Averaging over moduli in deformed WZW models
Junkai Dong, Thomas Hartman, Yikun Jiang
TL;DR
The paper studies an ensemble average over moduli of current-current deformations of SU$(N{+}1)_k$ WZW models, deriving a remarkably simple form for the averaged torus partition function that justamente takes the Poincaré-type expression $\big\langle Z_{\rm dWZW}(\tau)\big\rangle = \sum_{\gamma\in \Gamma_{\infty}\backslash PSL(2,\mathbb{Z})} \sum_{\lambda\in P_k^+} |c_0^\lambda(\gamma\tau)|^2$, hinting at a holographic dual described by a sum over 3-manifolds. The approach leverages averaging Siegel-Narain theta functions with twisted boundary conditions via the Siegel-Weil formula, decomposing the WZW model into parafermion and Narain sectors and expressing the average in terms of Eisenstein-series-like objects for congruence subgroups when the level is prime. The results yield a perturbative bulk picture as $U(1)^{2N}$ Chern-Simons theory coupled to topological matter dual to the parafermions and establish a clear connection between averaged affine characters and a Siegel-Weil-type structure. This work extends Narain ensemble holography to a broader class of exactly marginal deformations and opens avenues for exploring averaged correlators, higher-genus extensions, and possible string-theoretic embeddings of ensemble averages in AdS3/CFT3.
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.