Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New perspectives in Kac-Moody algebras associated to higher dimensional manifolds

Rutwig Campoamor-Stursberg, Alessio Marrani, Michel Rausch de Traubenberg

TL;DR

This work develops a general framework to construct Kac–Moody current algebras on higher-dimensional manifolds by promoting finite-dimensional Lie algebras to $\mathfrak{g}(\mathcal{M})$-type current algebras and coupling them to isometries. It leverages harmonic analysis (Peter–Weyl for compact, Plancherel for noncompact) to define explicit mode expansions, and it classifies compatible central extensions via 2-cocycles, yielding Virasoro-like structures in several settings. The authors provide concrete realizations for compact groups (e.g., $SU(2)$) and noncompact examples (e.g., $SL(2,\mathbb{R})$), including soft deformations of group manifolds, and analyze representation-theoretic and unitarity aspects. The framework connects to physics across domains—2D CFT/WZW, KK spectra, cosmological billiards, and supergravity—and outlines conjectures linking KM currents to holography and flux-induced anomalies, offering a unifying algebraic lens on higher-dimensional symmetries and compactifications.

Abstract

In this review, we present a general framework for the construction of Kac-Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on the circle $\mathbb{S}^{1}$, we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases and Killing vectors, we present the construction of generalized current algebras $\mathfrak{g}(\mathcal{M})$, their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and illustrate the framework through explicit realizations on $SU(2)$, $SU(2)/U(1)$, and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications.

New perspectives in Kac-Moody algebras associated to higher dimensional manifolds

TL;DR

This work develops a general framework to construct Kac–Moody current algebras on higher-dimensional manifolds by promoting finite-dimensional Lie algebras to -type current algebras and coupling them to isometries. It leverages harmonic analysis (Peter–Weyl for compact, Plancherel for noncompact) to define explicit mode expansions, and it classifies compatible central extensions via 2-cocycles, yielding Virasoro-like structures in several settings. The authors provide concrete realizations for compact groups (e.g., ) and noncompact examples (e.g., ), including soft deformations of group manifolds, and analyze representation-theoretic and unitarity aspects. The framework connects to physics across domains—2D CFT/WZW, KK spectra, cosmological billiards, and supergravity—and outlines conjectures linking KM currents to holography and flux-induced anomalies, offering a unifying algebraic lens on higher-dimensional symmetries and compactifications.

Abstract

In this review, we present a general framework for the construction of Kac-Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on the circle , we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases and Killing vectors, we present the construction of generalized current algebras , their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and illustrate the framework through explicit realizations on , , and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications.

Paper Structure

This paper contains 33 sections, 4 theorems, 329 equations.

Table of Contents

  1. Introduction
  2. Kac-Moody (KM) algebras on higher dimensional manifolds: Some generalities
  3. Relevant properties of manifolds
  4. A KM algebra associated to ${\cal M}$
  5. KM algebras on compact group manifolds
  6. Compact group manifolds
  7. Coset spaces of compact group manifolds
  8. KM algebras associated to compact Lie groups
  9. KM algebra associated to SU$(2)$ and SU$(2)/$U$(1)$
  10. Virasoro algebra associated to the two- and three- sphere
  11. KM algebras on non-compact Lie groups
  12. Some generalities
  13. A KM algebra associated to SL$(2,\mathbb R)$
  14. The group SL$(2,\mathbb R)$
  15. Matrix elements of $\mathfrak{sl}(2,\mathbb R)$ and Plancherel Theorem
  16. ...and 18 more sections

Key Result

Theorem 3.1

Let $\hat{\cal R}= \{{\cal R}_k, k \in \hat{G}_c\}$ be the set of all unitary irreducible representations of $G_c$, and let $D_{(k)}(g)\in {\cal R}_k$ for $g\in G_c$. Then the set of functions on $G_c$, forms a complete Hilbert basis of $L^2(G_c)$ with inner product

Theorems & Definitions (4)

  • Theorem 3.1: Peter-Weyl PW
  • Proposition 4.1
  • Proposition 6.1: mrm
  • Corollary 6.2