Jake Mannix
Apr9-04, 02:21 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>Hello all,\nThis post is partly directed to Prof. Baez, in reference to his\npaper (see the talk and links at the bottom of this page:\nhttp://math.ucr.edu/home/baez/gauge/ ), but also to people familiar\nwith the general physics folklore that "there exist no lagrangian\ndescriptions of self-dual 2-forms in 6 dimensions" (c.f. Witten\'s\npaper "The Effective Action of a 5-brane").\n\nIn Prof. Baez\'s paper, he describes (in more categorical language\nthan I\'ll use, as I\'m not sure what the field-theoretic relevance this\nhas for the time being) an action for a coupled system of a\nlie-algebra valued one form A, and a lie-algebra valued 2-form B (I\'ll\nbe dealing specifically with the case Baez calls the "automorphism\n2-group", so these two lie algebras can be identified).\nIn particular, to review, we have field strengths:\n\nF = dA + 1/2 A^A - B\nG = dB + A^B\n\nAnd he writes down a lagrangian density of the simplest possible\nform in these variables:\n\nL = |F|^2 + |G|^2\n\nMy first question is this: what are the gauge symmetries of this\nsystem?\n\nFrom my understanding of the origins of these things as\nconnections on gerbes, and from Baez\' description as being 2-group\nrelated - in this case there\'s a left Aut(G) action and compatible\nright G action on the fibers, and hence on the connection. But how\ndoes it show up as even infinitessimal variations on A, B?\n\nSecond, Baez mentions self-duality relations in *five*\ndimensions:\n\n*F = G\n\nWhich seems like it should be a string-particle duality given\nthat magnetic sources for strings in D=5 are particles.\n\nWhich brings me to my third question: string theorists like to\ntalk about 2-forms with self-dual field strength in D=6, which in this\nnotation would be just:\n\n*G = G\n\nBut what will this mean? The bianchi identities are, after a\nlittle rearrangement in more classical terms:\n\nD^2(A) = 0, DG = F^B,\n\nWhile the field equations are:\n\nD*F = (*DB)^B\nD*G = -*F,\n\nSo self-duality is requiring:\n\nF^B = - *F.\n\nAre there any other things that it requires? Are these\nconstraints solvable nontrivially? How many remaining degrees of\nfreedom are there once imposed?\n\nSome of this I\'m sure is obvious, but I haven\'t played around\nwith these enough yet to know what is going on here...\n\n-Jake Mannix\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>Hello all,
This post is partly directed to Prof. Baez, in reference to his
paper (see the talk and links at the bottom of this page:
http://math.ucr.edu/home/baez/gauge/ ), but also to people familiar
with the general physics folklore that "there exist no lagrangian
descriptions of self-dual 2-forms in 6 dimensions" (c.f. Witten's
paper "The Effective Action of a 5-brane").
In Prof. Baez's paper, he describes (in more categorical language
than I'll use, as I'm not sure what the field-theoretic relevance this
has for the time being) an action for a coupled system of a
lie-algebra valued one form A, and a lie-algebra valued 2-form B (I'll
be dealing specifically with the case Baez calls the "automorphism
2-group", so these two lie algebras can be identified).
In particular, to review, we have field strengths:
F = dA + 1/2 A^A - B[/itex]
G = dB + A^B
And he writes down a lagrangian density of the simplest possible
form in these variables:
L = |F|^2 + |G|^2
My first question is this: what are the gauge symmetries of this
system?
From my understanding of the origins of these things as
connections on gerbes, and from Baez' description as being 2-group
related - in this case there's a left Aut(G) action and compatible
right G action on the fibers, and hence on the connection. But how
does it show up as even infinitessimal variations on A, B?
Second, Baez mentions self-duality relations in *five*
dimensions:
*F = G
Which seems like it should be a string-particle duality given
that magnetic sources for strings in D=5 are particles.
Which brings me to my third question: string theorists like to
talk about 2-forms with self-dual field strength in D=6, which in this
notation would be just:
*G = G
But what will this mean? The bianchi identities are, after a
little rearrangement in more classical terms:
D^2(A) = 0, DG = F^B,
While the field equations are:
D*F = (*DB)^B
[itex]D*G = -*F,
So self-duality is requiring:
F^B = - *F.
Are there any other things that it requires? Are these
constraints solvable nontrivially? How many remaining degrees of
freedom are there once imposed?
Some of this I'm sure is obvious, but I haven't played around
with these enough yet to know what is going on here...
-Jake Mannix
This post is partly directed to Prof. Baez, in reference to his
paper (see the talk and links at the bottom of this page:
http://math.ucr.edu/home/baez/gauge/ ), but also to people familiar
with the general physics folklore that "there exist no lagrangian
descriptions of self-dual 2-forms in 6 dimensions" (c.f. Witten's
paper "The Effective Action of a 5-brane").
In Prof. Baez's paper, he describes (in more categorical language
than I'll use, as I'm not sure what the field-theoretic relevance this
has for the time being) an action for a coupled system of a
lie-algebra valued one form A, and a lie-algebra valued 2-form B (I'll
be dealing specifically with the case Baez calls the "automorphism
2-group", so these two lie algebras can be identified).
In particular, to review, we have field strengths:
F = dA + 1/2 A^A - B[/itex]
G = dB + A^B
And he writes down a lagrangian density of the simplest possible
form in these variables:
L = |F|^2 + |G|^2
My first question is this: what are the gauge symmetries of this
system?
From my understanding of the origins of these things as
connections on gerbes, and from Baez' description as being 2-group
related - in this case there's a left Aut(G) action and compatible
right G action on the fibers, and hence on the connection. But how
does it show up as even infinitessimal variations on A, B?
Second, Baez mentions self-duality relations in *five*
dimensions:
*F = G
Which seems like it should be a string-particle duality given
that magnetic sources for strings in D=5 are particles.
Which brings me to my third question: string theorists like to
talk about 2-forms with self-dual field strength in D=6, which in this
notation would be just:
*G = G
But what will this mean? The bianchi identities are, after a
little rearrangement in more classical terms:
D^2(A) = 0, DG = F^B,
While the field equations are:
D*F = (*DB)^B
[itex]D*G = -*F,
So self-duality is requiring:
F^B = - *F.
Are there any other things that it requires? Are these
constraints solvable nontrivially? How many remaining degrees of
freedom are there once imposed?
Some of this I'm sure is obvious, but I haven't played around
with these enough yet to know what is going on here...
-Jake Mannix