## Re: Twistors: the long-term goals of the program

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Robert C. Helling wrote:\n\n&gt; This problem only arises if I try to get the curve as the image of\n&gt; embedding a CP1 into CP3 via a quadratic map (as the Santa Barbara\n&gt; group does).\n\nI am not sure that I 100% understood what you are saying, but I think you\nare discussing the issue of whether the degenerate curves can arise as maps\nCP^1 --&gt; CP^{3|4}. Indeed, as you noted, they do _not_ arise as such maps,\nso if we realize the set of nondegenerate curves as the GL_2 quotient of\nthis map space (which we want to do, because that\'s the description in\nwhich we know the measure) then we have to compactify it if we want to\ndescribe the degenerate ones. One convenient compactification is given by\nthis "space of stable maps." The computations in sections 3 and 4 amount\nto choosing a particular coordinate system near a boundary divisor in this\nspace of stable maps, and studying the behavior of the integrand near that\nboundary divisor.\n\nIt\'s true that in some other descriptions, where you describe the curve by\nequations it satisfies inside the projective space, the\ncompactification looks sort of "automatic" -- to get the singular curve you\njust have to allow some discriminant to vanish. But as Lubos remarked, the\nmost general curve of degree d doesn\'t have this kind of description.\n\n--\nAndy Neitzke\nneitzke@fas.harvard.edu\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Robert C. Helling wrote:

> This problem only arises if I try to get the curve as the image of
> embedding a CP1 into CP3 via a quadratic map (as the Santa Barbara
> group does).

I am not sure that I 100% understood what you are saying, but I think you
are discussing the issue of whether the degenerate curves can arise as maps
$CP^1$ --> $CP^{3|4}$. Indeed, as you noted, they do _not_ arise as such maps,
so if we realize the set of nondegenerate curves as the $GL_2$ quotient of
this map space (which we want to do, because that's the description in
which we know the measure) then we have to compactify it if we want to
describe the degenerate ones. One convenient compactification is given by
this "space of stable maps." The computations in sections 3 and 4 amount
to choosing a particular coordinate system near a boundary divisor in this
space of stable maps, and studying the behavior of the integrand near that
boundary divisor.

It's true that in some other descriptions, where you describe the curve by
equations it satisfies inside the projective space, the
compactification looks sort of "automatic" -- to get the singular curve you
just have to allow some discriminant to vanish. But as Lubos remarked, the
most general curve of degree d doesn't have this kind of description.

--
Andy Neitzke
neitzke@fas.harvard.edu

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Hi Lubos, I think there has an example where two different kind of diagrams that contribute to a process both reduces to contribution from "boundary of the moduli space", and you actually only count one of them - I'm not sure if it is in any way related to your twistor proposal. This is in the context of the factorized S-matrices in scatterings in two-dimensional field theories, for example, in the work of the Zamolodchikovs. In these theories loop diagrams essentially reduces to only the contribution of on-shell intermediate propagators because of the special structure of the S-matrices. For example, there are two different diagrams of three particle scattering, which are related by the Yang-Baxter equation. Both can be regarded as 1-loop Feynman diagram, but reduces to only the contribution involving only on-shell propagators, and at the end you only count one of them. This sounds quite similar to what's happening in the twistor story. So I'd speculate: 1. The reduction of the twistor space amplitudes to the residue on the boundary moduli space is related to some integrability. and 2. One should probably only count one of the contributions from the various sector[s] of disconnected curves. Best, $-\Xi$ > In our paper we show > > http://www.arxiv.org/abs/http://www....hep-th/0404085 > > that the integrals over the connected, higher degree curves are equivalent > to the integrals over the families of the lines, because in fact in both > cases, the integral can be reduced, via the calculus of residues and > contour integrals, to an integral over the intersecting lines - which can > also be viewed as singular curves of higher degree. > > Let me write down the simplest example. A hyperbola $(xy=C)$ is a degree 2 > curve. It has 2 asymptotes $(x=0$ and $y=0,$ in this case), and actually if > you send C to zero, the hyperbola *degenerates* to the pair of > intersecting lines. We showed that the integral that encodes the > amplitude contains $a dC/C$ factor, and argued that it should be contour > integrated around $C=0,$ so that it is really the intersecting lines - > forming a tree - that contribute.

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