# Is there a rigorous proof that worldsheets are Riemann surfaces?

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Somewhat handwavingly, one can see that it is reasonable to think of worldsheets as Riemann surfaces by noting that on a worldsheet (due to the invariance of the Polyakov action under a Weyl transformation) and on a Riemann surface (due to the fact that confomal maps imply local scaling transformations) no physical lengths can be defined in both cases.

However, on a worldsheet the distance can always be defined by making use of the Minskowsky metric as

$-ds^2 = - d\tau^2 + d\sigma^2$

whereas on a Riemann surface making use of the the complex chart the distance is defined as

$ds^2 = ¦z¦^2 =dx^2 +dy^2$

Due to this sign difference it seems, that it can not  be proven that worldsheets are Riemann surfaces. Or has this been shown by now, maybe involving something like Wick rotations?

This issue might be related to an question that bugged me earlier, namely why it is that all possible Riemann surfaces can be constructed from string interactions (without there being Rieman surfaces that do not correspond to a certain history of interacting strings or strings producing worldsheets that are not Riemann surfaces)

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If we take as first tentative definition of the action on the string world-sheet the Nambu-Goto action then the metric on the world-sheet is not dynamical and it is simply the induced metric from the Minkowski space-time. In particular, if the strings really propagate in a time-like direction, this induced metric is Lorentzian, i.e. of signature (-+).

As we do not know how to make sense of the Nambu-Goto at the quantum mechanical level (because of the square root), we try to replace it by the Polyakov action. In this formulation, the world-sheet metric is a dynamical variable and is equal to the induced metric only when the equations of motion are satisfied. In any case, this metric should still be Lorentzian.

(Remark: the replacement of the Nambu-Goto action by the Polyakov action is generally justified by saying that the two actions define the same classical theory. But if the Nambu-Goto theory was defined at the quantum level, we would like to have an equivalence of quantum theories. Even if we don't know how to define the Nambu-Goto theory at the quantum level, it is possible to argue that if it was defined, the quantum equivalence should hold: see for example Chapter 9 of Polyakov book "Gauge fields and strings").

Now, to define a quantum theory from the Polyakov action, we should have to sum over the world-sheet topologies and to integrate over the space of Lorentzian metrics on a given world-sheet topology. In other words, we are trying to define some form of 2-dimensional quantum gravity. Using the Weyl invariance of the Polyakov action and assuming that the Weyl anomaly vanishes (critical dimension), we can reduce the integration to the space of conformal classes of Lorentzian metrics. The problem is that we don't know how to define such integral. One first problem is that the space of conformal classes, as the total space of metrics, is infinite dimensional in general (this is the case for a case as simple as a 2-torus). A second problem is that a Lorentzian metric on a world-sheet would be in general singular at the points where the strings break and join (the only compact orientable surface which admits everywhere defined Lorentzian metric is the torus). A still subtler fact is that Lorentzian geometry depends heavily of the degree of differentiability : for example two Lorentzian metric can be conformal equivalent by a 1 times differentiable map but not by a 2 times differentiable map, so it is not even clear what is the correct mathematical meaning of "conformal class" for the physical theory. These three problems : infinite dimensionality, singularities, dependance on regularity are typical of hyperbolic questions.

As we don't know how to define the integral over the space of Lorentzian metrics, we can try to do a Wick rotation and to integrate over the space of Euclidean metrics, i.e. of signature (++). Even if we don't know how to integrate over the space of  Lorentzian metrics, we would like that if it was defined, it would give the same result than the integration over the space of Euclidean metrics. An heuristic argument for that is given in 3.2. of Polchinski book. An easy but still convincing point is that the usual problem with the Wick rotation for quantum gravity in higher dimension (the wrong sign of the cinetic term of the conformal mode) does not appear in the two dimensional case considered.

So we are now trying to define the integration over the space of conformal classes of Euclidean metrics. The point is that the three problems of the Lorentzian case disappear: in general, the space of conformal classes of metrics will be finite dimensional, there exists many metrics without singularity and there is no regularity problem. These three facts: finite dimensionality, smoothness, regularity, are typical of elliptic questions.

To describe more precisely the space of conformal classes, we have to say what are the topologies considered. The goal of a perturbative theory is to define the scattering amplitudes, S-matrices, between states in the spectrum of the theory coming from and going to infinity. In particular, in the closed string case, the most obvious picture of world-sheet we want to consider is a oriented two dimensional surface with some holes in the middle and with some number of cylinders going to infinity, the external states being inserted in these cylinders. Now, as the theory living on the world-sheet is conformal, the state-operator correspondences implies that to insert an external state in a semi-infinite long cylinder is equivalent to suppress the cylinder, replace it by a marked point and insert an operator, called the vertex operator associated to the external state, at this marked point. After this reformulation, it is only necessary to consider compact surfaces with marked points and the space of conformal classes of metrics on such topology is finite dimensional. Once the measure to consider on this finite dimensional space is determined, perturbative string theory is well-defined. There is a similar story for the open string case.

What is usually called a Riemann surface in mathematics is a surface with a complex structure. By a well-known theorem, the data of a complex structure on a oriented surface is the same as the data of a conformal class of metrics. In other words, the space of conformal classes of metrics is really the same as the space of Riemann surfaces. (In physics, it happens that some people use the expression "Riemann surface" for just surface (=smooth two dimensional manifold) and I think it can be confusing).

I don't really understand the last part of the question because I don't know what is the meaning of "Riemann surfaces". If it is just smooth surface, the question (in the closed case) would be : why do we have to include surfaces with an arbitrarly number of holes? The answer is that from the moment you allow string to interact by joining and splitting, you can construct any surfaces by a sequence of these interactions. If the meaning of "Riemann surfaces" is surface with complex structure, the question would be : why do we have to integrate over the space of complex structures? The answer is that perturbative string theory is some version of 2d quantum gravity coupled to matter and so we have to integrate over all metrics, integration which reduces by conformal invariance to the space of complex structures.

Conclusion: the short answer is the same as in the  ahalanay's answer: No, simply because one side of the relation, quantum Lorentzian world-sheet, is not well-defined. Still, there is a general logic (that I have try to explained above) that to work with a quantum Euclidean world-sheet gives the same results that a quantum Lorentzian world-sheet would give.

Addendum: if one really looks to the details on how the integration on the moduli space of the conformal structures should be done, one finds that although the world-sheet has generically Eucildean signature, it has Lorentz signature when a string is going on-shell: see http://arxiv.org/abs/1307.5124

answered May 31, 2014 by (5,120 points)
edited Jun 1, 2014 by 40227
Wow thanks for these very nice and clear explanations :-)
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While the scale of distances behaves similarly, there is one difference. The world-sheet metric can be set to be proportional to the two-dimensional Minkowski metric , which defines distances as $-ds^2=-d\tau^2+d\sigma^2$ .On the other hand, on the complex plane the natural metric is Euclidean: $ds^2=|dz|^2=dx^2+dy^2$. Because of this sign difference, we cannot really prove that world-sheets can be treated as Riemann surfaces. But nothing stops us from trying to treat them as such, especially because it is known that much about a Minkowski theory can be learned from its Euclidean version. It turns out that thinking of world-sheets as Riemann surfaces leads to a consistent picture of string interactions.

(B.Zwiebach, A first course in String Theory, CUP 2004, p. 490)

My understanding is that Riemann surface is used as a synonym for compact oriented smooth surface. Topologically these are classified by the genus $g$ (the number of holes). The name is in one way misleading as does not imply the choosing of any particular Riemannian metric. On the contrary one is more concerned with the moduli space of complex structure on the surface (and implicitly conformal classes of hermitian metrics).

answered May 30, 2014 by (120 points)
edited May 30, 2014 by ahalanay

But worldsheets are not compact... More precisely worldsheets are not  "compact without boundary" 2D manifolds as Riemann surfaces are (they also are orientable). Perhaps we can assume that they are portions of Riemann surfaces and thus they admit complex structures (Euclidean conformal structures). As they are not "true" Riemann surfaces, it is not clear to me whether the space of complex structures is finite dimensional as it happens for two dimensional orientable compact (without boundary) real manifolds, i.e. Riemann surfaces.

They are compact for closed strings. When a the surface is no longer compact, but still orientable (e.g. a cylinder or the plane) it carries complex structures.

Yes, for closed strings, they are compact *if you include the boundaries*  made of a finite number of  circles (or closed curves with that topology). In that case, however, worldsheets have a boundary and thus the statement about the existence of a finite space of Euclidean conformal structures does not apply directly.  I think that conformal structures exist anyway, but I do not know  precise hypotheses (barring orientability) assuring this fact, also whether or not the space of the complex structures is finite dimensional (I suspect that the answer is negative), and if these hypotheses are filled by worldsheets. For instance, in string theory,  is it permitted  a wordsheet whose, say, future boundary is made of an infinite numbers of closed circles?

Indeed when one has non-empty boundary, then the moduli space of conformal metrics is infinite-dimensional. A tricky point is that is not true for punctured Riemann surface (though topologically there is an equivalence) in this case the Teichmueller space is finite-dimensional. I understood from a cursory glance in Zwiebach's book that to the world-sheet one applies various conformal maps in order to pass from the case with boundary to the punctured one.

Regarding you last question I think is allowed but to complicated.

Hm, would would a future boundery composed of an infinite number of closed strings not colapse to a black hole which can not be described perturbatively anyway? I like reading your and @ahalanay's comments.

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