Thomas Larsson
Apr27-04, 01:45 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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0404220635180.32710-100000@feynman.harvard.edu>...\n\nI really don\'t feel comfortable with posting to sps, so I prefer to\nmove this discussion to neutral territory.\n\n> On Tue, 20 Apr 2004, Thomas Larsson wrote:\n> > My point was that these extensions are not part of string theory,\n> > because if they were, GSW should have known about them.\n>\n> I think the point is that anomalies due to chiral fermions can be seen\n> already in the low energy effective action, so that we can say something\n> about them without knowing the full effective action/closed string field\n> theory.\n\nThe problem is that the Virasoro-like anomalies are functionals of a\nprivileged 1D curve, which can naturally interpreted as the observer\'s\ntrajectory. There is no way to see such an anomaly in a formalism where\nthe observer, or at least her trajectory, is not explicitly present.\nThis is the case in field and string theory. Lubos Motl is of course\nperfectly right when he says that GSW know all about the diff anomalies\nthat can arise in string theory. The Virasoro-like anomalies can not.\n\nHowever, I noted that Damour-Henneaux-Nicolai implicitly introduce what\nI call the observer\'s position. More about that below.\n\n> > A finite truncation of a graded Lie algebra violates the Jacobi identities,\n> > no? E.g., if we have A, B in g_1 and C in g_-1, [[A,B],C] is in g_1 so it\n> > should not be truncated, but it is because [A,B] is in g_2.\n>\n> Sorry for expressing myself badly. What can be truncated is the expansion\n> of the sigma-model dynamics on the E10/K(E10) group. This truncation\n> is consistent in the sense of a perturbative truncation at some given\n> order is consistent in that higher order terms won\'t invalidate the\n> approximate result at lower order.\n>\n> Of course you are perfectly right that there is no subalgebra of\n> a graded Lie algebra beyond level 0. But the truncation that I\n> was referring to is based on the fact that all elements of E10\n> (or any other KM algebra) at a given level transform under the\n> action of the level-0 subalgebra. Therefore every level can be\n> "understood" by decomposing it into irreps of the level-0\n> subalgebra.\n\nIOW, E10 admits a grading with g_0 = gl(10), all g_k are\nfinite-dimensional g_0 modules, and you can compute g_-k recursively up\nto any fixed order, say 30. Yes, I am pretty sure that this is true,\nand it is an useful description of E10. You can probably do the same\nwith any KM algebra which admits a grading such that g_0 is\nfinite-dimensional. You can definitely to it for all finite-dimensional\nLie algebras, where the grading terminates at some finite order. In\nfact, I have listed the nonlinear realizations that correspond to\nremoving one node in the Dynkin diagram. One day I may find the energy\nto put the list on the arxiv.\n\n>\n> In\n>\n> Damour, Henneaux, Nicolai\n> E10 and the \'small tension expansion\' of M Theory\n> hep-th/0207267\n>\n> a grading of E10 is used which is based on the \'exceptional\'\n> root (instead of on the over-extended root, in their\n> nomenclature). This way E10 is decomposed into irreps\n> of its SL(10) subalgebra, level by level.\n\nThis is a cool paper. Something which I wondered about for some time is\nwhy West talks about E11 and you talked about E10. I now realize that\nWest looks for M-theory in 11 spacetime dimensions and DHN works in 10\nspatial dimensions; their sigma-model is formulated in terms of a\ntime-dependent group element V(t) in E10, there is a lapse function,\netc.\n\n> The important\n> point now is that these irreps are of course nothing\n> but tensors, and that these tensors can be identified with\n> spatial modes of tensor fields of the bosonic sector\n> of 11d sugra (namely the metric and the 3-form field).\n\nThe given spatial point x that occurs in (12) of DHN is what I call\nthe observer\'s position, since the fields are "observed" from this\npoint. However, I allow the privileged point to vary with t, so I get a\ntrajectory x(t) (q(t) in my notation).\n\n>\n> So while the configuration point traces out a trajectory\n> on the E10/K(E10) \'manifold\' we can, even lacking a complete\n> understanding of what this full group really is, understand\n> the sugra configuration that this trajectory describes\n> order by order in the above notion of level (which corresponds\n> to an expansion in terms of spatial gradients on the\n> sugra side).\n\nOK, I understand that the geodesic Lagrangian on E10 is well-defined,\nand that it contains the field content of SUGRA plus much more.\nHowever, there are many questions, e.g.\n\n1. What so special about E10? You could almost surely repeat the same\nanalysis for any KM algebra, at least if it has a grading by finite-\ndimensional subspaces. This is true for infinitely many KM algebras, e.g.\n\n*\n|\no-o-o-o-o-...-o-o-o-o- ... -o-o-o-o-o\n\nWhat\'s wrong with these models? If you truncate the grading after finitely\nmany terms, the geodesic Lagrangian depends on finitely many dofs, i.e.\nyou have a quantum mechanics problem. There shouldn\'t be any difficulties\nhere. Whether nasty infinitites arise in the limit that you lift the\ntrunction is probably an open question.\n\n2. Diffeomorphisms! E10 is a fancy generalization of the gl(10) subalgebra\nof the diffeo algebra vect(10), but the remaining diffeos do not seem to\nfit in. In fact, not even translations seem to fit into E10. West seems\nto add translations outside E8; that\'s the P_a in (4.1) of hep-th/0104081.\nYou really need to include all diffeos, essentially for the same reason\nthat you need to go from SUSY to SUGRA. So you need some Lie algebra\nthat contains both E10 and vect(10) as subalgebras, such that their gl(10)\nsubalgebras are identified. Such algebras exist, but if E10 is BIG, they\nare GARGANTUAN.\n\n3. Let us assume that we manage to make the E10 symmetry local, i.e. we\ninclude diffeos in some way. Then vect(10) acts on the graded subspaces\nin some way. In particular, the spatial gradients in (13) of DHN, which\ntransform as symmetric n-tensors under gl(10), must transform as Taylor\ncoefficients, or n-jets, under vect(10). This means that they are\nmultiplied by matrices which depend on the base point x, and that x\nitself transforms non-trivially; for explicit formulas in multi-index\nnotation, see (4.1) of math-ph/9810003. It is not clear to me that\ninfinite towers in (13) really transform as jets.\n\n4. The Lagrangian (4) of DHN depends on the standard invariant bilinear\nform <.|.> on the KM algebra. However, there is no such form if you\ninclude arbitrary diffeos; if there were, you could use it to identify\nthe adjoint and coadjoint reps which are inequivalent. So how can the\ngeodesic Lagrangian be defined in the presence of diffeos?\n\n5. The experimental connection! 11D SUGRA is already on very shaky\nground, since there is virtually no experimental evidence for neither\nSUSY nor extra dimensions. To postulate an infinite, and exponentially\ngrowing, tower of new particles in this situation seems to me a bit bold,\nto put it mildly.\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>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0404220635180.32710-100000@feynman.harvard.edu>...
I really don't feel comfortable with posting to sps, so I prefer to
move this discussion to neutral territory.
> On Tue, 20 Apr 2004, Thomas Larsson wrote:
> > My point was that these extensions are not part of string theory,
> > because if they were, GSW should have known about them.
>
> I think the point is that anomalies due to chiral fermions can be seen
> already in the low energy effective action, so that we can say something
> about them without knowing the full effective action/closed string field
> theory.
The problem is that the Virasoro-like anomalies are functionals of a
privileged 1D curve, which can naturally interpreted as the observer's
trajectory. There is no way to see such an anomaly in a formalism where
the observer, or at least her trajectory, is not explicitly present.
This is the case in field and string theory. Lubos Motl is of course
perfectly right when he says that GSW know all about the diff anomalies
that can arise in string theory. The Virasoro-like anomalies can not.
However, I noted that Damour-Henneaux-Nicolai implicitly introduce what
I call the observer's position. More about that below.
> > A finite truncation of a graded Lie algebra violates the Jacobi identities,
> > no? E.g., if we have A, B in g_1 and C in g_-1, [[A,B],C] is in g_1 so it
> > should not be truncated, but it is because [A,B] is in g_2.
>
> Sorry for expressing myself badly. What can be truncated is the expansion
> of the \sigma-model dynamics on the E10/K(E10) group. This truncation
> is consistent in the sense of a perturbative truncation at some given
> order is consistent in that higher order terms won't invalidate the
> approximate result at lower order.
>
> Of course you are perfectly right that there is no subalgebra of
> a graded Lie algebra beyond level . But the truncation that I
> was referring to is based on the fact that all elements of E10
> (or any other KM algebra) at a given level transform under the
> action of the level-0 subalgebra. Therefore every level can be
> "understood" by decomposing it into irreps of the level-0
> subalgebra.
IOW, E10 admits a grading with g_0 = gl(10), all g_k are
finite-dimensional g_0 modules, and you can compute g_-k recursively up
to any fixed order, say 30. Yes, I am pretty sure that this is true,
and it is an useful description of E10. You can probably do the same
with any KM algebra which admits a grading such that g_0 is
finite-dimensional. You can definitely to it for all finite-dimensional
Lie algebras, where the grading terminates at some finite order. In
fact, I have listed the nonlinear realizations that correspond to
removing one node in the Dynkin diagram. One day I may find the energy
to put the list on the arxiv.
>
> In
>
> Damour, Henneaux, Nicolai
> E10 and the 'small tension expansion' of M Theory
> http://www.arxiv.org/abs/hep-th/0207267
>
> a grading of E10 is used which is based on the 'exceptional'
> root (instead of on the over-extended root, in their
> nomenclature). This way E10 is decomposed into irreps
> of its SL(10) subalgebra, level by level.
This is a cool paper. Something which I wondered about for some time is
why West talks about E11 and you talked about E10. I now realize that
West looks for M-theory in 11 spacetime dimensions and DHN works in 10
spatial dimensions; their \sigma-model is formulated in terms of a
time-dependent group element V(t) in E10, there is a lapse function,
etc.
> The important
> point now is that these irreps are of course nothing
> but tensors, and that these tensors can be identified with
> spatial modes of tensor fields of the bosonic sector
> of 11d sugra (namely the metric and the 3-form field).
The given spatial point x that occurs in (12) of DHN is what I call
the observer's position, since the fields are "observed" from this
point. However, I allow the privileged point to vary with t, so I get a
trajectory x(t) (q(t) in my notation).
>
> So while the configuration point traces out a trajectory
> on the E10/K(E10) 'manifold' we can, even lacking a complete
> understanding of what this full group really is, understand
> the sugra configuration that this trajectory describes
> order by order in the above notion of level (which corresponds
> to an expansion in terms of spatial gradients on the
> sugra side).
OK, I understand that the geodesic Lagrangian on E10 is well-defined,
and that it contains the field content of SUGRA plus much more.
However, there are many questions, e.g.
1. What so special about E10? You could almost surely repeat the same
analysis for any KM algebra, at least if it has a grading by finite-
dimensional subspaces. This is true for infinitely many KM algebras, e.g.
*
|
o-o-o-o-o-...-o-o-o-o- ... -o-o-o-o-o
What's wrong with these models? If you truncate the grading after finitely
many terms, the geodesic Lagrangian depends on finitely many dofs, i.e.
you have a quantum mechanics problem. There shouldn't be any difficulties
here. Whether nasty infinitites arise in the limit that you lift the
trunction is probably an open question.
2. Diffeomorphisms! E10 is a fancy generalization of the gl(10) subalgebra
of the diffeo algebra vect(10), but the remaining diffeos do not seem to
fit in. In fact, not even translations seem to fit into E10. West seems
to add translations outside E8; that's the P_a in (4.1) of http://www.arxiv.org/abs/hep-th/0104081.
You really need to include all diffeos, essentially for the same reason
that you need to go from SUSY to SUGRA. So you need some Lie algebra
that contains both E10 and vect(10) as subalgebras, such that their gl(10)
subalgebras are identified. Such algebras exist, but if E10 is BIG, they
are GARGANTUAN.
3. Let us assume that we manage to make the E10 symmetry local, i.e. we
include diffeos in some way. Then vect(10) acts on the graded subspaces
in some way. In particular, the spatial gradients in (13) of DHN, which
transform as symmetric n-tensors under gl(10), must transform as Taylor
coefficients, or n-jets, under vect(10). This means that they are
multiplied by matrices which depend on the base point x, and that x
itself transforms non-trivially; for explicit formulas in multi-index
notation, see (4.1) of http://www.arxiv.org/abs/math-ph/9810003. It is not clear to me that
infinite towers in (13) really transform as jets.
4. The Lagrangian (4) of DHN depends on the standard invariant bilinear
form <.|.> on the KM algebra. However, there is no such form if you
include arbitrary diffeos; if there were, you could use it to identify
the adjoint and coadjoint reps which are inequivalent. So how can the
geodesic Lagrangian be defined in the presence of diffeos?
5. The experimental connection! 11D SUGRA is already on very shaky
ground, since there is virtually no experimental evidence for neither
SUSY nor extra dimensions. To postulate an infinite, and exponentially
growing, tower of new particles in this situation seems to me a bit bold,
to put it mildly.
I really don't feel comfortable with posting to sps, so I prefer to
move this discussion to neutral territory.
> On Tue, 20 Apr 2004, Thomas Larsson wrote:
> > My point was that these extensions are not part of string theory,
> > because if they were, GSW should have known about them.
>
> I think the point is that anomalies due to chiral fermions can be seen
> already in the low energy effective action, so that we can say something
> about them without knowing the full effective action/closed string field
> theory.
The problem is that the Virasoro-like anomalies are functionals of a
privileged 1D curve, which can naturally interpreted as the observer's
trajectory. There is no way to see such an anomaly in a formalism where
the observer, or at least her trajectory, is not explicitly present.
This is the case in field and string theory. Lubos Motl is of course
perfectly right when he says that GSW know all about the diff anomalies
that can arise in string theory. The Virasoro-like anomalies can not.
However, I noted that Damour-Henneaux-Nicolai implicitly introduce what
I call the observer's position. More about that below.
> > A finite truncation of a graded Lie algebra violates the Jacobi identities,
> > no? E.g., if we have A, B in g_1 and C in g_-1, [[A,B],C] is in g_1 so it
> > should not be truncated, but it is because [A,B] is in g_2.
>
> Sorry for expressing myself badly. What can be truncated is the expansion
> of the \sigma-model dynamics on the E10/K(E10) group. This truncation
> is consistent in the sense of a perturbative truncation at some given
> order is consistent in that higher order terms won't invalidate the
> approximate result at lower order.
>
> Of course you are perfectly right that there is no subalgebra of
> a graded Lie algebra beyond level . But the truncation that I
> was referring to is based on the fact that all elements of E10
> (or any other KM algebra) at a given level transform under the
> action of the level-0 subalgebra. Therefore every level can be
> "understood" by decomposing it into irreps of the level-0
> subalgebra.
IOW, E10 admits a grading with g_0 = gl(10), all g_k are
finite-dimensional g_0 modules, and you can compute g_-k recursively up
to any fixed order, say 30. Yes, I am pretty sure that this is true,
and it is an useful description of E10. You can probably do the same
with any KM algebra which admits a grading such that g_0 is
finite-dimensional. You can definitely to it for all finite-dimensional
Lie algebras, where the grading terminates at some finite order. In
fact, I have listed the nonlinear realizations that correspond to
removing one node in the Dynkin diagram. One day I may find the energy
to put the list on the arxiv.
>
> In
>
> Damour, Henneaux, Nicolai
> E10 and the 'small tension expansion' of M Theory
> http://www.arxiv.org/abs/hep-th/0207267
>
> a grading of E10 is used which is based on the 'exceptional'
> root (instead of on the over-extended root, in their
> nomenclature). This way E10 is decomposed into irreps
> of its SL(10) subalgebra, level by level.
This is a cool paper. Something which I wondered about for some time is
why West talks about E11 and you talked about E10. I now realize that
West looks for M-theory in 11 spacetime dimensions and DHN works in 10
spatial dimensions; their \sigma-model is formulated in terms of a
time-dependent group element V(t) in E10, there is a lapse function,
etc.
> The important
> point now is that these irreps are of course nothing
> but tensors, and that these tensors can be identified with
> spatial modes of tensor fields of the bosonic sector
> of 11d sugra (namely the metric and the 3-form field).
The given spatial point x that occurs in (12) of DHN is what I call
the observer's position, since the fields are "observed" from this
point. However, I allow the privileged point to vary with t, so I get a
trajectory x(t) (q(t) in my notation).
>
> So while the configuration point traces out a trajectory
> on the E10/K(E10) 'manifold' we can, even lacking a complete
> understanding of what this full group really is, understand
> the sugra configuration that this trajectory describes
> order by order in the above notion of level (which corresponds
> to an expansion in terms of spatial gradients on the
> sugra side).
OK, I understand that the geodesic Lagrangian on E10 is well-defined,
and that it contains the field content of SUGRA plus much more.
However, there are many questions, e.g.
1. What so special about E10? You could almost surely repeat the same
analysis for any KM algebra, at least if it has a grading by finite-
dimensional subspaces. This is true for infinitely many KM algebras, e.g.
*
|
o-o-o-o-o-...-o-o-o-o- ... -o-o-o-o-o
What's wrong with these models? If you truncate the grading after finitely
many terms, the geodesic Lagrangian depends on finitely many dofs, i.e.
you have a quantum mechanics problem. There shouldn't be any difficulties
here. Whether nasty infinitites arise in the limit that you lift the
trunction is probably an open question.
2. Diffeomorphisms! E10 is a fancy generalization of the gl(10) subalgebra
of the diffeo algebra vect(10), but the remaining diffeos do not seem to
fit in. In fact, not even translations seem to fit into E10. West seems
to add translations outside E8; that's the P_a in (4.1) of http://www.arxiv.org/abs/hep-th/0104081.
You really need to include all diffeos, essentially for the same reason
that you need to go from SUSY to SUGRA. So you need some Lie algebra
that contains both E10 and vect(10) as subalgebras, such that their gl(10)
subalgebras are identified. Such algebras exist, but if E10 is BIG, they
are GARGANTUAN.
3. Let us assume that we manage to make the E10 symmetry local, i.e. we
include diffeos in some way. Then vect(10) acts on the graded subspaces
in some way. In particular, the spatial gradients in (13) of DHN, which
transform as symmetric n-tensors under gl(10), must transform as Taylor
coefficients, or n-jets, under vect(10). This means that they are
multiplied by matrices which depend on the base point x, and that x
itself transforms non-trivially; for explicit formulas in multi-index
notation, see (4.1) of http://www.arxiv.org/abs/math-ph/9810003. It is not clear to me that
infinite towers in (13) really transform as jets.
4. The Lagrangian (4) of DHN depends on the standard invariant bilinear
form <.|.> on the KM algebra. However, there is no such form if you
include arbitrary diffeos; if there were, you could use it to identify
the adjoint and coadjoint reps which are inequivalent. So how can the
geodesic Lagrangian be defined in the presence of diffeos?
5. The experimental connection! 11D SUGRA is already on very shaky
ground, since there is virtually no experimental evidence for neither
SUSY nor extra dimensions. To postulate an infinite, and exponentially
growing, tower of new particles in this situation seems to me a bit bold,
to put it mildly.