  Theta identities on genus g Riemann surfaces which decompose simple products
of fermion correlation functions with a constraint on their variables are
considered. This type of theta identities is, in a sense, dual to Fay s
formula, by which it is possible to sum over spin structures of certain part of
superstring amplitudes in NSR formalism without using Fay s formula nor Riemann
s theta formula in much simpler, more transparent way. Also, such identities
will help to cast correlation functions among arbitrary numbers of Kac-Moody
currents in a closed form. As for genus 1, the identities are reported before
in ref[1] [2]. Based on some notes on genus 1 case which were not reported in
ref[1] [2] and relating those to the results of the Dolan Goddard method ref[3]
on describing Kac-Moody currents in a closed form, we propose an idea of
generalizing genus 1 identities to the case of genus g surfaces. This is not a
complete derivation of the higher genus formula due to difficulties of
investigating singular part of derivatives of genus g Weierstrass Pe functions.
Mathematical issues remained unsolved for genus g >1 are described in the text.
