Re: questions on nonabelian p-form gauge theories and self-duality [Archive]

Danny Ross Lunsford

Apr11-04, 11:44 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\nJake Mannix wrote:\n\n> Hello all,\n> This post is partly directed to Prof. Baez, in reference to his\n> paper (see the talk and links at the bottom of this page:\n> http://math.ucr.edu/home/baez/gauge/ ), but also to people familiar\n> with the general physics folklore that "there exist no lagrangian\n> descriptions of self-dual 2-forms in 6 dimensions" (c.f. Witten\'s\n> paper "The Effective Action of a 5-brane").\n\nSure, there\'s a very simple notion of self-duality in 6-d because there\nis a cross-product of bivectors\n\nF1 F2 = P F3\n\nwhere P is the unit pseudoscalar. The even subalgebra amounts to\nscalars, bivectors, quadrivectors = pseudo-bivectors, and pseudoscalars.\nBy adding bivectors and quadrivectors one gets forms\n\nF + P G\n\n...and so it\'s very easy to define self-dual objects. This is exactly\nanalogous to what happens in 3-d (E + iB and all that).\n\nThe natural derivatives of these things should look like outer products\nof a vector and a derivative (a "bivectorial derivative") - a natural one is\n\nxm dn - xn dm\n\nwhich is more or less [x,p]. There should be a Lagrangian-based\nformalism that looks a lot like harmonic oscillators.\n\n-drl\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>Jake Mannix wrote:

> 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").

Sure, there's a very simple notion of self-duality in 6-d because there
is a cross-product of bivectors

F1 F2 = P F3

where P is the unit pseudoscalar. The even subalgebra amounts to
scalars, bivectors, quadrivectors = pseudo-bivectors, and pseudoscalars.
By adding bivectors and quadrivectors one gets forms

F + P G

...and so it's very easy to define self-dual objects. This is exactly
analogous to what happens in 3-d (E + iB and all that).

The natural derivatives of these things should look like outer products
of a vector and a derivative (a "bivectorial derivative") - a natural one is

xm dn - xn dm

which is more or less [x,p]. There should be a Lagrangian-based
formalism that looks a lot like harmonic oscillators.

-drl

Jake Mannix

Apr14-04, 03:17 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>> Sure, there\'s a very simple notion of self-duality in 6-d because there\n> is a cross-product of bivectors\n>\n> F1 F2 = P F3\n\nEep! I\'d definitely rather live in the "co-"world of forms, and just define\na product on 2-forms as F x F\' = *(F ^ F\'), but ok, this seems to be the\nsame thing.\n\n> where P is the unit pseudoscalar. The even subalgebra amounts to\n> scalars, bivectors, quadrivectors = pseudo-bivectors, and pseudoscalars.\n> By adding bivectors and quadrivectors one gets forms\n>\n> F + P G\n\nBut this doesn\'t tell me what to do with a combination (2-form, 3-form)\nin some self-dual way. The 3-form naturally decomposes as a self-dual\npart, and an anti-self dual part, but a 2-form in 6-d is as you say, like a\nvector in 3-d - which to me is rather unlike a 2-form in (3+1)-d (where\nE&M naturally lives). I don\'t have any 4-forms handy to combine with my\n2-form part of the curvature, so there\'s nothing naturally to require in\nthis way to be self-dual (other than what I think I mentioned in my other\npost)...\n\nHmm...\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>> Sure, there's a very simple notion of self-duality in 6-d because there
> is a cross-product of bivectors
>
> F1 F2 = P F3

Eep! I'd definitely rather live in the "co-"world of forms, and just define
a product on 2-forms as F x F' = *(F ^ F'), but ok, this seems to be the
same thing.

> where P is the unit pseudoscalar. The even subalgebra amounts to
> scalars, bivectors, quadrivectors = pseudo-bivectors, and pseudoscalars.
> By adding bivectors and quadrivectors one gets forms
>
> F + P G

But this doesn't tell me what to do with a combination (2-form, 3-form)
in some self-dual way. The 3-form naturally decomposes as a self-dual
part, and an anti-self dual part, but a 2-form in 6-d is as you say, like a
vector in 3-d - which to me is rather unlike a 2-form in (3+1)-d (where
E&M naturally lives). I don't have any 4-forms handy to combine with my
2-form part of the curvature, so there's nothing naturally to require in
this way to be self-dual (other than what I think I mentioned in my other
post)...

Hmm...

-jake mannix

Danny Ross Lunsford

Apr14-04, 08:36 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\nJake Mannix wrote:\n\n>>F1 F2 = P F3\n>\n> Eep! I\'d definitely rather live in the "co-"world of forms, and just define\n> a product on 2-forms as F x F\' = *(F ^ F\'), but ok, this seems to be the\n> same thing.\n\nYes :)\n\n>>By adding bivectors and quadrivectors one gets forms\n>>\n>>F + P G\n>\n> But this doesn\'t tell me what to do with a combination (2-form, 3-form)\n> in some self-dual way. The 3-form naturally decomposes as a self-dual\n> part, and an anti-self dual part, but a 2-form in 6-d is as you say, like a\n> vector in 3-d - which to me is rather unlike a 2-form in (3+1)-d (where\n> E&M naturally lives). I don\'t have any 4-forms handy to combine with my\n> 2-form part of the curvature, so there\'s nothing naturally to require in\n> this way to be self-dual (other than what I think I mentioned in my other\n> post)...\n\n3-form on what base? Are we talking 6-d? Let\'s be specific - I\'ll work\nwith Cl(3,3) with a representation (ASCII compromise: y = Dirac gamma,\nb=beta)\n\nb_mu = | ym 0 |\n| 0 -ym |\n\nb_5 = | 0 1 |\n| 1 0 |\n\nb_6 = | 0 -i |\n| i 0 |\n\nb_7 (unit pseudoscalar) = sqrt(det(g)) eps_1...6 b_1..b_6\n\n= | -iy5 0 |\n| 0 iy5 |\n\n{ b_m, b_n } = 2 diag (---+++)\n{ b_m, b_7 } = 0\n\nb_123 anti-Hermitian, b_456 Hermitian, b_7 anti-Hermitian\n\nGiven any ternary product\n\nb_i b_j b_k\n\nthen multiplication with b_7 will result in\n\ni b_l b_m b_n\n\nwhere ijklmn is some permutation of 123456.\n\nThis allows one to form self-dual ternary products of the form\n\n(Amnp + b_7 Bmnp) b_m b_n b_p\n\nwhile the anti-selfdual ones should just be something like\n\n(Amnp - b_7 Bmnp) b_m b_n b_p\n\nHitting one of these with (bm dm) will produce a 2-vector and a 4-vector\nand again these can be combined into self-dual and anti-self-dual forms.\nThe result should look a lot like Maxwell in E+iB form. I\'m going to\nwork it out now :) Of course there are slight but essential differences\nif one uses a different metric signature.\n\n-drl\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>Jake Mannix wrote:

>>F1 F2 = P F3
>
> Eep! I'd definitely rather live in the "co-"world of forms, and just define
> a product on 2-forms as F x F' = *(F ^ F'), but ok, this seems to be the
> same thing.

Yes :)

>>By adding bivectors and quadrivectors one gets forms
>>
>>F + P G
>
> But this doesn't tell me what to do with a combination (2-form, 3-form)
> in some self-dual way. The 3-form naturally decomposes as a self-dual
> part, and an anti-self dual part, but a 2-form in 6-d is as you say, like a
> vector in 3-d - which to me is rather unlike a 2-form in (3+1)-d (where
> E&M naturally lives). I don't have any 4-forms handy to combine with my
> 2-form part of the curvature, so there's nothing naturally to require in
> this way to be self-dual (other than what I think I mentioned in my other
> post)...

3-form on what base? Are we talking 6-d? Let's be specific - I'll work
with Cl(3,3) with a representation (ASCII compromise: y = Dirac \gamma,
b=\beta)

b_{mu} = | ym |[/itex]
| -ym |

b_5 = | 1 |
| 1 |

b_6 = | -i |
| i |

b_7 (unit pseudoscalar) = \sqrt(det(g)) eps_1...6 b_1..b_6

= | -iy5 |
| iy5 |

{ b_m, b_n } = 2 diag (---+++)
[itex]{ b_m, b_7 } =

b_{123} anti-Hermitian, b_{456} Hermitian, b_7 anti-Hermitian

Given any ternary product

b_i b_j b_k

then multiplication with b_7 will result in

i b_l b_m b_n

where ijklmn is some permutation of 123456.

This allows one to form self-dual ternary products of the form

(Amnp + b_7 Bmnp) b_m b_n b_p

while the anti-selfdual ones should just be something like

(Amnp - b_7 Bmnp) b_m b_n b_p

Hitting one of these with (bm dm) will produce a 2-vector and a 4-vector
and again these can be combined into self-dual and anti-self-dual forms.
The result should look a lot like Maxwell in E+iB form. I'm going to
work it out now :) Of course there are slight but essential differences
if one uses a different metric signature.

-drl