A General Construction of CFTs on Riemannian Manifolds

We predict a model of quantum gravity can be reached by taking the continuum limit of a conformal lattice action obeying the symmetries of the desired metric. To this end, we implement a procedure to construct a lattice conformal field theory (CFT) over a general curved Riemannian manifold. In this procedure, the target manifold is locally approximated with an affine-transformed periodic lattice equipped with a Ising action and tunable spin couplings. To ensure conformal symmetry, the action must be tuned to criticality, which is why in this work, we map the critical surface of the Ising model as a function of the anisotropy. We perform a parameter sweep of the spin coupling configuration space and locate the critical temperature by measuring the divergence in energy fluctuations using Monte Carlo methods. Multiple optimizations are employed to increase runtime and accuracy, including a multiple histogram reweighting algorithm used to extrapolate energy fluctuations to unsimulated configurations of the parameter space. We will use our results to measure spin correlations, which will be a necessary step towards our ultimate goal to associate spin couplings with the lattice bond lengths.