<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>Hey Squark,\n\n> 1) Page 30: It is mentioned the classical conformal invariance of 4D\n> Yang-Mills is broken on the quantum level since a cutoff has to be\n> introduced. N = 4 SUSY Yang-Mills, OTOH is a genuine CFT. Should it\n> be understood the later doesn\'t require renormalization at all? It\n> would make sense in the light of the fact it\'s supposed to be dual\n> to type IIB superstring theory in AdS_5 x S^5 which is finite order\n> by order in perturbation theory and thus requires no\n> renormalization. Do all CFTs have this property?\n\nI think that this is a good question, and many people should try to help\nto create a more complete answer. First of all, QCD is an example of a\nclassically conformal theory (there is conformal symmetry at the level of\nplaying with the classical expressions). However quantum effects - such as\nthe coupling constant running (quantified by the beta function) make the\ntheory non-conformal, and there is a priviliged scale (the QCD scale).\n\nOn the other hand, there is no running in the N=4 gauge theory, due to its\nexact conformal invariance. The vanishing of the beta function implies the\nabsence of certain logarithmic divergences. This effect is more general -\nthe absence of running (i.e. conformal symmetry) means that the\ncorresponding potential divergences cancel. In fact, the N=4 Yang-Mills is\n"finite" in the sense that the correlators of the "elementary fields" are\nfinite (after the cancellation between many contributions), aren\'t they?\n\nClearly, the correlators of composite fields lead to divergences,\nanomalous dimensions (quantum contribution to the dimension; which are\ndual to interaction corrections to the energy in AdS), and the need to\nrenormalize. The appearance of log divergences in the correlators of\ncomposite operators does not spoil the exact duality with the "finite"\ntype IIB string theory, I think, because these divergences can be absorbed\nin the normalization of these operators.\n\nI believe that N=4 has no higher divergences than log divergences.\n\nMost of the exactly conformal 4D gauge theories - like the orbifolds of\nN=4 - preserve some finiteness of N=4, but probably not all of it. Other\nexactly conformal theories are analogous, but the renormalization process\nwould be done differently and a special discussion may be needed.\n\n> 2) On page 45, beginning of section 2.2.2, it is explained massive\n> particles cannot ever reach the boundary of AdS along geodesics but\n> massless can. I believe it implies massive particles have to go\n> through acceleration of unbounded magnitude to reach the boundary.\n> Nevertheless, on page 46 it is explained fields have to be assigned\n> boundary conditions for energy conservation to hold which appear to\n> be as stringent for massive fields (m > 0) as for massless (m = 0).\n> Why is that?\n\nI don\'t know where you see on page 46 that the boundary conditions should\nbe m-independent. On the contrary, the fields behave as powers near the\nboundary, see (2.34), and the exponents (lambda) are computed from the\nmass (see (2.36)). In other coordinates, the decrease near the boundary is\nexponential. Of course, the sentence "energy is conserved" does not\ndirectly contain the words "mass of the particle", but it contains it\nindirectly via the definition of the energy of a scalar field.\n\n> 3) On page 74 orbifolds of AdS_5 x S^5 are discussed, where the\n> orbifolding is done by a discrete subgroup Gamma of the S^5\n> isometry group SO(6) (which is the R-symmetry group SU(4) of the\n> dual CFT).\n\nRight.\n\n> It is said that on both sides of the correspondence\n> the orbifolding cannot be done just by taking the\n> Gamma-invariant states, but on the field theory side we _can_ do\n> this if multiply the number of branes by dim(Gamma) first.\n\nThe number of elements of a discrete group is called rank(Gamma), not\ndim(Gamma). Also, sorry, on my copy of the text, Orbifolds are discussed\nfrom the page 113 on (not 74), so your specific links seem corrupt to me\n(maybe there is a sentence on orbifold on earlier pages, but they are\nsystematically explained later).\n\nWell, you must first add all images of your N D3-branes, which a priori\nextends the "locally seen" group U(N) to U(N.rank(Gamma)), but then you\nmust impose the condition that they are images of each other. This\nresults in a gauge group U(N)^{rank(Gamma)} or something along these\nlines, and some fields transform as bifundamental representations (as\ndescribed by quiver diagrams).\n\n> Do I assume correctly it is actually the number of elements #Gamma\n> which is meant?\n\nI only see dim(Gamma) in your posting - but definitely, the relevant\nmultiplication factor changing the *dimension* of the fundamental rep of\nthe gauge group is rank(Gamma).\n\n> 4) On page 115, the CFT dual to type IIB on AdS_5 x S^5 / Z_k.\n> The AdS/CFT correspondence is obtained, in this case, by\n> considering D3-branes at an R^4/Z_k orbifold singularity. Do I\n> understand correctly Z_k acts by multiplying x_1 + i x_2 and\n> x_3 + i x_4 by a k-th root of unity (x_i are the R^4\n> coordinates)?\n\nYes. This action must be done simultaneously on both x1+ix2 and x3+ix4.\n\n> Further below, it is claimed the dual CFT has k\n> complex parameters. On the string theory side, k-1 out of these\n> parameters are identified with the NS-NS and R-R 2-forms on the\n> k-1 2-cycles which vanish at the orbifold singularity (page 116).\n> The k-th parameter is said to be the dilaton. Isn\'t the dilaton\n> real?\n\nDepending on the terminology. Dilaton is usually the real part, but type\nIIB string theory has two such scalar fields - the dilaton and the axion\n(the RR-scalar) C_0 - that combine into a complex scalar (onto which the\nS-duality group SL(2,Z) acts).\n\n> 5) On page 121, superstring theory on AdS_5 x M_5 with M_5 a\n> 5-manifold no locally isomorphic to S^5 is discussed. It is said\n> the "effective cosmological constant" on the AdS_5 results from\n> the 5-form of type IIB SUGRA. It is then said the 5-form is self\n> dual and that is the reason M_5 has to be an Einstein manifold\n> with positive cosmological constant. Firstly, why is it\n> self-dual?\n\nThere are many reasons to see it. If you derive the physical Ramond-Ramond\nspectrum - think about the RR sector of a closed string - there are two\n(left-moving and right-moving) GSO projections. The "overall" Z_2\nprojection tells you that the field strengths in type IIB must be q-forms\nfor q odd. The other Z_2 projection reduces the number of polarizations\nto one half, to the self-dual ones. At the end, you obtain - as a\nconsistency check - 64 bosonic polarizations from the RR-sector.\n\nThis includes the RR-scalar (the axion, i.e. the other part of the\ncomplexified dilaton), 8.7/2.1 = 28 polarizations of the 2-form, and\none half of 8.7.6.5/4.3.2.1 = 70, i.e. 35 polarizations of the self-dual\nfive-form field strength (the transversely self-dual components of the\n4-form potential). 1+28+35=64, which combines with 64 NS-NS states into\n128 bosonic polarizations, which combine with 64+64 fermions (NS-R and\nR-NS) into the 256 states of the maximal supergravity.\n\nAlternatively, a 5-form field strength in general admits electric sources,\nas well as magnetic sources. In D=10, both of these sources would be\n3-branes, and because there exist one type of D3-branes in type IIB only,\nthey must be equivalent, and therefore the 5-form field strength must be\nself-dual.\n\n> Isn\'t it just the RR 5-form of type IIB superstring theory?\n\nI thought you were just asking whether the (RR) 5-form field strength of\ntype IIB string theory is self-dual - YES, IT IS - so why are you asking\nagain? Yes, the 5-form of type IIB SUGRA is (the low energy description\nof) "just the" RR 5-form of type IIB superstring theory. It\'s the same\nthing. Your question makes absolutely no sense to me. Yes, it is\nself-dual. If you need this to be repeated 26 times, feel free to ask,\nbut I hope that someone else will help me.\n\n> Secondly, how does the conclusion about M_5 follow?\n\nVia a careful analysis of signs in the Einstein equations. You realize\nthat the components of the stress energy tensor, such as T_{77}, along the\nS^5, if they are computed from the 5-form components such as F_{56789},\nare positive, and therefore the S^5 sphere must be positively curved, and\nby the same logic, the same component F_{01234}=F_{56789} (self-duality)\ncurves the AdS space, which must be negatively curved.\n\nIt is not hard to see that the sign of the scalar curvature of the AdS_5\nand of S^5 must be opposite. Continue to Euclidean time. Then you will\nhave the self-duality constraint F_{01234}=i.F_{56789} with i from the\nWick rotation, and therefore the stress energy tensors constructed in the\n01234 and 56789 directions will have opposite signs. It leads to opposite\ncurvatures, and the curvatures are unchanged by continuation.\n\nIt is useful to understand the whole solution. The near horizon geometry\nof the D3-branes simply *is* AdS5 x S5, and it solves the equations.\n\n> 6) On page 131, D-branes on AdS are discussed. It is explained\n> a D3-brane in AdS_5 x S^5 appears as a domain wall in the dual\n> CFT. Crossing this domain wall changes the gauge group from\n> SU(N) to SU(N+1)! How is this possibly?\n\nIt\'s explained there. Adding one D3-branes increases the flux through the\nS^5 by a single unit, and therefore the gauge group must also jump. Of\ncourse, such large D-branes going up to the boundary change the asymptotic\nstructure of the spacetime, so you should not consider this state to be a\nfinite perturbation of the original SU(N) spacetime. See papers by\nKarch-Randall and others to see how the domain walls in AdS/CFT work.\n\n> Symmetry breaking?\n\nI would say Yes, you can go to the Coulomb branch, and send one diagonal\ncomponent of the scalar in SU(N+1) to infinity, which effectively reduces\nit to SU(N).\n\n> Moreover, if we consider AdS_5 x RP^5 and a D5-brane or an\n> NS5-brane weapped on an RP^2 < RP^5, one gets a domain wall in\n> the dual CFT which switches between an SO(2N) and an Sp(N)\n> gauge group! None of these groups is a subgroup of the other.\n\nThat\'s right. Both of them are subgroups of U(2N), but it is not\nan infinitely important fact. Moreover, non-perturbatively there are\nreally two types of an Sp(N) theory, unless Edward Witten made an error\nseveral years ago.\n\n> Possibly the dual CFT is a gauge theory with a gauge group\n> bigger than both broken to different subgroups on the two\n> sides of the domain wall?\n\nYour implicit assumption that all gauge groups found anywhere in\nspacetime must be subgroups of a single, universal, and unified group is\nnot justified and probably wrong.\n\nMy favorite example is heterotic string theory on a circle. You can start\nwith a E_8 x E_8 gauge group in the bulk, and adjust the Wilson line\naround the circle so that you break it to U(1)^{16}. If you play with the\nWilson lines a little bit more, suddenly you enhance the gauge group to\nSO(32). SO(32) and E_8 x E_8 are not subgroups of one another - in fact,\nthey are both 496-dimensional groups, and the smallest group that may\ncontain both of them may be SO(496) - more or less obviously irrelevant.\n\nGauge symmetries come and go. They are getting broken, and other\nsymmetries are restored and enhanced at various points of the\nconfiguration space where new massless particles/objects occur.\n\nBest\nLubos\n________________________________________________ ______________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 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>Hey Squark,
> 1) Page 30: It is mentioned the classical conformal invariance of 4D
> Yang-Mills is broken on the quantum level since a cutoff has to be
> introduced. [itex]N = 4[/itex] SUSY Yang-Mills, OTOH is a genuine CFT. Should it
> be understood the later doesn't require renormalization at all? It
> would make sense in the light of the fact it's supposed to be dual
> to type IIB superstring theory in [itex]AdS_5 x S^5[/itex] which is finite order
> by order in perturbation theory and thus requires no
> renormalization. Do all CFTs have this property?
I think that this is a good question, and many people should try to help
to create a more complete answer. First of all, QCD is an example of a
classically conformal theory (there is conformal symmetry at the level of
playing with the classical expressions). However quantum effects - such as
the coupling constant running (quantified by the [itex]\beta[/itex] function) make the
theory non-conformal, and there is a priviliged scale (the QCD scale).
On the other hand, there is no running in the N=4 gauge theory, due to its
exact conformal invariance. The vanishing of the [itex]\beta[/itex] function implies the
absence of certain logarithmic divergences. This effect is more general -
the absence of running (i.e. conformal symmetry) means that the
corresponding potential divergences cancel. In fact, the N=4 Yang-Mills is
"finite" in the sense that the correlators of the "elementary fields" are
finite (after the cancellation between many contributions), aren't they?
Clearly, the correlators of composite fields lead to divergences,
anomalous dimensions (quantum contribution to the dimension; which are
dual to interaction corrections to the energy in AdS), and the need to
renormalize. The appearance of log divergences in the correlators of
composite operators does not spoil the exact duality with the "finite"
type IIB string theory, I think, because these divergences can be absorbed
in the normalization of these operators.
I believe that N=4 has no higher divergences than log divergences.
Most of the exactly conformal 4D gauge theories - like the orbifolds of
[itex]N=4 -[/itex] preserve some finiteness of [itex]N=4,[/itex] but probably not all of it. Other
exactly conformal theories are analogous, but the renormalization process
would be done differently and a special discussion may be needed.
> 2) On page 45, beginning of section 2.2.2, it is explained massive
> particles cannot ever reach the boundary of AdS along geodesics but
> massless can. I believe it implies massive particles have to go
> through acceleration of unbounded magnitude to reach the boundary.
> Nevertheless, on page 46 it is explained fields have to be assigned
> boundary conditions for energy conservation to hold which appear to
> be as stringent for massive fields (m > 0) as for massless [itex](m = 0)[/itex].
> Why is that?
I don't know where you see on page 46 that the boundary conditions should
be m-independent. On the contrary, the fields behave as powers near the
boundary, see (2.34), and the exponents [itex](\lambda)[/itex] are computed from the
mass (see (2.36)). In other coordinates, the decrease near the boundary is
exponential. Of course, the sentence "energy is conserved" does not
directly contain the words "mass of the particle", but it contains it
indirectly via the definition of the energy of a scalar field.
> 3) On page 74 orbifolds of [itex]AdS_5 x S^5[/itex] are discussed, where the
> orbifolding is done by a discrete subgroup [itex]\Gamma[/itex] of the [itex]S^5[/itex]
> isometry group SO(6) (which is the R-symmetry group SU(4) of the
> dual CFT).
Right.
> It is said that on both sides of the correspondence
> the orbifolding cannot be done just by taking the
> [itex]\Gamma-invariant[/itex] states, but on the field theory side [itex]we _can_[/itex] do
> this if multiply the number of branes by [itex]dim(\Gamma)[/itex] first.
The number of elements of a discrete group is called [itex]rank(\Gamma),[/itex] not
[itex]dim(\Gamma)[/itex]. Also, sorry, on my copy of the text, Orbifolds are discussed
from the page 113 on (not 74), so your specific links seem corrupt to me
(maybe there is a sentence on orbifold on earlier pages, but they are
systematically explained later).
Well, you must first add all images of your N D3-branes, which a priori
extends the "locally seen" group U(N) to U(N.[itex]rank(\Gamma)),[/itex] but then you
must impose the condition that they are images of each other. This
results in a gauge group [itex]U(N)^{rank(\Gamma)}[/itex] or something along these
lines, and some fields transform as bifundamental representations (as
described by quiver diagrams).
> Do I assume correctly it is actually the number of elements [itex]#\Gamma[/itex]
> which is meant?
I only see [itex]dim(\Gamma)[/itex] in your posting - but definitely, the relevant
multiplication factor changing the *dimension* of the fundamental rep of
the gauge group is [itex]rank(\Gamma)[/itex].
> 4) On page 115, the CFT dual to type IIB on [itex]AdS_5 x S^5 / Z_k[/itex].
> The [itex]AdS/CFT[/itex] correspondence is obtained, in this case, by
> considering D3-branes at an [itex]R^4/Z_k[/itex] orbifold singularity. Do I
> understand correctly [itex]Z_k[/itex] acts by multiplying [itex]x_1 + i x_2[/itex] and
> [itex]x_3 + i x_4[/itex] by [itex]a k-th[/itex] root of unity [itex](x_i[/itex] are the [itex]R^4[/itex]
> coordinates)?
Yes. This action must be done simultaneously on both [itex]x1+ix2[/itex] and [itex]x3+ix4[/itex].
> Further below, it is claimed the dual CFT has k
> complex parameters. On the string theory side, k-1 out of these
> parameters are identified with the NS-NS and R-R 2-forms on the
> k-1 2-cycles which vanish at the orbifold singularity (page 116).
> The [itex]k-th[/itex] parameter is said to be the dilaton. Isn't the dilaton
> real?
Depending on the terminology. Dilaton is usually the real part, but type
IIB string theory has two such scalar fields - the dilaton and the axion
(the RR-scalar) [itex]C_0 -[/itex] that combine into a complex scalar (onto which the
S-duality group SL(2,Z) acts).
> 5) On page 121, superstring theory on [itex]AdS_5 x M_5[/itex] with [itex]M_5 a[/itex]
> 5-manifold no locally isomorphic to [itex]S^5[/itex] is discussed. It is said
> the "effective cosmological constant" on the [itex]AdS_5[/itex] results from
> the 5-form of type IIB SUGRA. It is then said the 5-form is self
> dual and that is the reason [itex]M_5[/itex] has to be an Einstein manifold
> with positive cosmological constant. Firstly, why is it
> self-dual?
There are many reasons to see it. If you derive the physical Ramond-Ramond
spectrum - think about the RR sector of a closed string - there are two
(left-moving and right-moving) GSO projections. The "overall[itex]" Z_2[/itex]
projection tells you that the field strengths in type IIB must be q-forms
for q odd. The other [itex]Z_2[/itex] projection reduces the number of polarizations
to one half, to the self-dual ones. At the end, you obtain - as a
consistency check [itex]- 64[/itex] bosonic polarizations from the RR-sector.
This includes the RR-scalar (the axion, i.e. the other part of the
complexified dilaton), 8.7/2.[itex]1 = 28[/itex] polarizations of the 2-form, and
one half of 8.7.6.5/4.3.2.[itex]1 = 70, i[/itex].e. 35 polarizations of the self-dual
five-form field strength (the transversely self-dual components of the
4-form potential)[itex]. 1+28+35=64,[/itex] which combines with 64 NS-NS states into
128 bosonic polarizations, which combine with [itex]64+64[/itex] fermions [itex](NS-R[/itex] and
[itex]R-NS)[/itex] into the 256 states of the maximal supergravity.
Alternatively, a 5-form field strength in general admits electric sources,
as well as magnetic sources. In [itex]D=10,[/itex] both of these sources would be
3-branes, and because there exist one type of D3-branes in type IIB only,
they must be equivalent, and therefore the 5-form field strength must be
self-dual.
> Isn't it just the RR 5-form of type IIB superstring theory?
I thought you were just asking whether the (RR) 5-form field strength of
type IIB string theory is self-dual - YES, IT IS - so why are you asking
again? Yes, the 5-form of type IIB SUGRA is (the low energy description
of) "just the" RR 5-form of type IIB superstring theory. It's the same
thing. Your question makes absolutely no sense to me. Yes, it is
self-dual. If you need this to be repeated 26 times, feel free to ask,
but I hope that someone else will help me.
> Secondly, how does the conclusion about [itex]M_5[/itex] follow?
Via a careful analysis of signs in the Einstein equations. You realize
that the components of the stress energy tensor, such as [itex]T_{77},[/itex] along the
[itex]S^5,[/itex] if they are computed from the 5-form components such as [itex]F_{56789},[/itex]
are positive, and therefore the [itex]S^5[/itex] sphere must be positively curved, and
by the same logic, the same component [itex]F_{01234}=F_{56789}[/itex] (self-duality)
curves the AdS space, which must be negatively curved.
It is not hard to see that the sign of the scalar curvature of the [itex]AdS_5[/itex]
and of [itex]S^5[/itex] must be opposite. Continue to Euclidean time. Then you will
have the self-duality constraint [itex]F_{01234}=i[/itex].[itex]F_{56789}[/itex] with i from the
Wick rotation, and therefore the stress energy tensors constructed in the
01234 and 56789 directions will have opposite signs. It leads to opposite
curvatures, and the curvatures are unchanged by continuation.
It is useful to understand the whole solution. The near horizon geometry
of the D3-branes simply *is* AdS5 x S5, and it solves the equations.
> 6) On page 131, D-branes on AdS are discussed. It is explained
> a D3-brane [itex]in AdS_5 x S^5[/itex] appears as a domain wall in the dual
> CFT. Crossing this domain wall changes the gauge group from
> SU(N) to [itex]SU(N+1)![/itex] How is this possibly?
It's explained there. Adding one D3-branes increases the flux through the
[itex]S^5[/itex] by a single unit, and therefore the gauge group must also jump. Of
course, such large D-branes going up to the boundary change the asymptotic
structure of the spacetime, so you should not consider this state to be a
finite perturbation of the original SU(N) spacetime. See papers by
Karch-Randall and others to see how the domain walls in [itex]AdS/CFT[/itex] work.
> Symmetry breaking?
I would say Yes, you can go to the Coulomb branch, and send one diagonal
component of the scalar in [itex]SU(N+1)[/itex] to infinity, which effectively reduces
it to SU(N).
> Moreover, if we consider [itex]AdS_5 x RP^5[/itex] and a D5-brane or an
> NS5-brane weapped on an [itex]RP^2 < RP^5,[/itex] one gets a domain wall in
> the dual CFT which switches between an SO(2N) and an Sp(N)
> gauge group! None of these groups is a subgroup of the other.
That's right. Both of them are subgroups of U(2N), but it is not
an infinitely important fact. Moreover, non-perturbatively there are
really two types of an Sp(N) theory, unless Edward Witten made an error
several years ago.
> Possibly the dual CFT is a gauge theory with a gauge group
> bigger than both broken to different subgroups on the two
> sides of the domain wall?
Your implicit assumption that all gauge groups found anywhere in
spacetime must be subgroups of a single, universal, and unified group is
not justified and probably wrong.
My favorite example is heterotic string theory on a circle. You can start
with [itex]a E_8 x E_8[/itex] gauge group in the bulk, and adjust the Wilson line
around the circle so that you break it to [itex]U(1)^{16}[/itex]. If you play with the
Wilson lines a little bit more, suddenly you enhance the gauge group to
SO(32). SO(32) and [itex]E_8 x E_8[/itex] are not subgroups of one another - in fact,
they are both 496-dimensional groups, and the smallest group that may
contain both of them may be SO(496) - more or less obviously irrelevant.
Gauge symmetries come and go. They are getting broken, and other
symmetries are restored and enhanced at various points of the
configuration space where new massless particles/objects occur.
Best
Lubos
__{____________________________________________________________________ ________}
E-mail:
lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web:
http://lumo.matfyz.cz/
eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/496-8199[/itex] home: [itex]+1-617/868-4487[/itex] (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^