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Ground state representations of loop algebras

Yoh Tanimoto

TL;DR

The paper addresses the problem of classifying ground-state representations for the Schwartz-class loop algebra $\\mathscr{S}\\mathfrak{g}_{\\mathbb{C}}$ and linking them to vacuum loop-group representations. It establishes that translation-invariant 2-cocycles are unique up to a scalar, and that ground-state representations are completely determined by an integer level $c$, with $ ext{psi}=0$, yielding representations that extend to the vacuum representation of $L\\mathfrak{g}_{\\mathbb{C}}$. The approach combines an explicit cocycle analysis via locality and symmetry with Fourier-analytic techniques to characterize $n$-point functions, leading to a Lie-algebraic route to vacuum-state uniqueness in conformal nets. These results illuminate the relationship between Schwartz-class algebras and loop-group nets in chiral conformal field theory and provide a complementary perspective to operator-algebraic methods for the uniqueness of ground states at zero temperature.

Abstract

Let g be a simple Lie algebra, Lg be the loop algebra of g. Fixing a point in S^1 and identifying the real line with the punctured circle, we consider the subalgebra Sg of Lg of rapidly decreasing elements on R. We classify the translation-invariant 2-cocycles on Sg. We show that the ground state representation of Sg is unique for each cocycle. These ground states correspond precisely to the vacuum representations of Lg.

Ground state representations of loop algebras

TL;DR

The paper addresses the problem of classifying ground-state representations for the Schwartz-class loop algebra and linking them to vacuum loop-group representations. It establishes that translation-invariant 2-cocycles are unique up to a scalar, and that ground-state representations are completely determined by an integer level , with , yielding representations that extend to the vacuum representation of . The approach combines an explicit cocycle analysis via locality and symmetry with Fourier-analytic techniques to characterize -point functions, leading to a Lie-algebraic route to vacuum-state uniqueness in conformal nets. These results illuminate the relationship between Schwartz-class algebras and loop-group nets in chiral conformal field theory and provide a complementary perspective to operator-algebraic methods for the uniqueness of ground states at zero temperature.

Abstract

Let g be a simple Lie algebra, Lg be the loop algebra of g. Fixing a point in S^1 and identifying the real line with the punctured circle, we consider the subalgebra Sg of Lg of rapidly decreasing elements on R. We classify the translation-invariant 2-cocycles on Sg. We show that the ground state representation of Sg is unique for each cocycle. These ground states correspond precisely to the vacuum representations of Lg.

Paper Structure

This paper contains 10 sections, 18 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The loop group $LG$
  4. The loop algebra $L{\mathfrak g}$
  5. Preliminaries on the Schwartz class algebra $\mathscr{S}{\mathfrak g}$
  6. Uniqueness of translation invariant 2-cocycle
  7. Uniqueness of ground state representations
  8. Concluding remarks: ground states of conformal nets
  9. Ground state representations of Lie algebras and conformal nets
  10. Irreducibility and factoriality of representations

Key Result

Theorem 2.1

If $G$ is simple and simply connected, then there exists a family of central extensions of $LG$ which are parametrized by positive integers, and all such extensions come from the central extensions of $L{\mathfrak g}$ (see below).

Theorems & Definitions (36)

  • Theorem 2.1: Pressley and Segal
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: Pressley and Segal
  • Theorem 2.6
  • Theorem 2.7: Garland
  • Definition 3.1
  • Remark 3.2
  • Lemma 4.1
  • ...and 26 more