d=4 N=2 SYM [Archive] - Physics Forums

Jack Tremarco

Jul1-05, 04:29 AM

sander@brandeis.edu

Jul1-05, 11:17 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>Hi Jack,\n\nthe problem is, as you imply, that there is no easy way of writing the\nN=2 SYM Lagrangian in terms of N=2 superfields. Writing it in terms of\nN=1 superfields, however, disguises some symmetries. Namely, only half\nof the N=2 SUSYs and only a U(1)_J subgroup of the (classical) SU(2)_R\nX U(1)_R symmetry can be made manifest in a given N=1 formulation.\n\nBut you won\'t have to guess the requisite term in the N=1\nsuperpotential. One way of deriving this is as follows. A suitable 180\ndegree SU(2)_R rotation, call it R_0, interchanges the N=2 fermionic\nsuperspace coordinates. If you act with this rotation on a Lagrangian\nthat makes manifest the first half of the SUSYs, you must find an\nequivalent Lagrangian that is obviously invariant under the second\nhalf. This requirement is enough to fix the superpotential.\n\nTo do this in practice you need to know explicitly how R_0 acts on the\ncomponent fields. This can be worked out by decomposing the N=2 vector\nmultiplet into an N=1 vector and an N=1 chiral multiplet. That is,\nwrite down the N=2 V-plet SUSY transformations and identify what parts\nof it look like the N=1 V-plet transformations and N=1 chiral multiplet\ntransformations with respect to certain component fields. Of course,\nthe role of a given component field may be different under the\nrespective two sets of SUSY transformations. Now, R_0 interchanges the\ntwo sets of SUSY tranformations and therefore must interchange the\nfields accordingly. This observation tells you how R_0 acts on the\ncomponent fields.\n\nThe idea of requiring invariance under R_0 goes all the way back to P.\nFayet\'s pioneering work [Nucl.Phys.B113:135,1976], who incidentally\nreferred to extended supersymmetry as hypersymmetry. It\'s interesting\nhow some names stick and others don\'t...\n\nBest,\nTobias\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>Hi Jack,

the problem is, as you imply, that there is no easy way of writing the
N=2 SYM Lagrangian in terms of N=2 superfields. Writing it in terms of
N=1 superfields, however, disguises some symmetries. Namely, only half
of the N=2 SUSYs and only a U(1)_J subgroup of the (classical) SU(2)_RX U(1)_R symmetry can be made manifest in a given N=1 formulation.

But you won't have to guess the requisite term in the N=1
superpotential. One way of deriving this is as follows. A suitable 180
degree SU(2)_R rotation, call it R_0, interchanges the N=2 fermionic
superspace coordinates. If you act with this rotation on a Lagrangian
that makes manifest the first half of the SUSYs, you must find an
equivalent Lagrangian that is obviously invariant under the second
half. This requirement is enough to fix the superpotential.

To do this in practice you need to know explicitly how R_0 acts on the
component fields. This can be worked out by decomposing the N=2 vector
multiplet into an N=1 vector and an N=1 chiral multiplet. That is,
write down the N=2 V-plet SUSY transformations and identify what parts
of it look like the N=1 V-plet transformations and N=1 chiral multiplet
transformations with respect to certain component fields. Of course,
the role of a given component field may be different under the
respective two sets of SUSY transformations. Now, R_0 interchanges the
two sets of SUSY tranformations and therefore must interchange the
fields accordingly. This observation tells you how R_0 acts on the
component fields.

The idea of requiring invariance under R_0 goes all the way back to P.
Fayet's pioneering work [Nucl.Phys.B113:135,1976], who incidentally
referred to extended supersymmetry as hypersymmetry. It's interesting
how some names stick and others don't...

Best,
Tobias