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<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 Wed, 7 Jul 2004, Urs Schreiber wrote:\n\n> Well, at least I think so. Since we have free afternoon today I\'ll go back\n> to my hotel room and try to write this up in detail.\n\nOk, what I wrote before is essentially correct, but very intransparent, as\nyou may have noticed. ;-)\n\nThere is a much cleaner way to think about this:\n\n\nSo consider on loop space the nonabelian 2-form gauge covariant exterior\nderivative\n\n\nnabla = d + \\oint_A (B) ,\n\nwhere I use the notation as in Hofmann\'s paper hep-th/0207017, but, as I\nhave emphasized before, with the difference that I have A-holonomies\nU_A(\\sigma_1,\\sigma_2) on both sides of the insertion, so that explicitly\n\n\nnabla =\n\\int_0^{2\\pi}d\\sigma E^{\\dagger \\mu}(\\sigma) (\n\\partial_\\mu(\\sigma)\n+\nU_A(0,\\sigma) B_{\\mu\\nu}X^{\\prime \\nu}(\\sigma) U_A(\\sigma,0)\n)\n\nwhere E^{\\dagger} is the operator of exterior multiplication by the\nrespective differential form on loop space.\n\nNow, the nice thing about the loop space perspective is that here we are\ndealing simply with an ordinary connection. The worldsheet is a line in\nloop space and this line couples to the loop space 1-form \\oint_A (B) in\njust the usual way.\n\nThis means that gauge transformation of this connection will follow the\nusual rules. We can work them out and see what they imply for the target\nspace theory, instead of just trying to guess the latter.\n\nThe gauge transformation on the connection \\oint_A (B) is obtained as\nusual by specifying any group-valued function U on loop space and setting\n\n\n\\oint_A (B)\n\\mapsto\nU \\circ \\oint_A (B) \\circ U^\\dagger + U (dU^\\dagger) .\n\n\nNow first of all let\'s look at _global_ gauge transformations, i.e. those\nfor which U(dU^\\dagger) = 0, which is the case when U does not depend on\nthe embedding coordinates of the string/loop.\n\nGiven any such U, a little calculation demonstrates the following:\n\nGiven _any_ group-valued function V on the loop which coincides with U at\nthe arbitrary basepoint\n\nV = V(\\sigma)\n\nV(0) = U\n\nwe have the identity\n\nU \\circ \\oint_A (B) \\circ U^\\dagger = \\oint_{A\'} (B\')\n\nwhere\n\nA\' = V A V^\\dagger + V (d V^dagger)\n\nB\' = V B V^\\dagger .\n\n\nThis is the first of the gauge invariances of a 2-form connection, namely\nessentially the ordinary gauge transformation of the 1-form connection A\ntogether with the obvious action on B.\n\nSo this demonstrates that _global_ gauge transformations on loop space\nare equivalent to ordinary gauge transformations in target space.\n\nI should emphasize again that this crucially depends on that second U_A\nfactor in my definition of the loop space connection, which is derived\nfrom boundary state deformation theory. Without that factor the above does\nnot seem to have a meaningful analogue.\n\n\nThe next step is the more interesting one. There should be a further gauge\ntransformation associated to the cohomology equivalence classes of the\n2-form B, roughly.\n\nFor instance as stated in Lahiri\'s papers, e.g. hep-th/0109220, one\nexpects there to be a gauge invariance of the form\n\nB \\mapsto B + d_A A\n\nA \\mapsto A .\n\n\nAs far as I can see this is the invariance that is not respected by the\naction which is given in hep-th/0206130 .\n\n\nI am a little confused about the status of the proposal for this second\ngauge transformation. Why is this expected to be the correct form for\nnonabelian 2-forms? Is this a definition, a derived result, or a guess?\n\nThe reason I am asking this question is that I believe to have evidence\nthat this gauge transformation needs some correction terms. In order to\ndemonstrate this I\'ll just consider a local gauge transforation on loop\nspace, where I believe to know the correct form of the 2-form gauge\n_connection_, and simply see what the resulting effect on that connection\nis.\n\n\nSo consider a group-valued 1-form \\Lambda on loop space, and consider to\nfirst order the gauge transformation\n\nU = 1 + i \\oint_A (\\Lambda) + \\cdots ,\n\nin direct analogy with the respective expression for target space\ntheories.\n\nComputing the transformation of the connection \\oint_A (B) with respect to\nthis transformation gives, by the above formula, something like\n\n\n\\oint_A (B)\n\\mapsto\n\\oint_A (B + d_A B)\n+\ni\\oint_A (\\Lambda,B - d_A A) - i \\oint_A (B + d_A A,\\Lambda)\n\n\nInterestingly, the first line is precisely the expected gauge\ntransformation on B. But, due to the action of d on the U_A holonomies and\ndue to the commutation with \\oint (\\Lambda), there appear correction terms\nin the second line. As these involve path-ordered integrals of two\nquantities over the loop, they do not have any analogy on target space at\nall! (I think.)\n\nBy construction, the above is a valid gauge transformation on loop space.\nUnless I am confused this shows that we won\'t easily find any target space\naction in terms of a _local_ field theory of point particles which\nrespects this invariance.\n\nOn the other hand, it is of course possible to compute the gauge curvature\n\n\n(d + \\oint_A (B))^2\n\non loop space, take it\'s Hodge-square, trace over it and integrte it over\nloop space to obtain a YM-like theory whose parameter space is loop space.\nI\'d expect, but haven\'t checked, that this action, which manifestly has\nthe full loop space gauge symmetry, flows to some of the target space actions\nwhich have been written down before.\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 Wed, 7 Jul 2004, Urs Schreiber wrote:
> Well, at least I think so. Since we have free afternoon today I'll go back
> to my hotel room and try to write this up in detail.
Ok, what I wrote before is essentially correct, but very intransparent, as
you may have noticed. ;-)
There is a much cleaner way to think about this:
So consider on loop space the nonabelian 2-form gauge covariant exterior
derivative
nabla [itex]= d + \oint_A (B) ,[/itex]
where I use the notation as in Hofmann's paper http://www.arxiv.org/abs/hep-th/0207017, but, as I
have emphasized before, with the difference that I have A-holonomies
[itex]U_A(\sigma_1,\sigma_2)[/itex] on both sides of the insertion, so that explicitly
nabla =
[itex]\int_0^{2\pi}d\sigma E^{\dagger \mu}(\sigma) (\partial_\mu(\sigma)[/itex]
+
[itex]U_A(0,\sigma) B_{\mu\nu}X^{\prime \nu}(\sigma) U_A(\sigma,0)[/itex]
)
where [itex]E^{\dagger}[/itex] is the operator of exterior multiplication by the
respective differential form on loop space.
Now, the nice thing about the loop space perspective is that here we are
dealing simply with an ordinary connection. The worldsheet is a line in
loop space and this line couples to the loop space 1-form [itex]\oint_A (B)[/itex] in
just the usual way.
This means that gauge transformation of this connection will follow the
usual rules. We can work them out and see what they imply for the target
space theory, instead of just trying to guess the latter.
The gauge transformation on the connection [itex]\oint_A (B)[/itex] is obtained as
usual by specifying any group-valued function U on loop space and setting
[tex]\oint_A (B)\mapstoU \circ \oint_A (B) \circ U^\dagger + U (dU^\dagger) .[/tex]
Now first of all let's look at _global_ gauge transformations, i.e. those
for which [itex]U(dU^\dagger) = 0,[/itex] which is the case when U does not depend on
the embedding coordinates of the string/loop.
Given any such U, a little calculation demonstrates the following:
Given _any_ group-valued function V on the loop which coincides with U at
the arbitrary basepoint
[tex]V = V(\sigma)[/tex]
V(0) = U
we have the identity
[tex]U \circ \oint_A (B) \circ U^\dagger = \oint_{A'} (B')[/tex]
where
[tex]A' = V A V^\dagger + V (d V^{dagger})B' = V B V^\dagger .[/tex]
This is the first of the gauge invariances of a 2-form connection, namely
essentially the ordinary gauge transformation of the 1-form connection A
together with the obvious action on B.
So this demonstrates that _global_ gauge transformations on loop space
are equivalent to ordinary gauge transformations in target space.
I should emphasize again that this crucially depends on that second [itex]U_A[/itex]
factor in my definition of the loop space connection, which is derived
from boundary state deformation theory. Without that factor the above does
not seem to have a meaningful analogue.
The next step is the more interesting one. There should be a further gauge
transformation associated to the cohomology equivalence classes of the
2-form B, roughly.
For instance as stated in Lahiri's papers, e.g. http://www.arxiv.org/abs/hep-th/0109220, one
expects there to be a gauge invariance of the form
[tex]B \mapsto B + d_A AA \mapsto A .[/tex]
As far as I can see this is the invariance that is not respected by the
action which is given in http://www.arxiv.org/abs/hep-th/0206130 .
I am a little confused about the status of the proposal for this second
gauge transformation. Why is this expected to be the correct form for
nonabelian 2-forms? Is this a definition, a derived result, or a guess?
The reason I am asking this question is that I believe to have evidence
that this gauge transformation needs some correction terms. In order to
demonstrate this I'll just consider a local gauge transforation on loop
space, where I believe to know the correct form of the 2-form gauge
[itex]_connection_,[/itex] and simply see what the resulting effect on that connection
is.
So consider a group-valued 1-form [itex]\Lambda[/itex] on loop space, and consider to
first order the gauge transformation
[tex]U = 1 + i \oint_A (\Lambda) + \cdots ,[/tex]
in direct analogy with the respective expression for target space
theories.
Computing the transformation of the connection [itex]\oint_A (B)[/itex] with respect to
this transformation gives, by the above formula, something like
[tex]\oint_A (B)\mapsto\oint_A (B + d_A B)[/itex]
+
[itex]i\oint_A (\Lambda,B - d_A A) - i \oint_A (B + d_A A,\Lambda)[/tex]
Interestingly, the first line is precisely the expected gauge
transformation on B. But, due to the action of d on the [itex]U_A[/itex] holonomies and
due to the commutation with [itex]\oint (\Lambda),[/itex] there appear correction terms
in the second line. As these involve path-ordered integrals of two
quantities over the loop, they do not have any analogy on target space at
all! (I think.)
By construction, the above is a valid gauge transformation on loop space.
Unless I am confused this shows that we won't easily find any target space
action in terms of a _local_ field theory of point particles which
respects this invariance.
On the other hand, it is of course possible to compute the gauge curvature
[tex](d + \oint_A (B))^2[/tex]
on loop space, take it's Hodge-square, trace over it and integrte it over
loop space to obtain a YM-like theory whose parameter space is loop space.
I'd expect, but haven't checked, that this action, which manifestly has
the full loop space gauge symmetry, flows to some of the target space actions
which have been written down before.
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