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Résumé |
We propose a calculation of the density of states of the static patch in dS3 via canonical quantization. We
ask what is the
amplitude to find a particle with energy E and spin J in the static patch, imposing the boundary condition
that the horizon
is smooth. There is an SL(2,Z) family of such smooth horizons associated with lens spaces, which all
contribute to the
static patch density of states. To compute the amplitudes exactly, we must pick a quantization scheme.
We draw
inspiration from the SL(2,C) CS formulation of dS3. We find that there is a dual calculation of the amplitude
for each lens
space that involves two copies of complex Liouville string theory. The holographic direction in this 3d-2d
duality is the
spatial angle. The CLS calculation gives a robust answer for the Lens space amplitudes that agrees with
the semiclassical
path integral. For one of the lens spaces, the two copies of CLS are in turn holographically dual to two
copies of the DSSYK
GSigma model living on the observer's worldline. This goes towards establishing a 3d-1d worldline
hologram. |
