TITLE: What is a branched Riemann surface with cuts?
QUESTION [10 upvotes]: Edit: Let me restate the main claim being made in these two papers,

Consider the "branched" Riemann surface which has "n" sheets stuck along the intervals, $[z_i, z_{i+1}]$ for $i=1,..,2N$ then it is of genus $(n-1)(N-1)$ and is represented by the equation, $y^n = \prod_{i=1...N} \frac{z-z_{2i-1} }{z-z_{2i}}$ 
If the same intervals are labelled as, $[u_i,v_i]$ and $i=1,..,N$ then this is represented by the curve, $y^n = \prod_{i=1}^{N-1}(z-u_k)(z-v_k)^{n-1}(z-u_N)$ and has genus $(n-1)(N-1)$
(..and $v_N$ has somehow been sent to infinity using conformal maps..) 

I would like to understand why the above claims are true and their derivation and why these are the same things. (...to begin with I can't see how a "genus" can even be defined for such an object - it isn't something compact and orientable!..) 
I can't see how I can use any of the standard Riemann surface theory (like in the books by Griffiths or Jurgen Jost) on this weird structure! 
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The two paper references where this concept is mentioned which I would like to understand, 

http://arxiv.org/abs/1303.7221 (the branched Riemann surface that has been constructed at the top of page 6 and its algebraic curve and genus) 
http://arxiv.org/abs/1209.2428 (the construction around equation 3.16 on page 16) 

I guess both of them are constructing the same "branched Riemann surface" and writing down an algebraic curve for it and calculating its genus. I can't understand this construction and how the Riemann-Hurwitz formula is being used here. 
If I go back to my usual references of the books by Jurgen Jost or Phillip Griffiths then I don't see anything there called "n sheeted branched Riemann surface with N cuts" and how one calculates it genus or its algebraic curve. (or am I missing something and hence not able to recognize the concept?)

REPLY [6 votes]: Since no one has answered the genus computation, here goes:
The Riemann-Hurwitz formula states that for a map of Riemann surfaces $f : C_1 \to C_2$, that we have
$$
\chi(C_1) = n\chi(C_2) - \deg R
$$
where $n$ is the degree of the map, and $R$ the ramification divisor.
In the case you have, since we are covering $\mathbb{C}$, we are secretly covering $\mathbb{P}^1$, so the Euler characteristic of the base is 2. The degree is clearly $n$ (the number of sheets), and it is hopefully clear that the map is only ramified at the $2N$ points listed, with ramification index $(n - 1)$. Thus the Riemann-Hurwitz formula now reads
$$
2 - 2g(C) = 2n - 2N(n - 1)
$$
and solving for the genus gives you the desired answer.
As for why the genus is well-defined---well, here is the slight-of-hand that I used to say that it's secretly a map to $\mathbb{P}^1$. If you take the $n$ given sheets and glue them together you get a non-compact Riemann surface. To make it compact, you just have to add in a few points. The easiest way to do it is to replace the copies of $\mathbb{C}$ that you have with copies of $\mathbb{P}^1$. You can still perform the cutting-and-gluing construction to build a surface (this time with a map to $\mathbb{P}^1$). In this case, you will find that you have a compact surface. It is orientable, as at no point are we using anything but holomorphic transition functions.

REPLY [5 votes]: @user6818: To reiterate earlier replies, the Riemann surface constructed from slit planes is smooth and compact. Your authors' equation should be viewed as defining a locus in $S^2 \times S^2$, viewed as the componentwise compactification of $\mathbb{C}^2$. In this setting:

As Alexandre Eremenko and Paul Garrett noted, the "Riemann surface of $\sqrt{z}$" is the sphere $S^2$, equipped with the squaring mapping $f(w) = w^2$, namely, the (compact, smooth) locus of $z = w^2$ in $S^2 \times S^2$. The branch points (images of critical points of $f$) are $z = 0$ and $\infty$. Since $f$ is two-to-one, the classical construction is to take two copies of $\mathbb{C}$ (with coordinate $z$), make a slit between the branch points (e.g., by removing the negative real axis), and to "cross-glue" the edges of the slits. Here, the result is another copy of $\mathbb{C}$ (with coordinate $w$). Adding the branch point $z = \infty$ compactifies the $w$ plane to $S^2$.
Igor Khavine's example of the "(complex) circle" $w^2 + z^2 = 1$ (again, a compact, smooth locus in $S^2 \times S^2$) may be viewed as "the Riemann surface of $\sqrt{1 - z^2}$". The picture is identical to the squaring map (rotate the "$z$ sphere" to bring $0$ and $\infty$ to $-1$ and $-1$, say); the branch points are $\pm 1$, and the Riemann surface can be constructed by slitting two copies of $\mathbb{C}$ along the closed interval $[-1, 1]$ and "cross-gluing" the edges. In the diagram below, the slits are the upper and lower halves of the circle in the vertical plane above the real axis; the two "sheets" ($z$ planes) of the surface are the two "tilted planes". The locus is smooth even at the branch points $z = \pm 1$, i.e., $w = 0$. (The vertical axis is the real part of $w$. At risk of posting gratuitous eye candy, there's an animation loop at http://mathcs.holycross.edu/~ahwang/epix/misc/complex/quadric.gif that shows the result of rotating the $w$ plane while keeping $z$ fixed.)

Andy

REPLY [2 votes]: For convenience I've added some figures from those two papers:




Especially since you are talking about Entanglement Entropy, I thought you were asking what are Riemann surfaces in a more philosophical sense.