Quasi Riemann surfaces

A quasi Riemann surface is defined to be a certain kind of complete metric space Q whose integral currents are analogous to the integral currents of a Riemann surface. In particular, they have properties sufficient to express Cauchy-Riemann equations on Q. The prototypes are the spaces D 0 (Σ)m of integral 0-currents of total mass m in a Riemann surface Σ (usually called the integral 0-cycles of degree m). For M an oriented conformal 2n-manifold, there is a bundle Q(M) → B(M) of quasi Riemann surfaces naturally associated to M . For n odd, this is the bundle D n−1(M) ∂ − → ∂D n−1(M) of integral (n−1)-currents inM fibered over the integral (n−2)boundaries in M . For n even, the examples Q(M) are slightly more complicated. I suggest that complex analysis on quasi Rieman surfaces be developed by analogy with classical complex analysis on Riemann surfaces, based on the Cauchy-Riemann equations. I want to use complex analysis on quasi Rieman surfaces to construct a new class of quantum field theories in spacetimesM . The new quantum field theories are to be constructed on the quasi Riemann surfaces Q(M) by analogy with the construction of 2d conformal field theories on Riemann surfaces. The quasi Rieman surfaces Q(M) might also be useful in the study of the manifolds M .