Bubbling AdS space and 1/2 BPS geometries
Lubos Motl
least re-post a trivial introduction to a paper I like.
http://motls.blogspot.com/2005/01/bubbling-ads-space.html
I have not written anything about this paper by Lin, Lunin, and
Maldacena:
* hep-th/0409174
Because I believe that this is one of the best papers in the last 6
months, let me say a couple of words.
Take type IIB string theory on AdS5 x S5 or its pp-wave limit. Both of
them have the maximal number of 32 supercharges. Is there some interesting
generalization of these two geometries?
The answer is: yes, there is. Both of these geometries have at least the
SO(4) x SO(4) x R isometry. The pp-wave is a limit of the the anti de
Sitter space. Moreover, the pp-wave limit has a Z_2 symmetry exchanging
the two SO(4) factors - this symmetry is broken by the anti de Sitter
space. Is there some geometric heuristic picture how to visualize these
two geometries?
Yes, there is. You can imagine
* the AdS5 x S5 space as a black disk drawn on a white paper
* the pp-wave limit of it is a paper whose lower half-plane is black
Note that the lower half-plane is a limit of a very large disk. Also, the
two half-planes filled with different color have a Z_2 symmetry,
exchanging these two colors. This Z_2 symmetry does not exist for the disk
whose interior and exterior look different. So far, it sounds ridiculous,
of course. But the reason why I say it in this way is the following:
For any black-and-white picture that you can draw on the plane, there
exists a solution of type IIB string theory - more or less, it's a
geometry - with 16 supercharges. How do you construct it? Well,
parameterize the ten-dimensional space by the following coordinates:
* time "t"
* three coordinates labeling the three-sphere S^3 number one
* three coordinates labeling the three-sphere S^3 number two
* a coordinate "y" which is kind of "radial"
* two coordinates "x_1, x_2" spanning a plane that you imagine to be
analogous to the "x-p" phase space
You may count that the total number of dimensions is 10. One can see that
the AdS5 x S5 geometries satisfy this general Ansatz. How do we get the
black-and-white pictures to the game? Draw any black-and-white pictures on
the "x_1, x_2" plane. This picture expresses the behavior of the geometry
near "y=0". In the regions of the "x_1, x_2" plane that are black, the S^3
number one is filled and becomes locally a flat space R^4. In the white
regions of the "x_1, x_2" plane, the S^3 number two is filled and becomes
a R^4. One can see that there are only two ways how to regularize the
geometry near "y=0": black and white. Moreover, LLM have showed that one
not only obtains a nice smooth geometry in the bulk of the black region -
or, analogously, in the bulk of the white region. One also has a smooth
geometry at the boundary between them.
The black regions in the "x_1, x_2" plane represent the Fermi liquid known
from the matrix description of two-dimensional string theory. You may
imagine that the two-dimensional string theory is embedded into the
ten-dimensional type IIB string theory as a subsector. Analogous
constructions, although possibly slightly less exciting ones, exist for
other geometries - like the Anti de Sitter space solutions of M-theory.
So how many SUSY solutions of type IIB string theory did they obtain by
this construction? A huge number. First of all, for every different
topology of the black-and-white picture (a different number of "droplets"
etc.), one obtains a different topology of the spacetime. If all droplets
are large and their boundaries kind of straight, the curvature of the
spacetime will also be small. The spacetime curvature becomes large if the
droplets approach one another - a droplet eaten by a bigger droplet on the
black-and-white picture describes topology changing transitions.
Even if you fix the topology, the shape of the droplet can be anything you
want - and you obtain different geometries. In this sense, their Ansatz
has infinitely many parameters. If you describe a boundary of a droplet as
a function "x_2(x_1)" of one variable, for every function of one variable
you will obtain one solution. A huge number. Of course, all these
solutions have different asymptotics.
This continuously infinite number of parameters of the class of the
solutions is analogous to Mathur et al. who construct their revolutionary
solutions that are meant to describe the black holes, although they have
neither horizon nor singularity. In that case, the solutions are also
parameterized by a function of one variable - describing a shape of a
string - that is dualized by various dualities to obtain a solution that
looks like a black hole outside, but whose interior is very different.
Is any black-and-white picture allowed? One can see that the areas of all
droplets must be actually integers (in some proper units of areas on the
"x_1, x_2" plane). This requirement arises from quantization of the
fluxes. In the "AdS_5 x S_5" solution, for example, the black-and-white
picture is a black disk. Its area is proportional to "N", the five-form
flux through the five-sphere. The classical geometry is only appropriate
if the droplets are large and their curvature is small.
Therefore it sounds reasonable to imagine that the "x_1, x_2" plane is
noncommutative, like a phase space, and the quantum of the area is a
single cell of this phase space. The function "z" that equals +1/2 in the
black regions and -1/2 in the white regions could really be a function on
a non-commutative space that satisfies "z*z=1/4" where "*" is the
non-commutative star-product. Anyone has a way to see that such a
description is possible? There could be some "dual" object - like a
D3-brane that can wrap either of these spheres S^3. The coordinates "x_1,
x_2" would be fields living on the worldvolume of this dual object, and
one should be able to show that they don't commute and the commutator is
the right c-number. Such an object could be in various states, and a
lowest energy state would correspond to the field "z(x_1,x_2)" that
describes the black-and-white picture. Note that in the normal picture of
string theory, "z" parameterizes the geometry (and the RR field strengths)
and therefore we treat it as a closed string field. Near y=0, however,
there could be a dual way to describe physics in which geometry comes from
quantization of this "new kind of object" that sees a non-commutative
"x_1, x_2" plane.
Any comments related to this paper are welcome.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Aaron Bergman
> Is any black-and-white picture allowed? One can see that the areas of all
> droplets must be actually integers (in some proper units of areas on the
> "x_1, x_2" plane). This requirement arises from quantization of the
> fluxes. In the "AdS_5 x S_5" solution, for example, the black-and-white
> picture is a black disk. Its area is proportional to "N", the five-form
> flux through the five-sphere. The classical geometry is only appropriate
> if the droplets are large and their curvature is small.
Rather than ordinary ncg, this somewhat reminds me of some stuff I did
with Ori (0008030) where we (well mostly him) speculate that there could
be fixed area surfaces in a deformation of the (2,0) theory akin to the
dipole deformation of ordinary YM. But that's in 6D, of course.
Aaron
jcgon...@yahoo.com
I really like having two of the compact dimensions treated differently
than the other four whether it be as a 2-dim string theory worldsheet,
a Euclidean to Lorentz lattice thing, or a real to complex spacetime
thing. Can more than one way of thinking of these two dimensions be
correct?
Urs Schreiber
> than the other four
Note though that AdS5 x S5 is not a standard phenomenologically motivated
compactification and hence in particular not directly related to your wish
for (if I understand correctly) something like
realistic-spacetime4 x internal4 x treated-differently2 .
In general it is probably good to keep in mind that strings in AdS5 are not
supposed to be a phenomenologically realistic backgrounds in a direct naive
fashion but are interesting due to their AdS/CFT duality to 4D CFTs on the
boundary. In that respect of course they might again be phenomenologically
relevant - on the dual side - as emphasized again by the recent work by
Nastase.
jcgon...@yahoo.com
Urs Schreiber wrote:
> > I really like having two of the compact dimensions treated
differently
> > than the other four
>
> Note though that AdS5 x S5 is not a standard phenomenologically
motivated
> compactification and hence in particular not directly related to your
wish
Yes you have my wish described correctly and yes I guess having 5
compact dimensions instead of six should have been a hint to me that my
wish won't get much help from this background. I think I might have got
a little too hopeful by thinking of how Spin(5) de Sitter gravitons are
related to 4D spacetime.
Urs Schreiber
news:1106895745.2994...@z14g2000cwz.googlegroups.com...
> Yes you have my wish described correctly and yes I guess having 5
> compact dimensions instead of six should have been a hint to me that my
> wish won't get much help from this background. I think I might have got
> a little too hopeful by thinking of how Spin(5) de Sitter gravitons are
> related to 4D spacetime.
I am not really familiar with the frame of ideas that you are excited about
(i.e. Tony Smith's ideas), but it is maybe worth pointing out that the
splitting
10D = 8D + 2D treated differently
or
11D = 8D + 3D treated differently
is of course at the very heart of superstring theory, becoming manifest in
light cone gauge!
So the reason why superstrings live in 10D is simply because they have a 2D
worlsheet and their transversal excitations want to live in the nicest
number of dimensions best suited for susy. And this turns out to be 8
dimensions, because here triality of SO(8) is at work and ensures that
worldsheet susy is equivalent to target space susy and all kinds of magic
(like the cancelling of the conformal anomaly, which is of course the real
reason for the 10 target space dimenions).
The same applies if you replace superstrings by supermembranes. They want 8
transversal directions and hence live in 11 dimensional (M-)theory.
jcgon...@yahoo.com
Yes before worrying about compacting anything to get down to 4
dimensions things do seem pretty similar between Smith and
superstrings. Smith has bosonic strings so the M-theory is 24D + 3D
treated differently but Lee Smolin at least used to look at starting
here to include the bosonic side of heterotic strings. After some E6
orbifolding (inspired by you actually) there's a 10D spacetime of 4
physical + 4 internal + 2 conformal. The two conformal are the two
U(1)s one gets from the 26-dim E6/F4. So the two worldsheet dimensions
seem out of the picture and the two U(1)s for complex structure of
spacetime and fermions enter the picture. Superstrings have the E6
subgroup of E8xE8 to get standard model fermions and a compacting of 6
spacetime dimensions to get the 4 physical spacetime dimensions. Since
there's a heading towards the standard model and physical spacetime
going on I guess it makes sense that the string worldsheet goes away
but they don't just seem to go away, they seem to turn into
something. Two U(1)-like things for Smith and two more compact
dimensions for superstrings. Smith actually has spacetime created by
strings and I heard there are superstring ideas like this too but
I've never seen any details. It makes sense to me though that if you
start doing subalgebra kinds of things to your M-theory, you can get
not only standard model fermions but a standard model spacetime.
Math-wise Smith's spacetime is like the leptoquarks of SU(5) GUT.
Perhaps the superstring ideas for creating a spacetime from strings
would do something different for the two "something else"
dimensions down at the standard model level besides just having two
more compact dimensions? To add another slightly confusing way of
seeing things, Smith has the vector of the SO(8) triality as a
spacetime also (the two half spinors are for the fermions). If you
then go to SO(10) to get the two "something else" dimensions, you
kind of have (since SO(10) is D5) the U(1) that created complex
spacetime in the E6/F4 picture.
Urs Schreiber
> Urs Schreiber wrote
[...]
> Yes before worrying about compacting anything to get down to 4
> dimensions things do seem pretty similar between Smith and
> superstrings. Smith has bosonic strings so the M-theory is 24D + 3D
> treated differently but Lee Smolin at least used to look at starting
> here to include the bosonic side of heterotic strings. After some E6
> orbifolding (inspired by you actually)
Inspired by whom? By me? :-)
jcgon...@yahoo.com
Yes it really really was you! The following two quotes to be exact:
Urs Schreiber said "... the low energy effective worldsheet theory of a
single flat D3 brane of the bosonic string is, to lowest nontrivial
order, just U(1) gauge theory in 4D ...".
Urs Schreiber said "... I guess that's why you have to put the brane at
the singularity of an orbifold if you want to get rid of the scalars
... if the number of dimensions is not an issue the simplest thing
probably would be to consider the single space-filling D25 brane of the
bosonic string. This one does not have any transverse fluctuations and
there is indeed only the U(1) gauge field ...".
One of the reasons I've been getting a little more interested in
superstrings is that lots of superstring research ideas have been
slipping into Smith's model the last few years. It's still the same
Smith model, it just has more detailed ways of looking at it.
I found the paper I was told about where spacetime gets created by
string theory. It actually looks a lot like the paper we are
discussing here. Perhaps I will understand this a little bit better in
the not too distant future. The paper is at:
http://arxiv.org/abs/gr-qc/0410049
Urs Schreiber
news:1107219401.7110...@f14g2000cwb.googlegroups.com...
>
> Urs Schreiber wrote:
>> On Sun, 30 Jan 2005, jcgon...@yahoo.com wrote:
>>> After some E6
>> > orbifolding (inspired by you actually)
>>
>> Inspired by whom? By me? :-)
>
> Yes it really really was you! The following two quotes to be exact:
>
> Urs Schreiber said "... the low energy effective worldsheet theory of a
> single flat D3 brane of the bosonic string is, to lowest nontrivial
> order, just U(1) gauge theory in 4D ...".
>
> Urs Schreiber said "... I guess that's why you have to put the brane at
> the singularity of an orbifold if you want to get rid of the scalars
> ... if the number of dimensions is not an issue the simplest thing
> probably would be to consider the single space-filling D25 brane of the
> bosonic string. This one does not have any transverse fluctuations and
> there is indeed only the U(1) gauge field ...".
This is taken from a discussion about elements of how gauge theories arise
within string theory. I was just recalling some basic facts about effective
actions of D-branes. So you haven't really been inspired by me, but by
standard string theory results!
> One of the reasons I've been getting a little more interested in
> superstrings is that lots of superstring research ideas have been
> slipping into Smith's model the last few years. It's still the same
> Smith model, it just has more detailed ways of looking at it.
I am not sure, but it might be helpful to go the other way round: Instead of
trying to integrate string ideas in Smith's universe of ideas you or he
might be interested in seeing how his love for algebraic structures can be
satisfied in string theory. The advantage would be that M-theory, though
incompletely understood, is still much better understood than the 24+3
dimensional bosonic hypotheses that you talked about.
jcgon...@yahoo.com
> This is taken from a discussion about elements of how gauge theories arise
> within string theory. I was just recalling some basic facts about effective
> actions of D-branes. So you haven't really been inspired by me, but by
> standard string theory results!
> I am not sure, but it might be helpful to go the other way round: Instead of
> trying to integrate string ideas in Smith's universe of ideas you or he
> might be interested in seeing how his love for algebraic structures can be
> satisfied in string theory. The advantage would be that M-theory, though
> incompletely understood, is still much better understood than the 24+3
> dimensional bosonic hypotheses that you talked about.
I certainly appreciate that standard string theory includes areas that
are not too far away from the standard model. For me, besides being
hopelessly biased towards Smith's model :) I kind of have to start with
Smith's model simply beacuse it is what I know best. I have to start
with Smith's model even for the Standard Model or Einstein's gravity.
As for Smith, he calls his model a "work in progress", but I don't
think he has any reason to go away from the E7/E6xU(1) 27 complex
dimensions that he was using even before he put them into the context
of M-theory.
kneemo
> Smith has bosonic strings so the M-theory is 24D + 3D\ntreated
> differently but Lee Smolin at least used to look at starting\nhere to
> include the bosonic side of heterotic strings.
I recall Smith using the complexified exceptional Jordan algebra for
his model. This is a (complex) 27-dimensional algebra, over the
bioctonions. Now, is it Smith's intention to use the (complex) 11
dimensions of the complexified Jordan algebra, or use the 11 (real)
dimensions of the self-adjoint part (the exceptional Jordan algebra)
for space-time. This would seem to make a difference, as we would be
working with 3 complex dimensions, along with a bioctonionic space, in
the 11-dimensional complex case.
Would space-time as 11=8+3 complex dimensions be consistent? Or must
space-time dimensions be strictly real?
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Urs Schreiber
> As for Smith, he calls his model a "work in progress", but I don't
> think he has any reason to go away from the E7/E6xU(1) 27 complex
> dimensions that he was using even before he put them into the context
> of M-theory.
Probably once you try to understand quantized gravity in this context
further constraints will appear.
jcgon...@yahoo.com
I know how Smith dealt with the problems of bosonic strings but I don't
know what extra problems M-theory brings. Smith uses M-theory (and
F-theory) to describe interactions between brane universes.
jcgon...@yahoo.com
> I recall Smith using the complexified exceptional Jordan algebra for
> his model. This is a (complex) 27-dimensional algebra, over the
> bioctonions. Now, is it Smith's intention to use the (complex) 11
> dimensions of the complexified Jordan algebra, or use the 11 (real)
> dimensions of the self-adjoint part (the exceptional Jordan algebra)
> for space-time. This would seem to make a difference, as we would be
> working with 3 complex dimensions, along with a bioctonionic space,
in
> the 11-dimensional complex case.
>
> Would space-time as 11=8+3 complex dimensions be consistent? Or must
> space-time dimensions be strictly real?
Real vs. complex depends on whether you are talking about spacetime
within string theory or within M-theory (there's even an F-theory with
28 quaternionic dimensions). The +3 would imply M-theory, +2 would
imply string theory, +4 would imply F-theory. Spacetime itself is kind
of just the 8. Details from Smith's website about the algebra and what
these stringy spaces are used for are at:
http://www.valdostamuseum.org/hamsmith/StringMFbranegrav.html#Stheory