on gerbes and boundary states [Archive]

Charlie Stromeyer Jr.

Jun17-04, 03:41 AM

<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>Charlie Stromeyer Jr. wrote:\n\n> Urs, since the above Hashimoto paper is about D-branes and non-abelian\n> YM you might want to re-read what John Baez, Aaron Bergman, Thomas\n> Larsson, Squark, some mathematicians from Australia and I talked about\n> before in sci.physics.research on categorical non-abelian YM and the\n> relationship between p-branes and (non-abelian) n-gerbes and related\n> NA cohomology.\n\nUrs wrote:\n\n> Could you give a brief summary of that discusion, sketch some relevant\n> ideas?\n\nI\'m not sure if I can give you a good enough summary because posts\nover 5,000 bytes are currently not working for this newsgroup.\nHowever, others have written very good introductions to the concept of\ngerbe and n-gerbe. Since you are already an expert on differential\nforms and string theory you may want to start by looking at the paper\n"Introduction to Gerbes on Orbifolds" which is also intended to be a\n"gentle" intro for physicists:\n\nhttp://arxiv.org/abs/math/0402318\n\nYou might also look at "Introduction to the Language of Stacks and\nGerbes" because even though this paper is intended for topologists it\nalso has some useful introductory exposition:\n\nhttp://arxiv.org/abs/math/0212266\n\nAlso, you may first want to read the even easier intro to torsors\nwritten by John Baez:\n\nhttp://math.ucr.edu/home/baez/torsors.html\n\nJohn Baez has also written some other intro exposition about related\ntopics which can be found on his website and in TWF and in s.p.r.\n\nFor a very brief discussion about p-branes and n-gerbes see Section\n12, "Applications to string theory" on page 11 of this paper by J.\nPeter May:\n\nhttp://www.math.uchicago.edu/~may/NCATS/ForWeb.pdf\n\nand for more discussion about branes and gerbes see e.g. this paper\nwhich is less difficult than papers such as the one by Breen and\nMessing:\n\nhttp://arxiv.org/abs/hep-th/0002074\n\n\nSome of what we discussed earlier in s.p.r. is probably too\nmathematically abstract or not defined well enough yet to be of much\nuse for you.\n\nUrs wrote furthermore:\n\n> I haven\'t followed it in any detail, mainly because I couldn\'t\n> understand it at that time. If you can tell me about a concrete\n> relation between boundary states and n-gerbes you\'ll immediately\n> have my entire attention! :-)\n\nActually, you should instead save some of your attention for your\nsignificant other so that you do not end up making the same mistake\nthat Einstein and many others have made !-)\n\nSeriously, it will take me at least a few days to catch up more with\nthe topic of gerbes and to think more about a potential connection\nwith boundary states because I have not really thought much about\ngerbes in almost two years. (However, I would not be surprised if\nthese two concepts should be related somehow because I believe that\nall of the major aspects of string theory should be inter-related\nbecause otherwise the consistency and unity of string theory would\nimplode.)\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>Charlie Stromeyer Jr. wrote:

> Urs, since the above Hashimoto paper is about D-branes and non-abelian
> YM you might want to re-read what John Baez, Aaron Bergman, Thomas
> Larsson, Squark, some mathematicians from Australia and I talked about
> before in sci.physics.research on categorical non-abelian YM and the
> relationship between p-branes and (non-abelian) n-gerbes and related
> NA cohomology.

Urs wrote:

> Could you give a brief summary of that discusion, sketch some relevant
> ideas?

I'm not sure if I can give you a good enough summary because posts
over 5,000 bytes are currently not working for this newsgroup.
However, others have written very good introductions to the concept of
gerbe and n-gerbe. Since you are already an expert on differential
forms and string theory you may want to start by looking at the paper
"Introduction to Gerbes on Orbifolds" which is also intended to be a
"gentle" intro for physicists:

http://arxiv.org/abs/math/0402318

You might also look at "Introduction to the Language of Stacks and
Gerbes" because even though this paper is intended for topologists it
also has some useful introductory exposition:

http://arxiv.org/abs/math/0212266

Also, you may first want to read the even easier intro to torsors
written by John Baez:

http://math.ucr.edu/home/baez/torsors.html

John Baez has also written some other intro exposition about related
topics which can be found on his website and in TWF and in s.p.r.

For a very brief discussion about p-branes and n-gerbes see Section
12, "Applications to string theory" on page 11 of this paper by J.
Peter May:

http://www.math.uchicago.edu/~may/NCATS/ForWeb.pdf

and for more discussion about branes and gerbes see e.g. this paper
which is less difficult than papers such as the one by Breen and
Messing:

http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0002074

Some of what we discussed earlier in s.p.r. is probably too
mathematically abstract or not defined well enough yet to be of much
use for you.

Urs wrote furthermore:

> I haven't followed it in any detail, mainly because I couldn't
> understand it at that time. If you can tell me about a concrete
> relation between boundary states and n-gerbes you'll immediately
> have my entire attention! :-)

Actually, you should instead save some of your attention for your
significant other so that you do not end up making the same mistake
that Einstein and many others have made !-)

Seriously, it will take me at least a few days to catch up more with
the topic of gerbes and to think more about a potential connection
with boundary states because I have not really thought much about
gerbes in almost two years. (However, I would not be surprised if
these two concepts should be related somehow because I believe that
all of the major aspects of string theory should be inter-related
because otherwise the consistency and unity of string theory would
implode.)

Urs Schreiber

Jun17-04, 08:16 AM

<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>[I have a couple of TeX formulas in the following message. They become\nreadable by viewing this message from within PhysicsForums:\nhttp://www.physicsforums.com/showthread.php?t=31416. ]\n\n\n"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag\nnews:61773ed7.0406161445.3ae065ad-100000@posting.google.com...\n\n> and for more discussion about branes and gerbes see e.g. this paper\n> which is less difficult than papers such as the one by Breen and\n> Messing:\n>\n> http://arxiv.org/abs/hep-th/0002074\n\n\n[Yonatan Zunger: p-Gerbes and Extended Objects in String Theory]\n\nThanks, I am currently looking at this one.\n\nI have toyed around with some loop space (Omega^1(X)) notions in the past\n(hep-th/0401175, hep-th/0311064) because there is some nice magic going on\nwhen the super-Virasoro constraints are manifestly written as deformed\nDirac-Kaehler operators on loop space. In particular this allows a nice\nderivation of the boundary state for D-branes with gauge fields turned on\n(possibly other stuff, too).\n\nI am not a mathematicians but (at best) a physicist, therefore my approach\nto loop space was very physicist-like (i.e. naive and prone to failures but\ngood for practical calculation as long as no subtleties bite you ;-) in that\nI essentially noted that I can think of _parameterized_ loop space as a\nmanifold with infinitely many coordinate functions\n\nX^\\mu(\\sigma)\n\nfor all \\mu wrt the target space and all values \\sigma \\in (0,2\\pi) along a\nloop, i.e. that when I just introduce a multi-index I=(\\mu,\\sigma) my set of\ncoordinates reads\n\n{X^I} = {X^(\\mu,\\sigma)} = {X^\\mu(\\sigma)}\n\nand that using this I can pretty much adapt the standard formulas of\ndifferential geometry for finite dimensional manifolds to the infinite\ndimensional loop space.\n\nFor instance denoting by\n\n\\partial_I = \\partial_{(\\mu,\\sigma)} = \\frac{\\delta}{\\delta X^\\mu(\\sigma)}\n\nthe functional derivative by X^\\mu(\\sigma) and writing\n\nE^{\\dagger (\\mu,\\sigma)} and E^{(\\mu,\\sigma)}\n\nfor the operators of differential form creation/annihilation on loop space,\ni.e.\n\nE^{\\dagger I} f := dX^I /\\ f,\n\nthe exterior derivative on parameterized loop space is just\n\nd = E^{\\dagger I}\\partial_I .\n\nIn string theory the important object is actually the deformed exterior\nderivative\n\nd_K = d + i K-> = E^{\\dagger I}\\partial_I + i E_I K^I ,\n\nwhere K is a vector field on loop space with components\n\nK^I = T X^{\\prime I},\n\n(where the prime indicates the \\sigma-derivative)\n\nwhich is nothing but the Killing vector which generates reparameterizations.\n\n(All this applies to the space of unparameterized loops simply by\nrestricting everything above to rep invariant functions and p-forms on the\nparameterized loop space.)\n\n\n\nAnyway, this formalism makes it possible to translate some of the abstract\nnotation in the above paper into pretty simple-looking statements.\n\nFor instance consider the covariant 1-gerbe exterior derivative discussed on\np.5. (I consider just the cas p=1, because apparently all other cases can be\nreduced to that one by iteration.)\n\nSo let be B = (1/2) B_{\\mu\\nu}(x) dx^\\mu /\\ dx^\\nu be a 2-form on spacetime.\nWe can construct a corresponding 2 form on loop space simply by lifting:\n\nB_{(\\mu,\\sigma)(\\nu,\\kappa)} = \\delta(\\sigma,\\kappa)B_{\\mu\\nu}(X(\\sigma)) .\n\nBut that\'s not quite what we want, since by general abstract nonsense the\n2-form on spacetime X should give rise to a 1-form on loop space over X. So\nwe need to contract the abobe 2-form with something. The only "god given"\nvector field in sight is K. So we get a 1-form with component\n\nC_I = B_{IJ}K^J = B_{\\mu\\nu}(X(\\sigma))X^{\\prime \\nu}(\\sigma) .\n\nThis is a quick way to see what Zunger discusses in the last complete\nparagraph on p. 14.\n\nThe covariant exterior derivative in equation (8) of the above paper is now\nobtained precisely like any old covariant exterior derivative for gauge\nconnection C in finite dimensions, it is simply\n\n\\nabla = d + C = E^{\\dagger I}( \\partial_I + C_I ) .\n\nIncidentally, this are the first two terms in the combined left and right\nmoving worldsheet supercharge of the string in a B field background, (when\nthese are written in terms of canonical momenta P_I = -i \\partial_I). Note\nthat the connection term comes from conjugating the reparameterization inner\nproduct\n\nE_I K^I\n\nwith the exponential B :\n\nE^{\\dagger I} C_I = exp( B ) E_I K^I exp( -B ) .\n\n(The same conjugation with the first term in the deformed exterior\nderivative on loop space gives the B-field strength as on loop space.)\n\nSo next Yonatan Zunger discusses transformations of functions f(\\omega) \\to\nf(\\omega + \\delta \\omega) on loop space wrt that covariant exterior\nderivative. What is this \\delta \\omega ? Well, just using our naive notation\nknwon from finite dimensional differential geometry \\delta \\omega is nothing\nbut a vector at \\omega with components\n\n(\\delta \\omega)^{I} .\n\nThe covariant Lie derivative L wrt this vector is, by the usual formula\n\nL_v = {d,v\\rightharpoonup}\n\njust\n\n(\\delta \\omega)^I (\\partial_I + C_I) .\n\nWhy can Zunger write the second term as the integral of B over a 2-chain?\nSimply because\n\n(\\delta \\omega)^I C_I\n= (\\delta \\omega)^I B_{IJ}K^J\n= \\int d\\sigma d\\kappa\n\\delta{\\sigma,\\kappa} (\\delta \\omega)^\\mu(X(\\sigma)) B_{\\mu\\nu}X(\\sigma)\nX^{\\prime \\nu}(\\sigma)\n\nwhere summation over the continuou index \\sigma translates into integrals.\n\n\n\n\n\n\nWhat I don\'t understand yet is how for instance B-fields arise in string\ntheory which take values in a _non_-abelian group. Hm, how does that work?\nOf course B really mixes with the (non-abelian) gauge field A in that only\nthe combination B-(dA)/T is physically meaningful.But the non-abelian\nproperty of A is related to the fact that this comes from the open string\nsector, where stings carry Chan-Paton factors, while B comes from the closed\nstring sector without such factors. Hm. Apparently I am missing an\nelementary insight here.\n\n\n\n\n> Some of what we discussed earlier in s.p.r. is probably too\n> mathematically abstract or not defined well enough yet to be of much\n> use for you.\n\nSuppose a D-brane carries some B-field charge. Would that even affect the\nboundary condition of open strings propagating on that brane? I.e., would it\naffect the boundary state?\n\n> Actually, you should instead save some of your attention for your\n> significant other so that you do not end up making the same mistake\n> that Einstein and many others have made !-)\n\n\nWell, I would already consider myself lucky if I ever made mistakes like\nEinstein did. ;-)\n\n\n> Seriously, it will take me at least a few days to catch up more with\n> the topic of gerbes and to think more about a potential connection\n> with boundary states because I have not really thought much about\n> gerbes in almost two years. (However, I would not be surprised if\n> these two concepts should be related somehow because I believe that\n> all of the major aspects of string theory should be inter-related\n> because otherwise the consistency and unity of string theory would\n> implode.)\n\nSure they must be related. It is just that in order to get a hook into this\nvast topic I would like to see some concrete connection, in terms of a\nspecial case or something, between the new stuff and the stuff that I\nalready (more or less) understand.\n\nThat\'s why first of all I would like to understand if p-gerbe\ntheory/K-theory has any connection to boundary states of the respective\nbranes. (Probably a dumb question, but I need to start somewhere.)\n\n\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>[I have a couple of TeX formulas in the following message. They become
readable by viewing this message from within PhysicsForums:
http://www.physicsforums.com/showthread.php?t=31416. ]

"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag
news:61773ed7.0406161445.3ae065ad-100000@posting.google.com...

> and for more discussion about branes and gerbes see e.g. this paper
> which is less difficult than papers such as the one by Breen and
> Messing:
>
> http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0002074

[Yonatan Zunger: p-Gerbes and Extended Objects in String Theory]

Thanks, I am currently looking at this one.

I have toyed around with some loop space (\Omega^1(X)) notions in the past
(http://www.arxiv.org/abs/hep-th/0401175, http://www.arxiv.org/abs/hep-th/0311064) because there is some nice magic going on
when the super-Virasoro constraints are manifestly written as deformed
Dirac-Kaehler operators on loop space. In particular this allows a nice
derivation of the boundary state for D-branes with gauge fields turned on
(possibly other stuff, too).

I am not a mathematicians but (at best) a physicist, therefore my approach
to loop space was very physicist-like (i.e. naive and prone to failures but
good for practical calculation as long as no subtleties bite you ;-) in that
I essentially noted that I can think of _parameterized_ loop space as a
manifold with infinitely many coordinate functions

X^\mu(\sigma)

for all \mu wrt the target space and all values \sigma \in (0,2\pi) along a
loop, i.e. that when I just introduce a multi-index I=(\mu,\sigma) my set of
coordinates reads

{X^I} = {X^(\mu,\sigma)} = {X^\mu(\sigma)}

and that using this I can pretty much adapt the standard formulas of
differential geometry for finite dimensional manifolds to the infinite
dimensional loop space.

For instance denoting by

\partial_I = \partial_{(\mu,\sigma)} = \frac{\delta}{\delta X^\mu(\sigma)}

the functional derivative by X^\mu(\sigma) and writing

E^{\dagger (\mu,\sigma)}[/itex] and [itex]E^{(\mu,\sigma)}

for the operators of differential form creation/annihilation on loop space,
i.e.

E^{\dagger I} f := dX^I /\ f,

the exterior derivative on parameterized loop space is just

d = E^{\dagger I}\partial_I .

In string theory the important object is actually the deformed exterior
derivative

d_K = d + i K-> = E^{\dagger I}\partial_I + i E_I K^I ,

where K is a vector field on loop space with components

K^I = T X^{\prime I},

(where the prime indicates the \sigma-derivative)

which is nothing but the Killing vector which generates reparameterizations.

(All this applies to the space of unparameterized loops simply by
restricting everything above to rep invariant functions and p-forms on the
parameterized loop space.)

Anyway, this formalism makes it possible to translate some of the abstract
notation in the above paper into pretty simple-looking statements.

For instance consider the covariant 1-gerbe exterior derivative discussed on
p.5. (I consider just the cas p=1, because apparently all other cases can be
reduced to that one by iteration.)

So let be B = (1/2) B_{\mu\nu}(x) dx^\mu /\ dx^\nu be a 2-form on spacetime.
We can construct a corresponding 2 form on loop space simply by lifting:

B_{(\mu,\sigma)(\nu,\kappa)} = \delta(\sigma,\kappa)B_{\mu\nu}(X(\sigma)) .

But that's not quite what we want, since by general abstract nonsense the
2-form on spacetime X should give rise to a 1-form on loop space over X. So
we need to contract the abobe 2-form with something. The only "god given"
vector field in sight is K. So we get a 1-form with component

C_I = B_{IJ}K^J = B_{\mu\nu}(X(\sigma))X^{\prime \nu}(\sigma) .

This is a quick way to see what Zunger discusses in the last complete
paragraph on p. 14.

The covariant exterior derivative in equation (8) of the above paper is now
obtained precisely like any old covariant exterior derivative for gauge
connection C in finite dimensions, it is simply

\nabla = d + C = E^{\dagger I}( \partial_I + C_I ) .

Incidentally, this are the first two terms in the combined left and right
moving worldsheet supercharge of the string in a B field background, (when
these are written in terms of canonical momenta P_I = -i \partial_I). Note
that the connection term comes from conjugating the reparameterization inner
product

E_I K^I

with the exponential B :

E^{\dagger I} C_I = \exp( B ) E_I K^I \exp( -B ) .

(The same conjugation with the first term in the deformed exterior
derivative on loop space gives the B-field strength as on loop space.)

So next Yonatan Zunger discusses transformations of functions f(\omega) \tof(\omega + \delta \omega) on loop space wrt that covariant exterior
derivative. What is this \delta \omega ? Well, just using our naive notation
knwon from finite dimensional differential geometry \delta \omega is nothing
but a vector at \omega with components

(\delta \omega)^{I} .

The covariant Lie derivative L wrt this vector is, by the usual formula

L_v = {d,v\rightharpoonup}

just

(\delta \omega)^I (\partial_I + C_I) .

Why can Zunger write the second term as the integral of B over a 2-chain?
Simply because

(\delta \omega)^I C_I= (\delta \omega)^I B_{IJ}K^J= \int d\sigma d\kappa\delta{\sigma,\kappa} (\delta \omega)^\mu(X(\sigma)) B_{\mu\nu}X(\sigma)X^{\prime \nu}(\sigma)

where summation over the continuou index \sigma translates into integrals.

What I don't understand yet is how for instance B-fields arise in string
theory which take values in a _non_-abelian group. Hm, how does that work?
Of course B really mixes with the (non-abelian) gauge field A in that only
the combination B-(dA)/T is physically meaningful.But the non-abelian
property of A is related to the fact that this comes from the open string
sector, where stings carry Chan-Paton factors, while B comes from the closed
string sector without such factors. Hm. Apparently I am missing an
elementary insight here.

> Some of what we discussed earlier in s.p.r. is probably too
> mathematically abstract or not defined well enough yet to be of much
> use for you.

Suppose a D-brane carries some B-field charge. Would that even affect the
boundary condition of open strings propagating on that brane? I.e., would it
affect the boundary state?

> Actually, you should instead save some of your attention for your
> significant other so that you do not end up making the same mistake
> that Einstein and many others have made !-)

Well, I would already consider myself lucky if I ever made mistakes like
Einstein did. ;-)

> Seriously, it will take me at least a few days to catch up more with
> the topic of gerbes and to think more about a potential connection
> with boundary states because I have not really thought much about
> gerbes in almost two years. (However, I would not be surprised if
> these two concepts should be related somehow because I believe that
> all of the major aspects of string theory should be inter-related
> because otherwise the consistency and unity of string theory would
> implode.)

Sure they must be related. It is just that in order to get a hook into this
vast topic I would like to see some concrete connection, in terms of a
special case or something, between the new stuff and the stuff that I
already (more or less) understand.

That's why first of all I would like to understand if p-gerbe
theory/K-theory has any connection to boundary states of the respective
branes. (Probably a dumb question, but I need to start somewhere.)

Urs Schreiber

Jun17-04, 02:40 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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag\nnews:61773ed7.0406161445.3ae065ad-100000@posting.google.com...\n\n<On boundary states and p-gerbes>\n\n> However, I would not be surprised if\n> these two concepts should be related somehow\n\nSo how is the charge of the brane encoded in the boundary state? The answer\nis simply: In its normalization factor! (Not too surprising, actually...)\n\nHere are some nice lecture notes on boundary state formalism:\n\nP. Di Vecchia & A. Licardo:\nD-branes in string theory, I & II\nhep-th/9912161 & hep-th/9912275\n\nSection 3 of part II discusses the couplig of the boundary state of the\nbrane with form field excitations. There enters a normalization factor N(F)\n(F is the gauge field on the brane) in the definition (3.62) of the\nbosonic-matter part of the boundary state and another factor k(F) (equation\n(3.67)) in the fermionic-(RR)-matter sector.\n\nComputing the coupling of that boundary state to the RR form field as in\nequation (3.77) yields the result (3.78) from which one reads off the\nD-brane charge as\n\nQ = \\sqrt{2} T N(F) k(F) .\n\nQuantization of these charges is discussed in section 5, e.g. equation (5.9)\nin the context of bound states of F string with Dp branes, which is nothing\nbut the Dp-brane with a couple of units of electric charge turned on (as one\ncan easily check alternatively using some canonical gymnastics in the case\nof bound states with D1-branes:\nhttp://golem.ph.utexas.edu/string/archives/000288.html)\n\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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag
news:61773ed7.0406161445.3ae065ad-100000@posting.google.com...

<On boundary states and p-gerbes>

> However, I would not be surprised if
> these two concepts should be related somehow

So how is the charge of the brane encoded in the boundary state? The answer
is simply: In its normalization factor! (Not too surprising, actually...)

Here are some nice lecture notes on boundary state formalism:

P. Di Vecchia & A. Licardo:
D-branes in string theory, I & II
http://www.arxiv.org/abs/hep-th/9912161 & http://www.arxiv.org/abs/hep-th/9912275

Section 3 of part II discusses the couplig of the boundary state of the
brane with form field excitations. There enters a normalization factor N(F)
(F is the gauge field on the brane) in the definition (3.62) of the
bosonic-matter part of the boundary state and another factor k(F) (equation
(3.67)) in the fermionic-(RR)-matter sector.

Computing the coupling of that boundary state to the RR form field as in
equation (3.77) yields the result (3.78) from which one reads off the
D-brane charge as

Q = \sqrt{2} T N(F) k(F) .

Quantization of these charges is discussed in section 5, e.g. equation (5.9)
in the context of bound states of F string with Dp branes, which is nothing
but the Dp-brane with a couple of units of electric charge turned on (as one
can easily check alternatively using some canonical gymnastics in the case
of bound states with D1-branes:
http://golem.ph.utexas.edu/string/archives/000288.html)

Charlie Stromeyer Jr.

Jun17-04, 05:49 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:\n\n> Anyway, this formalism makes it possible to translate some of the abstract\n> notation in the above paper into pretty simple-looking statements.\n\nWhat you say here is interesting. Also, what is a simple intro to the\nconcept of Pohlmeyer invariants?\n\n> What I don\'t understand yet is how for instance B-fields arise in string\n> theory which take values in a _non_-abelian group. Hm, how does that work?\n> Of course B really mixes with the (non-abelian) gauge field A in that only\n> the combination B-(dA)/T is physically meaningful.But the non-abelian\n> property of A is related to the fact that this comes from the open string\n> sector, where stings carry Chan-Paton factors, while B comes from the closed\n> string sector without such factors. Hm. Apparently I am missing an\n> elementary insight here.\n\n> That\'s why first of all I would like to understand if p-gerbe\n> theory/K-theory has any connection to boundary states of the respective\n> branes. (Probably a dumb question, but I need to start somewhere.)\n\nQuite a few papers have been written addressing these questions, e.g.\n[hep-th/9907140] discusses the relationship between D9 boundary states\nand K-theory. There are also other papers [1] which examine how the\nboundary state formalism is related to K-Theory and gerbes.\n\nVarious different approaches or even answers have been proposed for\nyour question about the B-field and NA groups [2]. There are even NC\ngerbes defined via star products [3], and gerbes based upon quantum\ngroups [4]. I suggest that you first read the abstracts of each of\nthese papers to see which of them will interest you the most.\n\n\n[1] hep-th/0110041, 0107051, 0007175, 9903123\n\n[2] hep-th/0405074, 0110133, 0111018, 0002023, 9909089, 9910048,\n0106110, 0104153, 0008191, 0312154\n\n[3] hep-th/0206101\n\n[4] math/0308235\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:

> Anyway, this formalism makes it possible to translate some of the abstract
> notation in the above paper into pretty simple-looking statements.

What you say here is interesting. Also, what is a simple intro to the
concept of Pohlmeyer invariants?

> What I don't understand yet is how for instance B-fields arise in string
> theory which take values in a _non_-abelian group. Hm, how does that work?
> Of course B really mixes with the (non-abelian) gauge field A in that only
> the combination B-(dA)/T is physically meaningful.But the non-abelian
> property of A is related to the fact that this comes from the open string
> sector, where stings carry Chan-Paton factors, while B comes from the closed
> string sector without such factors. Hm. Apparently I am missing an
> elementary insight here.

> That's why first of all I would like to understand if p-gerbe
> theory/K-theory has any connection to boundary states of the respective
> branes. (Probably a dumb question, but I need to start somewhere.)

Quite a few papers have been written addressing these questions, e.g.
[http://www.arxiv.org/abs/hep-th/9907140] discusses the relationship between D9 boundary states
and K-theory. There are also other papers [1] which examine how the
boundary state formalism is related to K-Theory and gerbes.

Various different approaches or even answers have been proposed for
your question about the B-field and NA groups [2]. There are even NC
gerbes defined via star products [3], and gerbes based upon quantum
groups [4]. I suggest that you first read the abstracts of each of
these papers to see which of them will interest you the most.

[1] http://www.arxiv.org/abs/hep-th/0110041, 0107051, 0007175, 9903123

[2] http://www.arxiv.org/abs/hep-th/0405074, 0110133, 0111018, 0002023, 9909089, 9910048,
0106110, 0104153, 0008191, 0312154

[3] http://www.arxiv.org/abs/hep-th/0206101

[4] math/0308235

Charlie Stromeyer Jr.

Jun18-04, 01:06 AM

<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:\n\n> "Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag\n> news:61773ed7.0406161445.3ae065ad-100000@posting.google.com...\n>\n> <On boundary states and p-gerbes>\n>\n> > However, I would not be surprised if\n> > these two concepts should be related somehow\n>\n> So how is the charge of the brane encoded in the boundary state? The answer\n> is simply: In its normalization factor! (Not too surprising, actually...)\n\nRight, and also see my last post for more about the relationship\nbetween boundary states and gerbes. I could also refer you to further\nexamples if you want, however, what I really mean by n-gerbes or\np-gerbes should also include 3-gerbes and higher rather than merely\nonly 1- or 2-gerbes.\n\nIdeally, we should want a "tower" of non-abelian n-gerbes (including\nthe infinity-gerbe). I am going to think more about NA, NC or quantum\ngerbes in addition to reading at least some of these two papers by A.\nTsemo.\n\nIn the first paper he defines a tower of n-gerbes, but even though\nthis is abelian it might give us some hints or ideas for how to\nattempt something analogous for the non-abelian case. Since I\'m going\nto try to focus on the math perhaps you would also like to view this\nfirst paper to see if it reminds you of anything from\nmathematical/theoretical physics:\n\n\nhttp://arxiv.org/abs/math/0105201\n\nhttp://arxiv.org/abs/math/0301271\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:

> "Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag
> news:61773ed7.0406161445.3ae065ad-100000@posting.google.com...
>
> <On boundary states and p-gerbes>
>
> > However, I would not be surprised if
> > these two concepts should be related somehow
>
> So how is the charge of the brane encoded in the boundary state? The answer
> is simply: In its normalization factor! (Not too surprising, actually...)

Right, and also see my last post for more about the relationship
between boundary states and gerbes. I could also refer you to further
examples if you want, however, what I really mean by n-gerbes or
p-gerbes should also include 3-gerbes and higher rather than merely
only 1- or 2-gerbes.

Ideally, we should want a "tower" of non-abelian n-gerbes (including
the infinity-gerbe). I am going to think more about NA, NC or quantum
gerbes in addition to reading at least some of these two papers by A.
Tsemo.

In the first paper he defines a tower of n-gerbes, but even though
this is abelian it might give us some hints or ideas for how to
attempt something analogous for the non-abelian case. Since I'm going
to try to focus on the math perhaps you would also like to view this
first paper to see if it reminds you of anything from
mathematical/theoretical physics:

http://arxiv.org/abs/math/0105201

http://arxiv.org/abs/math/0301271

Urs Schreiber

Jun18-04, 03:22 AM

<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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag\nnews:61773ed7.0406171448.7e76f0da-100000@posting.google.com...\n\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n>\n> > Anyway, this formalism makes it possible to translate some of the\nabstract\n> > notation in the above paper into pretty simple-looking statements.\n>\n> What you say here is interesting. Also, what is a simple intro to the\n> concept of Pohlmeyer invariants?\n\n\nTo my mind the simplest introduction to their technical definition is\nhep-th/0403260 :-).\n\nWhen I originally learned about Pohlmeyer invariants it was from\nhep-th/0401172, which has a nice summary of existing literature. There the\nLax pair method to derive them is discussed. I find that unnecessarily heavy\nmachinery for such a simple task. After all, Pohlmeyer invariants are\nnothing but Wilson loops of large constant gauge field matrices along the\nclosed string. That essentially already says everything about them.\n\nHistorically, the Pohlmeyer invariants are connected with an attempt to find\nan \'alternative quantization\' of the string\n(http://golem.ph.utexas.edu/string/archives/000299.html,\nhttp://golem.ph.utexas.edu/string/archives/000338.html).\n\nI don\'t think that this is promising. Instead, I think the Pohlmeyer\noperators play a role in deformations of boundary states:\nhttp://golem.ph.utexas.edu/string/archives/000382.html .\n\nThis is unsurprising, given that boundary states for D-branes with gauge\nfields turned on are generally charcterized to the Wilson loop of these\ngauge fields along the closed string, e.g hep-th/0312260 .\n\n> > That\'s why first of all I would like to understand if p-gerbe\n> > theory/K-theory has any connection to boundary states of the respective\n> > branes. (Probably a dumb question, but I need to start somewhere.)\n>\n> Quite a few papers have been written addressing these questions, e.g.\n> [hep-th/9907140] discusses the relationship between D9 boundary states\n> and K-theory. There are also other papers [1] which examine how the\n> boundary state formalism is related to K-Theory and gerbes.\n\nThanks a lot for these references. I will have a look at them and then get\nback to you with further questions. :-)\n\n\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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag
news:61773ed7.0406171448.7e76f0da-100000@posting.google.com...

> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
>
> > Anyway, this formalism makes it possible to translate some of the
abstract
> > notation in the above paper into pretty simple-looking statements.
>
> What you say here is interesting. Also, what is a simple intro to the
> concept of Pohlmeyer invariants?

To my mind the simplest introduction to their technical definition is
http://www.arxiv.org/abs/hep-th/0403260 :-).

When I originally learned about Pohlmeyer invariants it was from
http://www.arxiv.org/abs/hep-th/0401172, which has a nice summary of existing literature. There the
Lax pair method to derive them is discussed. I find that unnecessarily heavy
machinery for such a simple task. After all, Pohlmeyer invariants are
nothing but Wilson loops of large constant gauge field matrices along the
closed string. That essentially already says everything about them.

Historically, the Pohlmeyer invariants are connected with an attempt to find
an 'alternative quantization' of the string
(http://golem.ph.utexas.edu/string/archives/000299.html,
http://golem.ph.utexas.edu/string/archives/000338.html).

I don't think that this is promising. Instead, I think the Pohlmeyer
operators play a role in deformations of boundary states:
http://golem.ph.utexas.edu/string/archives/000382.html .

This is unsurprising, given that boundary states for D-branes with gauge
fields turned on are generally charcterized to the Wilson loop of these
gauge fields along the closed string, e.g http://www.arxiv.org/abs/hep-th/0312260 .

> > That's why first of all I would like to understand if p-gerbe
> > theory/K-theory has any connection to boundary states of the respective
> > branes. (Probably a dumb question, but I need to start somewhere.)
>
> Quite a few papers have been written addressing these questions, e.g.
> [http://www.arxiv.org/abs/hep-th/9907140] discusses the relationship between D9 boundary states
> and K-theory. There are also other papers [1] which examine how the
> boundary state formalism is related to K-Theory and gerbes.

Thanks a lot for these references. I will have a look at them and then get
back to you with further questions. :-)

Urs Schreiber

Jun18-04, 04:00 AM

<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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag\nnews:61773ed7.0406171926.27f1172e-100000@posting.google.com...\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n>\n> > "Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag\n> > news:61773ed7.0406161445.3ae065ad-100000@posting.google.com...\n> >\n> > <On boundary states and p-gerbes>\n> >\n> > > However, I would not be surprised if\n> > > these two concepts should be related somehow\n> >\n> > So how is the charge of the brane encoded in the boundary state? The\nanswer\n> > is simply: In its normalization factor! (Not too surprising,\nactually...)\n>\n> Right, and also see my last post for more about the relationship\n> between boundary states and gerbes. I could also refer you to further\n> examples if you want, however, what I really mean by n-gerbes or\n> p-gerbes should also include 3-gerbes and higher rather than merely\n> only 1- or 2-gerbes.\n\nOk. I was focusing on 1-gerbes in my post because these I felt could most\nreadily be understood in rather elementary terms and also because I got the\nimpression that they already capture the essential ideas.\n\n\n> Ideally, we should want a "tower" of non-abelian n-gerbes (including\n> the infinity-gerbe).\n\nHm, would this still have any relation to string theory? Wouldn\'t string\ntheory stop at p=9 (p=25) gerbes? Indeed: Is there any way to understand the\ncritical dimension from within gerbes/K-theory? (All I know about K-theory\nis essentially what Aaron Bergman has written on s.p.r. over the times, e.g.\nhttp://groups.google.de/groups?selm=abergman-D673A1.20072726102000%40cnn.princeton.edu).\n\n\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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag
news:61773ed7.0406171926.27f1172e-100000@posting.google.com...
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
>
> > "Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag
> > news:61773ed7.0406161445.3ae065ad-100000@posting.google.com...
> >
> > <On boundary states and p-gerbes>
> >
> > > However, I would not be surprised if
> > > these two concepts should be related somehow
> >
> > So how is the charge of the brane encoded in the boundary state? The
answer
> > is simply: In its normalization factor! (Not too surprising,
actually...)
>
> Right, and also see my last post for more about the relationship
> between boundary states and gerbes. I could also refer you to further
> examples if you want, however, what I really mean by n-gerbes or
> p-gerbes should also include 3-gerbes and higher rather than merely
> only 1- or 2-gerbes.

Ok. I was focusing on 1-gerbes in my post because these I felt could most
readily be understood in rather elementary terms and also because I got the
impression that they already capture the essential ideas.

> Ideally, we should want a "tower" of non-abelian n-gerbes (including
> the infinity-gerbe).

Hm, would this still have any relation to string theory? Wouldn't string
theory stop at p=9 (p=25) gerbes? Indeed: Is there any way to understand the
critical dimension from within gerbes/K-theory? (All I know about K-theory
is essentially what Aaron Bergman has written on s.p.r. over the times, e.g.
http://groups.google.de/groups?selm=abergman-D673A1.20072726102000%40cnn.princeton.edu).

Charlie Stromeyer Jr.

Jun18-04, 12:31 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:\n\n> > Ideally, we should want a "tower" of non-abelian n-gerbes (including\n> > the infinity-gerbe).\n>\n> Hm, would this still have any relation to string theory? Wouldn\'t string\n> theory stop at p=9 (p=25) gerbes? Indeed: Is there any way to understand the\n> critical dimension from within gerbes/K-theory?\n\nI have only just started to think about this but I first want to\nconsider a "tower" approach for these four reasons:\n\n1) It might be easier and more consistent than trying to separately\ndefine concepts of 2-gerbes, 3-gerbes etc. which may be more ad hoc\nand much more difficult (especially without an adequate guiding\nprinciple from n-category theory).\n\n2) I just read the paper by E. Witten which I referred you to\n[hep-th/0007175] and so I am beginning to speculate about his\nconjecture which involves U(infinity)-valued Chan-Paton factors.\n\n3) There are other papers which I have not even begun to look at yet\nwhich discuss topics such as infinite towers of fields (or bound\nstates) or infinite number of charges.\n\n4) I am interested in any basic math ideas which involve an increasing\nsequence of X such that the "top" or "limit" is an infinite instance\nof X, e.g. pertaining to R. Bousso\'s question about a series of\nHilbert spaces.\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:

> > Ideally, we should want a "tower" of non-abelian n-gerbes (including
> > the infinity-gerbe).
>
> Hm, would this still have any relation to string theory? Wouldn't string
> theory stop at p=9 (p=25) gerbes? Indeed: Is there any way to understand the
> critical dimension from within gerbes/K-theory?

I have only just started to think about this but I first want to
consider a "tower" approach for these four reasons:

1) It might be easier and more consistent than trying to separately
define concepts of 2-gerbes, 3-gerbes etc. which may be more ad hoc
and much more difficult (especially without an adequate guiding
principle from n-category theory).

2) I just read the paper by E. Witten which I referred you to
[http://www.arxiv.org/abs/hep-th/0007175] and so I am beginning to speculate about his
conjecture which involves U(infinity)-valued Chan-Paton factors.

3) There are other papers which I have not even begun to look at yet
which discuss topics such as infinite towers of fields (or bound
states) or infinite number of charges.

4) I am interested in any basic math ideas which involve an increasing
sequence of X such that the "top" or "limit" is an infinite instance
of X, e.g. pertaining to R. Bousso's question about a series of
Hilbert spaces.

Urs Schreiber

Jun18-04, 01:43 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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag\nnews:61773ed7.0406180837.1b30a673-100000@posting.google.com...\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n>\n> > > Ideally, we should want a "tower" of non-abelian n-gerbes (including\n> > > the infinity-gerbe).\n> >\n> > Hm, would this still have any relation to string theory? Wouldn\'t string\n> > theory stop at p=9 (p=25) gerbes? Indeed: Is there any way to understand\nthe\n> > critical dimension from within gerbes/K-theory?\n>\n> I have only just started to think about this but I first want to\n> consider a "tower" approach for these four reasons:\n\n> 2) I just read the paper by E. Witten which I referred you to\n> [hep-th/0007175] and so I am beginning to speculate about his\n> conjecture which involves U(infinity)-valued Chan-Paton factors.\n\nOk, but this is the N \\to \\infty limit, where N is the number of branes, not\ntheir dimension. But a p-brane of p spatial dimensions corresponds to a\np-gerbe, and this necessarily stops at p=9 in susy string theory. So that\'s\nwhy I said that there seems to be no place for a (p=\\infty)-gerbe in string\ntheory, since that would correspond to a brane with an infinite number of\nspatial dimensions.\n\nOn the other hand, sending the number of branes N\\to \\infty certainly is a\nvery natural thing to do.\n\n> 3) There are other papers which I have not even begun to look at yet\n> which discuss topics such as infinite towers of fields (or bound\n> states) or infinite number of charges.\n\nOk, but not an infinite number of spacetime dimensions, I guess.\n\n> 4) I am interested in any basic math ideas which involve an increasing\n> sequence of X such that the "top" or "limit" is an infinite instance\n> of X, e.g. pertaining to R. Bousso\'s question about a series of\n> Hilbert spaces.\n\nI see, so then you are probably thinking about something along the lines of\np.6 of hep-th/0007175, I guess?\n\nOn the other hand, one must be careful, probably, to distinguish the Hilbert\nspaces here. The Hilbert space mentioned by Bousso is that of states of the\nTOE, whatever it is. The Hilbert space mentioned on that p.6, however, is\nthe Hilbert space on which we can imagine the matrices of the Matrix Model\nto be operators on. So here we do indeed have, as Bousso might seem to\nsuggest, a series of Hilber spaces of growing dimension as N\\to \\infty. But\nthe point is that the Hilbert space of states of our TOE is the Hilbert\nspace of states on the config space of these operators, i.e. something like\nthe space of L^2 functions over the space of operators on some seperable\nHilbert space. That one does not become finite when N is finite.\n\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>"Charlie Stromeyer Jr." <cstromey@hotmail.com> schrieb im Newsbeitrag
news:61773ed7.0406180837.1b30a673-100000@posting.google.com...
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
>
> > > Ideally, we should want a "tower" of non-abelian n-gerbes (including
> > > the infinity-gerbe).
> >
> > Hm, would this still have any relation to string theory? Wouldn't string
> > theory stop at p=9 (p=25) gerbes? Indeed: Is there any way to understand
the
> > critical dimension from within gerbes/K-theory?
>
> I have only just started to think about this but I first want to
> consider a "tower" approach for these four reasons:

> 2) I just read the paper by E. Witten which I referred you to
> [http://www.arxiv.org/abs/hep-th/0007175] and so I am beginning to speculate about his
> conjecture which involves U(infinity)-valued Chan-Paton factors.

Ok, but this is the N \to \infty limit, where N is the number of branes, not
their dimension. But a p-brane of p spatial dimensions corresponds to a
p-gerbe, and this necessarily stops at p=9 in susy string theory. So that's
why I said that there seems to be no place for a (p=\infty)-gerbe in string
theory, since that would correspond to a brane with an infinite number of
spatial dimensions.

On the other hand, sending the number of branes N\to \infty certainly is a
very natural thing to do.

> 3) There are other papers which I have not even begun to look at yet
> which discuss topics such as infinite towers of fields (or bound
> states) or infinite number of charges.

Ok, but not an infinite number of spacetime dimensions, I guess.

> 4) I am interested in any basic math ideas which involve an increasing
> sequence of X such that the "top" or "limit" is an infinite instance
> of X, e.g. pertaining to R. Bousso's question about a series of
> Hilbert spaces.

I see, so then you are probably thinking about something along the lines of
p.6 of http://www.arxiv.org/abs/hep-th/0007175, I guess?

On the other hand, one must be careful, probably, to distinguish the Hilbert
spaces here. The Hilbert space mentioned by Bousso is that of states of the
TOE, whatever it is. The Hilbert space mentioned on that p.6, however, is
the Hilbert space on which we can imagine the matrices of the Matrix Model
to be operators on. So here we do indeed have, as Bousso might seem to
suggest, a series of Hilber spaces of growing dimension as N\to \infty. But
the point is that the Hilbert space of states of our TOE is the Hilbert
space of states on the config space of these operators, i.e. something like
the space of L^2 functions over the space of operators on some seperable
Hilbert space. That one does not become finite when N is finite.

Charlie Stromeyer Jr.

Jun19-04, 05:16 AM

<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:\n\n> > 4) I am interested in any basic math ideas which involve an increasing\n> > sequence of X such that the "top" or "limit" is an infinite instance\n> > of X, e.g. pertaining to R. Bousso\'s question about a series of\n> > Hilbert spaces.\n>\n> I see, so then you are probably thinking about something along the lines of\n> p.6 of hep-th/0007175, I guess?\n\nRight. You are very aware that no one really understands what strings\nare, but whatever they are their mathematics has to make sense and it\nmay be preferable if this underlying math is fundamental and general\nenough so as to increase the likelihood of potential applications.\n\nI am starting to try to imagine what would happen if we attempt to\ngenralize the philosophy expressed in Witten\'s paper about beginning\nfrom an infinite case and condensing down to a more manageable case.\nFor this, my first priority will be to try to learn more about how to\nconstruct a general mathematical framework such that there is an\nincreasing sequence of X which converges to a limit that is an\ninfinite instance of X.\n\nRegarding Bousso\'s question, I might try to begin by considering the\nmost general possible concept of Hilbert space. For instance, it is\nknown that the concept of Hilbert space can be defined over R,C,H and\nO and this O case can be made to also obtain the usual associative\ncase (and of course R,C,H can be obtained from O).\n\nOr we might contemplate e.g. a mathematical framework that seems to\ninvolve only order structure and topology structure and then ask what\nseems to be the minimal amount of algebraic structure we need to add\nin order to obtain the concept of Hilbert space.\n\nI am not imagining ideas such as a D{infinity}-brane or the infinite\ncharges to be physically real but instead possibly as a conceptual\nstarting point for a perhaps more generalized way of thinking.\n\nSecondarily, I may also speculate some about potential applications of\nthis way of thinking, e.g. higher n-gerbes might be relevant for a\ntheory of higher dimensional duality which some believe would be\nuseful for better describing TQFT.\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:

> > 4) I am interested in any basic math ideas which involve an increasing
> > sequence of X such that the "top" or "limit" is an infinite instance
> > of X, e.g. pertaining to R. Bousso's question about a series of
> > Hilbert spaces.
>
> I see, so then you are probably thinking about something along the lines of
> p.6 of http://www.arxiv.org/abs/hep-th/0007175, I guess?

Right. You are very aware that no one really understands what strings
are, but whatever they are their mathematics has to make sense and it
may be preferable if this underlying math is fundamental and general
enough so as to increase the likelihood of potential applications.

I am starting to try to imagine what would happen if we attempt to
genralize the philosophy expressed in Witten's paper about beginning
from an infinite case and condensing down to a more manageable case.
For this, my first priority will be to try to learn more about how to
construct a general mathematical framework such that there is an
increasing sequence of X which converges to a limit that is an
infinite instance of X.

Regarding Bousso's question, I might try to begin by considering the
most general possible concept of Hilbert space. For instance, it is
known that the concept of Hilbert space can be defined over R,C,H and
O and this O case can be made to also obtain the usual associative
case (and of course R,C,H can be obtained from O).

Or we might contemplate e.g. a mathematical framework that seems to
involve only order structure and topology structure and then ask what
seems to be the minimal amount of algebraic structure we need to add
in order to obtain the concept of Hilbert space.

I am not imagining ideas such as a D{infinity}-brane or the infinite
charges to be physically real but instead possibly as a conceptual
starting point for a perhaps more generalized way of thinking.

Secondarily, I may also speculate some about potential applications of
this way of thinking, e.g. higher n-gerbes might be relevant for a
theory of higher dimensional duality which some believe would be
useful for better describing TQFT.