John Baez
May17-04, 05:04 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>\n\nIn article <12c1bbee.0404081648.95a7105@posting.google.com>,\ nJake Mannix <jake@rset.net> wrote:\n\n> In Prof. Baez\'s paper, he describes (in more categorical language\n>than I\'ll use, as I\'m not sure what the field-theoretic relevance this\n>has for the time being) an action for a coupled system of a\n>lie-algebra valued one form A, and a lie-algebra valued 2-form B (I\'ll\n>be dealing specifically with the case Baez calls the "automorphism\n>2-group", so these two lie algebras can be identified).\n> In particular, to review, we have field strengths:\n>\n> F = dA + 1/2 A^A - B\n> G = dB + A^B\n>\n> And he writes down a lagrangian density of the simplest possible\n>form in these variables:\n>\n> L = |F|^2 + |G|^2\n>\n> My first question is this: what are the gauge symmetries of this\n>system?\n\nNot as much as I\'d like, which is why I never tried to publish\nthis paper. It has the usual sort of gauge symmetry\n\nA -> gAg^{-1} + gdg^{-1}\nB -> gBg^{-1}\n\nbut I don\'t see any "higher gauge symmetry" showing up.\nIn other words, the Lagrangian is invariant under a group of\ngauge transformations, but not under a 2-group of them.\n\nBut, I am very confused about what exactly what "invariance of\na Lagrangian under a 2-group of symmetries" is supposed to mean!\nThe Lagrangian is an element of a mere set, after all, not an\nobject in a category, and groups act naturally on sets, while\n2-groups act on categories.\n\nI am going to Cambridge this summer and I plan to talk a bunch\nto Hendryk Pfeiffer. He\'s one of the people who has really taken\nthis 2-group gauge theory idea and run with it... though in the\n*lattice* context rather than the continuum. So, maybe we can\nfigure some things out. I also want to understand holonomies of\n2-connections better.\n\n[I\'m cc\'ing this to you since it\'s taken me an essentially infinite\ntime to reply, but please continue on s.p.r.]\n\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>In article <12c1bbee.0404081648.95a7105@posting.google.com>,
Jake Mannix <jake@rset.net> wrote:
> 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
> 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?
Not as much as I'd like, which is why I never tried to publish
this paper. It has the usual sort of gauge symmetry
A -> gAg^{-1} + gdg^{-1}B -> gBg^{-1}
but I don't see any "higher gauge symmetry" showing up.
In other words, the Lagrangian is invariant under a group of
gauge transformations, but not under a 2-group of them.
But, I am very confused about what exactly what "invariance of
a Lagrangian under a 2-group of symmetries" is supposed to mean!
The Lagrangian is an element of a mere set, after all, not an
object in a category, and groups act naturally on sets, while
2-groups act on categories.
I am going to Cambridge this summer and I plan to talk a bunch
to Hendryk Pfeiffer. He's one of the people who has really taken
this 2-group gauge theory idea and run with it... though in the
*lattice* context rather than the continuum. So, maybe we can
figure some things out. I also want to understand holonomies of
2-connections better.
[I'm cc'ing this to you since it's taken me an essentially infinite
time to reply, but please continue on s.p.r.]
Jake Mannix <jake@rset.net> wrote:
> 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
> 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?
Not as much as I'd like, which is why I never tried to publish
this paper. It has the usual sort of gauge symmetry
A -> gAg^{-1} + gdg^{-1}B -> gBg^{-1}
but I don't see any "higher gauge symmetry" showing up.
In other words, the Lagrangian is invariant under a group of
gauge transformations, but not under a 2-group of them.
But, I am very confused about what exactly what "invariance of
a Lagrangian under a 2-group of symmetries" is supposed to mean!
The Lagrangian is an element of a mere set, after all, not an
object in a category, and groups act naturally on sets, while
2-groups act on categories.
I am going to Cambridge this summer and I plan to talk a bunch
to Hendryk Pfeiffer. He's one of the people who has really taken
this 2-group gauge theory idea and run with it... though in the
*lattice* context rather than the continuum. So, maybe we can
figure some things out. I also want to understand holonomies of
2-connections better.
[I'm cc'ing this to you since it's taken me an essentially infinite
time to reply, but please continue on s.p.r.]