string theory and quark families [Archive]

alistair

May14-04, 05:29 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>Can string theory tell us why there are three families of quarks?\nOther theories cannot!\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>Can string theory tell us why there are three families of quarks?
Other theories cannot!

Lubos Motl

May15-04, 09:57 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 Fri, 14 May 2004, alistair wrote:\n\n> Can string theory tell us why there are three families of quarks?\n> Other theories cannot!\n\nString theory has not answered this question, but it has the capacity to\ndo so, unlike any other theory one can imagine.\n\nIn 1985, Candelas+Horowitz+Strominger+Witten showed that the Calabi-Yau\nspaces were important for string theory:\n\nhttp://www-spires.slac.stanford.edu/spires/find/hep/www?rawcmd=find+a+candelas+and+a+strominger\n\nby compactifying the (E_8 x E_8) heterotic string theory on a\nsix-dimensional Calabi-Yau manifold, we can reproduce the supersymmetric\nextension of Grand Unified Theories (GUT), which is an extension of the\nStandard Model.\n\nIn the same setup, one obtains the right spectrum of particles (plus some\nextra particles) including the right representations for the fermions\nwhich come out automatically. This setup also implies that there will be a\nrepetitive structure - the leptons and quarks will be organized into\nalmost identical generations (or families).\n\nIn the simplest case, the number of generations (minus the number of\nantigenerations) equals one half of the Euler character of the Calabi-Yau\nspace - i.e. equals the difference h_{1,1}-h_{1,2} of its two nontrivial\nHodge numbers. This result can be obtained as the number of the\nzero-energy solutions of the Dirac equation defined on the Calabi-Yau\nspace, which is roughly called the Dirac index.\n\nThe first known Calabi-Yau space was the quintic - the quintic\nhypersurface in CP^4; see the equation at\n\nhttp://en.wikipedia.org/wiki/Conifold\n\nThis quintic had the Euler character -200, i.e. it predicted 100\n(anti)generations - the difference between the generations and\nantigenerations is a matter of convention, and mirror symmetry in fact\nswaps them. Well, 100 is too many. For a couple of days (?), the authors\nbelieved that the quintic was the unique Calabi-Yau space, and therefore\nstring theory predicted a unique Universe (well, with a wrong number of\ngenerations).\n\nThe years that followed have revealed that there are at least thousands of\npossible (topological classes of) Calabi-Yau spaces. As far as I know,\nthere can still be infinitely many. Moreover, new structures - bundles,\nfluxes, and branes - defined on the Calabi-Yau space increased the number\nof possibilities. And there are also some other, non-Calabi-Yau stringy\npossibilities to obtain the real world - intersecting brane models;\nF-theory on SU(4) holonomy manifolds; M-theory on G_2-manifolds, and so\nforth. Each of them has a slightly different procedure to obtain the\nnumber of generations - for example, as the number of intersections of\nD-branes in the intersecting brane models, roughly speaking.\n\nAll these models are able to say something about the allowed number of\ngenerations, and correlate this number with other physics. It is now\nvigorously debated how many possible Universes string theory predicts -\nwhether its predictions can become almost unique, or whether they are so\nambiguous that the age of predictive science has ended, or something in\nbetween.\n\nFor example, is the number of generations "3" important for the existence\nof life, so that the anthropic principle could say something about this\nnumber? I don\'t think so. Until someone finds a convincingly unique\nrealization of the real world within string theory, we won\'t know whether\nthe number of generations - and other features of our Cosmos - can be\npredicted.\n\nAll the best\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 Fri, 14 May 2004, alistair wrote:

> Can string theory tell us why there are three families of quarks?
> Other theories cannot!

String theory has not answered this question, but it has the capacity to
do so, unlike any other theory one can imagine.

In 1985, Candelas+Horowitz+Strominger+Witten showed that the Calabi-Yau
spaces were important for string theory:

http://www-spires.slac.stanford.edu/spires/find/hep/www?rawcmd=find+a+candelas+and+a+strominger

by compactifying the (E_8 x E_8) heterotic string theory on a
six-dimensional Calabi-Yau manifold, we can reproduce the supersymmetric
extension of Grand Unified Theories (GUT), which is an extension of the
Standard Model.

In the same setup, one obtains the right spectrum of particles (plus some
extra particles) including the right representations for the fermions
which come out automatically. This setup also implies that there will be a
repetitive structure - the leptons and quarks will be organized into
almost identical generations (or families).

In the simplest case, the number of generations (minus the number of
antigenerations) equals one half of the Euler character of the Calabi-Yau
space - i.e. equals the difference h_{1,1}-h_{1,2} of its two nontrivial
Hodge numbers. This result can be obtained as the number of the
zero-energy solutions of the Dirac equation defined on the Calabi-Yau
space, which is roughly called the Dirac index.

The first known Calabi-Yau space was the quintic - the quintic
hypersurface in CP^4; see the equation at

http://en.wikipedia.org/wiki/Conifold

This quintic had the Euler character -200, i.e. it predicted 100
(anti)generations - the difference between the generations and
antigenerations is a matter of convention, and mirror symmetry in fact
swaps them. Well, 100 is too many. For a couple of days (?), the authors
believed that the quintic was the unique Calabi-Yau space, and therefore
string theory predicted a unique Universe (well, with a wrong number of
generations).

The years that followed have revealed that there are at least thousands of
possible (topological classes of) Calabi-Yau spaces. As far as I know,
there can still be infinitely many. Moreover, new structures - bundles,
fluxes, and branes - defined on the Calabi-Yau space increased the number
of possibilities. And there are also some other, non-Calabi-Yau stringy
possibilities to obtain the real world - intersecting brane models;
F-theory on SU(4) holonomy manifolds; M-theory on G_2-manifolds, and so
forth. Each of them has a slightly different procedure to obtain the
number of generations - for example, as the number of intersections of
D-branes in the intersecting brane models, roughly speaking.

All these models are able to say something about the allowed number of
generations, and correlate this number with other physics. It is now
vigorously debated how many possible Universes string theory predicts -
whether its predictions can become almost unique, or whether they are so
ambiguous that the age of predictive science has ended, or something in
between.

For example, is the number of generations "3" important for the existence
of life, so that the anthropic principle could say something about this
number? I don't think so. Until someone finds a convincingly unique
realization of the real world within string theory, we won't know whether
the number of generations - and other features of our Cosmos - can be
predicted.

All the 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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

alistair

May30-04, 09:08 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 said:\n\nThe years that followed have revealed that there are at least\nthousands of\npossible (topological classes of) Calabi-Yau spaces. As far as I know,\nthere can still be infinitely many. Moreover, new structures -\nbundles,\nfluxes, and branes - defined on the Calabi-Yau space increased the\nnumber\nof possibilities. And there are also some other, non-Calabi-Yau\nstringy\npossibilities to obtain the real world - intersecting brane models;\nF-theory on SU(4) holonomy manifolds; M-theory on G_2-manifolds, and\nso\nforth. Each of them has a slightly different procedure to obtain the\nnumber of generations - for example, as the number of intersections of\nD-branes in the intersecting brane models, roughly speaking.\n\nAlistair asks:\n\nDoes the number of generations come from a number of different types\nof intersection of D branes, or from a number of intersections of one\nkind?\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 said:

The years that followed have revealed that there are at least
thousands of
possible (topological classes of) Calabi-Yau spaces. As far as I know,
there can still be infinitely many. Moreover, new structures -
bundles,
fluxes, and branes - defined on the Calabi-Yau space increased the
number
of possibilities. And there are also some other, non-Calabi-Yau
stringy
possibilities to obtain the real world - intersecting brane models;
F-theory on SU(4) holonomy manifolds; M-theory on G_2-manifolds, and
so
forth. Each of them has a slightly different procedure to obtain the
number of generations - for example, as the number of intersections of
D-branes in the intersecting brane models, roughly speaking.

Alistair asks:

Does the number of generations come from a number of different types
of intersection of D branes, or from a number of intersections of one
kind?

Lubos Motl

May30-04, 02:06 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 Sun, 30 May 2004, alistair wrote:\n\n> Does the number of generations come from a number of different types\n> of intersection of D branes, or from a number of intersections of one\n> kind?\n\nFirst of all, the intersection of D-branes is extended in the whole\nspacetime as we know it; it is only localized in the "hidden" compact\ndimensions.\n\nEvery type of intersection adds some new particle species to the "zoo" of\nknown particles; these particles look like open strings whose one end is\nattached to the first D-brane, and the second end is attached to the\nsecond D-brane.\n\nSome intersections can predict new tachyons - particles whose squared mass\nis negative. Tachyons would signal instability of the whole configuration,\nof the whole Universe (instability with respect to the two D-branes\nmerging and/or annihilating in a certain way), and we usually want to\navoid them.\n\nAssuming that no tachyons appear, the most interesting particles are those\nmassless (or nearly massless) ones. One intersection of an appropriate\nkind can give rise to one new type of a quark, or one new type of a\nlepton, roughly speaking. Because there are 6 flavors of quarks, and each\nof them has 3 colors, you should imagine something like 18 different\nintersections of pairs of D-branes (well, 36 is a more reasonable\ndescription because fermions have two components - left-handed and\nright-handed) that give rise to quarks. There are other intersections that\nallow leptons (and perhaps other particles) to exist.\n\nYes, if we count the number of families of quarks (and leptons), we only\ncount the number of intersections of the right type that allows the quarks\n(or leptons).\n\nSome other type of intersections (with wrong angles) can contribute no\ninteresting particles at all. Nevertheless, all these things - the\nspectrum of new particles arising because of an intersection - is\ncalculable, and we know how to calculate it, at least at the\nnon-interacting level.\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 Sun, 30 May 2004, alistair wrote:

> Does the number of generations come from a number of different types
> of intersection of D branes, or from a number of intersections of one
> kind?

First of all, the intersection of D-branes is extended in the whole
spacetime as we know it; it is only localized in the "hidden" compact
dimensions.

Every type of intersection adds some new particle species to the "zoo" of
known particles; these particles look like open strings whose one end is
attached to the first D-brane, and the second end is attached to the
second D-brane.

Some intersections can predict new tachyons - particles whose squared mass
is negative. Tachyons would signal instability of the whole configuration,
of the whole Universe (instability with respect to the two D-branes
merging and/or annihilating in a certain way), and we usually want to
avoid them.

Assuming that no tachyons appear, the most interesting particles are those
massless (or nearly massless) ones. One intersection of an appropriate
kind can give rise to one new type of a quark, or one new type of a
lepton, roughly speaking. Because there are 6 flavors of quarks, and each
of them has 3 colors, you should imagine something like 18 different
intersections of pairs of D-branes (well, 36 is a more reasonable
description because fermions have two components - left-handed and
right-handed) that give rise to quarks. There are other intersections that
allow leptons (and perhaps other particles) to exist.

Yes, if we count the number of families of quarks (and leptons), we only
count the number of intersections of the right type that allows the quarks
(or leptons).

Some other type of intersections (with wrong angles) can contribute no
interesting particles at all. Nevertheless, all these things - the
spectrum of new particles arising because of an intersection - is
calculable, and we know how to calculate it, at least at the
non-interacting level.

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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Creighton Hogg

Jun1-04, 04:57 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 Sun, 30 May 2004, Lubos Motl wrote:\n\n> ...\n> Some intersections can predict new tachyons - particles whose squared mass\n> is negative. Tachyons would signal instability of the whole configuration,\n> of the whole Universe (instability with respect to the two D-branes\n> merging and/or annihilating in a certain way), and we usually want to\n> avoid them.\n\nI haven\'t really seen papers where intersecting D-brane models lead to\ntachyons, but then again I am not well read except for some approaches to\nstandard model like setups.\nI had thought, though, that the additional tachyons only occurred when you\nhave a D-brane and antiD-brane?\nAm I all washed up on that assumption?\n\n[Moderator\'s note: Yes, I also expressed the idea that the models with\nthe tachyons are not terribly interesting. If you want tachyons, it is\nnot necessary to have exactly parallel D-brane and anti-D-brane pair.\nIf you tilt them a bit, it is not hard to believe that the tachyon will\nstill be there. LM]\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 Sun, 30 May 2004, Lubos Motl wrote:

> ...
> Some intersections can predict new tachyons - particles whose squared mass
> is negative. Tachyons would signal instability of the whole configuration,
> of the whole Universe (instability with respect to the two D-branes
> merging and/or annihilating in a certain way), and we usually want to
> avoid them.

I haven't really seen papers where intersecting D-brane models lead to
tachyons, but then again I am not well read except for some approaches to
standard model like setups.
I had thought, though, that the additional tachyons only occurred when you
have a D-brane and antiD-brane?
Am I all washed up on that assumption?

[Moderator's note: Yes, I also expressed the idea that the models with
the tachyons are not terribly interesting. If you want tachyons, it is
not necessary to have exactly parallel D-brane and anti-D-brane pair.
If you tilt them a bit, it is not hard to believe that the tachyon will
still be there. LM]

Robert Helling

Jun9-04, 12:40 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 Tue, 1 Jun 2004 22:57:11 +0200, Creighton Hogg <wchogg@hep.wisc.edu>\nwrote:\n> On Sun, 30 May 2004, Lubos Motl wrote:\n>\n>> ...\n>> Some intersections can predict new tachyons - particles whose squared\nmass\n>> is negative. Tachyons would signal instability of the whole\nconfiguration,\n>> of the whole Universe (instability with respect to the two D-branes\n>> merging and/or annihilating in a certain way), and we usually want to\n>> avoid them.\n>\n> I haven\'t really seen papers where intersecting D-brane models lead to\n> tachyons, but then again I am not well read except for some approaches to\n> standard model like setups.\n> I had thought, though, that the additional tachyons only occurred when you\n> have a D-brane and antiD-brane?\n> Am I all washed up on that assumption?\n>\n> [Moderator\'s note: Yes, I also expressed the idea that the models with\n> the tachyons are not terribly interesting. If you want tachyons, it is\n> not necessary to have exactly parallel D-brane and anti-D-brane pair.\n> If you tilt them a bit, it is not hard to believe that the tachyon will\n> still be there. LM]\n\nThe generic (curved) brane set-up is not supersymmetric and is very\nlikely to contain tachyons. In fact, only if all branes are calibrated\nby the same form susy is preserved. This condition is easier to\nunderstand for two flat intersecting branes. Let\'s apply T-dualities\nso that we have two Dp branes (if that\'s not possible, one brane has\nto be of the wrong dimensionality for IIA/B anyway). Then the\nintersection can be described by p (principal) angles. To see this you\nneed some higher dimensional imagination: Lines intersect with one\nangle between them, two surfaces intersect with two angles etc. Now\nyou sum those p angles and if that\'s an integer multiple of pi, the\nintersection is supersymmetric. Otherwise it will probably contain\ntachyons.\n\nYou can convince yourself that this rule contains the more well known\ncheck for susy for branes intersecting at right angles (and being\nparallel in some other directions): Then the number of dimensions that\nis tangential to exactly one brane has to be a multiple of four.\n\nThe origin of the tachyonic mode is the following: If you start with\nan intersection that looks like\n\n/\n/\n/\n--------------/----------\n/\n/\n/\n\nyou can make the surface of the brane smaller by deforming it to\n\n/\n|\n\\\n------- ---------\n\\\n|\n/\n/\n\nand as you gain energy from this this mode is tachyonic. This process\nis known as brane recombination (unfortunate name).\n\nIt could be that this rippling off continiues forever or that it stops\nat some minimal surface. That depends on boundary conditions and the\nbackground manifold.\n\nThis process seems to always reduce the area but if the above angle\ncriterion is fulfilled this deformation doesn\'t change the area\n(basically because there are more dimension than the ones I drew and\nin those the area becomes larger.\n\nNote that for model building purposes the tachyonic mode is not\nlethal. As I said the process could stop for the D-brane wrapping some\nminimal surface. This is like the Higgs in the standard model: The\nphi^2 term has a negative coefficient so phi=0 is an unstable maximum\nof the potential (a tachyon) but there is also a true vacuum as at\nsome point the phi^4 term with its positive sign dominates. The same\ncould be true in string theory but it\'s just very hard to obtain\nreliable information about the full form of the potential. Ibanez and\nco have discussed this a lot and I have also written a couple of\npapers relating to this (search spires).\n\nRobert\n\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo\n..oO\nRobert C. Helling Department of Applied Mathematics and Theoretical\nPhysics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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 Tue, 1 Jun 2004 22:57:11 +0200, Creighton Hogg <wchogg@hep.wisc.edu>
wrote:
> On Sun, 30 May 2004, Lubos Motl wrote:
>
>> ...
>> Some intersections can predict new tachyons - particles whose squared
mass
>> is negative. Tachyons would signal instability of the whole
configuration,
>> of the whole Universe (instability with respect to the two D-branes
>> merging and/or annihilating in a certain way), and we usually want to
>> avoid them.
>
> I haven't really seen papers where intersecting D-brane models lead to
> tachyons, but then again I am not well read except for some approaches to
> standard model like setups.
> I had thought, though, that the additional tachyons only occurred when you
> have a D-brane and antiD-brane?
> Am I all washed up on that assumption?
>
> [Moderator's note: Yes, I also expressed the idea that the models with
> the tachyons are not terribly interesting. If you want tachyons, it is
> not necessary to have exactly parallel D-brane and anti-D-brane pair.
> If you tilt them a bit, it is not hard to believe that the tachyon will
> still be there. LM]

The generic (curved) brane set-up is not supersymmetric and is very
likely to contain tachyons. In fact, only if all branes are calibrated
by the same form susy is preserved. This condition is easier to
understand for two flat intersecting branes. Let's apply T-dualities
so that we have two Dp branes (if that's not possible, one brane has
to be of the wrong dimensionality for IIA/B anyway). Then the
intersection can be described by p (principal) angles. To see this you
need some higher dimensional imagination: Lines intersect with one
angle between them, two surfaces intersect with two angles etc. Now
you sum those p angles and if that's an integer multiple of \pi, the
intersection is supersymmetric. Otherwise it will probably contain
tachyons.

You can convince yourself that this rule contains the more well known
check for susy for branes intersecting at right angles (and being
parallel in some other directions): Then the number of dimensions that
is tangential to exactly one brane has to be a multiple of four.

The origin of the tachyonic mode is the following: If you start with
an intersection that looks like

/
/
/
--------------/----------
/
/
/

you can make the surface of the brane smaller by deforming it to

/
|
\
------- ---------
\
|
/
/

and as you gain energy from this this mode is tachyonic. This process
is known as brane recombination (unfortunate name).

It could be that this rippling off continiues forever or that it stops
at some minimal surface. That depends on boundary conditions and the
background manifold.

This process seems to always reduce the area but if the above angle
criterion is fulfilled this deformation doesn't change the area
(basically because there are more dimension than the ones I drew and
in those the area becomes larger.

Note that for model building purposes the tachyonic mode is not
lethal. As I said the process could stop for the D-brane wrapping some
minimal surface. This is like the Higgs in the standard model: The
\phi^2 term has a negative coefficient so \phi=0 is an unstable maximum
of the potential (a tachyon) but there is also a true vacuum as at
some point the \phi^4 term with its positive sign dominates. The same
could be true in string theory but it's just very hard to obtain
reliable information about the full form of the potential. Ibanez and
co have discussed this a lot and I have also written a couple of
papers relating to this (search spires).

Robert

--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo
..oO
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