John Gonsowski
Oct8-04, 05:45 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> wrote in message news:<Pine.LNX.4.31.0410061655180.1390-100000@feynman.harvard.edu>...\n> > Lee Smolin has tried to use a single exceptional group to\n> > get M-theory. Smolin interestingly started with a bosonic stringlike\n> > 26 dimensions and tried to use the 16 extra to get supersymmetry.\n>\n> A well-known non-stringy colleague of mine has recently rejected a paper\n> that claimed to have obtained a spacetime supersymmetric model from\n> bosonic string theory. If someone claims that there is a serious way to\n> get spacetime SUSY from bosonic string theory, I am eager to hear about it\n> (because the possible relations between the superstring and the bosonic\n> string have always excited me) - but be prepared that I think that\n> extraordinary claims require extraordinary evidence. ;-)\n>\n\nHere is the link to Lee Smolin\'s paper on this. He is doing very much\nthe same thing as Smith with added addition of looking for SUSY:\n\nhttp://xxx.lanl.gov/PS_cache/hep-th/pdf/0104/0104050.pdf\n\n> > In the old days didn\'t they try to use the extra 16 to get the\n> > standard model bosons?\n>\n> Not sure whether you mean successful research or unsuccessful\n> speculations. The success story is the heterotic string in which you can\n> derive the gauge group E_8 x E_8 (or SO(32)) from the 16=26-10 extra\n> chiral bosons, as long as you use the correct, modular invariant theory,\n> and you find all the wrapped/momentum states that produce the "W-bosons"\n> (those outside the Cartan subalgebra).\n\nThis from a 1986 Michael Green Scientific American article is what I\nhad in mind and it looks like you\'ve supplied the last sentences for\nthat story: "Accordingly, if the Yang-Mills forces, such as\nelectromagnetism, are included in a string theory, they must be\nunified with gravity in an intimate way. A kind of theory in which the\nYangMills forces can be associated with closed strings was formulated\nby David J. Gross, Jeffrey A. Harvey, Emil J. Martinec and Ryan Rohm\nof Princeton University. Such a theory is known as heterotic, and it\nis the most promising kind of superstring theory developed so far. Its\nconstruction is quite strange. The charges of the Yang-Mills forces\nare included by smearing them out over the entire heterotic string.\nWaves can travel around any closed string in two directions, but on a\nheterotic closed string the waves traveling clockwise are waves of a\n10-dimensional superstring theory; the waves traveling\ncounterclockwise are waves of the original, 26-dimensional string\ntheory. The extra 16 dimensions are interpreted as internal dimensions\nresponsible for the symmetries of the Yang-Mills forces."\n\n> > Smith uses the extra 16 to get standard model fermions.\n>\n> That does not really sound right. The extra 16 bosons simply must live on\n> the even self-dual lattice, and they uniquely lead to the two heterotic\n> string theories. Moreover, there are translation symmetries for them, so\n> the theory is guaranteed to have at least U(1)^{16} as the gauge group\n> from the very beginning. Does he make some orbifolds of the chiral bosons,\n> or chiral bosons with linear dilaton, or something like that? I\'ve been\n> thinking about such additional options recently, but certainly not in\n> connection with the Standard Model fermions. ;-)\n\nYes, this is how Tony Smith describes it on his website:\n\nThe following is my proposal to use the exceptional Lie algebra\nE6(-26), which I will for the rest of this message write as E6, to\nintroduce fermions into string theory in a new way, based on the\nexceptional E6 relations between bosonic vectors/bivectors and\nfermionic spinors, in which 16 of the 26 dimensions are seen as\norbifolds whose 8 + 8 singularities represent first-generation fermion\nparticles and antiparticles.\n\nThis structure allows string theory to be physically interpreted as a\ntheory of interaction among world-lines in the Many-Worlds.\n\nAccording to Soji Kaneyuki, in Graded Lie Algebras, Related Geometric\nStructures, and Pseudo-hermitian Symmetric Spaces, Analysis and\nGeometry on Complex Homogeneous Domains, by Jacques Faraut, Soji\nKaneyuki, Adam Koranyi, Qi-keng Lu, and Guy Roos (Birkhauser 2000), E6\nas a Graded Lie Algebra with 5 grades:\n\ng = E6 = g(-2) + g(-1) + g(0) + g(1) + g(2)\n\nsuch that\n\ng(0) = so(8) + R + R\ndimR g(-1) = dimR g(1) = 16 = 8 + 8\ndimR g(-2) = dimR g(2) = 8\n\n\nHere, step-by-step, is a description of the E6 structure:\n\n\n--------------------------------------------------------------------------------\nStep 1:\n\ng(0) = so(8) 28 gauge bosons\n\n_\n+ R + R |\n|\ndimR g(-1) = dimR g(1) = 16 = 8 + 8 |- 26-dim string spacetime\n| with J3(O)o structure\ndimR g(-2) = dimR g(2) = 8 _|\n\n\n\n\n--------------------------------------------------------------------------------\nStep 2:\n\nThe E6 GLA has an Even Subalgebra gE (Bosonic) and an Odd Part gO\n(Fermionic):\n\nBOSONIC gE = g(-2) + g(0) + g(2)\n\nFERMIONIC gO = g(-1) + g(1)\n\n\n\n\n--------------------------------------------------------------------------------\nStep 3:\n\nBOSONIC\n\ng(0) = so(8) 28 gauge bosons\n\n_\n+ R + R |\ndimR g(-2) = dimR g(2) = 8 |- 10-dim spacetime\n_|\n\n\nFERMIONIC\n\ndimR g(-1) = dimR g(1) = 16 = 8 8-dim orbifold\n+\n8 8-dim orbifold\n\nGiving the Fermionic sector orbifold structure gives each point of the\nstring/world-line a discrete value corresponding to one of the 8+8 =\n16 fundamental first-generation fermion particles or antiparticles.\n\n\n\n--------------------------------------------------------------------------------\nStep 4:\n\nBOSONIC\n\ng(0) = so(8) 28 gauge bosons\n\n_\n+ R + R |\ndimR g(-2) = dimR g(2) = 8 |- 10-dim spacetime\n_|\n\nFERMIONIC\n\ndimR g(-1) = dimR g(1) = 16 = 8 8 fermions\n+\n8 8 antifermions\n\n\n\n\n--------------------------------------------------------------------------------\nStep 5:\n\nBOSONIC\n\n16-dim conformal U(2,2)\ng(0) = so(8) +\n12-dim SU(3)xSU(2)xU(1)\n\n_\n+ R + R |\ndimR g(-2) = dimR g(2) = 4 |- 6-dim conformal spacetime\n_|\n+\n4 4-dim internal symmetry\nspace\n\n\nFERMIONIC\n\ndimR g(-1) = dimR g(1) = 16 = 8 8 fermions (3 gen)\n\n+ 8 8 antifermions (3 gen)\n\nDimensional reduction of spacetime breaks so(8) to U(2,2) and\nSU(3)xSU(2)xU(1) and also introduces 3 generations of fermion\nparticles and antiparticles.\n\n\n\n--------------------------------------------------------------------------------\nStep 6:\n\nBOSONIC\n\n16-dim conformal U(2,2)\ng(0) = so(8) +\n12-dim SU(3)xSU(2)xU(1)\n\n\n+ R + R 2 spacetime conformal dim\n\ndimR g(-2) = dimR g(2) = 4 4-dim physical spacetime\n+\n4 4-dim internal symmetry\nspace\n\n\nFERMIONIC\n\ndimR g(-1) = dimR g(1) = 16 = 8 8 fermions (3 gen)\n\n+ 8 8 antifermions (3 gen)\n\nThe 2 spacetime conformal dimensions R+R are related to complex\nstructure of\n\nspacetime g(-2) + g(2) and\nfermionic g(-1) + g(1).\nPhysical spacetime and internal symmetry space, and fermionic\nrepresentation spaces, are related to Shilov boundaries of the\ncorresponding complex domains.\n\n> Just a couple of trivialities: the couplings in the Standard Model are\n> chiral and distinguish left-handed and right-handed particles, and\n> therefore we need complex representations (those that are inequivalent to\n> their complex conjugates). E_6 is the only exceptional group whose compact\n> form has any complex representations (the "complex conjugation" follows\n> from the symmetry of the E_6 Dynkin diagram). Therefore E_6 is the only\n> viable group for Grand Unification.\n>\n> In the models based on heterotic string theory by David Gross et al., this\n> E_6 - or potentially smaller groups of it such as SO(16) - are obtained as\n> the subgroup of E_8 which appears in heterotic string theory\n> automatically. But the check is not just that E_6 is the correct subgroup\n> of E_8. Even the correct representations appear - 248 of E_8 decomposes\n> under E_6\\times SU(3) - where SU(3) is the centralizer of E_6 and vice\n> versa (in both cases inside E_8) - as\n>\n> 248 = (78,1) + (27,3) + (27*,3*), (1,8)\n>\n> If you compactify the heterotic string on Calabi-Yau, the adjoint (78) of\n> E_6 is of course broken to the unbroken group; the fermions appear from\n> the triplets - the triplets of SU(3) - 3 and 3* - are actually reduced to\n> "1" by the Calabi-Yau magic, but it is still important that the fermions\n> transform as 27 of E_6, which is a realistic (complex) representation of\n> E_6 for the fermions. (It contains the chiral complex spinor 16 under the\n> subgroup SO(10), and this 16 is exactly what we want to reproduce the 15\n> Weyl spinors of one generations of quarks and leptons, plus a single\n> extra "right-handed neutrino".) Well, everything goes through if I imagine\n> that E_8 is broken to E_6 via E_7 as an intermediate step - and therefore\n> a model with a E_7 starting group could give the right representations,\n> too, except that I don\'t know any good theory that only has E_7 at the\n> beginning.\n\nIt actually starts at a single E8 for Smith, he states it this way:\n\n78-dim E6 = 45-dim Adjoint of Spin(10) + 32-dim Spinor of Spin(10) +\nImaginary of C;\n133-dim E7 = 66-dim Adjoint of Spin(12) + 64-dim Spinor of Spin(12) +\nImaginaries of Q;\n248-dim E8 = 120-dim Adjoint of Spin(16) + 128-dim half-Spinor of\nSpin(16)\nPhysically,\n\nE6 corresponds to 26-dim String Theory, related to traceless J3(O)o\nand the symmetric space E6 / F4.\nE7 corresponds to 27-dim M-Theory, related to the Jordan algebra J3(O)\nand the symmetric space E7 / E6 x U(1).\nE8 corresponds to 28-dim F-Theory, related to the Jordan algebra J4(Q)\nand the symmetric space E8 / E7 x SU(2).\n\n> > It\'s amazing how the same math can be used so differently. I like\n> > Smith\'s interpretation but I\'m not sure even Smith would agree with my\n> > reasons. My reasons kind of fall in the Lubos Motl "mind of god"\n> > aestetics category. The root vector geometry for these 16 vertices\n> > (actually 32 vertices since the dimensions are complex) look more like\n> > fermions to me.\n>\n> I am not sure what is needed for a root vector to "look" like a fermion\n> ;-), but mathematically, the ground state in the heterotic string lattice\n> has the same statistics as the zero-momentum state. To change its\n> statistics, you must add some cocycles, and to make it consistent, you\n> must simultaneously redefine your spacetime rotations. Can he get the\n> superstring or something related by defining the physical Lorentz group as\n> a diagonal group acting on two groups of X\'s on the worldsheet? That\n> certainly sounds exciting and similar to many working things in physics,\n> but does it really work here? Is it really a conformal field theory that\n> leads to a realistic low-energy physics in spacetime?\n>\n\nI was talking about the 78 root vector vertices for the E6 string\ntheory itself but your question is much more interesting since Tony\nSmith does use a U(8) model with strings between D8 branes and his D8\nbrane is a superposition of the 8 E8 lattices. He describes it as\nfollows:\n\nStep 1:\n\nConsider the 26 Dimensions of String Theory as the 26-dimensional\ntraceless part J3(O)o\na O+ Ov\n\nO+* b O-\n\nOv* O-* -a-b\n(where Ov, O+, and O- are in Octonion space with basis\n{1,i,j,k,E,I,J,K} and a and b are real numbers with basis {1})\nof the 27-dimensional Jordan algebra J3(O) of 3x3 Hermitian Octonion\nmatrices.\n\n\n\n--------------------------------------------------------------------------------\n\n\nStep 2:\n\nTake Urs Schreiber\'s D3 brane to correspond to the Imaginary\nQuaternionic associative subspace spanned by {i,j,k} in the\n8-dimenisonal Octonionic Ov space.\n\n\n--------------------------------------------------------------------------------\n\n\nStep 3:\n\nCompactify the 4-dimensional co-associative subspace spanned by\n{E,I,J,K} in the Octonionic Ov space as a CP2 = SU(3)/U(2), with its 4\nworld-brane scalars corresponding to the 4 covariant components of a\nHiggs scalar.\nAdd this subspace to D3, to get D7.\n\n\n\n--------------------------------------------------------------------------------\n\n\nStep 4:\n\nOrbifold the 1-dimensional Real subspace spanned by {1} in the\nOctonionic Ov space by the discrete multiplicative group Z2 = {-1,+1},\nwith its fixed points {-1,+1} corresponding to past and future time.\nThis discretizes time steps and gets rid of the world-brane scalar\ncorresponding to the subspace spanned by {1} in Ov. It also gives our\nbrane a 2-level timelike structure, so that its past can connect to\nthe future of a preceding brane and its future can connect to the past\nof a succeeding brane.\nAdd this subspace to D7, to get D8.\n\nD8, our basic Brane, looks like two layers (past and future) of D7s.\n\nBeyond D8 our String Theory has 26 - 8 = 18 dimensions, of which 25 -\n8 have corresponding world-brane scalars:\n\n8 world-brane scalars for Octonionic O+ space;\n8 world-brane scalars for Octonionic O- space;\n1 world-brane scalars for real a space; and\n1 dimension, for real b space, in which the D8 branes containing\nspacelike D3s are stacked in timelike order.\n\n\n--------------------------------------------------------------------------------\n\n\nStep 5:\n\nTo use Urs Schreiber\'s idea to get rid of the world-brane scalars\ncorresponding to the Octonionic O+ space, orbifold it by the\n16-element discrete multiplicative group Oct16 =\n{+/-1,+/-i,+/-j,+/-k,+/-E,+/-I,+/-J,+/-K} to reduce O+ to 16 singular\npoints {-1,-i,-j,-k,-E,-I,-J,-K,+1,+i,+j,+k,+E,+I,+J,+K}.\n\nLet the 8 O+ singular points {-1,-i,-j,-k,-E,-I,-J,-K} correspond to\nthe fundamental fermion particles {neutrino, red up quark, green up\nquark, blue up quark, electron, red down quark, green down quark, blue\ndown quark} located on the past D7 layer of D8.\nLet the 8 O+ singular points {+1,+i,+j,+k,+E,+I,+J,+K} correspond to\nthe fundamental fermion particles {neutrino, red up quark, green up\nquark, blue up quark, electron, red down quark, green down quark, blue\ndown quark} located on the future D7 layer of D8.\nThis gets rid of the 8 world-brane scalars corresponding to O+, and\nleaves:\n\n8 world-brane scalars for Octonionic O- space;\n1 world-brane scalars for real a space; and\n1 dimension, for real b space, in which the D8 branes containing\nspacelike D3s are stacked in timelike order.\n\n\n\n\n--------------------------------------------------------------------------------\n\n\nStep 6:\n\nTo use Urs Schreiber\'s idea to get rid of the world-brane scalars\ncorresponding to the Octonionic O- space, orbifold it by the\n16-element discrete multiplicative group Oct16 =\n{+/-1,+/-i,+/-j,+/-k,+/-E,+/-I,+/-J,+/-K} to reduce O- to 16 singular\npoints {-1,-i,-j,-k,-E,-I,-J,-K,+1,+i,+j,+k,+E,+I,+J,+K}.\n\nLet the 8 O- singular points {-1,-i,-j,-k,-E,-I,-J,-K} correspond to\nthe fundamental fermion anti-particles {anti-neutrino, red up\nanti-quark, green up anti-quark, blue up anti-quark, positron, red\ndown anti-quark, green down anti-quark, blue down anti-quark} located\non the past D7 layer of D8.\nLet the 8 O- singular points {+1,+i,+j,+k,+E,+I,+J,+K} correspond to\nthe fundamental fermion anti-particles {anti-neutrino, red up\nanti-quark, green up anti-quark, blue up anti-quark, positron, red\ndown anti-quark, green down anti-quark, blue down anti-quark} located\non the future D7 layer of D8.\nThis gets rid of the 8 world-brane scalars corresponding to O-, and\nleaves:\n\n1 world-brane scalars for real a space; and\n1 dimension, for real b space, in which the D8 branes containing\nspacelike D3s are stacked in timelike order.\nHere is some discussion of some symmetries of fermion particles and\nantiparticles.\n\n\n\n--------------------------------------------------------------------------------\n\n\nStep 7:\n\nLet the 1 world-brane scalar for real a space correspond to a\nBohm-type Quantum Potential\nacting on strings in the stack of D8 branes.\n\nInterpret strings as world-lines in the Many-Worlds, short strings\nrepresenting virtual particles and loops.\n\n\n\n--------------------------------------------------------------------------------\n\n\nStep 8:\n\nFundamentally, physics is described on HyperDiamond Lattice\nstructures.\n\nThere are 7 independent E8 lattices, each corresponding to one of the\n7 imaginary octionions. They can be described as iE8, jE8, kE8, EE8,\nIE8, JE8, and KE8.\n\nFurther, an 8th naturally related, but dependent, E8 lattice\ncorresponds to the real octonions and can be described as 1E8.\n\nGive each D8 brane structure based on Planck-scale E8 lattices so that\neach D8 brane is a superposition/intersection/coincidence of the eight\nE8 lattices.\n\n\n\n--------------------------------------------------------------------------------\n\n\nStep 9:\n\nSince Polchinski says "... If r D-branes coincide ... there are r^2\nvectors, forming the adjoint of a U(r) gauge group ...", make the\nfollowing assignments:\n\na gauge boson emanating from D8 only from its 1E8 lattice is a U(1)\nphoton;\na gauge boson emanating from D8 only from its 1E8 and EE8 lattices is\na U(2) weak boson;\na gauge boson emanating from D8 only from its IE8, JE8, and KE8\nlattices is a U(3) gluon.\nNote that I do not consider it problematic to have U(2) and U(3)\ninstead of SU(2) and SU(3) for the weak and color forces,\nrespectively. Here is some further discussion of the global Standard\nModel group structure. Here is some discussion of the root vector\nstructures of the Standard Model groups.\n\n\n\n--------------------------------------------------------------------------------\n\n\nStep 10:\n\nSince Polchinski says "... there will also be r^2 massless scalars\nfrom the components normal to the D-brane. ... the collectives\ncoordinates ... X^u ... for the embedding of n D-branes in spacetime\nare now enlarged to nxn matrices. This \'noncummutative geometry\'\n....[may be]... an important hint about the nature of spacetime. ...",\nmake the following assignment:\n\nThe 8x8 matrices for the collective coordinates linking a D8 brane to\nthe next D8 brane in the stack are needed to connect\nthe eight E8 lattices of the D8 brane\nto the eight E8 lattices of the next D8 brane in the stack.\n\nWe have now accounted for all the scalars, and, since, as Lubos Motl\nnoted, "... string theory always contains gravity ...",\n\nwe have here at Step 10 a specific example of a String Theory\ncontaining gravity and the U(1)xSU(2)xSU(3) Standard Model.\n\n\n--------------------------------------------------------------------------------\n\n\nStep 11:\n\nWe can go a bit further by noting that we have not described gauge\nbosons emanating from D8 from its iE8, jE8, or kE8 lattices.\nTherefore, make the following assignment:\n\na gauge boson emanating from D8 only from its 1E8, iE8, jE8, and kE8\nlattices is a U(2,2) conformal gauge boson.\nWe have here at Step 10 a String Theory containing the Standard Model\nplus two forms of gravity:\n\nclosed-string gravity and\nconformal U(2,2) = Spin(2,4)xU(1) gravity plus conformal structures,\nbased on a generalized MacDowell-Mansouri mechanism.\nI conjecture that those two forms of gravity are not only consistent,\nbut that the structures of each will shed light on the structures of\nthe other, and that the conformal structures are related to the\nconformal gravity ideas of I. E. Segal.\n\n\n--------------------------------------------------------------------------------\n\n\nStep 12:\n\nGoing a bit further leads to consideration of the exceptional E-series\nof Lie algebras, as follows:\n\na gauge boson emanating from D8 only from its 1E8, iE8, jE8, kE8, and\nEE8 lattices is a U(5) gauge boson related to Spin(10) and Complex E6.\na gauge boson emanating from D8 only from its 1E8, iE8, jE8, kE8, EE8,\nand IE8 lattices is a U(6) gauge boson related to Spin(12) and\nQuaternionic E7.\n\na gauge boson emanating from D8 only from its 1E8, iE8, jE8, kE8, EE8,\nIE8, and JE8 lattices is a U(7) gauge boson related to Spin(14) and\npossibly to Sextonionic E(7+(1/2)).\n\na gauge boson emanating from D8 only from its 1E8, iE8, jE8, kE8, EE8,\nIE8, JE8, and KE8 lattices is a U(8) gauge boson related to Spin(16)\nand Octonionic E8.\n\n\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> wrote in message news:<Pine.LNX.4.31.0410061655180.1390-100000@feynman.harvard.edu>...
> > Lee Smolin has tried to use a single exceptional group to
> > get M-theory. Smolin interestingly started with a bosonic stringlike
> > 26 dimensions and tried to use the 16 extra to get supersymmetry.
>
> A well-known non-stringy colleague of mine has recently rejected a paper
> that claimed to have obtained a spacetime supersymmetric model from
> bosonic string theory. If someone claims that there is a serious way to
> get spacetime SUSY from bosonic string theory, I am eager to hear about it
> (because the possible relations between the superstring and the bosonic
> string have always excited me) - but be prepared that I think that
> extraordinary claims require extraordinary evidence. ;-)
>
Here is the link to Lee Smolin's paper on this. He is doing very much
the same thing as Smith with added addition of looking for SUSY:
http://xxx.lanl.gov/PS_cache/hep-th/pdf/0104/0104050.pdf
> > In the old days didn't they try to use the extra 16 to get the
> > standard model bosons?
>
> Not sure whether you mean successful research or unsuccessful
> speculations. The success story is the heterotic string in which you can
> derive the gauge group E_8 x E_8 (or SO(32)) from the 16=26-10 extra
> chiral bosons, as long as you use the correct, modular invariant theory,
> and you find all the wrapped/momentum states that produce the "W-bosons"
> (those outside the Cartan subalgebra).
This from a 1986 Michael Green Scientific American article is what I
had in mind and it looks like you've supplied the last sentences for
that story: "Accordingly, if the Yang-Mills forces, such as
electromagnetism, are included in a string theory, they must be
unified with gravity in an intimate way. A kind of theory in which the
YangMills forces can be associated with closed strings was formulated
by David J. Gross, Jeffrey A. Harvey, Emil J. Martinec and Ryan Rohm
of Princeton University. Such a theory is known as heterotic, and it
is the most promising kind of superstring theory developed so far. Its
construction is quite strange. The charges of the Yang-Mills forces
are included by smearing them out over the entire heterotic string.
Waves can travel around any closed string in two directions, but on a
heterotic closed string the waves traveling clockwise are waves of a
10-dimensional superstring theory; the waves traveling
counterclockwise are waves of the original, 26-dimensional string
theory. The extra 16 dimensions are interpreted as internal dimensions
responsible for the symmetries of the Yang-Mills forces."
> > Smith uses the extra 16 to get standard model fermions.
>
> That does not really sound right. The extra 16 bosons simply must live on
> the even self-dual lattice, and they uniquely lead to the two heterotic
> string theories. Moreover, there are translation symmetries for them, so
> the theory is guaranteed to have at least U(1)^{16} as the gauge group
> from the very beginning. Does he make some orbifolds of the chiral bosons,
> or chiral bosons with linear dilaton, or something like that? I've been
> thinking about such additional options recently, but certainly not in
> connection with the Standard Model fermions. ;-)
Yes, this is how Tony Smith describes it on his website:
The following is my proposal to use the exceptional Lie algebra
E6(-26), which I will for the rest of this message write as E6, to
introduce fermions into string theory in a new way, based on the
exceptional E6 relations between bosonic vectors/bivectors and
fermionic spinors, in which 16 of the 26 dimensions are seen as
orbifolds whose 8 + 8 singularities represent first-generation fermion
particles and antiparticles.
This structure allows string theory to be physically interpreted as a
theory of interaction among world-lines in the Many-Worlds.
According to Soji Kaneyuki, in Graded Lie Algebras, Related Geometric
Structures, and Pseudo-hermitian Symmetric Spaces, Analysis and
Geometry on Complex Homogeneous Domains, by Jacques Faraut, Soji
Kaneyuki, Adam Koranyi, Qi-keng Lu, and Guy Roos (Birkhauser 2000), E6
as a Graded Lie Algebra with 5 grades:
g = E6 = g(-2) + g(-1) + g(0) + g(1) + g(2)
such that
g(0) = so(8) + R + R
dimR g(-1) = dimR g(1) = 16 = 8 + 8
dimR g(-2) = dimR g(2) = 8
Here, step-by-step, is a description of the E6 structure:
--------------------------------------------------------------------------------
Step 1:
g(0) = so(8) 28 gauge bosons
_+ R + R |[/itex]
|
dimR g(-1) = dimR g(1) = 16 = 8 + 8 |- 26-dim string spacetime
| with J3(O)o structure
dimR g(-2) = dimR g(2) = 8 _|
--------------------------------------------------------------------------------
Step 2:
The E6 GLA has an Even Subalgebra gE (Bosonic) and an Odd Part gO
(Fermionic):
BOSONIC gE = g(-2) + g(0) + g(2)
FERMIONIC gO = g(-1) + g(1)
--------------------------------------------------------------------------------
Step 3:
BOSONIC
g(0) = so(8) 28 gauge bosons
_+ R + R |
dimR g(-2) = dimR g(2) = 8 |- 10-dim spacetime
_|
FERMIONIC
dimR g(-1) = dimR g(1) = 16 = 8 8-dim orbifold
+
8 8-dim orbifold
Giving the Fermionic sector orbifold structure gives each point of the
string/world-line a discrete value corresponding to one of the 8+8 =
16 fundamental first-generation fermion particles or antiparticles.
--------------------------------------------------------------------------------
Step 4:
BOSONIC
g(0) = so(8) 28 gauge bosons
_+ R + R |
dimR g(-2) = dimR g(2) = 8 |- 10-dim spacetime
_|
FERMIONIC
dimR g(-1) = dimR g(1) = 16 = 8 8 fermions
+
8 8 antifermions
--------------------------------------------------------------------------------
Step 5:
BOSONIC
16-dim conformal U(2,2)
g(0) = so(8) +12-dim SU(3)xSU(2)xU(1)
_+ R + R |
dimR g(-2) = dimR g(2) = 4 |- 6-dim conformal spacetime
_|
+
4 4-dim internal symmetry
space
FERMIONIC
dimR g(-1) = dimR g(1) = 16 = 8 8 fermions (3 gen)
+ 8 8 antifermions (3 gen)
Dimensional reduction of spacetime breaks so(8) to U(2,2) and
SU(3)xSU(2)xU(1) and also introduces 3 generations of fermion
particles and antiparticles.
--------------------------------------------------------------------------------
Step 6:
BOSONIC
16-dim conformal U(2,2)
g(0) = so(8) +12-dim SU(3)xSU(2)xU(1)
+ R + R 2 spacetime conformal dim
dimR g(-2) = dimR g(2) = 4 4-dim physical spacetime
+
4 4-dim internal symmetry
space
FERMIONIC
dimR g(-1) = dimR g(1) = 16 = 8 8 fermions (3 gen)
+ 8 8 antifermions (3 gen)
The 2 spacetime conformal dimensions R+R are related to complex
structure of
spacetime g(-2) + g(2) and
fermionic g(-1) + g(1).
Physical spacetime and internal symmetry space, and fermionic
representation spaces, are related to Shilov boundaries of the
corresponding complex domains.
> Just a couple of trivialities: the couplings in the Standard Model are
> chiral and distinguish left-handed and right-handed particles, and
> therefore we need complex representations (those that are inequivalent to
> their complex conjugates). E_6 is the only exceptional group whose compact
> form has any complex representations (the "complex conjugation" follows
> from the symmetry of the E_6 Dynkin diagram). Therefore E_6 is the only
> viable group for Grand Unification.
>
> In the models based on heterotic string theory by David Gross et al., this
> E_6 - or potentially smaller groups of it such as SO(16) - are obtained as
> the subgroup of E_8 which appears in heterotic string theory
> automatically. But the check is not just that E_6 is the correct subgroup
> of E_8. Even the correct representations appear - 248 of E_8 decomposes
> under E_6\times SU(3) - where SU(3) is the centralizer of E_6 and vice
> versa (in both cases inside E_8) - as
>
> 248 = (78,1) + (27,3) + (27*,3*), (1,8)
>
> If you compactify the heterotic string on Calabi-Yau, the adjoint (78) of
> E_6 is of course broken to the unbroken group; the fermions appear from
> the triplets - the triplets of SU(3) - 3 and 3* - are actually reduced to
> "1" by the Calabi-Yau magic, but it is still important that the fermions
> transform as 27 of E_6, which is a realistic (complex) representation of
> E_6 for the fermions. (It contains the chiral complex spinor 16 under the
> subgroup SO(10), and this 16 is exactly what we want to reproduce the 15
> Weyl spinors of one generations of quarks and leptons, plus a single
> extra "right-handed neutrino".) Well, everything goes through if I imagine
> that E_8 is broken to E_6 via E_7 as an intermediate step - and therefore
> a model with a E_7 starting group could give the right representations,
> too, except that I don't know any good theory that only has E_7 at the
> beginning.
It actually starts at a single E8 for Smith, he states it this way:
78-dim E6 = 45-dim Adjoint of Spin(10) + 32-dim Spinor of Spin(10) +
Imaginary of C;
133-dim E7 = 66-dim Adjoint of Spin(12) + 64-dim Spinor of Spin(12) +
Imaginaries of Q;
248-dim E8 = 120-dim Adjoint of Spin(16) + 128-dim half-Spinor of
Spin(16)
Physically,
E6 corresponds to 26-dim String Theory, related to traceless J3(O)o
and the symmetric space E6 / F4.
E7 corresponds to 27-dim M-Theory, related to the Jordan algebra J3(O)
and the symmetric space E7 / E6 x U(1).
E8 corresponds to 28-dim F-Theory, related to the Jordan algebra J4(Q)
and the symmetric space [itex]E8 / E7 x SU(2).
> > It's amazing how the same math can be used so differently. I like
> > Smith's interpretation but I'm not sure even Smith would agree with my
> > reasons. My reasons kind of fall in the Lubos Motl "mind of god"
> > aestetics category. The root vector geometry for these 16 vertices
> > (actually 32 vertices since the dimensions are complex) look more like
> > fermions to me.
>
> I am not sure what is needed for a root vector to "look" like a fermion
> ;-), but mathematically, the ground state in the heterotic string lattice
> has the same statistics as the zero-momentum state. To change its
> statistics, you must add some cocycles, and to make it consistent, you
> must simultaneously redefine your spacetime rotations. Can he get the
> superstring or something related by defining the physical Lorentz group as
> a diagonal group acting on two groups of X's on the worldsheet? That
> certainly sounds exciting and similar to many working things in physics,
> but does it really work here? Is it really a conformal field theory that
> leads to a realistic low-energy physics in spacetime?
>
I was talking about the 78 root vector vertices for the E6 string
theory itself but your question is much more interesting since Tony
Smith does use a U(8) model with strings between D8 branes and his D8
brane is a superposition of the 8 E8 lattices. He describes it as
follows:
Step 1:
Consider the 26 Dimensions of String Theory as the 26-dimensional
traceless part J3(O)o
a O+ Ov
O+* b O-
Ov* O-* -a-b
(where Ov, O+, and O- are in Octonion space with basis
{1,i,j,k,E,I,J,K} and a and b are real numbers with basis {1})
of the 27-dimensional Jordan algebra J3(O) of 3x3 Hermitian Octonion
matrices.
--------------------------------------------------------------------------------
Step 2:
Take Urs Schreiber's D3 brane to correspond to the Imaginary
Quaternionic associative subspace spanned by {i,j,k} in the
8-dimenisonal Octonionic Ov space.
--------------------------------------------------------------------------------
Step 3:
Compactify the 4-dimensional co-associative subspace spanned by
{E,I,J,K} in the Octonionic Ov space as a CP2 = SU(3)/U(2), with its 4
world-brane scalars corresponding to the 4 covariant components of a
Higgs scalar.
Add this subspace to D3, to get D7.
--------------------------------------------------------------------------------
Step 4:
Orbifold the 1-dimensional Real subspace spanned by {1} in the
Octonionic Ov space by the discrete multiplicative group Z2 = {-1,+1},
with its fixed points {-1,+1} corresponding to past and future time.
This discretizes time steps and gets rid of the world-brane scalar
corresponding to the subspace spanned by {1} in Ov. It also gives our
brane a 2-level timelike structure, so that its past can connect to
the future of a preceding brane and its future can connect to the past
of a succeeding brane.
Add this subspace to D7, to get D8.
D8, our basic Brane, looks like two layers (past and future) of D7s.
Beyond D8 our String Theory has 26 - 8 = 18 dimensions, of which 25 -
8 have corresponding world-brane scalars:
8 world-brane scalars for Octonionic O+ space;
8 world-brane scalars for Octonionic O- space;
1 world-brane scalars for real a space; and
1 dimension, for real b space, in which the D8 branes containing
spacelike D3s are stacked in timelike order.
--------------------------------------------------------------------------------
Step 5:
To use Urs Schreiber's idea to get rid of the world-brane scalars
corresponding to the Octonionic O+ space, orbifold it by the
16-element discrete multiplicative group Oct16 =
{+/-1,+/-i,+/-j,+/-k,+/-E,+/-I,+/-J,+/-K} to reduce O+ to 16 singular
points {-1,-i,-j,-k,-E,-I,-J,-K,+1,+i,+j,+k,+E,+I,+J,+K}.
Let the 8 O+ singular points {-1,-i,-j,-k,-E,-I,-J,-K} correspond to
the fundamental fermion particles {neutrino, red up quark, green up
quark, blue up quark, electron, red down quark, green down quark, blue
down quark} located on the past D7 layer of D8.
Let the 8 O+ singular points {+1,+i,+j,+k,+E,+I,+J,+K} correspond to
the fundamental fermion particles {neutrino, red up quark, green up
quark, blue up quark, electron, red down quark, green down quark, blue
down quark} located on the future D7 layer of D8.
This gets rid of the 8 world-brane scalars corresponding to O+, and
leaves:
8 world-brane scalars for Octonionic O- space;
1 world-brane scalars for real a space; and
1 dimension, for real b space, in which the D8 branes containing
spacelike D3s are stacked in timelike order.
--------------------------------------------------------------------------------
Step 6:
To use Urs Schreiber's idea to get rid of the world-brane scalars
corresponding to the Octonionic O- space, orbifold it by the
16-element discrete multiplicative group Oct16 =
{+/-1,+/-i,+/-j,+/-k,+/-E,+/-I,+/-J,+/-K} to reduce O- to 16 singular
points {-1,-i,-j,-k,-E,-I,-J,-K,+1,+i,+j,+k,+E,+I,+J,+K}.
Let the 8 O- singular points {-1,-i,-j,-k,-E,-I,-J,-K} correspond to
the fundamental fermion anti-particles {anti-neutrino, red up
anti-quark, green up anti-quark, blue up anti-quark, positron, red
down anti-quark, green down anti-quark, blue down anti-quark} located
on the past D7 layer of D8.
Let the 8 O- singular points {+1,+i,+j,+k,+E,+I,+J,+K} correspond to
the fundamental fermion anti-particles {anti-neutrino, red up
anti-quark, green up anti-quark, blue up anti-quark, positron, red
down anti-quark, green down anti-quark, blue down anti-quark} located
on the future D7 layer of D8.
This gets rid of the 8 world-brane scalars corresponding to O-, and
leaves:
1 world-brane scalars for real a space; and
1 dimension, for real b space, in which the D8 branes containing
spacelike D3s are stacked in timelike order.
Here is some discussion of some symmetries of fermion particles and
antiparticles.
--------------------------------------------------------------------------------
Step 7:
Let the 1 world-brane scalar for real a space correspond to a
Bohm-type Quantum Potential
acting on strings in the stack of D8 branes.
Interpret strings as world-lines in the Many-Worlds, short strings
representing virtual particles and loops.
--------------------------------------------------------------------------------
Step 8:
Fundamentally, physics is described on HyperDiamond Lattice
structures.
There are 7 independent E8 lattices, each corresponding to one of the
7 imaginary octionions. They can be described as iE8, jE8, kE8, EE8,
IE8, JE8, and KE8.
Further, an 8th naturally related, but dependent, E8 lattice
corresponds to the real octonions and can be described as 1E8.
Give each D8 brane structure based on Planck-scale E8 lattices so that
each D8 brane is a superposition/intersection/coincidence of the eight
E8 lattices.
--------------------------------------------------------------------------------
Step 9:
Since Polchinski says "... If r D-branes coincide ... there are r^2
vectors, forming the adjoint of a U(r) gauge group ...", make the
following assignments:
a gauge boson emanating from D8 only from its 1E8 lattice is a U(1)
photon;
a gauge boson emanating from D8 only from its 1E8 and EE8 lattices is
a U(2) weak boson;
a gauge boson emanating from D8 only from its IE8, JE8, and KE8
lattices is a U(3) gluon.
Note that I do not consider it problematic to have U(2) and U(3)
instead of SU(2) and SU(3) for the weak and color forces,
respectively. Here is some further discussion of the global Standard
Model group structure. Here is some discussion of the root vector
structures of the Standard Model groups.
--------------------------------------------------------------------------------
Step 10:
Since Polchinski says "... there will also be r^2 massless scalars
from the components normal to the D-brane. ... the collectives
coordinates ... X^u ... for the embedding of n D-branes in spacetime
are now enlarged to nxn matrices. This 'noncummutative geometry'
....[may be]... an important hint about the nature of spacetime. ...",
make the following assignment:
The 8x8 matrices for the collective coordinates linking a D8 brane to
the next D8 brane in the stack are needed to connect
the eight E8 lattices of the D8 brane
to the eight E8 lattices of the next D8 brane in the stack.
We have now accounted for all the scalars, and, since, as Lubos Motl
noted, "... string theory always contains gravity ...",
we have here at Step 10 a specific example of a String Theory
containing gravity and the U(1)xSU(2)xSU(3) Standard Model.
--------------------------------------------------------------------------------
Step 11:
We can go a bit further by noting that we have not described gauge
bosons emanating from D8 from its iE8, jE8, or kE8 lattices.
Therefore, make the following assignment:
a gauge boson emanating from D8 only from its 1E8, iE8, jE8, and kE8
lattices is a U(2,2) conformal gauge boson.
We have here at Step 10 a String Theory containing the Standard Model
plus two forms of gravity:
closed-string gravity and
conformal U(2,2) = Spin(2,4)xU(1) gravity plus conformal structures,
based on a generalized MacDowell-Mansouri mechanism.
I conjecture that those two forms of gravity are not only consistent,
but that the structures of each will shed light on the structures of
the other, and that the conformal structures are related to the
conformal gravity ideas of I. E. Segal.
--------------------------------------------------------------------------------
Step 12:
Going a bit further leads to consideration of the exceptional E-series
of Lie algebras, as follows:
a gauge boson emanating from D8 only from its 1E8, iE8, jE8, kE8, and
EE8 lattices is a U(5) gauge boson related to Spin(10) and Complex E6.
a gauge boson emanating from D8 only from its 1E8, iE8, jE8, kE8, EE8,
and IE8 lattices is a U(6) gauge boson related to Spin(12) and
Quaternionic E7.
a gauge boson emanating from D8 only from its 1E8, iE8, jE8, kE8, EE8,
IE8, and JE8 lattices is a U(7) gauge boson related to Spin(14) and
possibly to Sextonionic E(7+(1/2)).
a gauge boson emanating from D8 only from its 1E8, iE8, jE8, kE8, EE8,
IE8, JE8, and KE8 lattices is a U(8) gauge boson related to Spin(16)
and Octonionic E8.
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