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Lubos Motl
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Joined: 02 May 2005
Posts: 38

Posted: Fri Mar 11, 2005 11:19 pm    Post subject: Re: deconstruction

On Fri, 11 Mar 2005, Urs Schreiber wrote:

 Quote: Over on his blog Robert Helling made an imteresting comment on deconstruction. http://atdotde.blogspot.com/2005/03/deconstruction-help-sought_10.html . Does anyone have a good answer?

I suppose the question from Robert is

 Quote: So what is more in deconstruction that M(atrix)-T-duality?

obvious.

The models with T-duality he mentions always describe completely and
exactly continuous dimensions. There are really no "quivers" in these
models. The corresponding "quiver" is a smooth space. They're not
deconstruction.

The whole point of deconstruction is that the theory space (quiver or
moose) is *not* exactly continuous which makes a lot of difference. One
has two basic reasons why we consider discrete theory spaces:

1. First of all, there is a phenomenological reason: we may "deconstruct"
extra dimensions beyond the four dimensions we have, and obtain
effectively four-dimensional physics. This is done in the little Higgs
models and similar stuff where the quiver is pretty small (and as
different from a continuous extra dimensions as you can get) and these
four-dimensional theories with finite quivers are well-defined. If
the gauge groups are asymptotically free, the theories are defined
even non-perturbatively.

2. Even in the theoretical context, it's important to have a finite value
of N exactly because we don't know how to define a theory in which
something is explicitly infinite - but we know how to take the limit.
This allows you, for example, to write down an in-principle definition
of the (2,0) SCFT and (1,1) little string theory in 6D in such a way
that the locality at least in 4 dimensions is manifest

http://arxiv.org/abs/hep-th/0110146

The difference between this definition of these exotic 6D theories
and any other approach different from deconstruction is that
deconstruction can give the definition, but no other approach has
provided us with such a definition before. If Robert has another
definition of the (2,0) theory that allows calculations, at least in
principle, I am very interested in it.

Simply speaking, the "other side" that Robert says is equivalent to
deconstruction has nothing to do with deconstruction because all of its
dimensions are explicitly continuous. Deconstruction means that the "extra
dimensions" are really discrete.

All the best
Lubos
______________________________________________________________________________
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
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Urs Schreiber
science forum Guru Wannabe

Joined: 04 May 2005
Posts: 127

Posted: Sun Mar 13, 2005 2:59 pm    Post subject: Re: deconstruction

"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0503111908120.28181-100000@feynman.harvard.edu...

 Quote: This allows you, for example, to write down an in-principle definition of the (2,0) SCFT and (1,1) little string theory in 6D in such a way that the locality at least in 4 dimensions is manifest http://arxiv.org/abs/hep-th/0110146

By the way, can you identify n^3-scaling in the deconstrcuted constructions
of these 6D theories?
Lubos Motl
science forum beginner

Joined: 02 May 2005
Posts: 38

Posted: Mon Mar 14, 2005 1:22 pm    Post subject: Re: deconstruction

Urs wrote:

 Quote: By the way, can you identify n^3-scaling in the deconstrcuted constructions of these 6D theories?

Great question. The simple answer is NO (so far). The longer answer may be
YES, but the argument does not depend on the details of the deconstructed
theory too much - it is based on the identification of two different
scales that differ by a factor of "n" - in the intermediate interval, the
entropy gets an extra kick by a factor of "n" by linear scaling. If
someone knows how to do it directly from the quiver theory, I would love
to hear it.
______________________________________________________________________________
E-mail: lumo@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/
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Robert C. Helling
science forum beginner

Joined: 30 Apr 2005
Posts: 22

Posted: Fri Mar 18, 2005 11:28 am    Post subject: Re: deconstruction

On Fri, 11 Mar 2005 19:19:53 -0500, Lubos Motl <motl@feynman.harvard.edu> wrote:

 Quote: On Fri, 11 Mar 2005, Urs Schreiber wrote: Over on his blog Robert Helling made an imteresting comment on deconstruction. http://atdotde.blogspot.com/2005/03/deconstruction-help-sought_10.html . Does anyone have a good answer? I suppose the question from Robert is So what is more in deconstruction that M(atrix)-T-duality? I may misunderstand what is Robert asking about because the answer is obvious. The models with T-duality he mentions always describe completely and exactly continuous dimensions. There are really no "quivers" in these models. The corresponding "quiver" is a smooth space. They're not deconstruction. The whole point of deconstruction is that the theory space (quiver or moose) is *not* exactly continuous which makes a lot of difference. One has two basic reasons why we consider discrete theory spaces:

My question was about infinite N.

 Quote: 2. Even in the theoretical context, it's important to have a finite value of N exactly because we don't know how to define a theory in which something is explicitly infinite - but we know how to take the limit.

Really? As I said, the result for the limit at least resembles the
theory for a finite number of D3's say (D0 in the original reference)
with a periodic transversal direction. Wati Tayler explained how to
treat this case by including the inifinte number of image branes. The
resulting theory is a 3+1 dimensional theory with infinte matrices
but after a Fourier transform can be brought into the language of a
4+1 dimensional theory with finite matrices. What makes you confident
that the above limit you mention exists _in the quantum theory_?

 Quote: The difference between this definition of these exotic 6D theories and any other approach different from deconstruction is that deconstruction can give the definition, but no other approach has provided us with such a definition before. If Robert has another definition of the (2,0) theory that allows calculations, at least in principle, I am very interested in it.

No, of course I don't have an alternative description. I just mention
that this definition (assuming the limit exists) looks like another,
older version to write higher dimensional gauge theories in terms of
lower dimensional ones.

 Quote: Simply speaking, the "other side" that Robert says is equivalent to deconstruction has nothing to do with deconstruction because all of its dimensions are explicitly continuous. Deconstruction means that the "extra dimensions" are really discrete.

Discrete yes, but if there is an infinite number of discrete points,
the Fourier transform has the potential to relate this to a
continious, periodic dimension.

Robert

--
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Robert C. Helling School of Science and Engineering
International University Bremen
print "Just another Phone: +49 421-200 3574
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Lubos Motl
science forum beginner

Joined: 02 May 2005
Posts: 38

Posted: Fri Mar 18, 2005 12:07 pm    Post subject: Re: deconstruction

On Fri, 18 Mar 2005, Robert C. Helling wrote:

 Quote: My question was about infinite N.

That's OK, but in that case, your questions was *not* about deconstruction
because the point of deconstruction is to describe physics analogous to
continuous dimensions in terms of a theory with a *finite* N (number of
copies of the gauge group).

 Quote: 2. Even in the theoretical context, it's important to have a finite value of N exactly because we don't know how to define a theory in which something is explicitly infinite - but we know how to take the limit. Really? As I said, the result for the limit at least resembles the theory for a finite number of D3's say (D0 in the original reference) with a periodic transversal direction.

That's right. It's because it's the goal of deconstruction to resemble
physics in continuous dimensions - but resemble it using a controllable
theory where only a smaller number of dimensions are strictly continuous.

 Quote: Wati Tayler explained how to treat this case by including the inifinte number of image branes.

This is a very naive viewpoint. Wati Taylor has shown how to make Fourier
transformation in the space of continuous indices. One can either view a
D(p+1)-brane on a circle in terms of a new "continuous index" i.e. a new
dimension in its worldvolume theory, or as an array of Dp-branes in the
T-dual theory. T-duality is represented as a simple Fourier transform on
the space of gauge indices. This machinery says nothing about the
existence of the theories at quantum level.

 Quote: The resulting theory is a 3+1 dimensional theory with infinte matrices but after a Fourier transform can be brought into the language of a 4+1 dimensional theory with finite matrices.

What you obtain with this procedure is the naive 4+1-dimensional gauge
theory which is not renormalizable, and the whole quantum physics is
unknown. It's just a formal manipulation that happens to give the right
classical limit at low energies. But deconstruction gives you the quantum
consistent UV complete definition of the 4+1-dimensional gauge theory,
namely the (2,0) theory on a circle. You can't get these things just by
the manipulations you describe.

You know, the procedure that you seem to like so much was initially
thought to be relevant for the definition of M(atrix) theory on
*arbitrary* tori. It was soon realized that it is only good up to the
three-torus where the matrix model is the 3+1-dimensional maximally
supersymmetric gauge theory on a dual three-torus.

Beyond this point, one derives theories that can't be described in this
simple form. The "4+1-dimensional Yang-Mills" is a correct description of
the relevant matrix model for M-theory on T^4 at low energies only. The
appropriate quantum completion is the six-dimensional (2,0) theory on a
circle and a four-torus - more generally, on a five-torus. (The same
theory one gets by deconstruction.) One may derive this from T-dualities -
T-dualizing D0-branes four times gives D4-branes, but if one is careful
about the string coupling, it diverges. Therefore one really works with
M5-branes in M-theory, and the relevant regime that enters the definition
of the matrix model is the SCFT which is the (2,0) theory.

Similarly, M-theory on T^5 is not described just by "5+1-dimensional
Yang-Mills" that you obtain from your naive classical procedure. The right
UV completion is the (1,1) 6D little string theory on a five-torus - that,
incidentally, inherits the T-duality group SO(5,5) from the full IIB
string theory, which is interpreted as the U-duality of M-theory on T5.
The matrix description beyond T^5, namely for M-theory on T^6 and higher,
does not exist because gravity does not decouple, and one expects that
there is no non-gravitational UV completion of the gauge theory. The
toroidal compactification of BFSS is the main focus of Ashoke Sen's paper

http://arxiv.org/abs/hep-th/9709220

who identified the correct theories before they were described and detail
and called "little string theory", for example.

 Quote: What makes you confident that the above limit you mention exists _in the quantum theory_?

There are two main things that make me confident about that: the proof
that only relies on T-duality, the low-energy description of D3-branes on
orbifolds, and on the fact that the physics of cone with a small opening
angle approaches the physics of cylinder. Neither of these things makes
any unsatisfied assumption about the strength of the coupling, unlike your
classical procedure, and the reliability of the considerations is
strengthened by the large amount of preserved supersymmetry. The second
thing that makes me confident are consistency checks - the description has
the right limit "4+1-dimensional Yang-Mills" etc. by the usual
considerations, but it also has the right duality groups etc. so that it
is almost forced to be the advertised theory in all regimes - certainly
all asymptotic ones - at quantum level. It contains the right perturbative
excitations as well as monopoles, and I see no way how it could "fail" in
the middle, especially if there exists the proof.

 Quote: No, of course I don't have an alternative description. I just mention that this definition (assuming the limit exists) looks like another, older version to write higher dimensional gauge theories in terms of lower dimensional ones.

There is no other way how to write the little string theory using its
low-dimensional counterparts, and therefore the words "another, older" in
your sentence are not correct. Be sure that if you try to make a generic
latticization of 4+1-dimensional Yang-Mills, you won't obtain the (2,0)
theory on a circle or any other consistent quantum theory unless you
fine-tune an infinite number of coupling constants. The special virtue of
the deconstruction is that it preserves some supersymmetries which
together with the discrete symmetries in the continuum limit guarantee
that the limiting theory has all the required supercharges, and therefore
it is unique given the degrees of freedom. Why deconstruction is clearly
superior for "latticizing" supersymmetry - see e.g. a talk

http://arxiv.org/abs/hep-lat/0208046

 Quote: Discrete yes, but if there is an infinite number of discrete points, the Fourier transform has the potential to relate this to a continious, periodic dimension.

That's right. This is a fact that deconstruction uses heavily.
______________________________________________________________________________
E-mail: lumo@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/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
science forum Guru Wannabe

Joined: 04 May 2005
Posts: 127

Posted: Sun Mar 20, 2005 3:43 pm    Post subject: Re: deconstruction

Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.62.0503140918450.2188@feynman.harvard.edu>...
 Quote: Urs wrote: By the way, can you identify n^3-scaling in the deconstrcuted constructions of these 6D theories? Great question. The simple answer is NO (so far). The longer answer may be YES, but the argument does not depend on the details of the deconstructed theory too much - it is based on the identification of two different scales that differ by a factor of "n" - in the intermediate interval, the entropy gets an extra kick by a factor of "n" by linear scaling. If someone knows how to do it directly from the quiver theory, I would love to hear it.

Here is another question:

Given a field theory on some spacetime M with fields taking values in
some space F we know that it is in general not correct to describe
these fields as F-valued functions on M, but that they have to be
considered as sections of a (possibly non-trivial) bundle over M with
typical fiber F.

When deconstructing dimensions a quiver gets interpreted as (part of)
spacetime. There are fields living on the edges of the quiver which
are usually regarded as functions from these edges to some space of
homomorphisms.

Here is the question: Do we in the most general case have to allow
these functions to really be sections of a possibly non-trivial
"bundle" or something?
Urs Schreiber
science forum Guru Wannabe

Joined: 04 May 2005
Posts: 127

Posted: Tue Mar 22, 2005 7:55 am    Post subject: Re: deconstruction

Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<206f2305.0503200214.15aa53ed-100000@posting.google.com>...

 Quote: Given a field theory on some spacetime M with fields taking values in some space F we know that it is in general not correct to describe these fields as F-valued functions on M, but that they have to be considered as sections of a (possibly non-trivial) bundle over M with typical fiber F. When deconstructing dimensions a quiver gets interpreted as (part of) spacetime. There are fields living on the edges of the quiver which are usually regarded as functions from these edges to some space of homomorphisms. Here is the question: Do we in the most general case have to allow these functions to really be sections of a possibly non-trivial "bundle" or something?

Let me try to phrase the same question in a "dual" picture, where it
might be more transparent if it makes any sense:

So instead of thinking of the quiver as a deconstructed dimension, go
back to the D-brane picture and think of every vertex of the quiver as
a fractional brane on some C^3/G orbifold CY and as the edges as
string states between these fractional branes.

Consider a quiver which contains at least one non-contractible loop.

Now I am going to ask if it makes sense to consider the case where the
nature of the fractional branes gradually changes as we go along this
loop such that by going once around we have effectively traced out a
non-contractible loop in moduli space.

I'll try to make that a little more precise as follows:

Consider a flow Phi through the moduli space of the CY. Given any
D-brane E with respect to the CY at Phi(0) let Phi(t)(E) be the result
of transporting that D-brane with the flow Phi a distance t through
moduli space.

Since I want all my fractional branes to live in the same CY invert
this picture:

The brane which, when transported a distance t _to_ the CY at Phi(0),
looks like E there is given by Phi(-t)(E).

Consider for a moment a quiver with just one edge, i.e. a situation
where we have just two fractional branes E1 and E2 with strings
between them. Can we also have a physical situation with E1 and
Phi(-t)(E2) instead of E1 and E2, for sufficiently small but
nonvanishing Phi(-t)?

I have the vague idea that for small t this should not change anything
of the physics, but I don't really know. Does anyone?

If however this makes sense, then it should make sense to do the
following:

Given a quiver with a non-contractible loop, let the brane states
along that loop be labelled E0, E1, E2, ... En = E0.

Now, assuming the above is correct, it should make sense to gradually
go from

E0, E1 to Phi(-0 epsilon)(E0) , Phi(-1 epsilon)(E1)

and

E1, E2 to Phi(-1 epsilon)(E1) , Phi(-2 epsilon)(E2)

etc,

in a case where

Phi(-n epsilon) = Phi(0) ,

i.e. where the flow Phi is a loop in moduli space.

So in this case the fractional brane Ei is of the same kind as E(i+1),
but En is equal to E0 only op to the effect of going around a closed
loop in moduli space. If that loop has nontrivial monodromy, as it may
happen, we would get the situation that I am intersted in.

The reason I am interested in this is because I believe there would be
an interesting solution to make this setup consistent, if indeed this
setup makes any sense. A classical case of a solution looking for a
problem.
Aaron Bergman

Joined: 24 Mar 2005
Posts: 94

Posted: Wed Mar 23, 2005 8:55 am    Post subject: Re: deconstruction

Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:

 Quote: Consider a flow Phi through the moduli space of the CY. Given any D-brane E with respect to the CY at Phi(0) let Phi(t)(E) be the result of transporting that D-brane with the flow Phi a distance t through moduli space. Since I want all my fractional branes to live in the same CY invert this picture: The brane which, when transported a distance t _to_ the CY at Phi(0), looks like E there is given by Phi(-t)(E). Consider for a moment a quiver with just one edge, i.e. a situation where we have just two fractional branes E1 and E2 with strings between them. Can we also have a physical situation with E1 and Phi(-t)(E2) instead of E1 and E2, for sufficiently small but nonvanishing Phi(-t)?

I can't figure out what you're talking about here. But, as I said,
generally noncontractible loops in the moduli space give rise to
autoequivalences of the derived category when you look at the monodromy
around the loop.

Aaron
Urs Schreiber
science forum Guru Wannabe

Joined: 04 May 2005
Posts: 127

Posted: Wed Mar 23, 2005 9:08 am    Post subject: Re: deconstruction

On Wed, 23 Mar 2005, Aaron Bergman wrote:

 Quote: I can't figure out what you're talking about here.

I am trying to understand if the assignment of vector spaces and linear
maps to a quiver should just be a functor or rather really a 2-section
of a possibly nontrivial 2-bundle over the quiver.

The spacetime interpretation of the quiver in dimensional deconstruction
naively suggests the latter. I am trying to see if this would make sense
from the perspective of branes on orbifolds.

 Quote: But, as I said, generally noncontractible loops in the moduli space give rise to autoequivalences of the derived category when you look at the monodromy around the loop.

Yes. So the Picard group would naturally arise as the structure 2-group of
such a 2-bundle. Formally this idea looks quite suggestive. I am trying to
understand if it is physically meaningful.
Aaron Bergman

Joined: 24 Mar 2005
Posts: 94

 Posted: Thu Mar 24, 2005 2:27 pm    Post subject: Re: deconstruction In article , Urs Schreiber wrote: [...] Sorry, Urs. I wish I could help some more, but you've completely lost me. Aaron
Urs Schreiber
science forum Guru Wannabe

Joined: 04 May 2005
Posts: 127

Posted: Fri Mar 25, 2005 8:02 pm    Post subject: Re: deconstruction

Aaron Bergman wrote:

 Quote: [...] Sorry, Urs. I wish I could help some more, but you've completely lost me.

was telling him reminded him of the Klebanov-Strassler cascade
mechanism that he had reformulated in terms of systems of 5-branes and
3-branes.

So I guess when we blow up the C^3/G singlularity his setup applies and
we then know that the gauge group living on the branes is defined only
up to a certain shift. U(m)xU(n) to U(m-n)xU(m) or something.

But this gauge group is what is defined by the rep of the qiver. If
that's only well defined up to something we arrive essentially at my
idea that we should not think of the quiver rep as a function but as a
section. I think...

More later, I am still at Milano central station...
Aaron Bergman

Joined: 24 Mar 2005
Posts: 94

Posted: Sun Mar 27, 2005 5:05 pm    Post subject: Re: deconstruction

Urs <Urs.Schreiber@uni-essen.de> wrote:

 Quote: So I guess when we blow up the C^3/G singlularity his setup applies and we then know that the gauge group living on the branes is defined only up to a certain shift. U(m)xU(n) to U(m-n)xU(m) or something.

That's not really what's going on. The idea in KS is that there's a sort
of 'cascade' of Seiberg dualities. It's not that the gauge group is only
defined up to a shift; it's that in different energy scale regimes, the
has different effective descriptions. Sort of. That's not quite right,
but KS is a little funky.

But, it's certainly true that, even for a superconformal theory, the
assignment of a quiver to a particular geometry is not unique. This is
because the actual SCFT is the IR limit of the quiver gauge theory. You
can have lots of different quivers that have the same IR limit. The
procedure of obtaining quivers from derived categories depends on the
choice of an exceptional collection which is far from unique. Herzog
proves in hep-th/0405118 that many (all?) of these quivers are related
by Seiberg duality.

Aaron
Urs Schreiber
science forum Guru Wannabe

Joined: 04 May 2005
Posts: 127

Posted: Sun Mar 27, 2005 5:42 pm    Post subject: Re: deconstruction

On Sun, 27 Mar 2005, Aaron Bergman wrote:

 Quote: But, it's certainly true that, even for a superconformal theory, the assignment of a quiver to a particular geometry is not unique. This is because the actual SCFT is the IR limit of the quiver gauge theory. You can have lots of different quivers that have the same IR limit. The procedure of obtaining quivers from derived categories depends on the choice of an exceptional collection which is far from unique. Herzog proves in hep-th/0405118 that many (all?) of these quivers are related by Seiberg duality.

Thanks.

Familiy duties over easter holidays prevent me from doing
real work today, so once again just another followup question which in a
better world I would have time to look up in the literature:

When do we say that two quivers are equivalent? Or dual?

Given a representation R1 of a quiver Q1 which is equivalent or dual to
Q2, do we get an induced representation R2 of Q2 from R1?
Aaron Bergman

Joined: 24 Mar 2005
Posts: 94

Posted: Mon Mar 28, 2005 2:23 pm    Post subject: Re: deconstruction

In article <Pine.LNX.4.62.0503271437340.2929@feynman.harvard.edu>, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:

 Quote: On Sun, 27 Mar 2005, Aaron Bergman wrote: But, it's certainly true that, even for a superconformal theory, the assignment of a quiver to a particular geometry is not unique. This is because the actual SCFT is the IR limit of the quiver gauge theory. You can have lots of different quivers that have the same IR limit. The procedure of obtaining quivers from derived categories depends on the choice of an exceptional collection which is far from unique. Herzog proves in hep-th/0405118 that many (all?) of these quivers are related by Seiberg duality. Thanks. Familiy duties over easter holidays prevent me from doing real work today, so once again just another followup question which in a better world I would have time to look up in the literature: When do we say that two quivers are equivalent? Or dual? Given a representation R1 of a quiver Q1 which is equivalent or dual to Q2, do we get an induced representation R2 of Q2 from R1?

The quivers all have the same nodes, I think, so the dimension vectors
are the same. Then, what I think everyone believes is that the quiver
variety, ie, the moduli space of the gauge theories, is the same.

To see the action of Seiberg duality, I'd recommend checking out Chris's
paper above, Aspinwall's paper 0405134 and references therein.

Aaron
Urs Schreiber
science forum Guru Wannabe

Joined: 04 May 2005
Posts: 127

Posted: Tue Mar 29, 2005 2:24 pm    Post subject: Re: deconstruction

"Aaron Bergman" <abergman@physics.utexas.edu> schrieb im Newsbeitrag news:abergman-68A952.14112427032005@localhost...

 Quote: To see the action of Seiberg duality, I'd recommend checking out Chris's paper above, Aspinwall's paper 0405134 and references therein.

Ok. Beginning to do so I have come across

Berenstein & Douglas
Seiberg Duality for Quiver Gauge Theories
hep-th/0207027

They make explicit what I was going to ask as the next obvious question,
namely that these dualities are described by two-sided tilting complexes.

That is, a duality between a gauge theory described by a a quiver with path
algebra A and one with path algebra B is given by an object

T in D( A-mod-B )

of the derived category of left-A/right-B-bimodules which is weakly
invertible, i.e. such that there is a T' such that

T \otimes^L T' ~ A

and

T' \otimes^L T ~ B ,

where A and B here are the algebras themselves regarded as bimodules over
themselves and playing the role of the identity object.

The duality is then just given by \otimes^L-multiplying a D-brane state in

D(mod-A)

or

D(B-mod)

with T.

I note that a special case of D( A-mod-B ) this is the derived category of
A-bimodules

D( A-mod-A ).

describing autoequivalences (and hence, as you emphasized, in particular
monodromies in moduli space ) which contains a weak 2-group as a full
subcategory, namely that containing all the morphisms between weakly
invertible objects = tilting complexes.

I have asked Amnon Yekutieli if this 2-group structure here is discussed in
the literature. He didn't know about it, so I guess it doesn't. Which is
strange, because the group you get by taking isomorphism classes of objects
of this 2-group is the well-known derived Picard group. From the
category-theoretic viewpoint it seems strange to immediately pass to just
the isomorphism class of objects. There is much more information in the
2-group than just that.

Now, from Berenstein and Douglas discussion I have not quite managed to
extract yet if that "more" of information is physically useful or just
redundant.

Anyway, there is a small comment on subquivers in Berenstein&Douglas which
makes me wonder (p. 43). Would you know of any more detailed discussion
which crucially uses embeddings of gauge-theory quivers into larger quivers?
Somehow?

Sorry, a rather vague question once again.

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 Topic Author Forum Replies Last Post Similar Topics fermion doubling problem in deconstruction models Guest Research 0 Wed Jul 27, 2005 9:11 am