Abstract
We show how to formulate a lattice gauge theory whose naive continuum limit corresponds to two-dimensional (Euclidean) quantum gravity including a positive cosmological constant. More precisely the resultant continuum theory corresponds to gravity in a first-order formalism in which the local frame and spin connection are treated as independent fields. Recasting this lattice theory as a tensor network allows us to study the theory at strong coupling without encountering a sign problem. In two dimensions this tensor network is exactly soluble and we show that the system has a series of critical points that occur for pure imaginary coupling and are associated with first order phase transitions. We then augment the action with a Yang-Mills term which allows us to control the lattice spacing and show how to apply the TRG to compute the free energy and look for critical behavior. Finally we perform an analytic continuation in the gravity coupling in this extended model and show that its critical behavior in a certain scaling limit depends only on the topology of the underlying lattice. We also show how the lattice gauge theory can be naturally generalized to generate the Polyakov or Liouville action for two dimensional quantum gravity.
Highlights
The challenge of formulating a quantum theory of general relativity (GR) remains unmet, and many different approaches have been developed to surmount this challenge
It is common to lattice approaches such as causal and Euclidean dynamical triangulations [15,16,17,18,19,20,21]. The latter have revealed some intriguing aspects of quantum gravity like the emergence of fractal structure
As we show in this paper, the lattice model can be rewritten as a tensor network and new techniques such as the tensor renormalization group (TRG), which are insensitive to sign problems, can be brought to bear [45,46,47]
Summary
The challenge of formulating a quantum theory of general relativity (GR) remains unmet, and many different approaches have been developed to surmount this challenge. Tensor network formulations have had some success in analyzing many spin and lattice gauge models [49,50,51,52] but this paper constitutes the first attempt to use them to study quantum gravity. We are able to recover the results of the TRG analysis using an exact method based on analytic continuation and find that the critical exponent characterizing the behavior of the partition function close to the transition depends on the topology of the lattice This feature is in qualitative agreement with the Polyakov model of two-dimensional quantum gravity. X with possible extensions of the work by writing down a lattice theory of 4D quantum gravity that parallels the two-dimensional theory in this paper
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