# Ising CFT on a higher genus surface

I'm wondering if someone knows how to put the critical Ising model on an arbitrary Riemann surface of genus g. I know how to do it on a torus but I want to know how to find the correlator of the Ising fields $$<\psi(z) \psi(w) >$$ on any surface with either periodic or anti-periodic boundary conditions along the cycles of the torus.

One issue is that it possible to flatten a torus but not surfaces with genus $g \geq 2$. So I'm not sure how it would work but it seems like a fairly well posed problem that should have an answer in the string theory literature. Unfortunately I'm not well versed with this so I would appreciate either a self-contained answer or a good reference.

• Give yourself use of the full hyperbolic plane first, then do a quotient/images. – AHusain Sep 23 '16 at 2:16
• @AHusain Could you elaborate a bit please? My intuition is that the $\theta$-functions that show up on the torus will generalise to higher dimensional analogues, but again, I'm not an expert on this. Even a reference where this is done explicitly would be enormously helpful. – Aegon Sep 23 '16 at 2:25
• Maybe Dijkgraaf, Verkinde^2 but have to look at that one again to check if remembering correctly. – AHusain Sep 23 '16 at 3:26
• @AHusain Thank you for that reference, it was really helpful. One further question I have, if you don't mind answering, is how to extract the contribution coming from even and odd spin structures to the holomorphic correlation function $\sigma(z) \sigma(w)$ from the full correlator? It seems this answer is embedded in Eq. 4.16 of that paper but I'm having trouble extracting it. Thank you again! – Aegon Sep 26 '16 at 17:21
• @AHusain Here is the paper for convenience: projecteuclid.org/download/pdf_1/euclid.cmp/1104161089. Actually, I was mistaken in the previous comment - the paper, in Eq 4.16, has the correlation function of the spin fields whereas I want to understand the correlation function of the $\it{Majorana}$ fields. This doesn't seem to be contained within that paper and I'm not sure how to figure that out. – Aegon Sep 26 '16 at 22:46