Integrable hierarchy for multidimensional Toda equations and topological-anti-topological fusion

The negative symmetry flows are incorporated into the Riemann-Hilbert problem for the homogeneous A_m-hierarchy and its gl̂(m+1, C) extension. A loop group automorphism of order two is used to define a sub-hierarchy of gl̂(m+1, C) hierarchy containing only the odd symmetry flows. The positive and negative flows of the ±1 grade coincide with equations of the multidimensional Toda model and of topological-anti-topological fusion.