# What this parameter in the Riemann surface metric has to do with its complex structure?

As far as I'm aware, a complex manifold $$M$$ is a topological manifold together with an atlas $${\cal A}$$ of charts $$(U_i,\varphi_i)\in{\cal A}$$ such that the open sets $$U_i$$ cover $$M$$, the maps $$\varphi_i: U_i\to \mathbb{C}^n$$ are homeomorphisms and whenever $$U_i\cap U_j\neq\emptyset$$ the transition maps $$\varphi_i\circ\varphi_j^{-1}:\mathbb{C}^n\to \mathbb{C}^n$$ and $$\varphi_j\circ\varphi_i^{-1}:\mathbb{C}^n\to \mathbb{C}^n$$ are holomorphic. In that scenario, the atlas $${\cal A}$$ is called a complex structure.

Now I'm studying string theory and in the book I'm using (Becker, Becker and Schwarz) what the authors call a complex structure seems like something different. Quoting their book, pages 89 and 90:

Here $$\int Dh$$ means the sum over all Riemann surfaces $$(M,h)$$. However, this is a gauge theory, since $$S$$ is invariant under diffeomorphisms and Weyl transformations. So one should really sum over Riemann surfaces modulo diffeomorphisms and Weyl transformations. Worldsheet diffeomorphism symmetry allows one to choose a conformally flat worldsheet metric $$h_{\alpha\beta}=e^\psi \delta_{\alpha\beta}.\tag{3.109}$$ When this is done, one must add the Faddeev-Popov ghost fields $$b(z)$$ and $$c(z)$$ to the worldsheet theory to represent the relevant Jacobian factors in the path integral. Then the local Weyl symmetry $$(h_{\alpha\beta}\to \Lambda h_{\alpha\beta}$$) allows one to fix $$\psi$$ (locally) - say to zero. However, this is not possible globally, due to a topological obstruction: $$\psi=0\Longrightarrow R(h)=0\Longrightarrow \chi(M)=0.\tag{3.110}$$ So, such a choice is only possible for worldsheets that admit a flat metric. Among orientable Riemann surfaces without boundary, the only such case is $$n_{\rm h}=1$$ (the torus). For ach genus $$n_{\rm h}$$ there are particular $$\psi$$s compatible with $${\chi}(M)=2-2n_{\rm h}$$ that are allowed. A specific choice of such a $$\psi$$ corresponds to choosing a complex structure for $$M$$.

Now I'm puzzled with this last statement. This $$\psi$$ has to do with the metric, while the complex structure has to do with how we construct an atlas of holomorphic charts and therefore with how holomorphic functions are defined. I can't see how these two things are related.

What I imagined was this: we want to consider all possible Riemann surfaces $$(M,h)$$. Given any such surface the diffeomorphism symmetry allows the metric to be put in the form (3.109) and therefore be classified by $$\psi$$. Still, given a certain topology, not all metrics can be defined in the surface with that topology. Therefore we partition the space of such surfaces by genus $$g$$ and for each genus $$g$$ we have a class of possible $$\psi$$.

I'm conjecturing that the same happens with complex structure. In the same way that given a topology not all metrics can be defined in the surface with that topology, given a complex structure not all metrics are compatible with it.

Still I don't know if this is correct, and I think a more complete understanding is necessary here. So why does a specific choice of such a $$\psi$$ corresponds to choosing a complex structure for $$M$$?

• Is the metric defined on a surface? What is a worldsheet.... ? Feb 7, 2021 at 3:35
• Sorry, I've used Physics terminology and forgot to explain it. The worldsheet is the Riemann surface under consideration and the metric is defined on it
– Gold
Feb 7, 2021 at 14:02
• The boldface sentence is not just wrong, it is meaningless since, as the authors note, $\psi$ is not globally defined. There are many places where moduli space of conformal structures is treated rigorously, my suggestion is to read one of these. Feb 7, 2021 at 16:30
• @MoisheKohan thanks for pointing this out. As probably noticed, I'm new to the subject and I'm studying it because of string theory. Could you suggest such a reference? If there is one accessible to physicists it would be particularly nice, but any good reference would already help!
– Gold
Feb 7, 2021 at 17:45
• Besides the one suggested by Moishe, you may also try Riemann surface by J. Jost. Feb 7, 2021 at 19:05

As I said in the comment, what they wrote is simply nonsense. Here is a couple of references where at least you will not find any obvious nonsense:

1. As a mathematician, I like the first 6 chapters in

Ji, Lizhen; Looijenga, Eduard J. N., Introduction to moduli spaces of Riemann surfaces and tropical curves, Surveys of Modern Mathematics 14. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-353-1/pbk). iv, 221 p. (2017). ZBL1375.14002.

The tropical staff is (probably) irrelevant for you and is discussed in Part II of the book.

1. Also take a look at this, which is aimed to discuss integration over moduli spaces and is probably more relevant for your purposes:

Nag, Subhashis, Mathematics in and out of string theory, Kojima, Sadayoshi (ed.) et al., Topology and Teichmüller spaces. Proceedings of the 37th Taniguchi symposium, Katinkulta, Finland, July, 24–28, 1995. Singapore: World Scientific (ISBN 981-02-2686-1/hbk). 187-220 (1996). ZBL1050.32501.

A free version is available here, but there might be some differences with the published version.