NCG/SUSY/*-Product
eric alan forgy
I recently stumbled onto something that could potentially open up
mountains of research/reading and before starting the journey I was hoping
someone might say a few words summarizing what this *-product is that pops
up in noncommutative geometry. It looks suspiciously like something I have
cooked up on my own. If that is the case, then I have reinvented another
wheel :)
Does the *-product have anything at all to do with composition of strings
or paths. A quick lanl search in addition to a quick spr search turned up
tons of hits regarding NCG and string theory. I saw Aaron mention
something about string algebra and Prof Baez mention it in regard to
products on phase space.
What exactly is a string algebra? Is there some operation in string theory
that composes strings/paths? A path algebra is a very obvious
construction and I reinvented it on my own and I even called it "path
algebra". Is it at all related? Could path algebra be a kind of discrete
string algebra? A path algebra is simply a directed graph with product of
paths in the graph defined by concatenation when such a concatenation
makes sense. When it doesn't, i.e. the end of the first path doesn't
correspond to the beginning of the next path, then the product is zero.
It would be really fascinating to me if the work I have been doing on path
algebras is remotely related to string theory. I already know that my
stuff is related to NCG, and I have heard rumors that NCG and string
theory were related, so it would be quite cool if string theory came into
the picture. I'm fairly certain I would be the first electrical engineer
to write his thesis on string theory :)
Anyway, I'd really appreciate some words of wisdom from people who know
about this stuff.
Thank you very much,
Eric
Aaron Bergman
eric alan forgy <fo...@students.uiuc.edu> wrote:
> Hello,
>
> I recently stumbled onto something that could potentially open up
> mountains of research/reading and before starting the journey I was hoping
> someone might say a few words summarizing what this *-product is that pops
> up in noncommutative geometry.
There are a few different star products. The one you're probably
referring to is the so-called Moyal star product. Basically, given an
anti-symmetric matrix theta_{uv}, we have
(f * g)(z) = e^(i theta_{uv} d^x_u d^y_v) f(x) g(y) | x=y=z
In bigger words, this can be thought of as the deformation quantization
of R^{2n} wrt the symplectic form theta.
[...]
>
> Does the *-product have anything at all to do with composition of strings
> or paths.
There is also a star product for multiplying string fields (Witten '86).
In some sense, the Moyal star is a toy model of this (hep-th/0006071).
It has been more recently shown that, for some sectors of the string
field, the star product can be thought of as an uncountable direct sum
of Moyal *'s (hep-th/0202087).
[...]
> What exactly is a string algebra? Is there some operation in string theory
> that composes strings/paths?
One would like a way to multiply string fields. The obvious answer to
this is to just concatenate strings. Unfortunately, this is only
associative up to a reparametrization of the string field (Here is where
John Baez chimes in with what this algebraic structure is called. A_oo
algebra, maybe?). Rather than try to deal with this, Witten instead
decided to divide every string into halves and then, to multiply two
strings, you identify the right half of one with the left half of the
other. For more details, see the Witten paper referred to above.
Note that this only works nicely for open strings. Doing something
similar for closed strings in a nice way would be very nice. (There
exist bosonic closed string field theories, but they're not tremendously
aesthetically appealing and it's questionabl whether they actually
contain more information than just the perturbative expansion. One of
the really interesting discoveries in the past few years is that this
open string field theory defined by Witten really does contain more than
just the perturbation expansion, and it can be used to actually
calculate things like the potential for the open string tachyon in
bosonic string theory. See the archives for longer posts on this
subject.)
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
zirkus
>I was hoping
> someone might say a few words summarizing what this *-product is that pops
> up in noncommutative geometry.
You seem to be talking about the Moyal star product. Are you sure that
this is not discussed in Connes' textbook (which I have not read) ?
For an intro to star products in deformation quantization see page 2
of Kontsevich's (q-alg/9709040).
> Does the *-product have anything at all to do with composition of strings
> or paths.
Yes, for example, the Moyal star product is equivalent to Witten's
star product in string field theory. To see various ways in which the
star product makes up strings check out (hep-th/0202087, 0101219,
0012145 and 0107216).
Btw, perhaps it would be easier for you to learn about these topics if
you talk to some mathematician/string theorist at UIUC, instead of
trying to learn solely via what you find on the internet. An expert
would at least probably know what the best introductory sources are.
eric alan forgy
Thanks for your reply.
On Wed, 20 Feb 2002, Aaron Bergman wrote:
> eric alan forgy <fo...@students.uiuc.edu> wrote:
[snip of some good stuff about Moyal and Witten products]
> > What exactly is a string algebra? Is there some operation in string theory
> > that composes strings/paths?
>
> One would like a way to multiply string fields. The obvious answer to
> this is to just concatenate strings. Unfortunately, this is only
> associative up to a reparametrization of the string field (Here is where
> John Baez chimes in with what this algebraic structure is called. A_oo
> algebra, maybe?). Rather than try to deal with this, Witten instead
> decided to divide every string into halves and then, to multiply two
> strings, you identify the right half of one with the left half of the
> other. For more details, see the Witten paper referred to above.
After sending of the original post, I did a bit of reading and the
relation between the Moyal and Witten products is pretty fascinating and I
am beginning to think that some things I've been doing may be related.
However, everything I'm doing is on purely algebraic/combinatoric
background spaces. For instance, I have basically an abstract simplicial
complex where an edge (string?) is represented by a simple finite set,
e.g. [ij] = {{i},{j},{i,j}}. There is no parametrization.
Now, in LQG you start out with a manifold and then define a spin network
on it, do a bunch of fancy mathematics and when all is said and done, you
discover that you never really needed the manifold at all to begin with.
You could have started with this purely combinatorial spin network.
Is it possible something similar to this could happen in string theory?
That is the impression I get when I read about Witten product of strings.
Like you said, you first split this into two halves and then concatenate
the middle pieces like
(aL,aR) * (bL,bR) = (aL,bR)
Now if you distill out what the pure algebra of this is, you can almost
look at (aL,aR) as an abstract 1-simplex {{aL},{aR},{aR,aL}}. Notice how
the abstract 1-simplex is already in a sense composed of two "halves".
I'm wondering if there is some relation between string theory and
abstract graph theory where instead of continuously parameterized strings,
you have abstract sets like these abstract 1-simplices. Then the string
algebra seems closely related to the path algebra defined on a quiver.
I spent a couple of days in a swirl reading up on representation theory of
quivers and there is just tons of material on the subject. In fact, quite
a bit of the recent material has been in relation to string theory. I
won't pretend to understand the implications of it all, but at least I
know that the connections are there.
I might just be spewing nonsense, but from what I have seen so far is that
working with these abstract spaces automatically gets you this "background
independence" that is held so dear in this place :) Putting coordinates on
these abstract spaces is highly artificial but can be done. When you do
so, it is pretty straight forward to see that what 'labels' you choose to
lay on the space really is of no consequence to any physics.
Even general relatvity is not completely background independent because
you do have to specify a manifold before you can do anything. Then the
metric and everything is else free to wiggle as much as it likes but you
are still stuck with the manifold you originally chose. This is similar to
these abstract spaces. Ultimately, the physics will depend on the
underlying abstract space you began with, but all other quantities defined
ON the abstract space are free to vary as they wish. Even if I am not 100%
correct about my claims regarding GR, since I am no expert (not even a
novice) on the subject, whatever loopholes there are in GR for getting
away from the original manifold you started with, there is probably a
corresponding loophole for the abstract space approach. If topology can
change in GR, there is nothing saying topology of a graph cannot change
either.
> Note that this only works nicely for open strings. Doing something
> similar for closed strings in a nice way would be very nice.
I am totally out of my league here, but it seems as though any problems
with doing this may be arising because you are assuming a continuum
underlying space and you feel the need to "parameterize" strings. I can
imagine a million ways to define associative products of "abstract closed
strings".
Could graph theory come to the rescue here?
[snip]
Thinking out loud,
Eric
Aaron Bergman
zir...@hotmail.com (zirkus) wrote:
> eric alan forgy <fo...@students.uiuc.edu> wrote in message news:
>
> >I was hoping
> > someone might say a few words summarizing what this *-product is that pops
> > up in noncommutative geometry.
>
> You seem to be talking about the Moyal star product. Are you sure that
> this is not discussed in Connes' textbook (which I have not read) ?
Connes's book is really geared towards a different form of
noncommutative geometry.
> For an intro to star products in deformation quantization see page 2
> of Kontsevich's (q-alg/9709040).
Don't.
> > Does the *-product have anything at all to do with composition of strings
> > or paths.
> Yes, for example, the Moyal star product is equivalent to Witten's
> star product in string field theory.
No, it isn't. It's an interesting toy model, however.
> To see various ways in which the
> star product makes up strings check out (hep-th/0202087, 0101219,
> 0012145 and 0107216).
The first paper is relevant in relating the Witten and Moyal *. The
second is hardly an introduction. The third talks about the Moyal star
product and M(atrix) theory, and the last is, as far as I can tell,
utterly irrelevant.
> Btw, perhaps it would be easier for you to learn about these topics if
> you talk to some mathematician/string theorist at UIUC, instead of
> trying to learn solely via what you find on the internet. An expert
> would at least probably know what the best introductory sources are.
You're not really helping.
Thomas Larsson
news:abergman-FFACC1...@news.bellatlantic.net...
> There are a few different star products. The one you're probably
> referring to is the so-called Moyal star product. Basically, given an
> anti-symmetric matrix theta_{uv}, we have
>
> (f * g)(z) = e^(i theta_{uv} d^x_u d^y_v) f(x) g(y) | x=y=z
>
> In bigger words, this can be thought of as the deformation quantization
> of R^{2n} wrt the symplectic form theta.
The Moyal product has a rather neat geometrical interpretation.
Expand functions on the simplest symplectic space R^2 in a Fourier
series, so the basis functions are exp(i m.x) = exp(i m_i x^i),
m in Z^2, x in R^2. The usual commutative product
exp(i m.x) exp(i n.x) = exp(i (m+n).x)
can be illustrated by vector addition of the Fourier modes. If
exp(i m.x) is the vector m and exp(i n.x) is the vector n, then
exp(i (m+n).x) is of course the vector sum m+n.
m
----->
\ |
m+n \ |n
\ |
\|
V
The star-product is
exp(i m.x) * exp(i n.x) = exp(ih A(m,n)) exp(i (m+n).x)
where A(m,n) is the area enclosed by the vectors m, n and m+n and
h is Planck's constant/2 pi. When you add three vectors m, n and r,
the enclosed area can be split in two ways. This is somewhat difficult
to draw in ascii, but it is easy to convince oneself that
A(m,n) + A(m+n,r) = A(m,n+r) + A(n,r).
This gives associativity of the star product. I am pretty sure that
the Moyal product is the unique associative deformation of the
ordinary commutative product.
This interpretation is so simple that many people must have thought
about it, but I have actually never seen it in print. I once showed
it to Moshe Flato, who said that he sort of knew about it, but he
didn't sound completely convincing. It does not work in
higher-dimensional symplectic spaces, because the three vectors don't
need to lie in the same plane there.
zirkus
says...
>> Yes, for example, the Moyal star product is equivalent to Witten's
>> star product in string field theory.
>No, it isn't. It's an interesting toy model, however.
According to this paper by Itzhak Bars, they are equivalent:
http://arxiv.org/abs/hep-th/0106157
Perhaps you are thinking of something different (or older or newer).
>The first paper is relevant in relating the Witten and Moyal *. The
>second is hardly an introduction. The third talks about the Moyal star
>product and M(atrix) theory, and the last is, as far as I can tell,
>utterly irrelevant.
The last paper (hep-th/0107216) may not be what Eric had in mind but I
think it might be interesting for him or other readers because this paper
mentions how open strings or D-branes in the presence of a constant
background antisymmetric field implies noncommutative (NC) spacetime
coordinates. For the NC generalization of gravity, the *- product, in part,
means that the metric has to be complex. Since NC gravity manifestly
breaks Lorentz symmetry, the authors argue that *teleparallel gravity*
should be the basis for NC gravity (hence, this paper might interest
those who have read the threads on teleparallelism).
Teleparallel gravity also provides a natural treatment of Ashtekar's
complex variables [1], so could TPG be part of a bridge between string theory
and loop gravity approaches?
>You're not really helping.
Perhaps, but I am now eating enough M & Ms to at least help the manufacturer
a wee bit.
[1] E.W. Mielke, "Ashtekar's complex variables in general relativity and its
teleparallel equivalent", Annals of Physics, 219, pp. 78 - 108 (1992).
------------------------------------------------------------------------
Aaron Bergman
<Pine.GSO.4.31.020219...@ux11.cso.uiuc.edu>,
eric alan forgy <fo...@students.uiuc.edu> wrote:
> After sending of the original post, I did a bit of reading and the
> relation between the Moyal and Witten products is pretty fascinating and I
> am beginning to think that some things I've been doing may be related.
> However, everything I'm doing is on purely algebraic/combinatoric
> background spaces. For instance, I have basically an abstract simplicial
> complex where an edge (string?) is represented by a simple finite set,
> e.g. [ij] = {{i},{j},{i,j}}. There is no parametrization.
The thing is, in string theory, things live on the string. There is a
string field. It's a lot more than just combinatorics.
> Now, in LQG you start out with a manifold and then define a spin network
> on it, do a bunch of fancy mathematics and when all is said and done, you
> discover that you never really needed the manifold at all to begin with.
> You could have started with this purely combinatorial spin network.
>
> Is it possible something similar to this could happen in string theory?
In string theory, "spacetime" is really the 2D QFT living on the
worldsheet. It is part of the string field, in a sense.
> That is the impression I get when I read about Witten product of strings.
> Like you said, you first split this into two halves and then concatenate
> the middle pieces like
>
> (aL,aR) * (bL,bR) = (aL,bR)
>
> Now if you distill out what the pure algebra of this is, you can almost
> look at (aL,aR) as an abstract 1-simplex {{aL},{aR},{aR,aL}}. Notice how
> the abstract 1-simplex is already in a sense composed of two "halves".
One of the ideas to deal with string field theory is to split the string
into halves and deal with them. Look for articles on split strings.
>
> I'm wondering if there is some relation between string theory and
> abstract graph theory where instead of continuously parameterized strings,
> you have abstract sets like these abstract 1-simplices. Then the string
> algebra seems closely related to the path algebra defined on a quiver.
I don't know how to translate this into something meaningful. The string
field is there. You can look at it in any number of ways (you can almost
think of it as an infinite dimensional matrix and the Witten-* as matrix
multiplication as Witten points out in his paper), but it's there.
> I spent a couple of days in a swirl reading up on representation theory of
> quivers and there is just tons of material on the subject. In fact, quite
> a bit of the recent material has been in relation to string theory. I
> won't pretend to understand the implications of it all, but at least I
> know that the connections are there.
Quivers show up in string theory in a seemingly different context. When
you put D-branes at a singularity, it's interesting to study the gauge
theory that lives on them. This is in many cases given by a quiver
diagram. Each point in the quiver corresponds to a gauge group and each
line corresponds to bifundamental matter. For example, there is an
algorithm for going from any toric singularity to a quiver diagram. In
fact, you get multiple possibilities. It's been conjectured that all
these gauge theories are Seiberg dual. Anyways, the reason I included
"seemingly" is that, on an orbifold singularity, you can think of the
bifundamental matter as coming from strings twisted by the orbifold
group and the gauge fields as coming from untwisted strings.
> I might just be spewing nonsense, but from what I have seen so far is that
> working with these abstract spaces automatically gets you this "background
> independence" that is held so dear in this place :) Putting coordinates on
> these abstract spaces is highly artificial but can be done. When you do
> so, it is pretty straight forward to see that what 'labels' you choose to
> lay on the space really is of no consequence to any physics.
In the Witten open string field theory, the space you're living on is,
in a sense, the space of backgrounds, so you get a sort of background
independence. The problem is that there is a BRST operator in the action
and this, as I understand, depends explicitly on the background. (The *
product might also depend on the background; I've heard arguments about
this.)
[...]
> > Note that this only works nicely for open strings. Doing something
> > similar for closed strings in a nice way would be very nice.
>
> I am totally out of my league here, but it seems as though any problems
> with doing this may be arising because you are assuming a continuum
> underlying space and you feel the need to "parameterize" strings.
If you want to do anything with strings, you need to parametrize. It's
all very nice to mumble stuff about abstract nonsense, but when you
actually want to get down to the nitty gritty and say exactly what you
get when you multiply two string fields, the easiest thing to do is to
work in a specific reparametrization. It's basically gauge fixing. I'd
like to be able to quantize, say, Yang-Mills theory in a gauge covariant
way, but I don't know how to do it. What I do no how to do is to pick a
gauge, do the Fadeev-Popov trick and look at the BRST cohomology.
> I can
> imagine a million ways to define associative products of "abstract closed
> strings".
It can be done. The problem with the current closed bosonic field theory
is that it has an infinite number of interaction terms. It is altogether
quite icky.
> Could graph theory come to the rescue here?
I don't see how.
eric alan forgy
On Wed, 20 Feb 2002, Thomas Larsson wrote:
> The Moyal product has a rather neat geometrical interpretation.
Wow! VERY neat! :)
> Expand functions on the simplest symplectic space R^2 in a Fourier
> series, so the basis functions are exp(i m.x) = exp(i m_i x^i),
> m in Z^2, x in R^2. The usual commutative product
>
> exp(i m.x) exp(i n.x) = exp(i (m+n).x)
>
> can be illustrated by vector addition of the Fourier modes. If
> exp(i m.x) is the vector m and exp(i n.x) is the vector n, then
> exp(i (m+n).x) is of course the vector sum m+n.
>
> m
> ----->
> \ |
> m+n \ |n
> \ |
> \|
> V
>
> The star-product is
>
> exp(i m.x) * exp(i n.x) = exp(ih A(m,n)) exp(i (m+n).x)
>
> where A(m,n) is the area enclosed by the vectors m, n and m+n and
> h is Planck's constant/2 pi.
Wow! :)
That is pretty amazing :)
How on earth does it work out like that? Any ideas?
It is really fascinating that the area shows up that like. Do you know
what that looks like to me?
It reminds me of holonomies!!
You are parallel transporting a vector once along n, then along m. This
is related to parallel transporting the vector along m+n by the curvature,
which is proportional to the area enclosed! :) (at least for a small
enough triangle and/or constant curvature)
It's beautiful! :)
It also helps to confirm the fact that what I've been doing is somehow
related. If only I were smarter! :)
It might even hint of a relation to spin networks and LQG.
It would be even more suggestive if:
exp[i m.x] * exp[i n.x] * exp[-i(m+n).x] = exp[ih A(m,n)]
which I imagine is true. Then you are truly parallel transporting around a
closed loop.
Well, instead of imagining, let's go ahead and see :) I've already sunk
enough time into this post :)
First, note that
exp[i m.x] * exp[-i m.x]
= exp[ih A(m,-m)] exp[i (m-m).x]
= 1
since A(m,-m) denotes a degenerate triangle of zero area. (Just like the
loop group.)
Then,
[exp(i m.x) * exp(i n.x)] * exp[-i (m+n).x]
= exp[ih A(m,n)] exp[i (m+n).x] exp[-i (m+n).x]
= exp[ih A(m,n)]
Voila!
Ok. That wasn't so bad :)
> When you add three vectors m, n and r,
> the enclosed area can be split in two ways. This is somewhat difficult
> to draw in ascii, but it is easy to convince oneself that
>
> A(m,n) + A(m+n,r) = A(m,n+r) + A(n,r).
>
> This gives associativity of the star product. I am pretty sure that
> the Moyal product is the unique associative deformation of the
> ordinary commutative product.
Beautiful! :)
If you look at it as parallel transporting vectors around triangles, the
associativity becomes pretty obvious :)
exp(i m.x) * [exp(i n.x) * exp(i r.x)]
= exp[ih A(n,r)] exp(i m.x) * exp[i (n+r).x]
= exp[ih (A(n,r) + A(m,n+r))] exp[i (m+n+r).x]
[exp(i m.x) * exp(i n.x)] * exp(i r.x)
= exp[ih A(m,n)] exp[i (m+n).x] * exp(i r.x)
= exp[ih (A(m,n) + A(m+n,r))] exp[i (m+n+r).x]
Therefore, like you said, * is associative if
A(m,n) + A(m+n,r) = A(m,n+r) + A(n,r)
which it obviously is. However, I suspect that these A's are not simply
numbers but that they have an orientation associated with them such that
the overlapping areas with opposite orientation cancel leaving only the
hourglass figure (see below).
This can be seen (hopefully) by considering the figure below.
m
+--------->
|\ /|
| \ / |
| \ / |
| \ / |
m+n | / | n+r
| / \ |
| n/ \m+n+r
| / \ |
|/ \V
V--------->
r
The term A(m,n) corresponds to transporting around the loop:
m
+--------->
| /
|A(m,n) /
| (o) / Note: (o) designates clockwise orientation
| / (+) designates counter clockwise orientation
m+n | /
| /
| /n
| /
|/
V
The term A(m+n,r) corresponds to transporting around the loop:
+
|\
| \
| \
| \
m+n | \ m+n+r
| \
| (+) \
| \
|A(m+n,r)\
V--------->
r
When you compose the two, i.e. A(m,n) + A(m+n,r), you get cancellations
and the result is equivalent to transporting around the hourglass loop:
m
+--------->
\ /
\ (o) /
\ /
\ /
/
/ \
n/ \m+n+r
/ (+) \
/ \
V--------->
r
Note, this should be pretty obvious once you have the parallel transport
interpretation at your disposal because the hourglass loop may be written
as
exp(i m.x) * exp(i n.x) * exp(i r.x) * exp[-i (m+n+r).x].
This adds to my conviction that A(m,n) are oriented areas.
Similarly, the term A(m,n+r) corresponds to transporting around the loop:
m
+--------->
\A(m,n+r)|
\ |
\ (o) |
\ |
m+n+r \ | n+r
\ |
\ |
\ |
\|
V
Finally, the term A(n,r) corresponds to transporting around the loop:
+
/|
/ |
/ |
/ |
n / | n+r
/ |
/ (+) |
/ A(n,r)|
/ V
V--------->
r
Again, when you compose the two, i.e. A(m,n+r) + A(n,r) you get
cancellations and the result is equivalent to transporting around the
hourglass loop:
m
+--------->
\ /
\ (o) /
\ /
\ /
/
/ \
n/ \m+n+r
/ (+) \
/ \
V--------->
r
PHEW!! ASCII art is tough :)
This is another way to view the associativity of *
Nifty, eh?! :)
> This interpretation is so simple that many people must have thought
> about it, but I have actually never seen it in print. I once showed
> it to Moshe Flato, who said that he sort of knew about it, but he
> didn't sound completely convincing. It does not work in
> higher-dimensional symplectic spaces, because the three vectors don't
> need to lie in the same plane there.
Huh? Aren't they? m,n, and m+n are coplanar no matter what the dimension
is. I must be missing something :)
Thanks for a beautiful post! Wouldn't it be really fascinating if the
*-product were related to curvature for the more general cases as well?
When I first saw the *-product it reminded me of holonomies, which is why
I thought it was related to my stuff. Now that you explicitly show the
area popping up, which obviously (to me anyway) signifies curvature and
holonomies, then I am nearly convinced :)
I had a lot of fun writing this post! :)
Eric
PS: Hmm... with a bit of afterthought, this parallel transport idea is
extremely awesome. Quantization can be thought of phase space with
(constant?) curvature! :) Add that to the list of reinvented wheels :)
PPS: This algebra works equally well for open paths (open string?) and
closed loops (closed strings?).
PPPS: While I'm speculating, perhaps a string could somehow be a
carrier of parallel transport. A string hits a brane, sucks up a vector
and parallel transports it to another brane. Ok. Forgive this last one. It
is WAY past my bedtime :) Then again, I've heard somewhere that strings
can be interpretted as flux tubes. Hmmm... *incoherent rambling*
Aaron Bergman
zirkus<zir...@hotmail.com> wrote:
> In article <abergman-FA2510...@news.bellatlantic.net>, Aaron
> Bergman
> says...
>
> >> Yes, for example, the Moyal star product is equivalent to Witten's
> >> star product in string field theory.
>
> >No, it isn't. It's an interesting toy model, however.
>
> According to this paper by Itzhak Bars, they are equivalent:
>
> http://arxiv.org/abs/hep-th/0106157
>
> Perhaps you are thinking of something different (or older or newer).
There are a number of issues here. First of all, you'll note that Bars's
paper describes a map. In other words, he describes a field redefinition
of the bosonic variables on the string where the Witten-* can be written
as a infinite sum of Moyal-*s. Bars does not (with the exception of the
hope expressed on page 10) deal with the ghost sector at all. There are
also a number of subtle issues that Bars ignores.
> >The first paper is relevant in relating the Witten and Moyal *. The
> >second is hardly an introduction. The third talks about the Moyal star
> >product and M(atrix) theory, and the last is, as far as I can tell,
> >utterly irrelevant.
>
> The last paper (hep-th/0107216) may not be what Eric had in mind but I
> think it might be interesting for him or other readers because this paper
> mentions how open strings or D-branes in the presence of a constant
> background antisymmetric field implies noncommutative (NC) spacetime
> coordinates.
Actually, it imples that the worldvolume fields become noncommutative.
> For the NC generalization of gravity, the *- product, in part,
> means that the metric has to be complex.
There is no gravity on the brane.
A.J. Tolland
> zirkus wrote:
> > For an intro to star products in deformation quantization see page 2
> > of Kontsevich's (q-alg/9709040).
> Don't.
Hoo-boy, you said it! Kontsevich was just here at the math
department, 5 hours of lectures on "Mathematical Aspects of Quantum Field
Theory". I think Drinfeld was the only one who understood more than 2/3
of what he said.
--A.J.
alejandro.rivero
of the series from Cosmas Zachos.
I believe it is the right way to look at the product; just an
averaging of areas.
Thomas Larsson <Thomas....@hdd.se> wrote in message news:<E180A6F82759D211BFDA00E01890523D27F460@HDDNT01>...
John Baez
Thomas Larsson <Thomas....@hdd.se> wrote:
>The Moyal product has a rather neat geometrical interpretation.
>The star-product is
>
>exp(i m.x) * exp(i n.x) = exp(ih A(m,n)) exp(i (m+n).x)
>
>where A(m,n) is the area enclosed by the vectors m, n and m+n and
>h is Planck's constant/2 pi.
>This interpretation is so simple that many people must have thought
>about it, but I have actually never seen it in print.
This interpretation is actually built right into Belissard's theory
of the quantum Hall effect in terms of the noncommutative torus!
When you have a charged particle in a thin film in a constant transverse
magnetic field, it picks up a phase when you parallel translate it.
When you parallel translate it around the triangle enclosed by vectors
m, n, and m+n, it picks up a phase given by your formula - but with some
extra factors of the magnetic field and the charge floating around in
the exp(ih A(m,n)) term.
For a few more details and also some references, try:
Aaron Bergman
<Pine.SGI.4.40.020220...@hep.uchicago.edu>,
"A.J. Tolland" <a...@hep.uchicago.edu> wrote:
This is the guy who brought derived categories into all of this.
Something to do with something called "Fukaya categories" and mirror
symmetry.
Thomas Larsson
>
> It was, with the triangle included, in a recent preprint in hep, one
> of the series from Cosmas Zachos.
>
> I believe it is the right way to look at the product; just an
> averaging of areas.
Zachos probably knew about the construction ten years ago. Around 1990
he, together with D Fairlie and maybe someone else, rediscovered the
Moyal Lie algebra, i.e. the commutator version of the Moyal product.
For obvious reasons they called it the sine algebra, until they found
out about Moyal's work after a year or so. Their series of papers was
the place where I first heard about the Moyal algebra.
Robert C. Helling
> Hoo-boy, you said it! Kontsevich was just here at the math
>department, 5 hours of lectures on "Mathematical Aspects of Quantum Field
>Theory". I think Drinfeld was the only one who understood more than 2/3
>of what he said.
That's not too bad. Isn't there a saying that a decent math seminar talk
comes in thirds: The first can be understood by more or less everybody in
the audience, the second only by the experts in the field of the speeker,
the third only by the speaker and sometimes there is even a fourth third...
Sorry, couldn't resist.
Robert
--
.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Institut fuer Physik
Humboldt-Universitaet zu Berlin
print "Just another Fon +49 30 2093 7964
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
zirkus
>Bars does not (with the exception of the
> hope expressed on page 10) deal with the ghost sector at all. There are
> also a number of subtle issues that Bars ignores.
You're right, and it seems that trying to define string field theory
is very difficult (also see e.g. hep-th/0202087). Many times, the
history of physics and astronomy have taught us that when an idea gets
more and more complicated that it is probably wrong or incomplete, and
that a new way of thinking is needed to provide simplification and
usually generalization (e.g. GTR simplifies various problems that are
more complicated in a Newtonian framework). I get a feeling that those
who study string field theory may be missing something fundamental
(e.g. the paper hep-th/0202087 doesn't even try to account for ghosts
or the BRST operator).
>>open strings or D-branes in the presence of a constant
> > background antisymmetric field implies noncommutative (NC) spacetime
> > coordinates.
>
> Actually, it imples that the worldvolume fields become noncommutative.
>
> > For the NC generalization of gravity, the *- product, in part,
> > means that the metric has to be complex.
>
> There is no gravity on the brane.
What I said is taken from page 1 of (hep-th/0107216) and references 1
and 2 mentioned on page 1. However, it would not surprise me if all of
these authors are somehow wrong. This is because non-perturbative
string theory is poorly "defined", and because fundamental
understanding of string theory can change radically, as it has e.g.
with the second string revolution, D-branes, M-theory, NCG, etc.
alejandro.rivero
> In article <8c7d34cb.02021...@posting.google.com>,
> zir...@hotmail.com (zirkus) wrote:
> >
> > You seem to be talking about the Moyal star product. Are you sure that
> > this is not discussed in Connes' textbook (which I have not read) ?
>
> Connes's book is really geared towards a different form of
> noncommutative geometry.
encode Weyl quantization rule, which in turns isin the origin of
the * product. THere all also all the deformation theory, classified
in the book as a theory of asymptotic morphisms.
Said that, it is true that the xxx articles are better starting points
that the red book
Alejandro Rivero
http://dftuz.unizar.es/%7erivero/research/
R.X.
theory, namely mathematically as algebra of maps in the
derived category. In physics terms the maps are open strings and they
stretch between D-branes (they are the links of a quiver diagram).
However this has not much to do with non-commutativity in the sense
of background B-fields, Moyal products etc.
R.X.
>
> What exactly is a string algebra? Is there some operation in string theory
> that composes strings/paths? A path algebra is a very obvious
> construction and I reinvented it on my own and I even called it "path
> algebra". Is it at all related? Could path algebra be a kind of discrete
> string algebra? A path algebra is simply a directed graph with product of
> paths in the graph defined by concatenation when such a concatenation
> makes sense. When it doesn't, i.e. the end of the first path doesn't
> correspond to the beginning of the next path, then the product is zero.
>
>
http://lanl.arxiv.org/PS_cache/hep-th/pdf/0003/0003263.pdf
This comes close to what you said above: a discrete string algebra
realized concretely by links on a quiver diagram. However,
this doesn't have much to do with non-commutativity, background
B-fields, groupoids, etc; these subjects do not seem particularly
relevant.
For other, but ultimately related considerations of path algebras, see eg
chapter 3 in:
http://lanl.arxiv.org/PS_cache/hep-th/pdf/9908/9908036.pdf