Tobias Sander
Nov19-04, 03:38 PM
<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>Some time ago, one Rufus Anton asked in this forum about the seemingly\nmissing central charge of D6 branes:\n\nhttp://groups.google.com/groups?hl=en&lr=&threadm=a1c70df8.0409211643.364872-100000%40posting.google.com&rnum=2&prev=/groups%3Fq%3DRufus%2BAnton%26hl%3Den%26lr%3D%26gro up%3Dsci.physics.strings%26selm%3Da1c70df8.0409211 643.364872-100000%2540posting.google.\ncom%26rnum%3D2\n\nThis fact is known to many string theorists, but it continues to\npuzzle me a little bit. Let me explain -- here is the story:\n\n* Firstly, look at 11D SUGRA\n\nThe 11d SUSY algebra is schematically\n\n{Q,Q} = P_\\mu \\Gamma^\\mu +Z^2_{...} \\Gamma^{...} + Z^5_{...} \\Gamma^{...}\n\nHere P_\\mu is the 11d momentum, Z^2 is the central charge of the\nM2-brane and, Z^5 that of the M5-brane. (I have suppressed any spinor\nindices and the dots ... indicate contraction of the spacetime indices\nwith the appropriate \\Gamma-matrices.)\n\nNow, if you reduce on a circle, you get a scalar and a 10-vector from\nP_\\mu, which you can recognize as the D0-central charge and the\n10-momentum respectively. Similarly, you get a 1-form and a 2-form\nfrom Z_2, being the F1-charge and the D2-charge. And from Z_5 you get\na 4-form and a 5-form, the D4-charge and the NS5 charge.\n\nIn this scheme D(-1), D6, and D8 do not appear to have a charge. For\nD(-1) and D8 this is not surprising to me -- but for D6 it is at first\nsight. Lubos gave the answer to this in his reply to Rufus\' question,\npointing out that the D6 becomes a KK-monopole at strong coupling.\n\n[Moderator\'s note: You should first decide whether you talk about type\nIIA picture, or type IIB. It\'s no good to mix odd numbers like D(-1)\nwith even numbers like D6 and D8. Moreover, the D(-1)charge is not\na conserved operator - it is a feature of a spacetime configuration\n(D-instanton, localized both in space and the Euclidean time)\nthat does not survive in the Hilbert space formalism, I think. LM]\n\n* Secondly, let\'s look at this from the 10d perspective directly. For\nIIA there are 32 supercharges organized in two spinors Q and {\\bar Q}\nwhich must be in the 16 and 16\' of SO(9,1), respectively. The possible\ncentral charges are therefore in the product representations of 16 and\n16\', namely:\n\n(16 \\times 16\') = (16\' \\times 16) = [0] + [2] + [4] (=>\nD0,D2,D4-charges)\n(16 \\times 16)_s = [1] + [5]_+ (=> F1,NS5-charges)\n(16\' \\times 16\')_s = [1] + [5]_- (=> F1,NS5-charges)\n\n(The subscript _s makes clear that the anticommutator we consider has\ndefinite permuational symmetry with respect to its spinor indices,\nwhich explains the absence of tensors like [3].)\n\nAgain, D(-1), D6, D8-charges do not appear. Notice that if you count\nthe degrees of freedom of all the tensors on the RHS, you find 528.\nThis saturates the number of degrees of freedom that we expect for a\ntheory with 32 supercharges, because 32*33/2=528. That is, the algebra\nis complete without the missing D(-1), D6, D8-charges.\n\n* Thirdly, look at the same story in IIB-theory. Here you have two\nspinors of same chirality, let\'s call them Q_A and Q_B, both transform\nin either 16 or 16\'. The product representations are therefore\n\n(16_A \\times 16_B) = (16_B \\times 16_A) = [1] + [3] + [5]_+ (=>\nD1,D3,D5-charges)\n(16_A \\times 16_A)_s = (16_B \\times 16_B)_s = [1] + [5]_+ (=>\nF1,NS5-charges)\n\nNow, the D(-1), D7 and D9-branes do not appear to have charges. This\nis no surprise to me, given the nature of these objects. Again, the\ntotal number of degrees of freedom comes out all right (528).\n\n* Finally my question: The central charge of IIB\'s D5-brane *does*\nappear in the algebra, while that of IIA\'s D6-brane *does not*. But\nthe two are T-dual to one another, are they not? At the very least,\nthis seems to imply a quite non-trivial mapping of the central\ncharges, and perhaps the supercharges, under T-duality. Is this\nmapping known explicitly? Am I right to be confused about this?\n\nThanks,\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>Some time ago, one Rufus Anton asked in this forum about the seemingly
missing central charge of D6 branes:
http://groups.google.com/groups?hl=en&lr=&threadm=a1c70df8.0409211643.364872-100000%40posting.google.com&rnum=2&prev=/groups%3Fq%3DRufus%2BAnton%26hl%3Den%26lr%3D%26gro up%3Dsci.physics.strings%26selm%3Da1c70df8.0409211 643.364872-100000%2540posting.google.
com%26rnum%3D2
This fact is known to many string theorists, but it continues to
puzzle me a little bit. Let me explain -- here is the story:
* Firstly, look at 11D SUGRA
The 11d SUSY algebra is schematically
{Q,Q} = P_\mu \Gamma^\mu +Z^{2_}{...} \Gamma^{...} + Z^{5_}{...} \Gamma^{...}
Here P_\mu is the 11d momentum, Z^2 is the central charge of the
M2-brane and, Z^5 that of the M5-brane. (I have suppressed any spinor
indices and the dots ... indicate contraction of the spacetime indices
with the appropriate \Gamma-matrices.)
Now, if you reduce on a circle, you get a scalar and a 10-vector from
P_\mu, which you can recognize as the D0-central charge and the
10-momentum respectively. Similarly, you get a 1-form and a 2-form
from Z_2, being the F1-charge and the D2-charge. And from Z_5 you get
a 4-form and a 5-form, the D4-charge and the NS5 charge.
In this scheme D(-1), D6, and D8 do not appear to have a charge. For
D(-1) and D8 this is not surprising to me -- but for D6 it is at first
sight. Lubos gave the answer to this in his reply to Rufus' question,
pointing out that the D6 becomes a KK-monopole at strong coupling.
[Moderator's note: You should first decide whether you talk about type
IIA picture, or type IIB. It's no good to mix odd numbers like D(-1)
with even numbers like D6 and D8. Moreover, the D(-1)charge is not
a conserved operator - it is a feature of a spacetime configuration
(D-instanton, localized both in space and the Euclidean time)
that does not survive in the Hilbert space formalism, I think. LM]
* Secondly, let's look at this from the 10d perspective directly. For
IIA there are 32 supercharges organized in two spinors Q and {\bar Q}
which must be in the 16 and 16' of SO(9,1), respectively. The possible
central charges are therefore in the product representations of 16 and
16', namely:
(16 \times 16') = (16' \times 16) = [0] + [2] + [4] (=>
D0,D2,D4-charges)
(16 \times 16)_s = [1] + [5]_+ (=> F1,NS5-charges)
(16' \times 16')_s = [1] + [5]_- (=> F1,NS5-charges)
(The subscript _s makes clear that the anticommutator we consider has
definite permuational symmetry with respect to its spinor indices,
which explains the absence of tensors like [3].)
Again, D(-1), D6, D8-charges do not appear. Notice that if you count
the degrees of freedom of all the tensors on the RHS, you find 528.
This saturates the number of degrees of freedom that we expect for a
theory with 32 supercharges, because 32*33/2=528. That is, the algebra
is complete without the missing D(-1), D6, D8-charges.
* Thirdly, look at the same story in IIB-theory. Here you have two
spinors of same chirality, let's call them Q_A and Q_B, both transform
in either 16 or 16'. The product representations are therefore
(16_A \times 16_B) = (16_B \times 16_A) = [1] + [3] + [5]_+ (=>
D1,D3,D5-charges)
(16_A \times 16_A)_s = (16_B \times 16_B)_s = [1] + [5]_+ (=>
F1,NS5-charges)
Now, the D(-1), D7 and D9-branes do not appear to have charges. This
is no surprise to me, given the nature of these objects. Again, the
total number of degrees of freedom comes out all right (528).
* Finally my question: The central charge of IIB's D5-brane *does*
appear in the algebra, while that of IIA's D6-brane *does not*. But
the two are T-dual to one another, are they not? At the very least,
this seems to imply a quite non-trivial mapping of the central
charges, and perhaps the supercharges, under T-duality. Is this
mapping known explicitly? Am I right to be confused about this?
Thanks,
Tobias
missing central charge of D6 branes:
http://groups.google.com/groups?hl=en&lr=&threadm=a1c70df8.0409211643.364872-100000%40posting.google.com&rnum=2&prev=/groups%3Fq%3DRufus%2BAnton%26hl%3Den%26lr%3D%26gro up%3Dsci.physics.strings%26selm%3Da1c70df8.0409211 643.364872-100000%2540posting.google.
com%26rnum%3D2
This fact is known to many string theorists, but it continues to
puzzle me a little bit. Let me explain -- here is the story:
* Firstly, look at 11D SUGRA
The 11d SUSY algebra is schematically
{Q,Q} = P_\mu \Gamma^\mu +Z^{2_}{...} \Gamma^{...} + Z^{5_}{...} \Gamma^{...}
Here P_\mu is the 11d momentum, Z^2 is the central charge of the
M2-brane and, Z^5 that of the M5-brane. (I have suppressed any spinor
indices and the dots ... indicate contraction of the spacetime indices
with the appropriate \Gamma-matrices.)
Now, if you reduce on a circle, you get a scalar and a 10-vector from
P_\mu, which you can recognize as the D0-central charge and the
10-momentum respectively. Similarly, you get a 1-form and a 2-form
from Z_2, being the F1-charge and the D2-charge. And from Z_5 you get
a 4-form and a 5-form, the D4-charge and the NS5 charge.
In this scheme D(-1), D6, and D8 do not appear to have a charge. For
D(-1) and D8 this is not surprising to me -- but for D6 it is at first
sight. Lubos gave the answer to this in his reply to Rufus' question,
pointing out that the D6 becomes a KK-monopole at strong coupling.
[Moderator's note: You should first decide whether you talk about type
IIA picture, or type IIB. It's no good to mix odd numbers like D(-1)
with even numbers like D6 and D8. Moreover, the D(-1)charge is not
a conserved operator - it is a feature of a spacetime configuration
(D-instanton, localized both in space and the Euclidean time)
that does not survive in the Hilbert space formalism, I think. LM]
* Secondly, let's look at this from the 10d perspective directly. For
IIA there are 32 supercharges organized in two spinors Q and {\bar Q}
which must be in the 16 and 16' of SO(9,1), respectively. The possible
central charges are therefore in the product representations of 16 and
16', namely:
(16 \times 16') = (16' \times 16) = [0] + [2] + [4] (=>
D0,D2,D4-charges)
(16 \times 16)_s = [1] + [5]_+ (=> F1,NS5-charges)
(16' \times 16')_s = [1] + [5]_- (=> F1,NS5-charges)
(The subscript _s makes clear that the anticommutator we consider has
definite permuational symmetry with respect to its spinor indices,
which explains the absence of tensors like [3].)
Again, D(-1), D6, D8-charges do not appear. Notice that if you count
the degrees of freedom of all the tensors on the RHS, you find 528.
This saturates the number of degrees of freedom that we expect for a
theory with 32 supercharges, because 32*33/2=528. That is, the algebra
is complete without the missing D(-1), D6, D8-charges.
* Thirdly, look at the same story in IIB-theory. Here you have two
spinors of same chirality, let's call them Q_A and Q_B, both transform
in either 16 or 16'. The product representations are therefore
(16_A \times 16_B) = (16_B \times 16_A) = [1] + [3] + [5]_+ (=>
D1,D3,D5-charges)
(16_A \times 16_A)_s = (16_B \times 16_B)_s = [1] + [5]_+ (=>
F1,NS5-charges)
Now, the D(-1), D7 and D9-branes do not appear to have charges. This
is no surprise to me, given the nature of these objects. Again, the
total number of degrees of freedom comes out all right (528).
* Finally my question: The central charge of IIB's D5-brane *does*
appear in the algebra, while that of IIA's D6-brane *does not*. But
the two are T-dual to one another, are they not? At the very least,
this seems to imply a quite non-trivial mapping of the central
charges, and perhaps the supercharges, under T-duality. Is this
mapping known explicitly? Am I right to be confused about this?
Thanks,
Tobias