Theta Functions on Riemann Surfaces

Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

“…where the LHS is the sum over all residues on a fundamental domain, the poles z i are all the poles of ω as well as the pole at q = p. This identity is obtained by moving the integration contours on the surface, and taking carefully into account discontinuities along the nontrivial cycles (see [24,25]). …”

“…Its expression in terms of theta-functions can be found in [24,25], it depends only on the complex structure of E.…”

Non-renormalization of mass of some non-supersymmetric string states

“…To see that it does, we need to consider in somewhat detail the boundary of the moduli space of the higher genus Riemann surfaces. This is a rather well-studied subject [15,16,17]. For our purposes, we can regard the boundary of the moduli space as consisting of two distinct kind of degenerations of the Riemann surface:…”

Beyond the elliptic genus

“…Note that we can write h δ = Nĥ δ where h δ , h δ = |N| 2 . It well known [29,30] that the square of the holomorphic spinor may be taken to be…”