Abstract
AbstractWe study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra $$[x_i,x_j]=2\imath \lambda _p \epsilon _{ijk}x_k$$
[
x
i
,
x
j
]
=
2
ı
λ
p
ϵ
ijk
x
k
modulo setting $$\sum _i x_i^2$$
∑
i
x
i
2
to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric $$3 \times 3$$
3
×
3
matrices g and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature $$ \frac{1}{2}(\mathrm{Tr}(g^2)-\frac{1}{2}\mathrm{Tr}(g)^2)/\det (g)$$
1
2
(
Tr
(
g
2
)
-
1
2
Tr
(
g
)
2
)
/
det
(
g
)
. As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure.
Funder
CONACyT
Fundacion Alberto y Dolores Andrade
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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