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We develop a weak coupling approach to superconductivity in twisted bilayergraphene, starting from the Fermi liquid regime. A key observation is that nearhalf filling, the fermiology consists of well nested Fermi pockets derived fromopposite valleys, leading to enhanced valley fluctuation, which in turn canmediate superconductivity. This scenario is studied within the random phaseapproximation. We find that inter-valley electron pairing with either chiral($d+i d$ mixed with $p-i p$) or helical form factor is the dominantinstability. An approximate SO(4) spin-valley symmetry implies a neardegeneracy of spin-singlet and triplet pairing. On increasing interactions,commensurate inter-valley coherence wave (IVCW) order can arise, withsimultaneous condensation at the three M points in the Brillouin Zone, and a$2\times2$ pattern in real space. In simple treatments though, this leads to afull gap at fillings $\pm (1/2+1/8)$, slightly away from half-filling. An SO(4)symmetry breaking "anti-Hunds" coupling favors the spin-singlet order both forthe IVCW and the superconductor, consistent with observations. Mott insulatorsderived from phase fluctuating superconductors are also discussed, whichexhibit both symmetry protected and intrinsic topological orders.

Superconductivity from Valley Fluctuations and Approximate SO(4) Symmetry in a Weak Coupling Theory of Twisted Bilayer Graphene