Representation theory of the groups appearing in String Theory? [Archive]

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Rene Meyer

May27-04, 01:08 PM

Lubos Motl

May27-04, 01:26 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>On Thu, 27 May 2004, Rene Meyer wrote:\n\n> is there any reference giving a somewhat detailed\n> calculation of those group representations like SO(8)_v x SO(8)_s\n> appearing in String Theory. The standard text books always like to go\n> over these topics with a phrase like "is well known" or something like\n> this.\n\nHi Rene,\n\nit might be useful if you ask a more specific question. Let me say a\ncouple of words about the representations of spin(8).\n\nThe Dynkin diagram of spin(8) i.e D_4 looks like the Mercedes-Benz logo\nwith rank=4 nodes and 3 simple legs, and therefore it has a S_3 symmetry\nexchanging the three legs. This symmetry is known as triality, and it also\ninterchanges various representations of SO(8). Other SO(2k) groups -\nDynkin diagrams - only have a Z_2 symmetry, that acts as parity in\nspacetime (and exchanges the left-handed and right-handed spinors, for\nexample).\n\nSO(8) has obviously an 8-dimensional real vector (fundamental\nrepresentation), and then it has 2 Weyl 8-dimensional real spinor\nrepresentations, 8_s and 8_c - the total non-chiral Dirac spinor has\n8+8=16=2^{8/2} components. These three, 8_v, 8_s, 8_c, are related by\ntriality. The tensor product of 8_s and 8_c is a 64-dimensional\nrepresentation that decomposes into 56 + 8_v; here 8_v is the third\neight-dimensional representation while 56 is a 3-form, 8 x 7 x 6 / 3 x 2 x 1.\n\nTwo similar statements are obtained by triality, i.e. permutations of 8_v,\n8_s, 8_c, and the triality-related 56-dimensional representations are the\ngravitinos - the tensor product of a vector 8_v and a spinor 8_s (or 8_c),\nwith the constraint that it contains no component of 8_c.\n\nOn the other hand, the tensor product of 8_v with 8_v is a 64-dimensional\nrep. that decomposes into the 35-dimensional symmetric traceless tensor\n(8x9/2 - 1, the components of the graviton), the 28-dimensional\nantisymmetric tensor (the B-field, adjoint of SO(8)), and a scalar (the\ndilaton).\n\nThe two triality-related representations to the symmetric traceless tensor\n35 (graviton) are the self-dual and anti-selfdual four-forms,\nrespectively, whose dimension is also (1/2) x (8 x 7 x 6 x 5) / 4! = 35.\n\nIn the physical spectrum of type IIA and type IIB theory you have\n\n(8v + 8s) (x) (8v + 8c) type IIA\n(8v + 8s) (x) (8v + 8s) type IIB\n\nNote that the left-moving and right-moving factors have the opposite\nchirality (s/c) in type IIA, and the same chirality in type IIB. This\ntensor product contains 8v x 8v, which is the graviton, B-field, dilaton\nexplained above (the NS NS sector). This part is the same in type IIB. The\nmixed (NS-R and R-NS) products 8s x 8v (or 8v x 8c) describe the (56)\ngravitino and (8) dilatino, also explained above - in type IIA you find\nboth mirror versions of it, in type IIB you find twice as many fermions\nwith the same chirality.\n\nFinally, the 8s x 8c (or 8s x 8s in type IIB) gives you the Ramond-Ramond\nstates which can be decomposed as differential p-forms where p is even\n(0,2,4selfdual) in type IIB or odd (1,3) in type IIA. In the case of type\nIIB, the forms have dimensions 64=1+28+35(selfdual 4-form), while in type\nIIA they have 64=8+56 where 56 is the three-form that I started with.\n\nBest\nLubos\n____________________________ __________________________________________________ \nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\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>On Thu, 27 May 2004, Rene Meyer wrote:

> is there any reference giving a somewhat detailed
> calculation of those group representations like SO(8)_v x SO(8)_s
> appearing in String Theory. The standard text books always like to go
> over these topics with a phrase like "is well known" or something like
> this.

Hi Rene,

it might be useful if you ask a more specific question. Let me say a
couple of words about the representations of spin(8).

The Dynkin diagram of spin(8) i.e D_4 looks like the Mercedes-Benz logo
with rank=4 nodes and 3 simple legs, and therefore it has a S_3 symmetry
exchanging the three legs. This symmetry is known as triality, and it also
interchanges various representations of SO(8). Other SO(2k) groups -
Dynkin diagrams - only have a Z_2 symmetry, that acts as parity in
spacetime (and exchanges the left-handed and right-handed spinors, for
example).

SO(8) has obviously an 8-dimensional real vector (fundamental
representation), and then it has 2 Weyl 8-dimensional real spinor
representations, 8_s and 8_c - the total non-chiral Dirac spinor has
8+8=16=2^{8/2} components. These three, 8_v, 8_s, 8_c, are related by
triality. The tensor product of 8_s and 8_c is a 64-dimensional
representation that decomposes into 56 + 8_v; here 8_v is the third
eight-dimensional representation while 56 is a 3-form, 8 x 7 x 6 / 3 x 2 x 1.

Two similar statements are obtained by triality, i.e. permutations of 8_v,8_s, 8_c, and the triality-related 56-dimensional representations are the
gravitinos - the tensor product of a vector 8_v and a spinor 8_s (or 8_c),
with the constraint that it contains no component of 8_c.

On the other hand, the tensor product of 8_v with 8_v is a 64-dimensional
rep. that decomposes into the 35-dimensional symmetric traceless tensor
(8x9/2 - 1, the components of the graviton), the 28-dimensional
antisymmetric tensor (the B-field, adjoint of SO(8)), and a scalar (the
dilaton).

The two triality-related representations to the symmetric traceless tensor
35 (graviton) are the self-dual and anti-selfdual four-forms,
respectively, whose dimension is also (1/2) x (8 x 7 x 6 x 5) / 4! = 35.

In the physical spectrum of type IIA and type IIB theory you have

(8v + 8s) (x) (8v + 8c) type IIA
(8v + 8s) (x) (8v + 8s) type IIB

Note that the left-moving and right-moving factors have the opposite
chirality (s/c) in type IIA, and the same chirality in type IIB. This
tensor product contains 8v x 8v, which is the graviton, B-field, dilaton
explained above (the NS NS sector). This part is the same in type IIB. The
mixed (NS-R and R-NS) products 8s x 8v (or 8v x 8c) describe the (56)
gravitino and (8) dilatino, also explained above - in type IIA you find
both mirror versions of it, in type IIB you find twice as many fermions
with the same chirality.

Finally, the 8s x 8c (or 8s x 8s in type IIB) gives you the Ramond-Ramond
states which can be decomposed as differential p-forms where p is even
(0,2,4selfdual) in type IIB or odd (1,3) in type IIA. In the case of type
IIB, the forms have dimensions 64=1+28+35(selfdual 4-form), while in type
IIA they have 64=8+56 where 56 is the three-form that I started with.

Best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Urs Schreiber

May28-04, 06:06 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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0405271311030.4448-100000@lamb.physics.harvard.edu...\n> On Thu, 27 May 2004, Rene Meyer wrote:\n>\n> > is there any reference giving a somewhat detailed\n> > calculation of those group representations like SO(8)_v x SO(8)_s\n> > appearing in String Theory. The standard text books always like to go\n> > over these topics with a phrase like "is well known" or something like\n> > this.\n\n> The Dynkin diagram of spin(8) i.e D_4 looks like the Mercedes-Benz logo\n> with rank=4 nodes and 3 simple legs, and therefore it has a S_3 symmetry\n> exchanging the three legs. This symmetry is known as triality, and it also\n> interchanges various representations of SO(8).\n\nSince this somewhat abstract property of SO(8) becomes a little more\nmanifest when a suitable representation of the Clifford algebra in 8\ndimensions is considered, I\'ll give Rene a couple of links:\n\nFirst one notes that in 7 dimensions there is a cross product of vectors\nassociated with the multiplication table of the octonion. Let me quote from\n\nhttp://groups.google.de/groups?selm=3E928455.BAA92AF1%40uni-essen.de :\n\n-----\n\n[Consider] a generalization of the usual\n3-d vector product so that it maps two vectors to another vector in d <>\n3 (while also satisfying most of the usual algebraic relations).\n\nThe result is that such exists only in d=7 and that the structure\nconstants of this generalized vector product have to be those of the\nimaginary octonions.\n\nThe natural question is: Can we realize this octonionic vector product\nin terms of Clifford algebra, somehow? Robert Helling pointed me to the\npaper\n\nB. De Wit & H. Nicolai, The parallelizing S^7 torsion in gauged N=8\nSupergravity, Nucl. Phys. B231 (1984), 506-532\n\nwhich has a brief discussion of this topic on p. 513:\n\nThe point is that in 7 dimensions there is a representation of the\nClifford algebra (Cl(7)) by 8x8 matrices y^m with entries (y^m)_np such\nthat\n\n(y^m)_n8 = i delta_nm\n\n(y^m)_np = i a_mnp (for n,p <> 8),\n\nwhere a_mnp is the multiplication table of the octonions and hence, for\nm,n,p < 8, the multiplication table of the generalized vector product in\n7 dimensions that is discussed in the math.la/0204357 paper: Let\nv_1,v_2,...,v_7 be an orthonormal basis for R^7 then that generalized\nvector product reads in terms of a_mnp:\n\nv_m x v_n = sum over p of a_mnp v_p .\n\nIn other words: Let y(v) be the Clifford generator associated with the\nvector v and let psi(w) be the spinor associated with w by triality\n(i.e. as column vectors psi(w) and w have the same first seven entries\nin the above representation), then\n\ny(v) psi(w) = -psi(v x w).\n\n----\n\nNext one constructs the corresponding Clifford matrices in 8 dimensions in\nthe usual way, as explained by Robert Helling in the footnote to this post\n\nhttp://groups.google.de/groups?selm=86i2hu%24o7o%241%40rosencrantz.stcloud state.edu :\n\n----\n\nTake for concreteness d = 10.\nHere we go to a light-cone frame by using coordinates\n\nx+ = x^0 + x^9 and\n\nx- = x^0 - x^1.\n\nThen we write the Gamma_m as block matrices where Gamma+ and Gamma- have\nthe +/- unit matrix as blocks and the others have gamma_i as blocks\nwhere gamma_i are the SO(8) Dirac matrices (i=1,...,9). But they are\nintimately related to the octonions. Remember there is triality in SO(8)\nwhich means that we can treat left-handed spinors, right-handed spinors\nand vectors on an equal basis (see "week61", "week90", and "week91").\nNow I write out all three indices of gamma_i. Because of triality I can\nuse i,j,k for spinor, dotted spinor and vector indices. Then it is\nknown that\n\ngamma_{ijk} = c_{ijk} for i,j,k < 8\n\ndelta_{ij }for k=8 (and ijk permuted)\n\n0 for more than 2 of ijk equal 8.\n\nis a representation of Cliff(8) if c_ijk} are the structure constants of\nthe octonions (i.e. e_i e_j = c_{ijk} ek for the 7 roots of -1 in the\noctonions).\n\n----\n\nThe crucial fact here is that in D dimensions a Clifford gamma is 2^{D/2} x\n2^{D/2} and hence its "Weyl blocks" are 2^{D/2-1} x 2^{D/2 -1} . Precisely\nfor D = 8 we have\n\nD = 2^{D/2 -1}\n\nso that here we have a chance for a correspondence between vectors and\nspinors, because both will have the same number of components in the\nrepresentation. The gamma matrix with its one vector index, one dotted\nspinor index and one undotted spinor index hence precisely encodes the\nrelationship between the vector representation 8_v, and the two chiral Weyl\nspinor representations 8_s and 8_c.\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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0405271311030.4448-100000@lamb.physics.harvard.edu...
> On Thu, 27 May 2004, Rene Meyer wrote:
>
> > is there any reference giving a somewhat detailed
> > calculation of those group representations like SO(8)_v x SO(8)_s
> > appearing in String Theory. The standard text books always like to go
> > over these topics with a phrase like "is well known" or something like
> > this.

> The Dynkin diagram of spin(8) i.e D_4 looks like the Mercedes-Benz logo
> with rank=4 nodes and 3 simple legs, and therefore it has a S_3 symmetry
> exchanging the three legs. This symmetry is known as triality, and it also
> interchanges various representations of SO(8).

Since this somewhat abstract property of SO(8) becomes a little more
manifest when a suitable representation of the Clifford algebra in 8
dimensions is considered, I'll give Rene a couple of links:

First one notes that in 7 dimensions there is a cross product of vectors
associated with the multiplication table of the octonion. Let me quote from

http://groups.google.de/groups?selm=3E928455.BAA92AF1%40uni-essen.de :

-----

[Consider] a generalization of the usual
3-d vector product so that it maps two vectors to another vector in d <>
3 (while also satisfying most of the usual algebraic relations).

The result is that such exists only in d=7 and that the structure
constants of this generalized vector product have to be those of the
imaginary octonions.

The natural question is: Can we realize this octonionic vector product
in terms of Clifford algebra, somehow? Robert Helling pointed me to the
paper

B. De Wit & H. Nicolai, The parallelizing S^7 torsion in gauged N=8
Supergravity, Nucl. Phys. B231 (1984), 506-532

which has a brief discussion of this topic on p. 513:

The point is that in 7 dimensions there is a representation of the
Clifford algebra (Cl(7)) by 8x8 matrices y^m with entries (y^m)_np such
that

(y^m)_n8 = i \delta_nm(y^m)_np = i a_{mnp}[/itex] (for n,p <> 8),

where a_{mnp} is the multiplication table of the octonions and hence, for
m,n,p < 8, the multiplication table of the generalized vector product in
7 dimensions that is discussed in the math.la/0204357 paper: Let
v_1,v_2,...,v_7 be an orthonormal basis for R^7 then that generalized
vector product reads in terms of a_{mnp}:v_m x v_n = sum over p of a_{mnp} v_p .

In other words: Let y(v) be the Clifford generator associated with the
vector v and let \psi(w) be the spinor associated with w by triality
(i.e. as column vectors \psi(w) and w have the same first seven entries
in the above representation), then

y(v) \psi(w) = -\psi(v x w).

----

Next one constructs the corresponding Clifford matrices in 8 dimensions in
the usual way, as explained by Robert Helling in the footnote to this post

http://groups.google.de/groups?selm=86i2hu%24o7o%241%40rosencrantz.stcloud state.edu :

----

Take for concreteness d = 10.
Here we go to a light-cone frame by using coordinates

x+ = x^0 + x^9 and

x- = x^0 - x^1.

Then we write the \Gamma_m as block matrices where \Gamma+ and \Gamma- have
the +/- unit matrix as blocks and the others have \gamma_i as blocks
where \gamma_i are the SO(8) Dirac matrices (i=1,...,9). But they are
intimately related to the octonions. Remember there is triality in SO(8)
which means that we can treat left-handed spinors, right-handed spinors
and vectors on an equal basis (see "week61", "week90", and "week91").
Now I write out all three indices of \gamma_i. Because of triality I can
use i,j,k for spinor, dotted spinor and vector indices. Then it is
known that

\gamma_{ijk} = c_{ijk} for i,j,k < 8

\delta_{ij }for k=8 (and ijk permuted)

for more than 2 of ijk equal 8.

is a representation of Cliff(8) if c_{ijk}} are the structure constants of
the octonions (i.e. e_i e_j = c_{ijk} ek for the 7 roots of -1 in the
octonions).

----

The crucial fact here is that in D dimensions a Clifford \gamma is 2^{D/2} x2^{D/2} and hence its "Weyl blocks" are 2^{D/2-1} x 2^{D/2 -1} . Precisely
for D = 8 we have

[itex]D = 2^{D/2 -1}

so that here we have a chance for a correspondence between vectors and
spinors, because both will have the same number of components in the
representation. The \gamma matrix with its one vector index, one dotted
spinor index and one undotted spinor index hence precisely encodes the
relationship between the vector representation 8_v, and the two chiral Weyl
spinor representations 8_s and 8_c.

Robert C. Helling

May28-04, 06:26 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>On Thu, 27 May 2004 13:08:37 -0400, Rene Meyer <meyr2@web.de> wrote:\n> Hi,\n>\n> is there any reference giving a somewhat detailed\n> calculation of those group representations like SO(8)_v x SO(8)_s\n> appearing in String Theory. The standard text books always like to go\n> over these topics with a phrase like "is well known" or something like\n> this.\n\nThis tensor product is, as Lubos already explained, easy to work\nout. But let me give another perspective. There is a general procedure\nto work out thensor products of spinor representations called\nFierzing. The idea is the following: An element of the above tensor\nproduct is something with two spinor indices (one dotted and one\nundotted in your case). It is a fact of life that anti-symmetrized\ngamma matrices form a basis for such "spinorial matrices". That is,\nany such matix is a linear combination of\n\nI (the identity in spinor space), gamma^i, gamma^ij, gamma^ijk,...\n\nand this ends at some point because you can only anti-symmetrize d\nindices in d dimensions. In fact, it ends earlier because of Hode\ndualtiy: In Weyl-represenations in even dimensions (as above, those\nare the ones with dotted and undotted indices) or in odd dimensions,\nyou can reexpress gammas with k indices as gammas with d-k indices\nusing the epsilon tensor. So your expansion above ends already with\nd/2 anti-symmetrized indices. (In even dimensions, this reduction with\nepsilon implies that the product with exactly d/2 indices is either\nselfdual or antiselfdual).\n\nFinally, a gamma matrix in even dimensions has one dotted and one\nundotted index. Thus, in the above expansion, the products with an\neven number of indices have two spinor indices of the same type and\nthe odd ones with two different types.\n\nAnd I forgot to mention, objects with antisymmetrized vector indices\n(and (anti-)selfduality imposed in the middel dimension) form\nirreducible representations of SO(d).\n\nSo let\'s do this in your case. You have d=8, so you will have at most\nfour indices. You want 8s x 8c, so only odd numbers of indices\nappear. Hence you know, your bi-spinor can be written as the sum of\na gamma matrix with one and one with three indices. Something with one\nindex is obviously a vector and something with three indices is a\nthreeform.\n\nYou should be able to do this for yourself for 8s x 8s and check with\nLubos\' result.\n\nIIRC, you didn\'t ask for an explanation but for a reference. The\nclassic reference for decompositions of tensor products of\nrepresenations is\n\nGROUP THEORY FOR UNIFIED MODEL BUILDING.\nBy R. Slansky (Los Alamos),. LA-UR-80-3495, (Received Jan 1981). 262pp.\nPublished in Phys.Rept.79:1-128,1981\n\n(esp. the appendices are extremely useful). There is also a computer\nprogramm (freely available) that can do these kinds of calculations:\nhttp://young.sp2mi.univ-poitiers.fr/~marc/LiE/\n\nYou can either download it or use the online version. Select "Tensor\nproduct decomposition" and D4 (maths speak for SO(8)) for your group.\n\nThen you have to enter highest weights for your represenations. In\neven dimensions, [1,0,0,...,0] is the vector, [0,0,....,0,1] is the\nchiral spinor and [0,0,,....,0,1,0] is the anti-chiral spinor. So you\nselect [0,0,0,1] and [0,0,1,0]. The result is\n1X[0,0,1,1] +1X[1,0,0,0]\nThe second factor is the vector. To find out that the first factor is\nthe 3-form, some more work is needed. For example with "dimension of\nmodule" you find out that it\'s 56 dimensional and once you guess it\'s\nthe threeform you can check that with "tensor power" by computing the\nthird alternating tensor power of the vector.\n\nRobert\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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 Thu, 27 May 2004 13:08:37 -0400, Rene Meyer <meyr2@web.de> wrote:
> Hi,
>
> is there any reference giving a somewhat detailed
> calculation of those group representations like SO(8)_v x SO(8)_s
> appearing in String Theory. The standard text books always like to go
> over these topics with a phrase like "is well known" or something like
> this.

This tensor product is, as Lubos already explained, easy to work
out. But let me give another perspective. There is a general procedure
to work out thensor products of spinor representations called
Fierzing. The idea is the following: An element of the above tensor
product is something with two spinor indices (one dotted and one
undotted in your case). It is a fact of life that anti-symmetrized
\gamma matrices form a basis for such "spinorial matrices". That is,
any such matix is a linear combination of

I (the identity in spinor space), \gamma^i, \gamma^ij, \gamma^ijk,...

and this ends at some point because you can only anti-symmetrize d
indices in d dimensions. In fact, it ends earlier because of Hode
dualtiy: In Weyl-represenations in even dimensions (as above, those
are the ones with dotted and undotted indices) or in odd dimensions,
you can reexpress gammas with k indices as gammas with d-k indices
using the \epsilon tensor. So your expansion above ends already with
d/2 anti-symmetrized indices. (In even dimensions, this reduction with
\epsilon implies that the product with exactly d/2 indices is either
selfdual or antiselfdual).

Finally, a \gamma matrix in even dimensions has one dotted and one
undotted index. Thus, in the above expansion, the products with an
even number of indices have two spinor indices of the same type and
the odd ones with two different types.

And I forgot to mention, objects with antisymmetrized vector indices
(and (anti-)selfduality imposed in the middel dimension) form
irreducible representations of SO(d).

So let's do this in your case. You have d=8, so you will have at most
four indices. You want 8s x 8c, so only odd numbers of indices
appear. Hence you know, your bi-spinor can be written as the sum of
a \gamma matrix with one and one with three indices. Something with one
index is obviously a vector and something with three indices is a
threeform.

You should be able to do this for yourself for 8s x 8s and check with
Lubos' result.

IIRC, you didn't ask for an explanation but for a reference. The
classic reference for decompositions of tensor products of
represenations is

GROUP THEORY FOR UNIFIED MODEL BUILDING.
By R. Slansky (Los Alamos),. LA-UR-80-3495, (Received Jan 1981). 262pp.
Published in Phys.Rept.79:1-128,1981

(esp. the appendices are extremely useful). There is also a computer
programm (freely available) that can do these kinds of calculations:
http://young.sp2mi.univ-poitiers.fr/~marc/LiE/

You can either download it or use the online version. Select "Tensor
product decomposition" and D4 (maths speak for SO(8)) for your group.

Then you have to enter highest weights for your represenations. In
even dimensions, [1,0,0,...,0] is the vector, [0,0,....,0,1] is the
chiral spinor and [0,0,,....,0,1,0] is the anti-chiral spinor. So you
select [0,0,0,1] and [0,0,1,0]. The result is
1X[0,0,1,1] +1X[1,0,0,0]
The second factor is the vector. To find out that the first factor is
the 3-form, some more work is needed. For example with "dimension of
module" you find out that it's 56 dimensional and once you guess it's
the threeform you can check that with "tensor power" by computing the
third alternating tensor power of the vector.

Robert

--
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Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling

rick1138

Jun9-04, 04:31 AM

A book that is rich in representations, including those of groups and supergroups associated to SST, is Frappat, Sciarrino and Sorba's "Dictionary on Lie Algebras and Superalgebras".