Abstract
We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form Q. We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by Q. The resulting partition function can be written as a modular form, expressed as a sum over the partition functions of Chern-Simons theories on lens spaces. For odd lattices, the dual bulk theory is a spin Chern-Simons theory, and we identify several novel phenomena in this case. We also discuss the holographic duality prior to averaging in terms of Maxwell-Chern-Simons theories.
Highlights
Where IIp,p denotes the even, self-dual lattice associated with the compactification on the p-dimensional torus Tp
Our analysis shows that the partition function after the ensemble average contains spin Chern-Simons invariants for the handlebody geometries, giving further support to the appearance of the Chern-Simons term in the holographic dual
We find that the ensemble average of the CFT partition function is equal to an Eisenstein series associated with Q, which can be interpreted as a sum over geometries in the three-dimensional Chern-Simons theories
Summary
We consider free boson CFTs with momenta valued in a (p + q)-dimensional integral lattice Λ = Zp+q ⊂ Rp+q, equipped with an even quadratic form p+q. The point of the moduli space MQ is again specified by decomposing the quadratic form into left and right-moving parts QL and QR. As in the case of the circle compactification, one can define a positive quadratic form, the Hamiltonian H( ) := QL( )+QR( ), which can be used as another parametrization of the moduli space. The incompatibility between a general O(p, q; R) transformation and the integrality of the lattice Λ means that we have VL ∩ Λ = VR ∩ Λ = ∅ at a generic point in the moduli space. The cosets MQ are submanifolds of MIIp,p where only restricted sets of exactly marginal operators are turned on
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