Algebraic Realisation of the Zamolodchikov Metric in Narain Theories

TL;DR

This paper provides an algebraic approach to Narain conformal field theories, expressing the Zamolodchikov metric using Lie algebra data and exploring implications for ensemble averages and holography.

Contribution

It introduces a novel Lie algebraic realization of the Zamolodchikov metric in Narain theories, linking lattice structures to moduli space geometry.

Findings

Lie algebraic data encodes Narain momenta and partition functions.

Constructs a realisation of the Zamolodchikov metric using Cartan matrices.

Discusses ensemble averaging and holographic duals of these CFTs.

Abstract

We revisit Narain conformal field theories from an algebraic perspective based on finite dimensional Lie algebras $\mathbf{g}$ and representations $\mathcal{R}_{\mathbf{g}}$, and show how the root and weight lattices can encode the momenta and subsequently the partition functions of Narain theories. In this framework, we construct a realisation of the Zamolodchikov metric of the moduli space $\mathcal{M}_{\mathbf{g}}$ in terms of Lie algebraic data namely the Cartan matrix K$_{\mathbf{g}}$ and its inverse K$_{\mathbf{g}}^{-1}$. Properties regarding the ensemble averaging of these CFTs and their holographic dual are also derived. Additionally, we discuss possible generalisations to NCFTs having dis-symmetric central charges $(\mathrm{c}_{L},\mathrm{c}_{R})=(\mathrm{s},% \mathrm{r})$ with $s>r$ and highlight further features of the partition function Z$_{\mathbf{g}}^{(r,r)}$.