Background Independence

**d70yxj** · Aug10-04, 12:43 AM

Dear Forum,

As far as I can tell, the usual statement of background independence in string theory is that even though one chooses a specific metric (and other target space fields) in the string theory path integral, a particular choice of background field is exactly equivalent to any other choice - so long as one inserts appropriate vertex operators in the path integral. For example, the path integral with a curved metric is the same as the path integral with a flat target space AND a certain coherent state of gravitons inserted.

In this sense, two different background fields do not give two different theories, but can be thought of as two different states of the same string theory. Hopefully this is OK so far, and agrees with the usual interpretations.

If so, my question is as follows; suppose I consider a target space manifold with, say, a flat metric for simplicity, but with non-trivial homology groups. Can I think of the string theory path integral with this target space as just being the flat-space R^10 path integral but with an insertion of some appropriate string vertex operators?

In other words, is there an obvious way to think of two topologically-different target manifolds as corresponding to different states of the same string theory, just as there is for two different choices of background fields?

Sorry, I possibly haven't said this too clearly, but if anyone can put me straight it's just something that's been bugging me....

Aug12-04, 01:44 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>"d70yxj" <redbull_j@yahoo.com> schrieb im Newsbeitrag\nnews:d70yxj.1aqwm7-100000@physicsforums.com...\n\n> If so, my question is as follows; suppose I consider a target space\n> manifold with, say, a flat metric for simplicity, but with non-trivial\n> homology groups. Can I think of the string theory path integral with\n> this target space as just being the flat-space R^10 path integral but\n> with an insertion of some appropriate string vertex operators?\n\nNo, if the metric is flat, no vertex operators are inserted.\n\n> In other words, is there an obvious way to think of two\n> topologically-different target manifolds as corresponding to different\n> states of the same string theory, just as there is for two different\n> choices of background fields?\n\nI believe this is best thought of in terms of covering spaces. work in the\ncovering space (e.g. flat space for toroidal compactifications) and take\ncare of the correct identifications.\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>"d70yxj" <redbull_j@yahoo.com> schrieb im Newsbeitrag
news:d70yxj.1aqwm7-100000@physicsforums.com...

> If so, my question is as follows; suppose I consider a target space
> manifold with, say, a flat metric for simplicity, but with non-trivial
> homology groups. Can I think of the string theory path integral with
> this target space as just being the flat-space $LaTeX Code: R^{10}$ path integral but
> with an insertion of some appropriate string vertex operators?

No, if the metric is flat, no vertex operators are inserted.

> In other words, is there an obvious way to think of two
> topologically-different target manifolds as corresponding to different
> states of the same string theory, just as there is for two different
> choices of background fields?

I believe this is best thought of in terms of covering spaces. work in the
covering space (e.g. flat space for toroidal compactifications) and take
care of the correct identifications.

**d70yxj** · Aug15-04, 04:18 PM

Originally Posted by Urs Schreiber

> > In other words, is there an obvious way to think of two
> > topologically-different target manifolds as corresponding to different
> > states of the same string theory, just as there is for two different
> > choices of background fields?[/color]

> I believe this is best thought of in terms of covering spaces. work in the
> covering space (e.g. flat space for toroidal compactifications) and take
> care of the correct identifications.

So I do the path integral with a cover of my original manifold, and then do something (what?) at the end to get to the correct answer? This doesn't seem as obviously `stringy' as the way different background fields can be seen as string states sitting on a flat background.

Also, a covering space for a given manifold needn't be topologically trivial anyway. So it's not obvious to me that I can start with any target manifold and relate it back to R^10 using what you say.

Aug17-04, 01:00 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>( Redbull d70 wrote: )\n\n> In this sense, two different background fields do not give two\n> different theories, but can be thought of as two different states of\n> the same string theory. Hopefully this is OK so far, and agrees with\n> the usual interpretations.\n\nIt think that\'s fine so far.\n\n> If so, my question is as follows; suppose I consider a target space\n> manifold with, say, a flat metric for simplicity, but with non-trivial\n> homology groups. Can I think of the string theory path integral with\n> this target space as just being the flat-space R^10 path integral but\n> with an insertion of some appropriate string vertex operators?\n>\n> In other words, is there an obvious way to think of two\n> topologically-different target manifolds as corresponding to different\n> states of the same string theory, just as there is for two different\n> choices of background fields?\n\nWell, let\'s back off for a second. The Polyakov path integral\nprescription tells you to gauge fix as follows: Divide the space of\nall metrics that needs to be summed over in the path integral into\nequivalence classes, then (arbitrarily) choose a fiducial metric in\neach equivalence class and sum over those fiducial metrics instead.\nThis procedure mods out the diff x Weyl gauge group volumes (at the\nexpense of introducing ghosts).\nNow, if you need to include a topologically non-trivial sector into\nyour path integral, this will necessarily involve an equivalence class\ndifferent from the one that contains the topologically trivial R^10.\nHence you will have to choose a different fiducial metric in that\nsector.\nWithin each equivalence class different choices of the fiducial metric\ncorrespond to different insertions of vertex operators. But you can\nnever get from one equivalence class to another by any insertion of\nvertex operators. This should be clear as the vertex operators are\nlocal insertions (on the worldsheet) and as such cannot account for a\nglobal change of topology. The trick of thinking of a curved target\nspace metric in terms of a coherent states of gravitons is cute, but\nit should be pointed out that it only works as long as the *global*\ntopology of the curved metric is that of the flat R^10, that is, the\nhomology groups must be the same.\n\nThe statement of background independence is preserved as follows.\nBefore gauge fixing nothing depends on a specific metric because you\nsum over all of them. (An integral is always independent of the\nintegration variable...) After gauge fixing you have a sum over\nfiducial metrics, so superficially you might worry that this seems to\ndepend on the choice of fiducial metrics you had to make. But, by\nconstruction, the coice of fiducial metrics is irrelevant. The sum\nover them is equivalent to the initial sum over everything and hence\nbackground independent.\n\nHope that helps.\n\nBest,\nRufus\n\n[Moderator\'s note: have not you confused the worldsheet metric and the\ntarget manifold\'s metric, Rufus? In string theory, the target space\nmetric is fixed at the beginning, and one calculates the S-matrix on\nthis particular background. Physics at other backgrounds of the same\ntopology can be obtained by inserting vertex operators to the action,\nby deforming the worldsheet action with perturbative string states.\nIt\'s because an infinitesimal change of geometry corresponds to a\ncondensate of closed strings. However, if you want to obtain a manifold\nwith completely different homology groups, you must make a true\ntopology change transition, and switching from one topology to another\ntopology (branch) is represented by a condensation of non-perturbative\nstates, e.g. massless D3-branes (see chapter 13 of The Elegant Universe\nfor an elementary introduction). Massless D3-branes are not really\nlocal vertex operators on the worldsheet; in some sense, they can be\ndescribed by nonlocal vertex operators that add a boundary to the\nworldsheet. At any rate, the stringy perturbative expansion breaks\ndown once you change the geometry in such a way that some wrapped\nD3-branes become massless. For a more technical description of the\nconifold transition, see e.g. http://arxiv.org/abs/hep-th/9504145 LM]\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>( Redbull d70 wrote: )

> In this sense, two different background fields do not give two
> different theories, but can be thought of as two different states of
> the same string theory. Hopefully this is OK so far, and agrees with
> the usual interpretations.

It think that's fine so far.

> If so, my question is as follows; suppose I consider a target space
> manifold with, say, a flat metric for simplicity, but with non-trivial
> homology groups. Can I think of the string theory path integral with
> this target space as just being the flat-space $LaTeX Code: R^{10}$ path integral but
> with an insertion of some appropriate string vertex operators?
>
> In other words, is there an obvious way to think of two
> topologically-different target manifolds as corresponding to different
> states of the same string theory, just as there is for two different
> choices of background fields?

Well, let's back off for a second. The Polyakov path integral
prescription tells you to gauge fix as follows: Divide the space of
all metrics that needs to be summed over in the path integral into
equivalence classes, then (arbitrarily) choose a fiducial metric in
each equivalence class and sum over those fiducial metrics instead.
This procedure mods out the diff x Weyl gauge group volumes (at the
expense of introducing ghosts).
Now, if you need to include a topologically non-trivial sector into
your path integral, this will necessarily involve an equivalence class
different from the one that contains the topologically trivial $LaTeX Code: R^{10}$ .
Hence you will have to choose a different fiducial metric in that
sector.
Within each equivalence class different choices of the fiducial metric
correspond to different insertions of vertex operators. But you can
never get from one equivalence class to another by any insertion of
vertex operators. This should be clear as the vertex operators are
local insertions (on the worldsheet) and as such cannot account for a
global change of topology. The trick of thinking of a curved target
space metric in terms of a coherent states of gravitons is cute, but
it should be pointed out that it only works as long as the *global*
topology of the curved metric is that of the flat $LaTeX Code: R^{10},$ that is, the
homology groups must be the same.

The statement of background independence is preserved as follows.
Before gauge fixing nothing depends on a specific metric because you
sum over all of them. (An integral is always independent of the
integration variable...) After gauge fixing you have a sum over
fiducial metrics, so superficially you might worry that this seems to
depend on the choice of fiducial metrics you had to make. But, by
construction, the coice of fiducial metrics is irrelevant. The sum
over them is equivalent to the initial sum over everything and hence
background independent.

Hope that helps.

Best,
Rufus

[Moderator's note: have not you confused the worldsheet metric and the
target manifold's metric, Rufus? In string theory, the target space
metric is fixed at the beginning, and one calculates the S-matrix on
this particular background. Physics at other backgrounds of the same
topology can be obtained by inserting vertex operators to the action,
by deforming the worldsheet action with perturbative string states.
It's because an infinitesimal change of geometry corresponds to a
condensate of closed strings. However, if you want to obtain a manifold
with completely different homology groups, you must make a true
topology change transition, and switching from one topology to another
topology (branch) is represented by a condensation of non-perturbative
states, e.g. massless D3-branes (see chapter 13 of The Elegant Universe
for an elementary introduction). Massless D3-branes are not really
local vertex operators on the worldsheet; in some sense, they can be
described by nonlocal vertex operators that add a boundary to the
worldsheet. At any rate, the stringy perturbative expansion breaks
down once you change the geometry in such a way that some wrapped
D3-branes become massless. For a more technical description of the
conifold transition, see e.g. http://arxiv.org/abs/http://www.arxi...hep-th/9504145 LM]

**d70yxj** · Aug17-04, 03:01 PM

> [Moderator's note: have not you confused the worldsheet metric and the
> target manifold's metric, Rufus? In string theory, the target space
> metric is fixed at the beginning, and one calculates the S-matrix on
> this particular background. Physics at other backgrounds of the same
> topology can be obtained by inserting vertex operators to the action,
> by deforming the worldsheet action with perturbative string states.
> It's because an infinitesimal change of geometry corresponds to a
> condensate of closed strings.
> However, if you want to obtain a manifold
> with completely different homology groups, you must make a true
> topology change transition, and switching from one topology to another
> topology (branch) is represented by a condensation of non-perturbative
> states,

Thanks moderator, I think this is exactly the question I was asking.

> in some sense, they can be
> described by nonlocal vertex operators that add a boundary to the
> worldsheet.

Is there a good reference to see it in this way? The Greene et al paper doesn't seem to talk about it in quite this language.

> At any rate, the stringy perturbative expansion breaks
> down once you change the geometry in such a way that some wrapped
> D3-branes become massless.

So I specify infinite string coupling in order that the tension of the D3-brane goes to zero, is that right? Maybe I've misinterpreted that, but it somehow seems a bit odd; perturbative string theory on a target space with non-trivial homology groups can only be thought of as an infinite coupling string theory on flat space (with D3-branes added). Whereas perturbative string theory in manifold with just different geometry can always be thought of as perturbative string theory on flat space (with gravitons). Is that odd? Maybe it's what I should expect, I'm not sure. Thanks for the help, anyway.

I have a slightly different question, partly related to this, and also to the recent Hawking material. In his talk here he emphasised that in AdS/CFT one should sum over all supergravity configurations compatible with the appropriate boundary conditions. (I guess this is also emphasised in the Maldacena paper he references, and in 9803131 too). So at the level of partition functions you have
Z_cft = Z_s where Z_s is obtained by integrating or summing the sugra action over all appropriate metrics. In particular, in those above examples the interesting thing is that both AdS and AdS-Schwartzschild need to be included when comparing bulk results for correlation functions etc to results in the CFT.

So what happens when I move from just supergravity in the bulk, to string theory? Do I also need to sum over all appropriate target space metrics, or can I just choose one and string theory somehow does the rest for me?

(And is AdS-Schwartschild target space related to an AdS target space but with some kind of condensation of non-perturbative states?)

**arivero** · Aug17-04, 03:02 PM

Originally Posted by d70yxj

Dear Forum,

As far as I can tell, the usual statement of background independence in string theory is that even though one chooses a specific metric (and other target space fields) in the string theory path integral, a particular choice of background field is exactly equivalent to any other choice ...

At the epoch of the T-duality revolution (about 1995, I mean) , there was around another concept of background independence; some lecturers presented the target space as a temporary concept, to be substituted in the future by some other calculation technique where only "spaces of strings" were to be consirered, and "spaces of points" were to be not needed anymore. I have not idea is such conception has been preserved in the publications.

Aug18-04, 01:17 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>Lubos Motl wrote:\n\n> Moderator\'s note: have not you confused the worldsheet metric and the\n> target manifold\'s metric, Rufus? In string theory, the target space\n> metric is fixed at the beginning, and one calculates the S-matrix on\n> this particular background.\n\nThanks for your remarks, Lubos. Apparently I didn\'t make this very\nclear, but the point is that the set of equivalence classes of\n*worldsheet metrics* depends in an essential way on the homology\ngroups of the *target space*. For every nontrivial cycle on the target\nspace there will be a set of equivalence classes of worldsheet metrics\ncoming from world sheets that wrap this cycle. Toroidal\ncompactification is possibly the simplest example for this: The\npartition function breaks into a sum of topologically distinct sectors\nlabeled by winding numbers.\n\n[Moderator\'s note: Yes, these configurations are called the worldsheet\ninstantons. LM]\n\n> Physics at other backgrounds of the same\n> topology can be obtained by inserting vertex operators to the action,\n> by deforming the worldsheet action with perturbative string states.\n> It\'s because an infinitesimal change of geometry corresponds to a\n> condensate of closed strings. However, if you want to obtain a manifold\n> with completely different homology groups, you must make a true\n> topology change transition, and switching from one topology to another\n> topology (branch) is represented by a condensation of non-perturbative\n> states, e.g. massless D3-branes (see chapter 13 of The Elegant Universe\n> for an elementary introduction). Massless D3-branes are not really\n> local vertex operators on the worldsheet; in some sense, they can be\n> described by nonlocal vertex operators that add a boundary to the\n> worldsheet. At any rate, the stringy perturbative expansion breaks\n> down once you change the geometry in such a way that some wrapped\n> D3-branes become massless. For a more technical description of the\n> conifold transition, see e.g. http://arxiv.org/abs/hep-th/9504145 LM\n\nWhy do you think this is in conflict with what I said? If you insert\nsomething nonlocal into the path integral, so that you add another\nboundary to the worldsheet as you say, then you end up in a different\nequivalence class within the space of worldsheet metrics. But if you\nstarted with a *complete* set of such classes, as I suggested, it was\nalready there to begin with. This is the very definition of the\n(first-quantized) approach to string theory as we understand it and it\ncontains both perturbative and non-perturbative states. So all you do\nis pretending that you can transcend the classification of metrics\ninto equivalence classes by allowing for non-local vertex operator\ninsertions.\n\n[Moderator\'s note: I just don\'t quite understand the role that you want to\nassign to worldsheet instantons in answering the question. No doubt,\nworldsheet instantons DO play a very important role in topology change,\nas Witten showed in the example of the flop transition, see\nhttp://arxiv.org/abs/hep-th/9301042 - but the role seems to be different\nthan your comments. The worldsheet instantons contribute to various\nobservables such as the particle masses (or Yukawa couplings) and\nguarantee that the total value is continuous throughout the transition,\neven though the contribution of the classical Yukawa couplings has\na discontinuity. Nevertheless, different target space manifolds have\ndifferent spectrum of possible worldsheet instantons - they have\ndifferent second homology - and therefore the division of the worldsheet\nconfigurations into topological classes (the division into worldsheet\ninstantons) DOES depend on the target space manifold\'s topology. You\nseem to propose a way how to mask this difference and use a unified\ntreatment, but I just don\'t understand how it works and whether there\nis anything physical about such a description. LM]\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>Lubos Motl wrote:

> Moderator's note: have not you confused the worldsheet metric and the
> target manifold's metric, Rufus? In string theory, the target space
> metric is fixed at the beginning, and one calculates the S-matrix on
> this particular background.

Thanks for your remarks, Lubos. Apparently I didn't make this very
clear, but the point is that the set of equivalence classes of
*worldsheet metrics* depends in an essential way on the homology
groups of the *target space*. For every nontrivial cycle on the target
space there will be a set of equivalence classes of worldsheet metrics
coming from world sheets that wrap this cycle. Toroidal
compactification is possibly the simplest example for this: The
partition function breaks into a sum of topologically distinct sectors
labeled by winding numbers.

[Moderator's note: Yes, these configurations are called the worldsheet
instantons. LM]

> Physics at other backgrounds of the same
> topology can be obtained by inserting vertex operators to the action,
> by deforming the worldsheet action with perturbative string states.
> It's because an infinitesimal change of geometry corresponds to a
> condensate of closed strings. However, if you want to obtain a manifold
> with completely different homology groups, you must make a true
> topology change transition, and switching from one topology to another
> topology (branch) is represented by a condensation of non-perturbative
> states, e.g. massless D3-branes (see chapter 13 of The Elegant Universe
> for an elementary introduction). Massless D3-branes are not really
> local vertex operators on the worldsheet; in some sense, they can be
> described by nonlocal vertex operators that add a boundary to the
> worldsheet. At any rate, the stringy perturbative expansion breaks
> down once you change the geometry in such a way that some wrapped
> D3-branes become massless. For a more technical description of the
> conifold transition, see e.g. http://arxiv.org/abs/http://www.arxi...hep-th/9504145 LM

Why do you think this is in conflict with what I said? If you insert
something nonlocal into the path integral, so that you add another
boundary to the worldsheet as you say, then you end up in a different
equivalence class within the space of worldsheet metrics. But if you
started with a *complete* set of such classes, as I suggested, it was
already there to begin with. This is the very definition of the
(first-quantized) approach to string theory as we understand it and it
contains both perturbative and non-perturbative states. So all you do
is pretending that you can transcend the classification of metrics
into equivalence classes by allowing for non-local vertex operator
insertions.

[Moderator's note: I just don't quite understand the role that you want to
assign to worldsheet instantons in answering the question. No doubt,
worldsheet instantons DO play a very important role in topology change,
as Witten showed in the example of the flop transition, see
http://arxiv.org/abs/http://www.arxi...hep-th/9301042 - but the role seems to be different
than your comments. The worldsheet instantons contribute to various
observables such as the particle masses (or Yukawa couplings) and
guarantee that the total value is continuous throughout the transition,
even though the contribution of the classical Yukawa couplings has
a discontinuity. Nevertheless, different target space manifolds have
different spectrum of possible worldsheet instantons - they have
different second homology - and therefore the division of the worldsheet
configurations into topological classes (the division into worldsheet
instantons) DOES depend on the target space manifold's topology. You
seem to propose a way how to mask this difference and use a unified
treatment, but I just don't understand how it works and whether there
is anything physical about such a description. LM]

**d70yxj** · Sep7-04, 04:04 PM

Rufus, do you mean that by summing correctly over all equivalence classes of worldsheet metrics, one is actually summing over all possible topologies and geometries of the target space, almost without realizing it?

I don't think I've seen this expressed in quite this way in the usual sources, but perhaps I've missed something. Is there a good reference?

Sep9-04, 02:51 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>d70yxj <redbull_j@yahoo.com> wrote in message news:<d70yxj.1c8a8r-100000@physicsforums.com>...\n\n> Rufus, do you mean that by summing correctly over all equivalence\n> classes of worldsheet metrics, one is actually summing over all\n> possible topologies and geometries of the target space, almost without\n> realizing it?\n\nLet me first say that what I am saying is not half as outlandish as\nyou apparently think and it certainly is nothing new. It may just be a\nview point that is slightly different from the standard lore.\n\nD-branes were first discovered as soliton solutions of the low-energy\neffective equations of motion of (super-) string theory. Polchinski\'s\naccomplishment (in \'96) was to show that these solitons of the\neffective theory correspond to nonperturbative microscopic states in\nthe full-fledged string theory. Nevertheless, Polchinski\'s treatment\nstill pertains to a description in terms of classical target space\nproperties. While such a description provides intuitive pictures of\nthe low-energy description and is useful for a number of important\nissues (including the structure of moduli space), it is not quite\nsufficient. After all, the great thing about string theory is that it\nis a theory of quantum gravity and, hence, we should not expect to be\nable to describe it solely in terms of classical geometry. Maybe in\nthe future a more natural and/or complete understanding of the quantum\ngeometry of target space will emerge, but for now we can study the\nquantum regime by studying the CFT on the world sheet. The spectrum of\nthis CFT includes both perturbative and non-perturbative states and\nencodes the (quantum) geometry of target space. Surely if you\nformulate the CFT on the world-sheet general enough it will have\nexcitations in its spectrum that correspond to target spaces of\ndifferent topolgy in the low-energy effective description.\n\nHow does this relate to Lubos\' remarks concerning, e.g., Witten\'s\nwork? Well, if you insist on describing the target space in classical\nterms, some hard work is needed to show that physical properties\nremain continous even if the topology changes. The reason why one\nwould choose the classical description is, of course, that CFTs are\nnot sufficiently under control to calculate all the things we want to\nknow. It is simply too hard to solve them.\n\nHowever, even if we can\'t calculate many things explicitly, we can\nstill learn a great deal about the conceptual workings of string\ntheory. Despite the fact that the most general CFT that is consistent\nwith all classical string backgrounds hasn\'t been found yet, it\npresumably does exist. The original question in this thread was about\nbackground independence. This question has a particularly simple and\ngeneral answer in the abstract CFT language. That\'s what I tried to\nexplain using familiar terms.\n\nBest,\nRufus\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>d70yxj <redbull_j@yahoo.com> wrote in message news:<d70yxj.1c8a8r-100000@physicsforums.com>...

> Rufus, do you mean that by summing correctly over all equivalence
> classes of worldsheet metrics, one is actually summing over all
> possible topologies and geometries of the target space, almost without
> realizing it?

Let me first say that what I am saying is not half as outlandish as
you apparently think and it certainly is nothing new. It may just be a
view point that is slightly different from the standard lore.

D-branes were first discovered as soliton solutions of the low-energy
effective equations of motion of (super-) string theory. Polchinski's
accomplishment (in '96) was to show that these solitons of the
effective theory correspond to nonperturbative microscopic states in
the full-fledged string theory. Nevertheless, Polchinski's treatment
still pertains to a description in terms of classical target space
properties. While such a description provides intuitive pictures of
the low-energy description and is useful for a number of important
issues (including the structure of moduli space), it is not quite
sufficient. After all, the great thing about string theory is that it
is a theory of quantum gravity and, hence, we should not expect to be
able to describe it solely in terms of classical geometry. Maybe in
the future a more natural

complete understanding of the quantum
geometry of target space will emerge, but for now we can study the
quantum regime by studying the CFT on the world sheet. The spectrum of
this CFT includes both perturbative and non-perturbative states and
encodes the (quantum) geometry of target space. Surely if you
formulate the CFT on the world-sheet general enough it will have
excitations in its spectrum that correspond to target spaces of
different topolgy in the low-energy effective description.

How does this relate to Lubos' remarks concerning, e.g., Witten's
work? Well, if you insist on describing the target space in classical
terms, some hard work is needed to show that physical properties
remain continous even if the topology changes. The reason why one
would choose the classical description is, of course, that CFTs are
not sufficiently under control to calculate all the things we want to
know. It is simply too hard to solve them.

However, even if we can't calculate many things explicitly, we can
still learn a great deal about the conceptual workings of string
theory. Despite the fact that the most general CFT that is consistent
with all classical string backgrounds hasn't been found yet, it
presumably does exist. The original question in this thread was about
background independence. This question has a particularly simple and
general answer in the abstract CFT language. That's what I tried to
explain using familiar terms.

Best,
Rufus

Sep9-04, 03:22 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>"Rufus Anton" <rufusanton@gmx.de> schrieb im Newsbeitrag\nnews:a1c70df8.0409091040.108b4b3d-100000@posting.google.com...\n> d70yxj <redbull_j@yahoo.com> wrote in message\nnews:<d70yxj.1c8a8r-100000@physicsforums.com>...\n\n> Surely if you\n> formulate the CFT on the world-sheet general enough it will have\n> excitations in its spectrum that correspond to target spaces of\n> different topolgy in the low-energy effective description.\n\n[...]\n\n>Despite the fact that the most general CFT that is consistent\n> with all classical string backgrounds hasn\'t been found yet, it\n> presumably does exist.\n\nI am wondering what you mean by that. I\'d say that every CFT (sort for: SCFT\nwith the correct central charge) corresponds to precisely one classical\nbackground of string theory. This indeed is precisely the general definition\nof "classical background" in string theory, isn\'t it?\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>"Rufus Anton" <rufusanton@gmx.de> schrieb im Newsbeitrag
news:a1c70df8.0409091040.108b4b3d-10....google.com...
> d70yxj <redbull_j@yahoo.com> wrote in message
news:<d70yxj.1c8a8r-100000@physicsforums.com>...

> Surely if you
> formulate the CFT on the world-sheet general enough it will have
> excitations in its spectrum that correspond to target spaces of
> different topolgy in the low-energy effective description.

[...]

>Despite the fact that the most general CFT that is consistent
> with all classical string backgrounds hasn't been found yet, it
> presumably does exist.

I am wondering what you mean by that. I'd say that every CFT (sort for: SCFT
with the correct central charge) corresponds to precisely one classical
background of string theory. This indeed is precisely the general definition
of "classical background" in string theory, isn't it?

Sep9-04, 04:36 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>On Thu, 9 Sep 2004, Urs Schreiber wrote:\n\n> I am wondering what you mean by that. I\'d say that every CFT (sort for: SCFT\n> with the correct central charge) corresponds to precisely one classical\n> background of string theory. This indeed is precisely the general definition\n> of "classical background" in string theory, isn\'t it?\n\nMy guess is that Rufus might have meant a general (S)CFT with some\nparameters and other defining features whose choice correspond to a\nselection of a particular classical background - in other words, Rufus\nwanted to find a map of the whole perturbative beach of the landscape,\nwhich means the universal description of all backgrounds (at least all\nperturbative backgrounds); do I understand you well, Rufus?\n__________________________________________ ____________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\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>On Thu, 9 Sep 2004, Urs Schreiber wrote:

> I am wondering what you mean by that. I'd say that every CFT (sort for: SCFT
> with the correct central charge) corresponds to precisely one classical
> background of string theory. This indeed is precisely the general definition
> of "classical background" in string theory, isn't it?

My guess is that Rufus might have meant a general (S)CFT with some
parameters and other defining features whose choice correspond to a
selection of a particular classical background - in other words, Rufus
wanted to find a map of the whole perturbative beach of the landscape,
which means the universal description of all backgrounds (at least all
perturbative backgrounds); do I understand you well, Rufus?
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax:

Web: http://lumo.matfyz.cz/
eFax:

work:

home:

(call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Sep10-04, 04:08 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>> do I understand you well, Rufus?\n\nYes, you do. Thanks for clarifying. In the early 90s some people\nthought that finding the "master (S)CFT" would be the key to making\nprogress. But then spacetime rather worldsheet symmetries took over as\nthe most important tools for understanding more features of string\ntheory. Maybe the original plan should not be completely forgotten...\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>> do I understand you well, Rufus?

Yes, you do. Thanks for clarifying. In the early 90s some people
thought that finding the "master (S)CFT" would be the key to making
progress. But then spacetime rather worldsheet symmetries took over as
the most important tools for understanding more features of string
theory. Maybe the original plan should not be completely forgotten...

Sep10-04, 04:28 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0409091632270.1610 3-100000@feynman.harvard.edu...\n> On Thu, 9 Sep 2004, Urs Schreiber wrote:\n>\n> > I am wondering what you mean by that. I\'d say that every CFT (sort for:\nSCFT\n> > with the correct central charge) corresponds to precisely one classical\n> > background of string theory. This indeed is precisely the general\ndefinition\n> > of "classical background" in string theory, isn\'t it?\n>\n> My guess is that Rufus might have meant a general (S)CFT with some\n> parameters and other defining features whose choice correspond to a\n> selection of a particular classical background - in other words, Rufus\n> wanted to find a map of the whole perturbative beach of the landscape,\n> which means the universal description of all backgrounds (at least all\n> perturbative backgrounds); do I understand you well, Rufus?\n\nOk. And background *independence* (which is desireable as opposed to\nbackground *freedom* which may be problematic) means that each such SCFT can\nbe obtained from any other one by turning on some sort of coherent state in\nthe latter, roughly.\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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0409091632270.161...harvard.edu...
> On Thu, 9 Sep 2004, Urs Schreiber wrote:
>
> > I am wondering what you mean by that. I'd say that every CFT (sort for:
SCFT
> > with the correct central charge) corresponds to precisely one classical
> > background of string theory. This indeed is precisely the general
definition
> > of "classical background" in string theory, isn't it?
>
> My guess is that Rufus might have meant a general (S)CFT with some
> parameters and other defining features whose choice correspond to a
> selection of a particular classical background - in other words, Rufus
> wanted to find a map of the whole perturbative beach of the landscape,
> which means the universal description of all backgrounds (at least all
> perturbative backgrounds); do I understand you well, Rufus?

Ok. And background *independence* (which is desireable as opposed to
background *freedom* which may be problematic) means that each such SCFT can
be obtained from any other one by turning on some sort of coherent state in
the latter, roughly.

Sep10-04, 08:26 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>On Thu, 9 Sep 2004 15:22:28 -0400, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> I am wondering what you mean by that. I\'d say that every CFT (sort for: SCFT\n> with the correct central charge) corresponds to precisely one classical\n> background of string theory. This indeed is precisely the general definition\n> of "classical background" in string theory, isn\'t it?\n\nWhat exactly do you mean by "classical"? And targets related by\nT-duality (or mirror symmetry if you like) have the same CFT but with\ntwo classical backgrounds described by it. Furthermore, I am not sure\nthat the converse is true: That would mean that each CFT has at least\none point in its moduli space ("large volume") where it is described\nby a sigma model.\n\nRobert\n\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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>On Thu, 9 Sep 2004 15:22:28

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

> I am wondering what you mean by that. I'd say that every CFT (sort for: SCFT
> with the correct central charge) corresponds to precisely one classical
> background of string theory. This indeed is precisely the general definition
> of "classical background" in string theory, isn't it?

What exactly do you mean by "classical"? And targets related by
T-duality (or mirror symmetry if you like) have the same CFT but with
two classical backgrounds described by it. Furthermore, I am not sure
that the converse is true: That would mean that each CFT has at least
one point in its moduli space ("large volume") where it is described
by $LaTeX Code: a \\sigma$ model.

Robert

--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone:

stupid . $LaTeX Code: sig\\n$ "; http://www.aei-potsdam.mpg.de/~helling

Sep10-04, 11:56 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im\nNewsbeitrag news:2qdasnFuimkpU1-100000@uni-berlin.de...\n> On Thu, 9 Sep 2004 15:22:28 -0400, Urs Schreiber\n<Urs.Schreiber@uni-essen.de> wrote:\n>\n> > I am wondering what you mean by that. I\'d say that every CFT (sort for:\nSCFT\n> > with the correct central charge) corresponds to precisely one classical\n> > background of string theory. This indeed is precisely the general\ndefinition\n> > of "classical background" in string theory, isn\'t it?\n>\n> What exactly do you mean by "classical"?\n\nFor instance that the (S)CFT is a classical solution of the corresponding\nstring field theory action.\n\nThat\'s what a background is, isn\'t it? A classical solution, i.e. a saddle\npoint of the full string field action that we\'d like to compute the path\nintegral of (if only we could) but which we can only perturb about.\n\n(From the classical solution Phi to the string field theory one gets a\ndeformed BRST operator Q_Phi which is the BRST operator of the new (S)CFT.)\n\nThere is some fine print here, of course, but up to that defining a (S)CFT\nmeans defining the (generalized) classical background that the string whose\nworldsheet dynamics is described by that CFT.\n\nI guess the reason why you feel uncomfortable with me saying "classical\nbackground" is that the most general such background is far from being a\n"classical spacetime" with smooth space and everything. It may be an exotic\nquantum gravitic thingy. But it is still the saddle point solution on which\nthe perturbative string propagates. Wouldn\'t you agree?\n\n> And targets related by\n> T-duality (or mirror symmetry if you like) have the same CFT but with\n> two classical backgrounds described by it.\n\nOk, right. I should have said that the SCFT describes the background up to\nsymmetries like gauge symmetries and dualities. But that\'s more a matter of\nlanguage, depending on if you consider two spacetime theories related by\nduality to be "different".\n\n> Furthermore, I am not sure\n> that the converse is true: That would mean that each CFT has at least\n> one point in its moduli space ("large volume") where it is described\n> by a sigma model.\n\nOh, no, that\'s not what I mean. By "classical background" I don\'t\nnecessarily mean one described by a CFT which is a sigma model.\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>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im
Newsbeitrag news:2qdasnFuimkpU1-100000@uni-berlin.de...
> On Thu, 9 Sep 2004 15:22:28

Urs Schreiber
<Urs.Schreiber@uni-essen.de> wrote:
>
> > I am wondering what you mean by that. I'd say that every CFT (sort for:
SCFT
> > with the correct central charge) corresponds to precisely one classical
> > background of string theory. This indeed is precisely the general
definition
> > of "classical background" in string theory, isn't it?
>
> What exactly do you mean by "classical"?

For instance that the (S)CFT is a classical solution of the corresponding
string field theory action.

That's what a background is, isn't it? A classical solution, i.e. a saddle
point of the full string field action that we'd like to compute the path
integral of (if only we could) but which we can only perturb about.

(From the classical solution $LaTeX Code: \\Phi$ to the string field theory one gets a
deformed BRST operator $LaTeX Code: Q_{Phi}$ which is the BRST operator of the new (S)CFT.)

There is some fine print here, of course, but up to that defining a (S)CFT
means defining the (generalized) classical background that the string whose
worldsheet dynamics is described by that CFT.

I guess the reason why you feel uncomfortable with me saying "classical
background" is that the most general such background is far from being a
"classical spacetime" with smooth space and everything. It may be an exotic
quantum gravitic thingy. But it is still the saddle point solution on which
the perturbative string propagates. Wouldn't you agree?

> And targets related by
> T-duality (or mirror symmetry if you like) have the same CFT but with
> two classical backgrounds described by it.

Ok, right. I should have said that the SCFT describes the background up to
symmetries like gauge symmetries and dualities. But that's more a matter of
language, depending on if you consider two spacetime theories related by
duality to be "different".

> Furthermore, I am not sure
> that the converse is true: That would mean that each CFT has at least
> one point in its moduli space ("large volume") where it is described
> by $LaTeX Code: a \\sigma$ model.

Oh, no, that's not what I mean. By "classical background" I don't
necessarily mean one described by a CFT which is $LaTeX Code: a \\sigma$ model.

Sep11-04, 08:43 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>Rufus Anton <rufusanton@gmx.de> wrote in message news:<a1c70df8.0409091859.63fd8b9e-100000@posting.google.com>...\n\n> > do I understand you well, Rufus?\n>\n> Yes, you do. Thanks for clarifying. In the early 90s some people\n> thought that finding the "master (S)CFT" would be the key to making\n> progress. But then spacetime rather worldsheet symmetries took over as\n> the most important tools for understanding more features of string\n> theory. Maybe the original plan should not be completely forgotten...\n\nWorld-sheet concepts like CFT are pretty much useless for describing\nnon-perturbative backgrounds, such as eg F-theory compactifications\nwhich are non-perturbative backgrounds of the type IIB string\n(involving eg mutually non-local 7-branes; I wouldn\'t know of any\nCFT description of this situation). In the space of all consistent\nground states, it seems that only a small subset has a\nperturbative decription in terms of weakly coupled worldsheet\ntheories, like sigma models on gently curved "classical" manifolds.\nSuch backgrounds are fine as toy models, but if one wants to gain\na better understanding of the whole space of string backgrounds,\none definitely needs to go beyond on-shell worldsheet physics.\n\n[Moderator\'s note: I agree, of course, but my feeling was that the\ndiscussion was focussed on weakly-coupled perturbative backgrounds\nthat still do not have to be geometric sigma-models, such as various\nGepner-like models. One can still ask whether they can be always\nconnected with geometric backgrounds. Various islands, orbifolds\nby T-dualities and other CFTs with possibly frozen "size" modulus\nsuggest that the answer is "No", at least if we want them to connect\non-shell. LM]\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>Rufus Anton <rufusanton@gmx.de> wrote in message news:<a1c70df8.0409091859.63fd8b9e-1...google.com>...

> > do I understand you well, Rufus?
>
> Yes, you do. Thanks for clarifying. In the early 90s some people
> thought that finding the "master (S)CFT" would be the key to making
> progress. But then spacetime rather worldsheet symmetries took over as
> the most important tools for understanding more features of string
> theory. Maybe the original plan should not be completely forgotten...

World-sheet concepts like CFT are pretty much useless for describing
non-perturbative backgrounds, such as eg F-theory compactifications
which are non-perturbative backgrounds of the type IIB string
(involving eg mutually non-local 7-branes; I wouldn't know of any
CFT description of this situation). In the space of all consistent
ground states, it seems that only a small subset has a
perturbative decription in terms of weakly coupled worldsheet
theories, like $LaTeX Code: \\sigma$ models on gently curved "classical" manifolds.
Such backgrounds are fine as toy models, but if one wants to gain
a better understanding of the whole space of string backgrounds,
one definitely needs to go beyond on-shell worldsheet physics.

[Moderator's note: I agree, of course, but my feeling was that the
discussion was focussed on weakly-coupled perturbative backgrounds
that still do not have to be geometric $LaTeX Code: \\sigma-models,$ such as various
Gepner-like models. One can still ask whether they can be always
connected with geometric backgrounds. Various islands, orbifolds
by T-dualities and other CFTs with possibly frozen "size" modulus
suggest that the answer is "No", at least if we want them to connect
on-shell. LM]