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Notes on $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ representations

Alexei Kitaev

TL;DR

These notes survey representations of the universal cover of SL(2,R), emphasizing applications to physics through spinors on the hyperbolic plane and AdS2 and connections to the SYK model. The work classifies unitary and non-unitary representations, develops explicit Casimir eigenfunctions, and builds a concrete Fourier analysis (matrix elements and Plancherel measure) on the noncompact group, linking abstract representation theory to geometric spinor fields. Key contributions include intertwiners between non-unitary and unitary series, detailed Clebsch–Gordan coefficients for tensor products, and a spinor formalism that realizes representations as fields on H^2 and AdS2, enabling practical computations in low-dimensional holography and related quantum models.

Abstract

These notes describe representations of the universal cover of $\mathrm{SL}(2,\mathbb{R})$ with a view toward applications in physics. Spinors on the hyperbolic plane and the two-dimensional anti-de Sitter space are also discussed.

Notes on $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ representations

TL;DR

These notes survey representations of the universal cover of SL(2,R), emphasizing applications to physics through spinors on the hyperbolic plane and AdS2 and connections to the SYK model. The work classifies unitary and non-unitary representations, develops explicit Casimir eigenfunctions, and builds a concrete Fourier analysis (matrix elements and Plancherel measure) on the noncompact group, linking abstract representation theory to geometric spinor fields. Key contributions include intertwiners between non-unitary and unitary series, detailed Clebsch–Gordan coefficients for tensor products, and a spinor formalism that realizes representations as fields on H^2 and AdS2, enabling practical computations in low-dimensional holography and related quantum models.

Abstract

These notes describe representations of the universal cover of with a view toward applications in physics. Spinors on the hyperbolic plane and the two-dimensional anti-de Sitter space are also discussed.

Paper Structure

This paper contains 24 sections, 124 equations, 4 figures.

Table of Contents

  1. The group $\mathfrak{G}\cong\mathop{\mathrm{PSL}}\nolimits(2,\mathbb{R})$ and its universal cover
  2. Irreducible unitary representations
  3. Non-unitary continuous and discrete series
  4. The individual representations
  5. Isomorphisms and intertwiners
  6. Fourier transform on $\widetilde{\mathfrak{G}}$ and the Plancherel measure
  7. Fourier transform on a group
  8. The L- and R-actions on $\widetilde{\mathfrak{G}}$ in explicit coordinates
  9. Casimir eigenfunctions
  10. General solutions:
  11. Nonsingular solutions:
  12. Normalizability:
  13. The matrix elements and Plancherel measure for the irreps of $\widetilde{\mathfrak{G}}$
  14. Spinors on the hyperbolic plane and anti-de Sitter space
  15. The spaces $\mathop{\mathrm{H}}\nolimits^2$, $\mathop{\mathrm{AdS}}\nolimits_2$, and their complex embeddings
  16. ...and 9 more sections

Figures (4)

  • Figure 1: The structure of representations $\mathcal{F}^\mu_\lambda$ for some $\lambda$ and $\mu$. The full circles indicate those basis vectors $f_{\lambda,m}$ that span an invariant subspace, whereas the empty circles correspond to quotients.
  • Figure 2: Schwarzschild patch of the anti-de Sitter space.
  • Figure 3: The action of $L_{-1}$, $L_{1}$ on Casimir eigenfunctions for $\lambda=1,\frac{3}{2},2,\ldots$ and $\nu\in\lambda+\mathbb{Z}$. A circle with label $m\in\nu+\mathbb{Z}$ represents the basis function ${\psi}_{\lambda,m}^{\,\nu,+}$ if $m\geqslant\nu$ and ${\psi}_{\lambda,m}^{\,\nu,-}$ if $m\leqslant\nu$.
  • Figure 4: The spectra of the operators $Q$ and $\frac{1}{2}(Q+\nu^2)$ as functions of the spin value $\nu$.