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Hidden $SL_2(\mathbb{Z})$-symmetry in $BC_l^{(2)}$

K. Iohara, Y. Saito

TL;DR

The paper reveals a hidden $\mathrm{SL}_2(\mathbb{Z})$ symmetry in the affine Lie algebra of type $BC_l^{(2)}$ by extending the $\Gamma_{\theta}$-action to a full modular action on the vector space spanned by (twisted) characters at even level. It builds this symmetry via two coordinate realizations, theta-series, and Poisson resummation, and expresses normalized characters as ratios of twisted/untwisted theta-denominator invariants. A key finding is that $\chi_\Lambda$ and $\chi_\Lambda^{\psi}$ can be realized as modular objects tied to $A_{\lambda+\rho}$ and its twisted counterpart, establishing an $\mathrm{SL}_2(\mathbb{Z})$-action on the corresponding character space. The work also uncovers connections to the affine Lie superalgebra $B^{(1)}(0,l)$, hinting at a broader hidden symmetry for non-reduced affine root systems and potential extensions to related algebras.

Abstract

A well-known \(Γ_θ\)-action on the characters of integrable highest weight modules over the affine Lie algebra of type \(BC_l^{(2)}\) at a positive level is extended to an \(\mathrm{SL}_2(\mathbb{Z})\)-action at a positive even level by supplementing their twisted characters.

Hidden $SL_2(\mathbb{Z})$-symmetry in $BC_l^{(2)}$

TL;DR

The paper reveals a hidden symmetry in the affine Lie algebra of type by extending the -action to a full modular action on the vector space spanned by (twisted) characters at even level. It builds this symmetry via two coordinate realizations, theta-series, and Poisson resummation, and expresses normalized characters as ratios of twisted/untwisted theta-denominator invariants. A key finding is that and can be realized as modular objects tied to and its twisted counterpart, establishing an -action on the corresponding character space. The work also uncovers connections to the affine Lie superalgebra , hinting at a broader hidden symmetry for non-reduced affine root systems and potential extensions to related algebras.

Abstract

A well-known -action on the characters of integrable highest weight modules over the affine Lie algebra of type \(BC_l^{(2)}\) at a positive level is extended to an \(\mathrm{SL}_2(\mathbb{Z})\)-action at a positive even level by supplementing their twisted characters.

Paper Structure

This paper contains 21 sections, 17 theorems, 124 equations.

Table of Contents

  1. Kac-Moody Lie algebras of type $BC_l^{(2)}$
  2. Root datum
  3. Kac-Moody Lie algebras and their roots
  4. The affine Weyl group $W$
  5. The complex domain $\mathcal{Y}$ and its coordinate systems
  6. Weight lattice
  7. Characters
  8. Algebra $\mathcal{E}$, formal characters
  9. Nice functions on the Weyl group $W$ and twisted character
  10. Theta functions
  11. Theta functions
  12. Normalized (twisted-)characters
  13. Actions of $\mathrm{SL}_2(\mathbb{Z})$
  14. Poisson's resummation formula
  15. Modular transformations on twisted-invariants
  16. ...and 6 more sections

Key Result

Lemma 1.4

(1) Both $t_\mu$ and $s_\beta$ belong to $\mathrm{O}(\mathfrak{h}^\ast,\widehat{I})$. (2) For every $\mu_1,\mu_2\in F$, one has $t_{\mu_1}t_{\mu_2}=t_{\mu_1+\mu_2}$. (3) For every non-isotropic $\beta\in F$, one has $s_{\delta-\beta}s_\beta=t_{\beta^\vee}$. (4) For $w\in W$ and $\mu\in F$, one has $

Theorems & Definitions (31)

  • Remark 1.1
  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Lemma 1.6
  • proof
  • Lemma 1.7
  • Lemma 1.8
  • ...and 21 more