Weak gravity conjecture in the asymptotical dS and AdS background

The cosmological observations provide a strong evidence that there is a positive cosmological constant in our universe and thus the spacetime is asymptotical de Sitter space. The conjecture of gravity as the weakest force in the asymptotical dS space leads to a lower bound on the U(1) gauge coupling g, or equivalently, the positive cosmological constant gets an upper bound rho(V) <= g(2)M(p)(4) in order that the U(1) gauge theory can survive in four dimensions. This result has a simple explanation in string theory, i. e. the string scale root alpha' should not be greater than the size of the cosmic horizon. Our proposal in string theory can be generalized to U(N) gauge theory and gives a guideline to the microscopic explanation of the de Sitter entropy. The similar results are also obtained in the asymptotical anti-de Sitter space.