Abstract
WZW models live on a moduli space parameterized by current-current deformations. The moduli space defines an ensemble of conformal field theories, which generically have N abelian conserved currents and central charge c > N. We calculate the average partition function and show that it can be interpreted as a sum over 3-manifolds. This suggests that the ensemble-averaged theory has a holographic dual, generalizing recent results on Narain CFTs. The bulk theory, at the perturbative level, is identified as U(1)2N Chern-Simons theory coupled to additional matter fields. From a mathematical perspective, our principal result is a Siegel-Weil formula for the characters of an affine Lie algebra.
Highlights
WZW models live on a moduli space parameterized by current-current deformations
There are indications that the sum over topologies in the gravitational path integral is related to an ensemble average over microscopic theories [1,2,3,4,5,6,7]
The second is that the sum over topologies is an uncontrolled artifact of the low-energy description, and the true microscopic theory is ordinary quantum mechanics without disorder
Summary
There are indications that the sum over topologies in the gravitational path integral is related to an ensemble average over microscopic theories [1,2,3,4,5,6,7]. We generalize the Narain duality to an ensemble of CFTs defined by the moduli space of the SU(N + 1)k WZW. We will not have much to say about the bulk theory, but based on the partition function we conclude that perturbatively, it should be a theory of U(1)2N Chern-Simons fields coupled to topological matter dual to the parafermions. There are non-conserved primary operators coming from the parafermion sector that survive the averaging procedure, so the bulk theory has matter fields This allows for the study of averaged correlation functions, we will not pursue this here. By summing over topologies, we end up averaging over the apparent global symmetry group of the bulk theory It is an interesting question whether this pattern applies more generally to theories of quantum gravity with apparent global symmetries.
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