Two Two - Dimensional Supergravity Theories from Calabi - Yau Four - Folds

We consider two-dimensional supergravity theories with four supercharges constructed from compactification of Type II string theory on a generic Calabi-Yau four-fold. In Type IIA and Type IIB cases, respectively, new superspace formulations of N = (2, 2) and N = (0, 4) dilaton supergravities are found and their coupling to matter multiplets is discussed. 1. Introduction For a long time, compactification of heterotic string theory on Calabi-Yau manifolds was the primary candidate for constructing realistic models in four dimensions with N = 1 supersymmetry. This was also a strong motivation to study Type II superstrings on Calabi-Yau three-folds which share many common properties with the corresponding heterotic compactifications. At the same time substantial progress has been made in understanding the mathematical aspects of Calabi-Yau three-folds, such as quantum cohomology and mirror symmetry [1]. It was not until the discovery of F-theory [2] that it was realized that N = 1 four-dimensional heterotic string vacua can be equivalently described as F-theory compactifications on elliptically fibered Calabi-Yau four-folds. Since then, the study of Calabi-Yau four-fold compactifications has become of particular importance for physical applications. Compactification of F-theory on an elliptically fibered Calabi-Yau four-fold is closely related to the corresponding compactifications of Type IIA and M-theory. Namely, when the area of the elliptic fiber shrinks to zero, M-theory compactification on a Calabi-Yau four-fold is well described by F-theory compactification on the same Calabi-Yau mani-fold. On the other hand, Type IIA string theory is related to M-theory via compactification on an extra circle (of small radius). One of the most striking outcomes in the study of compactifications on Calabi-Yau three-folds is the great success in understanding non-perturbative phenomena in N = 2 field theories in four dimensions, see e.g. [3] for introduction and references. On the other hand, understanding of Calabi-Yau four-fold compactifications still is quite far from that stage, so we shall not discuss non-perturbative phenomena in this paper. Instead, we consider classical supergravity theories interacting with two-dimensional non-linear sigma-models that can be constructed from Calabi-Yau four-folds. Surprisingly, it turns out that manifestly supersymmetric formulations of such theories has not been given previously. On general grounds, a compactification of Type IIA (IIB) string theory on a Calabi-Yau four-fold leads to a N = (2, 2) (resp. N = (0, 4)) effective field theory in two dimensions. In the low-energy limit the theory is described by supergravity coupled to matter. For example, from the …