AdS3 gravity and RCFT ensembles with multiple invariants

Abstract

We use the Poincaré series method to compute gravity partition functions associated to SU(N)1 WZW models with arbitrarily large numbers of modular invariants. The result is an average over these invariants, with the weights being given by inverting a matrix whose size is of order the number of invariants. For the chosen models, this matrix takes a special form that allows us to invert it for arbitrary size and thereby explicitly calculate the weights of this average. For the identity seed we find that the weights are positive for all N, consistent with each model being dual to an ensemble average over CFT’s.

Chern-Simons Invariants from Ensemble Averages

“…Let us start with the case of 1 = 2 = 0. In this case, the definition (70) coincides with the previous definition in the even case (13), except here Q is not necessarily even. We can cure this problem by writing (70) in terms of 2Q and τ /2:…”

“…is the Siegel-Eisenstein series (henceforth referred to simply as Eisenstein series) associated with the quadratic form Q, and δ h = 1 for h ∈ Λ, and δ h = 0 for h / ∈ Λ. Note that the constant term is expected for h ∈ Λ since in the limit τ 2 → ∞, we still have a contribution from the origin = 0 of the lattice Λ in the sum (13). The factor γ Q,h (c, d) is given by a version of the quadratic Gauss sum…”

“…This holographic duality was studied further in e.g. [8][9][10][11][12][13]. In this paper we consider the generalization where the associated CFT moduli space is a more general type of Narain moduli space associated with an indefinite quadratic form Q of rank p + q and signature (p, q) 1 :…”

“…In the literature the level of the Chern-Simons theory is often denoted by Q/2, not Q. Our normalization here is useful when we discuss spin Chern-Simons theory in section 3 13. As in footnote 12, we choose a normalization where the minimal value of the level for the non-spin (even) Chern-Simons theory is k = 2.…”

Wormholes and spectral statistics in the Narain ensemble

“…13 The gravitational description of the Narain ensemble was further studied in [61][62][63], and the discrete average of holomorphic free boson CFTs was discussed in [64]. Aspects of ensemble averages over other classes of rational two-dimensional conformal field theories have also been discussed [65][66][67][68][69]. 14 In fact, the subleading behaviour in (1.6) -and in particular the subleading term which scales like O(∆ 0 ) -requires the inclusion of the full "SL(2, Z)" family of Euclidean black holes.…”