Orbit method for quantum corner symmetries

The classification of the unitary irreducible representations of symmetry groups is a cornerstone of modern quantum physics, as it provides the fundamental building blocks for constructing the Hilbert spaces of theories admitting these symmetries. In the context of gravitational theories, several arguments point toward the existence of a universal symmetry group associated with corners, whose structure is the same for every diffeomorphism-invariant theory in any dimension. Recently, the representations of the maximal central extension of this group in the two-dimensional case have been classified using purely algebraic techniques. In this work, we present a complementary and independent derivation based on Kirillov’s orbit method. We study the coadjoint orbits of the group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mover accent="true"> <a:mrow> <a:mi>SL</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>2</a:mn> <a:mo>,</a:mo> <a:mi mathvariant="double-struck">R</a:mi> <a:mo stretchy="false">)</a:mo> </a:mrow> <a:mrow> <a:mo stretchy="true">˜</a:mo> </a:mrow> </a:mover> <a:mo>⋉</a:mo> <a:msub> <a:mrow> <a:mi mathvariant="double-struck">H</a:mi> </a:mrow> <a:mrow> <a:mn>3</a:mn> </a:mrow> </a:msub> </a:mrow> </a:math> , where <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi mathvariant="double-struck">H</i:mi> <i:mn>3</i:mn> </i:msub> </i:math> is the Heisenberg group of a quantum particle in one dimension. Our main result is that, despite the non-Abelian nature of the normal subgroup in the semidirect product, these orbits admit a simple description. In a coordinate system associated with modified Lie algebra generators, the orbits factorize into a product of coadjoint orbits of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>SL</l:mi> <l:mo stretchy="false">(</l:mo> <l:mn>2</l:mn> <l:mo>,</l:mo> <l:mi mathvariant="double-struck">R</l:mi> <l:mo stretchy="false">)</l:mo> </l:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi mathvariant="double-struck">H</q:mi> <q:mn>3</q:mn> </q:msub> </q:math> . The subsequent geometric quantization of these factorized orbits successfully reproduces the known representations.