[Math] a branched Riemann surface with cuts

ag.algebraic-geometryalgebraic-curvesquantum-field-theoryriemann-surfacesstring-theory

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,

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?)

Best Answer

There are two different things which are called "Riemann surface" in the literature.

  1. The modern notion (introduced by Hermann Weyl): complex 1-dimensional manifold. In older literature this is sometimes called "Abstract Riemann surface".

  2. "Riemann surface spread over the plane" (or over the sphere, or over some other surface). Surface de Riemann etalee in French, Uberlagerungsflache in German. This is what many old authors (and physicists) call a Riemann surface.

In the work of Riemann you can see both notions, but they are not clearly distinguished.

The relation between these two things is the following. A Riemann surface spread over the sphere is a pair $(F,p)$ where $F$ is an abstract surface, and $p:F\to S$ is a holomorphic map to the Riemann sphere. Critical points of $p$ are called "branch points" of the surface $(F,p)$, and so on. If you have an analytic germ, and perform an analytic continuation over all paths on which continuation is possible, you obtain a Riemann surface spread over the sphere $(F,p)$ whose $F$-part is an abstract Riemann surface. To visualize a Riemann surface spread over the sphere, you make cuts and paste it from sheets. The cuts and sheets are arbitrary to some extent (they are not intrinsically connected with $(F,p)$; the fact that physicists frequently ignore).

Most of the modern literature uses the first meaning. Some modern mathematical works (besides the elementary complex variables textbooks) which deal with surfaces spread over the plane or sphere are the papers of J. Ecalle on resurgent functions, or my survey on Geometric theory of meromorphic functions, which can be found in http://www.math.purdue.edu/~eremenko/surveys.html, or the work of H. Stahl, arXiv:1205.3811.

For many people with modern education (who think that a Riemann surface is a "complex 1-dimensional manifold"), the expression "the Riemann surface of $\sqrt{z}$" is meaningless, because this is just the same as the sphere.

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