Gravitational actions in two dimensions and the Mabuchi functional
Introduction
When a matter quantum field theory is coupled to gravity, the metric dependence of the partition function induces an effective gravitational action. On a two-dimensional Euclidean space–time, when the quantum field theory is conformally invariant, this effective action is always proportional to the Liouville action [1]. In the so-called conformal gauge, for which the metric g is proportional to a fixed background metric , the Liouville action takes the simple form where is the Ricci scalar for the metric . The resulting Liouville models for two-dimensional quantum gravity have been extensively studied in the literature.
The gravitational coupling of non-conformally invariant matter has been much less studied (see e.g. [2]). In this case, one expects a complicated and model-dependent gravitational action. It is unclear a priori if new interesting and simple actions, yielding models with nice geometrical and physical features, can emerge.
Recently, the authors have explained that two-dimensional quantum gravity could be naturally formulated in the context of Kähler geometry [3], [4]. In this context, mathematicians have much used and studied natural functionals defined on the space of Kähler potentials, the most salient being the Mabuchi action [5], also called the Mabuchi energy in the mathematical literature. Our aim in the present work is to investigate whether such actions could contribute to the gravitational effective action in some simple physical models. We shall see that they do, yielding new models of two-dimensional quantum gravity with potentially very rich mathematical and physical features.
Our space–time will always be a compact Riemann surface of genus h. An arbitrary Riemannian metric on can be parametrized by a finite set of complex moduli τ and the conformal factor , with a background metric in (1.1) of the form . In the Kähler framework, we write the conformal factor in terms of the area A (which is the unique Kähler modulus in two dimensions) and the Kähler potential ϕ, where and are the area and the positive Laplacian for the metric respectively. The lemma shows that the differential equation (1.3) can always be solved for A and ϕ in terms of σ, and the solution is unique up to constant shifts in ϕ.
There are many reasons to use the Kähler potential ϕ in the context of two-dimensional quantum gravity. In particular it is possible to rigorously define in terms of ϕ a regularized version of the gravitational path integrals [3], [4]. For our present purposes, the variable ϕ is useful because interesting functionals of the metric are most naturally written down in terms of ϕ. A famous example is the Mabuchi action [5], This action and its higher dimensional generalizations play a central rôle in Kähler geometry. It is well defined on the space or metrics, because it is unchanged if ϕ is shifted by a constant. It is bounded from below and convex, which makes it a good candidate for an action to be used in a path integral. It satisfies the so-called cocycle conditions, which can be checked straightforwardly from the definition (1.4). The same cocycle identities are also satisfied by the Liouville action. As we shall review below, these identities actually are fundamental consistency conditions that any effective gravitational action must satisfy. Finally, the critical points of the Mabuchi action are the metrics of constant scalar curvature, another property shared in two dimensions with the Liouville action. This property is also valid for the higher dimensional generalizations of (1.4), which goes a long way in explaining the central rôle played by the Mabuchi action in the study of such metrics on general Kähler manifolds. For more details and references on the profound geometrical properties of the space of metrics on a Kähler manifold, we refer the reader to [4], [6], [7] and the comprehensive recent review [8].
We focus in the following on the simple model of a massive scalar field X with action Our main result is to show that the Mabuchi action and other simple functionals of the Kähler potential ϕ, like the so-called Aubin–Yau action, contribute to the gravitational effective action for this model in a small mass expansion. We also study the gravitational dressing of operators of the form and show that it involves the Aubin–Yau action on top of the familiar dressing factors found in the conformal field theory limit.
A striking feature is that it is always possible to cancel out the familiar Liouville term in the effective action, for example by coupling with a suitable spectator conformal field theory. The resulting pure Mabuchi theories are entirely new and intriguing two-dimensional quantum gravity models.
We start in Section 2 with a brief review on two-dimensional quantum gravity, gravitational effective actions and gravitational dressing. In Section 3, we discuss the basic properties of various actions, in particular Mabuchiʼs and Aubin–Yauʼs. We derive fundamental identities relating these actions to the variations of various functionals associated with the Laplace operator. In Section 4, we apply the results of Section 3 to compute the gravitational effective action and the gravitational dressing of the operators for the model (1.7). We also compute the trace of the stress–energy tensor. Finally, in Section 4, we conclude and discuss possible extensions of our work. We have tried to make the presentation as elementary and self-contained as possible. In particular we have included Appendix A On the derivatives of the Greenʼs function, Appendix B Greenʼs function and containing simple derivations of results used in the main text.
To a metric on we associate its Kähler form, In two dimensions, the Kähler form coincides with the volume form and thus in particular the area is given by The positive Laplacian for the metric g, whose determinant we also denote by g, is defined as usual by In terms of the standard Dolbeault operators ∂ and we have from which it can be seen that the relation (1.3) is equivalent to The space of Kähler potentials ϕ is simply the condition on ϕ ensuring the strict positivity of the metrics, . The scalar curvature, or Ricci scalar R, is such that The integral of the scalar curvature is a topological invariant, This is a useful formula, from which for instance the invariance of (1.4) under constant shifts of ϕ is derived. Finally, the Laplacians Δ and and scalar curvatures R and of two metrics g and related by a formula of the form (1.1) are themselves linked by the simple relations
Section snippets
Gravitational effective actions
The partition function of a matter quantum field theory, defined on the two-dimensional Euclidean space–time endowed with a fixed metric g, with fields X, couplings λ and action , is defined by the QFT path integral The quantum gravity partition function Z is obtained by integrating further over the space of metrics on , where μ is the bare cosmological constant. It is also interesting to consider the partition function
Functionals and variations
We are now going to derive a set identities for the variation under change of metric of various functionals. All these identities will then be applied in Section 4 to compute the gravitational effective action and gravitational dressing of the model (1.7). The technical derivations that we present below and in Appendix A On the derivatives of the Greenʼs function, Appendix B Greenʼs function and are not necessary in understanding Section 4 and thus may be skipped on a first reading.
Application to the massive scalar field
We are now going to apply the results of the previous section to study the model (1.7). In order to compute both the effective gravitational action and the gravitational dressing of the operators (1.8) at the same time, we consider the generalized partition function where k and x denote collectively all the s and s. This generalized partition function contains all the information we need. Indeed, the usual matter partition function is simply
Conclusion
Studies of two-dimensional gravity have focused almost entirely on the Liouville model. This model is singled out since it universally describes the coupling of a CFT to gravity. Our aim in the present work was to motivate the study of different models, based on different actions like the Mabuchi functional, that are singled out by their nice geometrical features and their fundamental importance in Kähler geometry. Our main result was to show that these models do appear naturally in simple
Acknowledgements
This work is supported in part by the Belgian Fonds de la Recherche Fondamentale Collective (grant 2.4655.07), the Belgian Institut Interuniversitaire des Sciences Nucléaires (grant 4.4505.86), the Interuniversity Attraction Poles Programme (Belgian Science Policy), the Russian RFFI grant 11-01-00962 and the American NSF grant DMS-0904252.
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On leave of absence from ITEP, Moscow, Russia.