Discussion:
generalized geometry
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Urs Schreiber
2005-05-04 17:52:42 UTC
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I am a little confused concerning the question in which sense Hitchin's
generalized geometry yields any generalization of previously known string
backgrounds. I.e. does it yield new physics in addition to the new
math? Can anyone provide me with more details?
Aaron Bergman
2005-05-05 02:27:13 UTC
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In article <***@news.dfncis.de>,
Urs Schreiber <***@uni-essen.de> wrote:

> I am a little confused concerning the question in which sense Hitchin's
> generalized geometry yields any generalization of previously known string
> backgrounds. I.e. does it yield new physics in addition to the new
> math? Can anyone provide me with more details?

Check out hep-th/0503083 for some relevance for Hitchin's construction.

Aaron
Urs Schreiber
2005-05-05 10:51:24 UTC
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"Aaron Bergman" <***@physics.utexas.edu> schrieb im Newsbeitrag
news:abergman-***@localhost...
> In article <***@news.dfncis.de>,
> Urs Schreiber <***@uni-essen.de> wrote:
>
>> I am a little confused concerning the question in which sense Hitchin's
>> generalized geometry yields any generalization of previously known string
>> backgrounds. I.e. does it yield new physics in addition to the new math?
>> Can anyone provide me with more details?
>
> Check out hep-th/0503083 for some relevance for Hitchin's construction.

This is actually where my confusion came from. At first sight the result of
this paper makes it seem as if the B-string lives on generalized CY instead
of on ordinary CYs. But can this be true? From what Andy and Lubos say on
the "reference frame" it seems not to be clear at all:

http://motls.blogspot.com/2005/04/generalized-geometry.html

I also see that Kapustin is arguing that some generalized complex geometry
should describe "noncommutative CYs" in hep-th/0310057 and hep-th/0502212.

Somewhat amusingly to me, with hindsight, is that (and Eric had to remind me
of this on http://golem.ph.utexas.edu/string/archives/000563.html#c002287)
the degree of freedom that he tries to give a physical interpretation to is
essentially the one that I called C in section (2.8) of this entry

http://golem.ph.utexas.edu/string/archives/000278.html

and which I think comes from the Kalb-Ramond B by a transformation which
switches the sign of the dilaton.
Andy Neitzke
2005-05-06 07:54:30 UTC
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Urs Schreiber wrote:

> This is actually where my confusion came from. At first sight the result
> of this paper makes it seem as if the B-string lives on generalized CY
> instead of on ordinary CYs. But can this be true?

I think this is the correct interpretation -- and actually it was known
before the result of Pestun and Witten -- already in the very early
literature on the B model it is emphasized that the space of observables
contains more than just the deformations of the complex moduli, although
this was sometimes forgotten later on. In retrospect, this makes it sort
of obvious that the target space field theory of the B model ought to
include extra fields describing these generalized CY structures, so that
our conjecture about the B model being equivalent to the ordinary Hitchin
functional was doomed from the start!

I'm not an expert on the generalized-geometry literature by a long shot, but
the definitions of the generalized A/B models -- including the conditions
on the twisted generalized Kahler manifold which are necessary for the
topological models to exist -- seem to appear in

Kapustin, Anton and Li, Yi. "Topological sigma-models with H-flux and
twisted generalized complex manifolds," hep-th/0407249,

and perhaps other places as well.

-Andy
Urs Schreiber
2005-05-06 12:23:21 UTC
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"Andy Neitzke" <***@fas.harvard.edu> schrieb im Newsbeitrag
news:d5f1gg$29d$***@us23.unix.fas.harvard.edu...
> Urs Schreiber wrote:
>
>> This is actually where my confusion came from. At first sight the result
>> of this paper makes it seem as if the B-string lives on generalized CY
>> instead of on ordinary CYs. But can this be true?
>
> I think this is the correct interpretation -- and actually it was known
> before the result of Pestun and Witten -- already in the very early
> literature on the B model it is emphasized that the space of observables
> contains more than just the deformations of the complex moduli, although
> this was sometimes forgotten later on.


Thanks. Good to know.


> Kapustin, Anton and Li, Yi. "Topological sigma-models with H-flux and
> twisted generalized complex manifolds," hep-th/0407249,


Thanks again, very interesting.

I wasn't aware before of the fact discussed in section 3.3 of that paper and
proved in the following sections, that the BRST cohomology of the
topological string coincides with the cohomology of the corresponding
(1-)algebroid that comes with the generalized complex structure of the
target.

That's a powerful statement, it seems.

It is kind of tempting to speculate how this could be a special case of a
much more general statement:

We can generalize the given 1-algebroid to a p-algebroid, roughly by
including 3-form twists, 4-form twists, and so on. All of them have a dual
description in terms of a differential graded algebra, as I have recently
summarized here:

http://golem.ph.utexas.edu/string/archives/000565.html

Hence to all of them we can associate a nilpotent Q and compute its
cohomology.

I wonder if we can associate BRST operators of more general topological
field theories with all these. Since 2-algebroids are to 1-algebroids as
membranes are to strings, maybe there should be a topological membrane
theory?


Here is a more down-to-earth question:

As far as I am aware the generalization to generalized complex geometry in
string physics so far takes place only after we have switched to the
topological string in the frist place.

Is there any way to identify the degrees of freedom that the generalized
complex structure comes from for the physical (untwisted) string?
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