Abstract
We analyze the divergences of the three-loop partition function at fixed area in 2D quantum gravity. Considering the Liouville action in the Kähler formalism, we extract the coefficient of the leading divergence ∼AΛ2(lnAΛ2)2. This coefficient is non-vanishing. We discuss the counterterms one can and must add and compute their precise contribution to the partition function. This allows us to conclude that every local and non-local divergence in the partition function can be balanced by local counterterms, with the only exception of the maximally non-local divergence (lnAΛ2)3. Yet, this latter is computed and does cancel between the different three-loop diagrams. Thus, requiring locality of the counterterms is enough to renormalize the partition function. Finally, the structure of the new counterterms strongly suggests that they can be understood as a renormalization of the measure action.
Highlights
The coupling of conformal matter and two-dimensional quantum gravity is a subject which has been deeply studied, with a broad variety of approaches, from the discrete – triangulations [1] and matrix models [2] [3] [4] [5] – to the continuum approaches [6] [7] [8]
The same situation repeats itself at three loops where the counterterms generate exactly the necessary terms to cancel all divergences of the three-loop partition function, except for a ln AΛ2 3 divergence which, if present, cannot be cancelled by a local counterterm. This divergence is present in individual three-loop diagrams but we show that the different contributions cancel among themselves. This is an encouraging result meaning that all the non-local divergences appearing through the computation of the partition function may be offset by local counterterms
When summed, they vanish! This is similar to what happened for the ln AΛ2 2 divergence in the two-loop partition function, and one expects the ln AΛ2 L divergence to cancel in the L-loop partition function
Summary
The coupling of conformal matter and two-dimensional quantum gravity is a subject which has been deeply studied, with a broad variety of approaches, from the discrete – triangulations [1] and matrix models [2] [3] [4] [5] – to the continuum approaches [6] [7] [8]. The other renormalization constant had no particular reason to be fixed to the KPZ value, allowing a one-parameter family of quantization schemes which could eventually open the possibility to go beyond the c = 1 barrier The presence of this free parameter is intriguing and a natural question is whether the structure of the counterterm action introduced at two-loops is enough to cancel the divergences at three (and higher) loops or whether new counterterms, with additional undetermined finite renormalization constants are required. This divergence is present in individual three-loop diagrams but we show that the different contributions cancel among themselves This is an encouraging result meaning that all the non-local divergences appearing through the computation of the partition function may be offset by local counterterms. Remain after imposing cancellation of the divergences, of any regulator dependence and requiring the strong locality condition
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