View Full Version : rational curve in CY
<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>Is there any classification of singularities on local Calabi-Yau 3-fold\nthat can be resolved by a P^1 with O(1)+O(-3) normal bundle? The only\nexample I know is Laufer\'s example,\n\nz_1 = x^3 y_1 + y_2^2 + x^2 y^_2^{2n+1},\nz_2 = x^{-1} y_2,\n\nwhere x paramterizes the base P^1, (y_1, y_2) and (z_1, z_2) parameterize\nthe fiber on two patches of the P^1.\nI\'d be very happy if there are other examples.\n\nThanks,\nXi\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"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Is there any classification of singularities on local Calabi-Yau 3-fold
that can be resolved by a P^1 with O(1)+O(-3) normal bundle? The only
example I know is Laufer's example,
z_1 = x^3 y_1 + y_2^2 + x^2 y^{_2}^{2n+1},z_2 = x^{-1} y_2,
where x paramterizes the base P^1, (y_1, y_2) and (z_1, z_2) parameterize
the fiber on two patches of the P^1.
I'd be very happy if there are other examples.
Thanks,
\Xi
Volker Braun
Nov16-04, 11:19 AM
<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>On Mon, 15 Nov 2004 11:27:48 -0500, Xi Yin wrote:\n\n> Is there any classification of singularities on local Calabi-Yau 3-fold\n> that can be resolved by a P^1 with O(1)+O(-3) normal bundle?\n\nLocally, the singularity must be like O(1)+O(-3)->P^1 with the base P^1\ncontracted?\n\nBut why would you be interested in that? Since the base curve is not\nrigid, the localization/topological vertex argument should not work. So\njust looking at a local CY is not good enough, and you need the global\ngeometry to compute, say, GW invariants.\n\n-Volker\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"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 15 Nov 2004 11:27:48 -0500, \Xi Yin wrote:
> Is there any classification of singularities on local Calabi-Yau 3-fold
> that can be resolved by a P^1 with O(1)+O(-3) normal bundle?
Locally, the singularity must be like O(1)+O(-3)->P^1 with the base P^1
contracted?
But why would you be interested in that? Since the base curve is not
rigid, the localization/topological vertex argument should not work. So
just looking at a local CY is not good enough, and you need the global
geometry to compute, say, GW invariants.
-Volker
<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>Volker Braun wrote:\n\n> Locally, the singularity must be like O(1)+O(-3)->P^1 with the base P^1\n> contracted?\n>\n> But why would you be interested in that? Since the base curve is not\n> rigid, the localization/topological vertex argument should not work. So\n> just looking at a local CY is not good enough, and you need the global\n> geometry to compute, say, GW invariants.\n\nAh, that\'s exactly the point. I\'m not trying to compute GW invariants. I\'m\ntrying to geometrically engineer N=1 gauge theories with two adjoint\nflavors. That\'s why I need O(1)+O(-3) normal bundle instead of O(0)+O(-2)\nor O(-1)+O(-1). In the Laufer example, you can wrap N D5-branes on the P^1\nand get an N=1 U(N) gauge theory with two adjoint falvors X and Y, with a\nsuperpotential W(X,Y) which was computed by Sheldon Katz years ago. This\ntheory has been studied by Cachazo, Katz and Vafa quite explicitly. I\'m\nwondering if there are generalizations of this example.\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"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Volker Braun wrote:
> Locally, the singularity must be like O(1)+O(-3)->P^1 with the base P^1
> contracted?
>
> But why would you be interested in that? Since the base curve is not
> rigid, the localization/topological vertex argument should not work. So
> just looking at a local CY is not good enough, and you need the global
> geometry to compute, say, GW invariants.
Ah, that's exactly the point. I'm not trying to compute GW invariants. I'm
trying to geometrically engineer N=1 gauge theories with two adjoint
flavors. That's why I need O(1)+O(-3) normal bundle instead of O(0)+O(-2)
or O(-1)+O(-1). In the Laufer example, you can wrap N D5-branes on the P^1
and get an N=1 U(N) gauge theory with two adjoint falvors X and Y, with a
superpotential W(X,Y) which was computed by Sheldon Katz years ago. This
theory has been studied by Cachazo, Katz and Vafa quite explicitly. I'm
wondering if there are generalizations of this example.
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