M-theory torus duality (${\tau }_{T^2}=C_0+i/g_B$)

M-theory on a two-torus and type IIB superstring theory on a circle describe the same nine-dimensional physics. Torus shape supplies the axion-dilaton; torus area supplies the circle-radius modulus after matching metrics. An M2-brane on a one-cycle gives a (p,q) string; an M5-brane on the whole torus gives a D3-brane. Matching brane charges and tensions establishes the dictionary in The Power of M Theory.