Dark Energy and Dark Matter in Emergent Gravity

We suggest that dark energy and dark matter may be a cosmic ouroboros of quantum gravity due to the coherent vacuum structure of spacetime. We apply the emergent gravity to a large $N$ matrix model by considering the vacuum in the noncommutative (NC) Coulomb branch satisfying the Heisenberg algebra. We observe that UV fluctuations in the NC Coulomb branch are always paired with IR fluctuations and these UV/IR fluctuations can be extended to macroscopic scales. We show that space-like fluctuations give rise to the repulsive gravitational force while time-like fluctuations generate the attractive gravitational force. When considering the fact that the fluctuations are random in nature and we are living in the (3+1)-dimensional spacetime, the ratio of the repulsive and attractive components will end in $\frac{3}{4}: \frac{1}{4}=75:25$ and this ratio curiously coincides with the dark composition of our current Universe. If one includes ordinary matters which act as the attractive force, the emergent gravity may explain the dark sector of our Universe more precisely.Comment: v3; two appendices and references are added, 20 page

Lie algebra cohomology and group structure of gauge theories

We explicitly construct the adjoint operator of coboundary operator and obtain the Hodge decomposition theorem and the Poincar\'e duality for the Lie algebra cohomology of the infinite-dimensional gauge transformation group. We show that the adjoint of the coboundary operator can be identified with the BRST adjoint generator $Q^{\dagger}$ for the Lie algebra cohomology induced by BRST generator $Q$. We also point out an interesting duality relation - Poincar\'e duality - with respect to gauge anomalies and Wess-Zumino-Witten topological terms. We consider the consistent embedding of the BRST adjoint generator $Q^{\dagger}$ into the relativistic phase space and identify the noncovariant symmetry recently discovered in QED with the BRST adjoint N\"other charge $Q^{\dagger}$.Comment: 24 pages, RevTex, Revised version submitted to J. Math. Phy