R Gravity in (2 + ǫ)–Dimensional Quantum Gravity

We study R2 gravity in (2 + ǫ)–dimensional quantum gravity. Taking care of the oversubtraction problem in the conformal mode dynamics, we perform a full order calculation of string susceptibility in the ǫ → 0 limit. The result is consistent with that obtained through Liouville approach. E-mail address : [email protected], JSPS Research Fellow. E-mail address : [email protected] E-mail address : [email protected] Two-dimensional quantum gravity [1] has been studied intensively through Liouville theory [2] and the matrix model [3] these several years, and indeed its progress has provided us with much insight into quantum gravity and string theory. There are two ways to proceed further; one is to go beyond the so-called c = 1 barrier and the other is to explore higher dimensional quantum gravity. As for the former, R gravity, which has been studied for years [4, 5], can be considered as a possible trial. Recently Kawai and Nakayama [6] have succeeded in treating it in the framework of Liouville theory. It has been discovered that the surface becomes locally smooth allowing a central charge greater than one. Although R gravity is not acceptable as a realistic string theory due to loss of positivity, it is still worth studying as a new type of universality class of two-dimensional quantum gravity. In contrast to ordinary gravity, R gravity is difficult to study in the matrix model and so far we do not have any other results which can be compared with theirs. It seems, therefore, desirable to investigate the theory in other approaches. As for exploring higher-dimensional quantum gravity, numerical simulations provide a powerful tool; yet we should develop analytic approaches at the same time. One of the possibilities is (2 + ǫ)–dimensional quantum gravity [7, 8, 9]. The formalism, however, contains some subtlety concerning the oversubtraction problem in the conformal mode dynamics, as was discovered by Kawai, Kitazawa and Ninomiya [8]. Considering the situation, we feel that it is worth while acquiring a deeper insight into the formalism by applying it to other theories. In this letter, therefore, we study R gravity in (2+ ǫ)–dimensional quantum gravity. We obtain results consistent with ref. [6] by taking the ǫ → 0 limit in the strong coupling regime. This provides another success in treating R gravity. We define a (2 + ǫ)–dimensional system corresponding to R gravity by the following action, S = μ G ∫ dx √ gR + μ 4m2 ∫ dx √ gR + Λμ ∫ dx √ g + μ c ∑