An Overview of the History of Projective Representations of Groups

An overview of the history of projective representations (= spin representations) of groups, preceded by the prehistory of studies on the theory of quaternion due to Rodrigues and Hamilton. Beginning with Schur, we cover many mathematicians until today, and also physicists Pauli and Dirac.

Conformal bootstrap in dS/CFT and topological quantum gravity

We show that the correspondence among [Formula: see text], the 1D Schwarzian Model, Sachdev–Ye–Kitaev model and [Formula: see text] Topological Quantum Gravity can be extended to the case of [Formula: see text]. The [Formula: see text]-matrix, related to the gravitational scattering amplitude near the horizon of [Formula: see text] black hole, corresponds (on the side of the holographic projection) to a crossing kernel in the Schwarzian Model. The [Formula: see text]-matrix is related to the 6j-symbol of SU[Formula: see text]. We also find that in the Euclidean [Formula: see text] a new Kac–Moody symmetry of instantons emerges out. We dub these new solutions Kac–Moodions. A one-to-one correspondence of Kac–Moodion levels and SU[Formula: see text] spin representations is established. Every instanton then corresponds to spin representations deployed in Topological Quantum Gravity. The instantons are directly connected to the Black Hole entropy as punctures on its horizon. This strongly supports the recent proposal, in arXiv:1707.00347, that a Kac–Moody symmetry of gravitational instantons is related to the black hole information processing. We also comment on a further correspondence that can be established between the Schwarzian Model and noncommutative spacetimes in [Formula: see text]D, passing through the equivalence with Topological Quantum Gravity with cosmological constant, in the limit when the latter vanishes.