2D Quantum Gravity on Compact Riemann Surfaces and Two-Loop Partition Function: Circumventing the c=1 Barrier?

Abstract

We study two-dimensional quantum gravity on arbitrary genus Riemann surfaces in the Kaehler formalism where the basic quantum field is the (Laplacian of the) Kaehler potential. We do a careful first-principles computation of the fixed-area partition function $Z[A]$ up to and including all two-loop contributions. This includes genuine two-loop diagrams as determined by the Liouville action, one-loop diagrams resulting from the non-trivial measure on the space of metrics, as well as one-loop diagrams involving various counterterm vertices. Contrary to what is often believed, several such counterterms, in addition to the usual cosmological constant, do and must occur. We consistently determine the relevant counterterms from a one-loop computation of the full two-point Green's function of the Kaehler field. Throughout this paper we use the general spectral cutoff regularization developed recently and which is well-suited for multi-loop computations on curved manifolds. At two loops, while all "unwanted" contributions to $\ln (Z[A]/Z[A_0])$ correctly cancel, it appears that the finite coefficient of $\ln (A/A_0)$ does depend on the finite parts of certain counterterm coefficients, i.e. on the finite renormalization conditions one has to impose. There exists a choice that reproduces the famous KPZ-scaling, but it seems to be only one consistent choice among others. Maybe, this hints at the possibility that other renormalization conditions could eventually provide a way to circumvent the famous $c=1$ barrier.

54 pages

References in corpus (6)

• Duality and KPZ in Liouville Quantum Gravity
• Gravitational Actions in Two Dimensions and the Mabuchi Functional
• Multi-Loop Zeta Function Regularization and Spectral Cutoff in Curved Spacetime
• Random Geometry, Quantum Gravity and the Kähler Potential
• 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions
• 2D quantum gravity at three loops: a counterterm investigation

Cited by in corpus (4)

• Finite cut-off JT and Liouville quantum gravities on the disk at one loop
• The two-sphere partition function in two-dimensional quantum gravity
• 2D quantum gravity at three loops: a counterterm investigation
• Gravitational action for a massive Majorana fermion in 2d quantum gravity