higher derivative corrections. These correction terms occur individually with finite coefficients but are of similar structure to those that occur with divergent coefficients in the corresponding quantized supergravity [1,2,3]. In this sense, the string theory may be viewed as a regularization of the corresponding supergravity. The string tension α plays the rôle of the dimensionful cutoff parameter and gives the scale at which the microscopic physics of string theory begins to take over from the effective field theory of the massless modes. At the (α ) 3 level in type II theories, one encounters corrections quartic in Riemann curvatures; these are the first such corrections whose variations do not vanish subject to the leading order effective supergravity field equations, so they play a particularly important rôle in the onset of string-theory effects. This subject has had a revival of interest as appreciation has grown of the importance of noncompact Calabi-Yau spaces or G 2 manifolds as the underlying Ricci-flat geometries of brane spacetimes. In the compact cases, the relation between the quartic curvature corrections and Calabi-Yau compactifications has been studied in Ref. [4], while the non-compact case has been studied in [5].This review, based on Refs [6,7,8,9], summarizes the geometrical impact that the quartic corrections have on Calabi-Yau, G 2 and Spin 7 holonomy manifolds and on the preservation of supersymmetry in the face of the α corrections, extending the earlier analysis given in Refs [10,11]. It then proceeds to show how this supersymmetry preservation is expressed in the language of generalized structure groups and generalized holonomy for the α -corrected Killing spinor operator.

String corrected spacetimes andSU(N)-structure manifolds

“…The corrected geometry has SU (3)-structure, rather than SU (3)-holonomy, as we show explicitly in the following. See [16] and references therein for related discussions.…”

“…A detailed analysis of the effect that these corrections have on a Calabi-Yau background was performed in the sigma model by [8] (see also [9] and [10]). The novelty of our work, is that it makes contact to the more recent literature about string theory compactification on SU (N )-structure manifolds (see [11], [12], [13], [14], [15], [16] and [17] for a partial list of references, and see especially [18] for another work which studies string corrections in the language of G-structures, though in a somewhat different context). The approach taken here is closely related to the effective field theory approach for M-theory and type II theory compactifications on G 2 -structure manifolds and Spin(7)structure manifolds performed recently in [19].…”