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<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>Dear Squark,\n\nhere are answers to some of your questions from the\nsecond batch.\n\nIlarion\n\n> 5) On page 5 Aspinwall defines the fermions of N = (2,2)\n> non-linear sigma-model. If phi is the bosonic part of the\n> sigma-model, the fermions are sections of the following\n> bundles on the worldsheet Sigma:\n>\n> a) psi^i_+ of K^1/2 (x) phi*T_X\n> b) psi^jbar_+ of K^1/2 (x) phi*Tbar_X\n> c) psi^i_- of Kbar^1/2 (x) phi*T_X\n> d) psi^jbar_- of Kbar^1/2 (x) phi*Tbar_X\n>\n> X is the target space, K is the holomorphic cotangent\n> bundle on Sigma. The question is, what are conditions for\n> K^1/2 to exist and how unique is it? What is the physical\n> meaning of these conditions?\n\nRemember, that the discussion in Aspinwall\'s notes is\nrestricted to zero genus worldsheet. There all is well, and\nK^(1/2) (or equivalently, a spin structure) exists and is\nunique. For a Riemann surface of higher genus there is no\nlonger a unique spin structure, and I imagine (I hope the\nexperts will correct me) that K^(1/2) has the same kind of\nambiguity. Basically, think spin manifold thoughts. The\ncondition for a manifold to be spin is that the second\nSteifel-Whitney class vanishes, so 2, and 3 dimensional\nmanifolds are automatically spin, but spin structures are not\nnecessarily unique and can be characterized by H^1(M, Z_2).\n\n> 7) In section 6, starting on page 63, Aspinwall talks about\n> stability of A and B branes. Is it correct to assume that\n> when he speaks here of the decay procces of a brane M into\n> branes N1 ... Nk, the decay products may actually contain\n> closed strings as well?\n\nI don\'t think so, or at least not directly. Of course, when\nreal branes (i.e. not topological and not at g_s = 0) decay,\nyou are, of course, correct. However,\nI think that the production of closed strings would occur at\nhigher order in g_s that is considered (namely g_s^0 !) in this\nframework. The closed strings only show up through the background\ngeometry.\n\n> 8) On page 63 Aspinwall claims BPS branes in the untwisted\n> theory descend to D-branes in A and B models. Why is true?\n> Intuitively it appears BPS D-branes should correspond to\n> "classical" boundary conditions on the world sheet, which\n> are not affected quantum corrections, therefore, they can\n> be painlessly transferred to the twisted case. Can one\n> make this more rigorous? Now, it seems possible that BPS\n> D3-branes would become A-branes and BPS D-2k-branes would\n> become B-branes. What about BPS D1-branes and BPS\n> D5-branes? Or can\'t these exist for some reason**? What\n> about NS5-branes?\n\nI know of no reason why the boundary conditions are not\naffected by quantum corrections. In fact, one would assume\nthey are so. The picture of branes as boundary conditions is\nonly valid in the large radius limit, where it clearly emerges\nfrom worldsheet or spacetime (Becker^2, Strominger) considerations.\nHowever, quantum effects modify this quite a bit. The framework\nthat Aspinwall presents in his notes allows to track these objects\naway from the large radius limit in the topological field theory.\n\nNS5: these are hard to get hold of in a controlled TQFT.\nD5: these can appear as so-called co-isotropic branes. (look\nat Kapustin\'s papers on the subject).\nD1: these require a non-trivial H_1 for the target space, which\nis not oft considered.\n\n> 10) On page 68, it is claimed the central charge of an\n> A-brane L is given by (integral over L) Omega where Omega\n> is again the unique-up-to-constant (3,0)-form on the\n> target space Y. The problem is, Omega is only defined up\n> to a constant whereas the central charge should be\n> absolutely defined (rather than its rations only), no?\n\nI am not a big expert on A-brane business (or B-brane business\nfor that matter :) ), but I would say that the story is much\nthe same as for the grade defined in Eq. (162). The grade\nof a given Lagrangian is fairly meaningless precisely for\nthe reason you gave. However, if you compare two Lagrangians,\nit makes sense! Note that on the B-brane side the central\ncharge is given by ratios of periods of Omega, so that this\nambiguity is removed. My understanding (again, it would be\ngreat for the experts to correct) is that this has to do with\nmeasuring that central charge with respect to the 0-brane.\n\n> 15) On page 77, Aspinwall defines the parameters ksi\n> which controll B-brane stability as a function of the\n> Kahler form on the target space. These are defined as the\n> arguments of the central charges which are in turn\n> computed via the so-called Picard-Fuchs equations and the\n> small alpha\' approximation given in equation 191, page\n> 76. Is there a more elegant way to define these\n> quantities or at least have a grasp of their physical\n> meaning (besides just saying they\'re the given by the\n> A-brane ksi-s in the mirror)?\n\nWell, one can talk in analogies about lattices of charges in\nN=2 theories and various Zs lining up, etc. However, that\nwon\'t give you Eq. (191)\'s generalization. For that one\nreally has to use mirror symmetry, as it is an entirely\na non-trivial matter!\n\n> 17) On page 84, Pi-stability for quintic threefold is\n> analyzed. The periods (integrals over elements of the\n> homology group) of the holomorphic 3-form Omega are\n> claimed to be given by equation 210:\n>\n> omega_j = (-1/5)(Sum over m = 1 to infinity)\n> alpha^(2+j)m Gamma(m/5) z^-m/5 / Gamma(m) Gamma(1 - m/5)^4\n>\n> where the quintic is given by the equation\n>\n> x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 +\n> z^5 x_0 x_1 x_2 x_3 x_5 = 0\n>\n> in homogeneous coordinates. What is alpha in this equation?\n> Also, what dzeta in equation 214 on page 85?\n\nFirst, I hope that you now know what the ... in Eq.(36) stands\nfor. Second, the Phi_i given are a particular set of periods\nwith the requisite asymptotic properties (namely, those that\nmatch (216)). The alpha is a fifth root of unity, zeta is\nthe Riemann zeta function!\n\n> 19) On page 103, the Bridge-Kind-Reid (BKR) theorem is\n> stated: "Suppose X is a smooth resolution of the orbifold\n> C^d / G with G a finite subgroup of SU(d) and d < 4. Then\n> the derived category D(X) is equivalent to the derived\n> category of G-equivariant sheaves on C^d". Is it known\n> what happens for more general orbifold resolutions? Also,\n> footnote 42 says that "the only reason why it should fail\n> for d > 3 is that smooth resolutions need not exist". What\n> if a smooth resolution happens to exist after all?\n\nHmm, how more general would you like? Bridgeland-King-Reid\nalready tells you that for SUSY orbifolds of C^3 the result\nis true. Now you can (and people do) consider non-SUSY\norbifolds. In principle, this is not so bad. The sigma\nmodel still has N=(2,2) SUSY, and the only thing to go wrong\nis the integrality of the R-charges no longer holds, so that\nspectral flow is gone. This is almost bearable, but much of\nthe machinery described in the notes does not apply at all!\nAs for the footnote, I suppose that means if you can show\nthat your C^4/G orbifold admits a smooth resolution, then\nBKR holds and you can start deriving.\n\n> 22) On the same page 207, Aspinwall calims the blow-up of\n> the oribifold singularity is produced by "twisted closed\n> string marginal operators". What does that "twisted" part\n> mean?\n\nWell, when one constructs an orbifold CFT, one finds oneself\nwith twisted sectors. These may contain marginal operators.\nWhen one is orbifolding a CFT with a nice geometric interpretation,\none finds that the twisted marginal ops are in one to one correspondence\nwith the blow-up modes of the orbifold, and turning them on changes\nthe closed string background by blowing up the orbifold singularity.\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>Dear Squark,
here are answers to some of your questions from the
second batch.
Ilarion
> 5) On page 5 Aspinwall defines the fermions of [itex]N = (2,2)[/itex]
> non-linear [itex]\sigma-model[/itex]. If [itex]\phi[/itex] is the bosonic part of the
> [itex]\sigma-model,[/itex] the fermions are sections of the following
> bundles on the worldsheet [itex]\Sigma:[/itex]
>
> [itex]a) \psi^i_+[/itex] of [itex]K^1/2[/itex] (x) [itex]\phi*T_X[/itex]
> [itex]b) \psi^jbar_+[/itex] of [itex]K^1/2[/itex] (x) [itex]\phi*Tbar_X[/itex]
> [itex]c) \psi^i_-[/itex] of [itex]Kbar^1/2 (x) \phi*T_X[/itex]
> [itex]d) \psi^jbar_-[/itex] of [itex]Kbar^1/2 (x) \phi*Tbar_X[/itex]
>
> X is the target space, K is the holomorphic cotangent
> bundle on [itex]\Sigma[/itex]. The question is, what are conditions for
> [itex]K^1/2[/itex] to exist and how unique is it? What is the physical
> meaning of these conditions?
Remember, that the discussion in Aspinwall's notes is
restricted to zero genus worldsheet. There all is well, and
[itex]K^(1/2) (or[/itex] equivalently, a spin structure) exists and is
unique. For a Riemann surface of higher genus there is no
longer a unique spin structure, and I imagine (I hope the
experts will correct me) that [itex]K^(1/2)[/itex] has the same kind of
ambiguity. Basically, think spin manifold thoughts. The
condition for a manifold to be spin is that the second
Steifel-Whitney class vanishes, so 2, and 3 dimensional
manifolds are automatically spin, but spin structures are not
necessarily unique and can be characterized by [itex]H^1(M, Z_2)[/itex].
> 7) In section 6, starting on page 63, Aspinwall talks about
> stability of A and B branes. Is it correct to assume that
> when he speaks here of the decay procces of a brane M into
> branes N1 ... Nk, the decay products may actually contain
> closed strings as well?
I don't think so, or at least not directly. Of course, when
real branes (i.e. not topological and not [itex]at g_s = 0)[/itex] decay,
you are, of course, correct. However,
I think that the production of closed strings would occur at
higher order in [itex]g_s[/itex] that is considered (namely [itex]g_s^0 !)[/itex] in this
framework. The closed strings only show up through the background
geometry.
> 8) On page 63 Aspinwall claims BPS branes in the untwisted
> theory descend to D-branes in A and B models. Why is true?
> Intuitively it appears BPS D-branes should correspond to
> "classical" boundary conditions on the world sheet, which
> are not affected quantum corrections, therefore, they can
> be painlessly transferred to the twisted case. Can one
> make this more rigorous? Now, it seems possible that BPS
> D3-branes would become A-branes and BPS D-2k-branes would
> become B-branes. What about BPS D1-branes and BPS
> D5-branes? Or can't these exist for some reason**? What
> about NS5-branes?
I know of no reason why the boundary conditions are not
affected by quantum corrections. In fact, one would assume
they are so. The picture of branes as boundary conditions is
only valid in the large radius limit, where it clearly emerges
from worldsheet or spacetime [itex](Becker^2,[/itex] Strominger) considerations.
However, quantum effects modify this quite a bit. The framework
that Aspinwall presents in his notes allows to track these objects
away from the large radius limit in the topological field theory.
NS5: these are hard to get hold of in a controlled TQFT.
D5: these can appear as so-called co-isotropic branes. (look
at Kapustin's papers on the subject).
D1: these require a non-trivial [itex]H_1[/itex] for the target space, which
is not oft considered.
> 10) On page 68, it is claimed the central charge of an
> A-brane L is given by (integral over [itex]L) \Omega[/itex] where [itex]\Omega[/itex]
> is again the unique-up-to-constant (3,0)-form on the
> target space Y. The problem is, [itex]\Omega[/itex] is only defined up
> to a constant whereas the central charge should be
> absolutely defined (rather than its rations only), no?
I am not a big expert on A-brane business (or B-brane business
for that matter :) ), but I would say that the story is much
the same as for the grade defined in Eq. (162). The grade
of a given Lagrangian is fairly meaningless precisely for
the reason you gave. However, if you compare two Lagrangians,
it makes sense! Note that on the B-brane side the central
charge is given by ratios of periods of [itex]\Omega,[/itex] so that this
ambiguity is removed. My understanding (again, it would be
great for the experts to correct) is that this has to do with
measuring that central charge with respect to the 0-brane.
> 15) On page 77, Aspinwall defines the parameters ksi
> which controll B-brane stability as a function of the
> Kahler form on the target space. These are defined as the
> arguments of the central charges which are in turn
> computed via the so-called Picard-Fuchs equations and the
> small [itex]\alpha'[/itex] approximation given in equation 191, page
> 76. Is there a more elegant way to define these
> quantities or at least have a grasp of their physical
> meaning (besides just saying they're the given by the
> A-brane ksi-s in the mirror)?
Well, one can talk in analogies about lattices of charges in
N=2 theories and various Zs lining up, etc. However, that
won't give you Eq. (191)'s generalization. For that one
really has to use mirror symmetry, as it is an entirely
a non-trivial matter!
> 17) On page 84, [itex]\Pi-stability[/itex] for quintic threefold is
> analyzed. The periods (integrals over elements of the
> homology group) of the holomorphic 3-form [itex]\Omega[/itex] are
> claimed to be given by equation 210:
>
> [itex]\omega_j = (-1/5)(Sum[/itex] over [itex]m = 1[/itex] to infinity)
> [itex]\alpha^(2+j)m \Gamma(m/5) z^-m/5 / \Gamma(m) \Gamma(1 - m/5)^4[/itex]
>
> where the quintic is given by the equation
>
> [itex]x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 +[/itex]
> [itex]z^5 x_0 x_1 x_2 x_3 x_5 =[/itex]
>
> in homogeneous coordinates. What is [itex]\alpha[/itex] in this equation?
> Also, what dzeta in equation 214 on page 85?
First, I hope that you now know what the ... in Eq.(36) stands
for. Second, the [itex]\Phi_i[/itex] given are a particular set of periods
with the requisite asymptotic properties (namely, those that
match (216)). The [itex]\alpha[/itex] is a fifth root of unity, [itex]\zeta[/itex] is
the Riemann [itex]\zeta[/itex] function!
> 19) On page 103, the Bridge-Kind-Reid (BKR) theorem is
> stated: "Suppose X is a smooth resolution of the orbifold
> [itex]C^d / G[/itex] with G a finite subgroup of SU(d) and d < 4. Then
> the derived category D(X) is equivalent to the derived
> category of G-equivariant sheaves on [itex]C^d[/itex]". Is it known
> what happens for more general orbifold resolutions? Also,
> footnote 42 says that "the only reason why it should fail
> for d > 3 is that smooth resolutions need not exist". What
> if a smooth resolution happens to exist after all?
Hmm, how more general would you like? Bridgeland-King-Reid
already tells you that for SUSY orbifolds of [itex]C^3[/itex] the result
is true. Now you can (and people do) consider non-SUSY
orbifolds. In principle, this is not so bad. The [itex]\sigma[/itex]
model still has [itex]N=(2,2)[/itex] SUSY, and the only thing to go wrong
is the integrality of the R-charges no longer holds, so that
spectral flow is gone. This is almost bearable, but much of
the machinery described in the notes does not apply at all!
As for the footnote, I suppose that means if you can show
that your [itex]C^4/G[/itex] orbifold admits a smooth resolution, then
BKR holds and you can start deriving.
> 22) On the same page 207, Aspinwall calims the blow-up of
> the oribifold singularity is produced by "twisted closed
> string marginal operators". What does that "twisted" part
> mean?
Well, when one constructs an orbifold CFT, one finds oneself
with twisted sectors. These may contain marginal operators.
When one is orbifolding a CFT with a nice geometric interpretation,
one finds that the twisted marginal ops are in one to one correspondence
with the blow-up modes of the orbifold, and turning them on changes
the closed string background by blowing up the orbifold singularity.
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