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Algebraic Realisation of the Zamolodchikov Metric in Narain Theories

El Hassan Saidi, Rajae Sammani

TL;DR

The paper develops an algebraic program that classifies Narain CFT2s through finite Lie algebras $\mathbf{g}$ and their representations, encoding left-right momenta via root and weight lattices and realising the Zamolodchikov metric on $\mathcal{M}_{\mathbf{g}}$ in terms of the Cartan matrix $K_{\mathbf{g}}$ and its inverse. It provides explicit constructions for rank-2 and higher-rank cases, expressing the genus-one partition functions with Siegel–Narain theta functions and deriving the moduli-space metric from $K_{\mathbf{g}}$; ensemble averaging yields moduli-independent constants linked to the bulk dual, supporting a holographic picture. The work offers a systematic classification into ADE, orthogonal, and exceptional classes, with detailed treatments of $\mathrm{su}_{r+1}$, $\mathrm{so}_{2r}$, and $\mathbf{e}_r$ families, and discusses generalisations to NCFTs with asymmetric central charges and non-unimodular lattices. Together, these results deepen the connection between Lie-algebraic data and Narain moduli, enabling precise computations of partition functions and their averaged properties, with potential implications for holography and Swampland considerations.

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$_{\mathbf{g}}^{(r,r)}$.

Algebraic Realisation of the Zamolodchikov Metric in Narain Theories

TL;DR

The paper develops an algebraic program that classifies Narain CFT2s through finite Lie algebras and their representations, encoding left-right momenta via root and weight lattices and realising the Zamolodchikov metric on in terms of the Cartan matrix and its inverse. It provides explicit constructions for rank-2 and higher-rank cases, expressing the genus-one partition functions with Siegel–Narain theta functions and deriving the moduli-space metric from ; ensemble averaging yields moduli-independent constants linked to the bulk dual, supporting a holographic picture. The work offers a systematic classification into ADE, orthogonal, and exceptional classes, with detailed treatments of , , and families, and discusses generalisations to NCFTs with asymmetric central charges and non-unimodular lattices. Together, these results deepen the connection between Lie-algebraic data and Narain moduli, enabling precise computations of partition functions and their averaged properties, with potential implications for holography and Swampland considerations.

Abstract

We revisit Narain conformal field theories from an algebraic perspective based on finite dimensional Lie algebras and representations , 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 in terms of Lie algebraic data namely the Cartan matrix K and its inverse K. 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 with and highlight further features of the partition function Z.

Paper Structure

This paper contains 30 sections, 161 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Narain CFT$_{2}$ and Lie algebras
  3. Standard Narain CFT$_{2}^{\mathtt{su}_{2}}$
  4. Refined Narain CFT$_{2}^{\mathtt{su}_{2}}$
  5. NCFT$_{2}^{\mathtt{su}_{2}}$ at the point $R=1/2$
  6. A) Two comments
  7. B) Quadratic forms and su(2) Lie algebra
  8. NCFT$_{2}^{\mathtt{su}_{2}}$ at generic values of radius R
  9. Results for NCFT$_{2}^{\mathtt{su}_{2}}$
  10. Rank two Narain CFT$_{2}^{\mathbf{g}}$
  11. NCFT$_{2}^{su_{3}}$ as an extension of NCFT$_{2}^{su_{2}}$
  12. Sitting at the point $x_{i}=1$ in $\mathcal{M}_{\mathbf{su}_{3}}$
  13. NCFT$_{2}^{su_{3}}$ at generic moduli $\left\{ x_{1},x_{2}\right\}$
  14. Results for diagonal NCFT$_{2}^{\mathtt{su}_{3}}$
  15. Metric of $\mathcal{M}_{\mathbf{su}_{3}}$ and partition function $Z_{\mathbf{su}_{3}}$
  16. ...and 15 more sections

Figures (4)

  • Figure 1: Graphic representations of the intersection matrices $\mathbf{K}^{\mathrm{su}_{3}}$ and $\mathbf{\tilde{K}}^{\mathrm{su}_{3}}$. The two nodes of $\mathbf{K}^{\mathrm{su}_{3}}$ and $\mathbf{\tilde{K}}^{\mathrm{su}_{3}}$ represent the two circles used in the compactification. The $x_{i}=1/(2R_{i})$ gives the real moduli of the 2-torus and its dual. For $x_{i}=1,$ the $\mathbf{K}^{\mathrm{su}_{3}}$leads to the Dynkin diagram of su(3).
  • Figure 2: Graphic representation of the intersection matrix $\mathbf{K}_{su_{r+1}}$. For the cases of su(4) and su(5), they are explicitly given by eqs(\ref{['40']}-\ref{['41']}).
  • Figure 3: Graphic representations of the intersection matrices $\mathbf{K}^{\mathrm{so}_{8}}$ and $\mathbf{K}^{\mathrm{so}_{10}}$.
  • Figure 4: Graphic representation of the intersection matrix $\mathbf{K}^{\mathrm{E}_{6}}$.