Lubos Motl
Sep15-04, 11:31 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>On Wed, 15 Sep 2004, Urs Schreiber wrote (in the "Background\nindependence" thread):\n\n> But for something different: You mentioned recently the failure of SFT to\n> capture certain non-perturbative degrees of freedom. Is it conceivable that\n> one can somehow "augment" SFT in a nice way to include these?\n\nThis is a very interesting question, I think. Let me say a couple of\nrelated opinions of mine plus facts.\n\nSFT used to be a very natural candidate for a full formulation of string\ntheory. It is the closest thing to "field theory" that one can have - a\nfield theory with infinitely many fields, if you decompose the string\nfield into component fields. From this point of view, it looks "background\nindependent" to many critics - and it is "off-shell", which means that it\nsort of *has* local Green\'s functions, not just the S-matrix, which is\nsomething that has extensively been used in the study of tachyon\ncondensation.\n\nThis ambitious program has only been partially successful. First of all, a\nnon-perturbatively complete theory should be defined with an exact form of\nthe action etc., not just a perturbative approximation of it. However,\nclosed string field theory requires to add correcting terms to the action\nat each order (as well as the BV machinery). The action is therefore an\nexpansion itself, and it can have various non-perturbative completions.\nNot a good starting point for a non-perturbatively complete theory.\n\nAll these reasons led the people to focus on the open string field theory\nwhose action can be well-defined - e.g. the cubic (polynomial) Witten\'s\naction; it is enough to get the full amplitudes and cover the full Riemann\nsurface moduli spaces. Can one see all physics of string theory in it?\nWell, the first problem are the closed string states. They can be seen as\npoles in open string scattering, but as far as I know, no one has made a\nconvincing construction of the closed strings as composites of the open\nstring fields so far. The understanding of the closed strings would have\nto improve a lot so that one could also construct non-trivial geometric\nconfigurations including NS21-branes (or NS5-branes) etc. in open SFT.\n\nAnother question are the D-branes. Using the modern perspective, the open\nstrings themselves describe dynamics of a spacetime filling D-brane. Sen\'s\ninsights made it expected that one can construct the lower dimensional\nbranes as classical solutions of open string field theory.\n\nString field theory has nevertheless been made less natural by the results\nof the Duality revolution - its degrees of freedom are made of strings,\nbut at a generic point in the moduli space, there should be brane\ndemocracy and strings are equally (non-)fundamental as other objects. If\nstring field theory becomes a good starting point for a full formulation,\none must ask several obvious questions.\n\nAre the S-dualities and the strong coupling limit derivable from this\ngeneralized SFT? For example, can one derive that the strong coupling\nlimit of a type IIA string theory is eleven-dimensional, and type IIB is\nS-self-dual?\n\nThe answers must be yes if the generalized SFT is gonna become\nnon-perturbatively successful. Well, there are still two basic pictures\nhow this could happen:\n\n1. One would still be using the same string fields, even at strong\ncoupling, and there are non-trivial functions or transformations\nof these string fields that can be used to define the S-dual strings,\nor the 11-dimensional physics, and so on.\n\nI think that this viewpoint has become a bit obsolete after the\nstrong coupling revolutions of the 1990s. At strong coupling, the\noriginal degrees of freedom are strongly coupled, physics becomes\nobscure if we use them. They are not too useful, and moreover we\nhave learned that there can be better degrees of freedom that\nbecome weakly coupled. They are typically infinitely heavy in the\nweakly coupled limit, and therefore they are absent.\n\nIn field theory it is legitimate to imagine - for example in QCD\n- that the fundamental UV fields are the gluons and quarks, and\nthe IR physics is whatever is implied. The gluons are superuseful\nin the UV - because of the asymptotic freedom - but their physics\ncan be extrapolated to low energies. But we know that there is an\nasymmetry - the IR can be derived from the UV, but not quite the\nother way around. Therefore the analogy with strings, that are good\nvariables at the weak coupling, is not quite perfect - because\nthe strong and weak coupling may be totally equivalent.\n\nThe lessons of the 1990s seem to indicate that we should not try to\npush the validity of some degrees of freedom to too strong couplings.\n\n2. Of course, the second choice is that at generic coupling, there could\nbe new generalized degrees of freedom, whose structure itself is\ndetermined, together with the action or whatever replaces it, by some\nself-consistent rules. These degrees of freedom, determined by the\ndeeper rules, would have to reduce to the usual perturbative strings\nin each weakly-coupled stringy limit.\n\nWhile this second option is highly unusual, I believe that it is\nplausible and attractive. It is unusual because we have not constructed\na single theory whose degrees of freedom are themselves determined by\ndeeper rules, dynamically. We always start with some well-defined\ndegrees of freedom, with a well-defined action or something equivalent.\nSuch theories can have many interesting regimes and behaviors, but\nthey cannot be quite universal.\n\nIn the perturbative limit we kind of know what are the rules that\ndetermine the allowed degrees of freedom and the action: the rules are\nthe usual axioms of conformal field theory. The conformal symmetry\nconstrains both the worldsheet field content as well as the action.\nBut is there a non-perturbative generalization of this nice structure?\n\nWhat happens with the worldsheet as you increase the coupling? Well,\nit transmutes into a M-volume, which is the worldvolume of M, which\nis the non-perturbative generalization of a string. ;-) The worldsheet\nbecomes a bit fuzzy, non-local, its dimension may effectively grow\n(strings become membranes, but don\'t imagine quite local membranes). I\nthink that its internal dynamics is itself target space dynamics of\nsome other string theory; I have the N=2 and N=(2,1) string in mind.\n\nWe know that this complicated structure of the worldsheet theory *does*\noccur in some context: the worldsheet of a D-string at weak coupling,\nin which the D-string is superheavy, is described by open string theory\n- all open strings attached to the D-string with the whole Hagedorn\ntower of excitations are relevant. Nevertheless this D-string can be\ncontinued to something we call the fundamental string.\n\nThere should be some more general\ndescription of the allowed worldvolume theories of objects, including\nnon-geometrical ones - and the rules would non-perturbatively\ngeneralize conformal field theory.\n\nI\'ve spent some time with thinking about the form of such a possible\ngeneralization. Try to think about a more general theory that has a\nBRST operator and the state-operator correspondence, but you relax\nthe assumption that it is a local two-dimensional theory. It can be\na theory of any dimension, with fuzzy dynamics, matrices, whatever\nyou want. Just try to require that something as strong as the\nrequirement of conformal symmetry applies, and the conformal symmetry\nitself appears as a limit of this requirement for the special case\nof weakly coupled backgrounds...\n\n.... One more comment. There have been some Japanese papers that studied\nthe behavior of the boundary states under the closed-string\nKyoto-group-like SFT star product; the boundary states act as projectors,\nroughly speaking. This sort of thinking, even though it is formal, looks\nlike an important step towards obtaining the non-perturbative\ngeneralization of CFT mentioned above. Today, our consistency requirements\nfor closed strings and open strings follow similar logic, but technically\nthey are different.\n\nThe allowed spectrum of D-branes must follow Cardy\'s rules, and so forth.\nWhat I would like to see is to derive Cardy\'s rules as something like the\n(generalized) closed string (M) equations of motion applied on the closed\nstring field whose vev happens to be the (total) boundary state. Adding a\nD-brane is a deformation of the background, and it does correspond to a\nchange of the two-dimensional CFT. Well, the change is that we allow some\nnew boundaries. Formally, it is analogous to adding the vertex operator\nfor the boundary state into the 2D action although I realize that there\nare technical difficulties in making this procedure well-defined (but\ndefinitely, this is how the D-brane is seen from far away, as a\ndeformation of the closed string background; in this case, we can\nrestrict the boundary state to its lowest components).\n\nNow imagine that the coupling becomes larger. Adding a D1-brane in type\nIIB becomes equivalent to adding a light string if the coupling is really\nlarge - by S-duality. But adding a fundamental string is a local change of\nthe 2D action. Recall that adding the D-brane was a non-local change: we\nallowed the worldsheets to have boundaries. The goal is to describe a\nstructure that interpolates between this local modification of the 2D\naction (adding a fundamental string) and a non-local modification\n(allowing D-branes). The worldsheet itself should become fuzzy; the\ndistinction between local and non-local must go away at the generic\ncoupling. But what is exactly the theory at the generic point, and how do\nyou constrain it?\n\nThis is a sort of bootstrap thinking, but maybe not so impossible - it may\nbe just a generalization of CFTs.\n\nAll the best\nLubos\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</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 Wed, 15 Sep 2004, Urs Schreiber wrote (in the "Background
independence" thread):
> But for something different: You mentioned recently the failure of SFT to
> capture certain non-perturbative degrees of freedom. Is it conceivable that
> one can somehow "augment" SFT in a nice way to include these?
This is a very interesting question, I think. Let me say a couple of
related opinions of mine plus facts.
SFT used to be a very natural candidate for a full formulation of string
theory. It is the closest thing to "field theory" that one can have - a
field theory with infinitely many fields, if you decompose the string
field into component fields. From this point of view, it looks "background
independent" to many critics - and it is "off-shell", which means that it
sort of *has* local Green's functions, not just the S-matrix, which is
something that has extensively been used in the study of tachyon
condensation.
This ambitious program has only been partially successful. First of all, a
non-perturbatively complete theory should be defined with an exact form of
the action etc., not just a perturbative approximation of it. However,
closed string field theory requires to add correcting terms to the action
at each order (as well as the BV machinery). The action is therefore an
expansion itself, and it can have various non-perturbative completions.
Not a good starting point for a non-perturbatively complete theory.
All these reasons led the people to focus on the open string field theory
whose action can be well-defined - e.g. the cubic (polynomial) Witten's
action; it is enough to get the full amplitudes and cover the full Riemann
surface moduli spaces. Can one see all physics of string theory in it?
Well, the first problem are the closed string states. They can be seen as
poles in open string scattering, but as far as I know, no one has made a
convincing construction of the closed strings as composites of the open
string fields so far. The understanding of the closed strings would have
to improve a lot so that one could also construct non-trivial geometric
configurations including NS21-branes (or NS5-branes) etc. in open SFT.
Another question are the D-branes. Using the modern perspective, the open
strings themselves describe dynamics of a spacetime filling D-brane. Sen's
insights made it expected that one can construct the lower dimensional
branes as classical solutions of open string field theory.
String field theory has nevertheless been made less natural by the results
of the Duality revolution - its degrees of freedom are made of strings,
but at a generic point in the moduli space, there should be brane
democracy and strings are equally (non-)fundamental as other objects. If
string field theory becomes a good starting point for a full formulation,
one must ask several obvious questions.
Are the S-dualities and the strong coupling limit derivable from this
generalized SFT? For example, can one derive that the strong coupling
limit of a type IIA string theory is eleven-dimensional, and type IIB is
S-self-dual?
The answers must be yes if the generalized SFT is gonna become
non-perturbatively successful. Well, there are still two basic pictures
how this could happen:
1. One would still be using the same string fields, even at strong
coupling, and there are non-trivial functions or transformations
of these string fields that can be used to define the S-dual strings,
or the 11-dimensional physics, and so on.
I think that this viewpoint has become a bit obsolete after the
strong coupling revolutions of the 1990s. At strong coupling, the
original degrees of freedom are strongly coupled, physics becomes
obscure if we use them. They are not too useful, and moreover we
have learned that there can be better degrees of freedom that
become weakly coupled. They are typically infinitely heavy in the
weakly coupled limit, and therefore they are absent.
In field theory it is legitimate to imagine - for example in QCD
- that the fundamental UV fields are the gluons and quarks, and
the IR physics is whatever is implied. The gluons are superuseful
in the UV - because of the asymptotic freedom - but their physics
can be extrapolated to low energies. But we know that there is an
asymmetry - the IR can be derived from the UV, but not quite the
other way around. Therefore the analogy with strings, that are good
variables at the weak coupling, is not quite perfect - because
the strong and weak coupling may be totally equivalent.
The lessons of the 1990s seem to indicate that we should not try to
push the validity of some degrees of freedom to too strong couplings.
2. Of course, the second choice is that at generic coupling, there could
be new generalized degrees of freedom, whose structure itself is
determined, together with the action or whatever replaces it, by some
self-consistent rules. These degrees of freedom, determined by the
deeper rules, would have to reduce to the usual perturbative strings
in each weakly-coupled stringy limit.
While this second option is highly unusual, I believe that it is
plausible and attractive. It is unusual because we have not constructed
a single theory whose degrees of freedom are themselves determined by
deeper rules, dynamically. We always start with some well-defined
degrees of freedom, with a well-defined action or something equivalent.
Such theories can have many interesting regimes and behaviors, but
they cannot be quite universal.
In the perturbative limit we kind of know what are the rules that
determine the allowed degrees of freedom and the action: the rules are
the usual axioms of conformal field theory. The conformal symmetry
constrains both the worldsheet field content as well as the action.
But is there a non-perturbative generalization of this nice structure?
What happens with the worldsheet as you increase the coupling? Well,
it transmutes into a M-volume, which is the worldvolume of M, which
is the non-perturbative generalization of a string. ;-) The worldsheet
becomes a bit fuzzy, non-local, its dimension may effectively grow
(strings become membranes, but don't imagine quite local membranes). I
think that its internal dynamics is itself target space dynamics of
some other string theory; I have the N=2 and N=(2,1) string in mind.
We know that this complicated structure of the worldsheet theory *does*
occur in some context: the worldsheet of a D-string at weak coupling,
in which the D-string is superheavy, is described by open string theory
- all open strings attached to the D-string with the whole Hagedorn
tower of excitations are relevant. Nevertheless this D-string can be
continued to something we call the fundamental string.
There should be some more general
description of the allowed worldvolume theories of objects, including
non-geometrical ones - and the rules would non-perturbatively
generalize conformal field theory.
I've spent some time with thinking about the form of such a possible
generalization. Try to think about a more general theory that has a
BRST operator and the state-operator correspondence, but you relax
the assumption that it is a local two-dimensional theory. It can be
a theory of any dimension, with fuzzy dynamics, matrices, whatever
you want. Just try to require that something as strong as the
requirement of conformal symmetry applies, and the conformal symmetry
itself appears as a limit of this requirement for the special case
of weakly coupled backgrounds...
.... One more comment. There have been some Japanese papers that studied
the behavior of the boundary states under the closed-string
Kyoto-group-like SFT star product; the boundary states act as projectors,
roughly speaking. This sort of thinking, even though it is formal, looks
like an important step towards obtaining the non-perturbative
generalization of CFT mentioned above. Today, our consistency requirements
for closed strings and open strings follow similar logic, but technically
they are different.
The allowed spectrum of D-branes must follow Cardy's rules, and so forth.
What I would like to see is to derive Cardy's rules as something like the
(generalized) closed string (M) equations of motion applied on the closed
string field whose vev happens to be the (total) boundary state. Adding a
D-brane is a deformation of the background, and it does correspond to a
change of the two-dimensional CFT. Well, the change is that we allow some
new boundaries. Formally, it is analogous to adding the vertex operator
for the boundary state into the 2D action although I realize that there
are technical difficulties in making this procedure well-defined (but
definitely, this is how the D-brane is seen from far away, as a
deformation of the closed string background; in this case, we can
restrict the boundary state to its lowest components).
Now imagine that the coupling becomes larger. Adding a D1-brane in type
IIB becomes equivalent to adding a light string if the coupling is really
large - by S-duality. But adding a fundamental string is a local change of
the 2D action. Recall that adding the D-brane was a non-local change: we
allowed the worldsheets to have boundaries. The goal is to describe a
structure that interpolates between this local modification of the 2D
action (adding a fundamental string) and a non-local modification
(allowing D-branes). The worldsheet itself should become fuzzy; the
distinction between local and non-local must go away at the generic
coupling. But what is exactly the theory at the generic point, and how do
you constrain it?
This is a sort of bootstrap thinking, but maybe not so impossible - it may
be just a generalization of CFTs.
All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
independence" thread):
> But for something different: You mentioned recently the failure of SFT to
> capture certain non-perturbative degrees of freedom. Is it conceivable that
> one can somehow "augment" SFT in a nice way to include these?
This is a very interesting question, I think. Let me say a couple of
related opinions of mine plus facts.
SFT used to be a very natural candidate for a full formulation of string
theory. It is the closest thing to "field theory" that one can have - a
field theory with infinitely many fields, if you decompose the string
field into component fields. From this point of view, it looks "background
independent" to many critics - and it is "off-shell", which means that it
sort of *has* local Green's functions, not just the S-matrix, which is
something that has extensively been used in the study of tachyon
condensation.
This ambitious program has only been partially successful. First of all, a
non-perturbatively complete theory should be defined with an exact form of
the action etc., not just a perturbative approximation of it. However,
closed string field theory requires to add correcting terms to the action
at each order (as well as the BV machinery). The action is therefore an
expansion itself, and it can have various non-perturbative completions.
Not a good starting point for a non-perturbatively complete theory.
All these reasons led the people to focus on the open string field theory
whose action can be well-defined - e.g. the cubic (polynomial) Witten's
action; it is enough to get the full amplitudes and cover the full Riemann
surface moduli spaces. Can one see all physics of string theory in it?
Well, the first problem are the closed string states. They can be seen as
poles in open string scattering, but as far as I know, no one has made a
convincing construction of the closed strings as composites of the open
string fields so far. The understanding of the closed strings would have
to improve a lot so that one could also construct non-trivial geometric
configurations including NS21-branes (or NS5-branes) etc. in open SFT.
Another question are the D-branes. Using the modern perspective, the open
strings themselves describe dynamics of a spacetime filling D-brane. Sen's
insights made it expected that one can construct the lower dimensional
branes as classical solutions of open string field theory.
String field theory has nevertheless been made less natural by the results
of the Duality revolution - its degrees of freedom are made of strings,
but at a generic point in the moduli space, there should be brane
democracy and strings are equally (non-)fundamental as other objects. If
string field theory becomes a good starting point for a full formulation,
one must ask several obvious questions.
Are the S-dualities and the strong coupling limit derivable from this
generalized SFT? For example, can one derive that the strong coupling
limit of a type IIA string theory is eleven-dimensional, and type IIB is
S-self-dual?
The answers must be yes if the generalized SFT is gonna become
non-perturbatively successful. Well, there are still two basic pictures
how this could happen:
1. One would still be using the same string fields, even at strong
coupling, and there are non-trivial functions or transformations
of these string fields that can be used to define the S-dual strings,
or the 11-dimensional physics, and so on.
I think that this viewpoint has become a bit obsolete after the
strong coupling revolutions of the 1990s. At strong coupling, the
original degrees of freedom are strongly coupled, physics becomes
obscure if we use them. They are not too useful, and moreover we
have learned that there can be better degrees of freedom that
become weakly coupled. They are typically infinitely heavy in the
weakly coupled limit, and therefore they are absent.
In field theory it is legitimate to imagine - for example in QCD
- that the fundamental UV fields are the gluons and quarks, and
the IR physics is whatever is implied. The gluons are superuseful
in the UV - because of the asymptotic freedom - but their physics
can be extrapolated to low energies. But we know that there is an
asymmetry - the IR can be derived from the UV, but not quite the
other way around. Therefore the analogy with strings, that are good
variables at the weak coupling, is not quite perfect - because
the strong and weak coupling may be totally equivalent.
The lessons of the 1990s seem to indicate that we should not try to
push the validity of some degrees of freedom to too strong couplings.
2. Of course, the second choice is that at generic coupling, there could
be new generalized degrees of freedom, whose structure itself is
determined, together with the action or whatever replaces it, by some
self-consistent rules. These degrees of freedom, determined by the
deeper rules, would have to reduce to the usual perturbative strings
in each weakly-coupled stringy limit.
While this second option is highly unusual, I believe that it is
plausible and attractive. It is unusual because we have not constructed
a single theory whose degrees of freedom are themselves determined by
deeper rules, dynamically. We always start with some well-defined
degrees of freedom, with a well-defined action or something equivalent.
Such theories can have many interesting regimes and behaviors, but
they cannot be quite universal.
In the perturbative limit we kind of know what are the rules that
determine the allowed degrees of freedom and the action: the rules are
the usual axioms of conformal field theory. The conformal symmetry
constrains both the worldsheet field content as well as the action.
But is there a non-perturbative generalization of this nice structure?
What happens with the worldsheet as you increase the coupling? Well,
it transmutes into a M-volume, which is the worldvolume of M, which
is the non-perturbative generalization of a string. ;-) The worldsheet
becomes a bit fuzzy, non-local, its dimension may effectively grow
(strings become membranes, but don't imagine quite local membranes). I
think that its internal dynamics is itself target space dynamics of
some other string theory; I have the N=2 and N=(2,1) string in mind.
We know that this complicated structure of the worldsheet theory *does*
occur in some context: the worldsheet of a D-string at weak coupling,
in which the D-string is superheavy, is described by open string theory
- all open strings attached to the D-string with the whole Hagedorn
tower of excitations are relevant. Nevertheless this D-string can be
continued to something we call the fundamental string.
There should be some more general
description of the allowed worldvolume theories of objects, including
non-geometrical ones - and the rules would non-perturbatively
generalize conformal field theory.
I've spent some time with thinking about the form of such a possible
generalization. Try to think about a more general theory that has a
BRST operator and the state-operator correspondence, but you relax
the assumption that it is a local two-dimensional theory. It can be
a theory of any dimension, with fuzzy dynamics, matrices, whatever
you want. Just try to require that something as strong as the
requirement of conformal symmetry applies, and the conformal symmetry
itself appears as a limit of this requirement for the special case
of weakly coupled backgrounds...
.... One more comment. There have been some Japanese papers that studied
the behavior of the boundary states under the closed-string
Kyoto-group-like SFT star product; the boundary states act as projectors,
roughly speaking. This sort of thinking, even though it is formal, looks
like an important step towards obtaining the non-perturbative
generalization of CFT mentioned above. Today, our consistency requirements
for closed strings and open strings follow similar logic, but technically
they are different.
The allowed spectrum of D-branes must follow Cardy's rules, and so forth.
What I would like to see is to derive Cardy's rules as something like the
(generalized) closed string (M) equations of motion applied on the closed
string field whose vev happens to be the (total) boundary state. Adding a
D-brane is a deformation of the background, and it does correspond to a
change of the two-dimensional CFT. Well, the change is that we allow some
new boundaries. Formally, it is analogous to adding the vertex operator
for the boundary state into the 2D action although I realize that there
are technical difficulties in making this procedure well-defined (but
definitely, this is how the D-brane is seen from far away, as a
deformation of the closed string background; in this case, we can
restrict the boundary state to its lowest components).
Now imagine that the coupling becomes larger. Adding a D1-brane in type
IIB becomes equivalent to adding a light string if the coupling is really
large - by S-duality. But adding a fundamental string is a local change of
the 2D action. Recall that adding the D-brane was a non-local change: we
allowed the worldsheets to have boundaries. The goal is to describe a
structure that interpolates between this local modification of the 2D
action (adding a fundamental string) and a non-local modification
(allowing D-branes). The worldsheet itself should become fuzzy; the
distinction between local and non-local must go away at the generic
coupling. But what is exactly the theory at the generic point, and how do
you constrain it?
This is a sort of bootstrap thinking, but maybe not so impossible - it may
be just a generalization of CFTs.
All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^