2019/12/17 by Tuite, Michael P.
#Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1912.07947
For a Riemann surface of genus g≥ 2 there exists a unique Green's function GN(x,y) which transforms as a weight N≥ 2 form in x and a weight 1-N form in y and is meromorphic in x, with a unique simple pole at x=y, but is not meromorphic in y. For a Schottky uniformized Riemann surface we consider meromorphic extensions of GN(x,y) called Green's Functions with Extended Meromorphicity or GEM forms. GEM forms are meromorphic in both x and y with a unique simple pole at x=y, transform as weight N≥ 2 forms in x but as weight 1-N quasiperiodic forms in y. We give a reformulation of the bijective Bers map and describe a choice of GEM form with an associated canonical basis of normalized holomorphic N-forms. We describe an explicit differential operator constructed from N=2 GEM forms giving the variation with respect to moduli space parameters of a punctured Riemann surface. We also describe a new expression for the inverse Bers map.