Aspinwall's "D-Branes on Calabi-Yau Manifolds"

Squark

<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>Hello everyone.\n\nI\'ve been reading Paul Aspinwall\'s "D-Branes on Calabi-Yau Manifolds"\n(hep-th/0403166) and I\'ve got a few questions.\n\n1) Page 15, equation 35: I must be getting slow. What does the "..."\nstand for?\n\n2) Page 22: An A-brane carrying a line bundle is represented by a\nLagrangean submanifold of the Calabi-Yau, or, more generally,\nsymplectic manifold Y. If transformed along a Hamiltonian flow, one\ngets a physically equivalent A-brane. Now consider an A-brane\ncarrying a rank n > 1 bundle. It might be viewed as a collection of\nn line-bundle A-branes with coincident locations. Now, move each\nof those components along a different Hamiltonian flow. The result\nis no longer a higher rank A-brane. Is it still physically\nequivalent? If so, where highes the additional gauge symmetry in\nthis representation?\n\n3) Page 22, equation 46: Do I understand correctly this equation is\nvalid on tree level only? Or, do we get no loop corrections for some\nreason?\n\n4) Page 26, equation 53: It\'s late, I have fever and cannot quite\nfollow the reasoning behind this equation :-) At any rate, what\nhappens when mue(pi_b) < mue(pi_a) + 1?\n\nBest regards,\nSquark.\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>Hello everyone.

I've been reading Paul Aspinwall's "D-Branes on Calabi-Yau Manifolds"
(http://www.arxiv.org/abs/hep-th/0403166) and I've got a few questions.

1) Page 15, equation 35: I must be getting slow. What does the "..."
stand for?

2) Page 22: An A-brane carrying a line bundle is represented by a
Lagrangean submanifold of the Calabi-Yau, or, more generally,
symplectic manifold Y. If transformed along a Hamiltonian flow, one
gets a physically equivalent A-brane. Now consider an A-brane
carrying a rank n > 1 bundle. It might be viewed as a collection of
n line-bundle A-branes with coincident locations. Now, move each
of those components along a different Hamiltonian flow. The result
is no longer a higher rank A-brane. Is it still physically
equivalent? If so, where highes the additional gauge symmetry in
this representation?

3) Page 22, equation 46: Do I understand correctly this equation is
valid on tree level only? Or, do we get no loop corrections for some
reason?

4) Page 26, equation 53: It's late, I have fever and cannot quite
follow the reasoning behind this equation :-) At any rate, what
happens when [itex]mue(\pi_b) < mue(\pi_a) + 1[/itex]?

Best regards,
Squark.

Kris Kennaway

<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 2004-05-03, Squark <fiis5d@yahoo.com> wrote:\n\n> 2) Page 22: An A-brane carrying a line bundle is represented by a\n> Lagrangean submanifold of the Calabi-Yau, or, more generally,\n> symplectic manifold Y. If transformed along a Hamiltonian flow, one\n> gets a physically equivalent A-brane. Now consider an A-brane\n> carrying a rank n > 1 bundle. It might be viewed as a collection of\n> n line-bundle A-branes with coincident locations. Now, move each\n> of those components along a different Hamiltonian flow. The result\n> is no longer a higher rank A-brane. Is it still physically\n> equivalent? If so, where highes the additional gauge symmetry in\n> this representation?\n\nI\'m too lazy to download the paper right now, but it sounds like\nyou\'re just moving the N coincident branes apart along a flat\ndirection. This breaks the gauge symmetry U(N) -> U(1)^N just as in\nflat space.\n\nKris\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 [itex]2004-05-03,[/itex] Squark <fiis5d@yahoo.com> wrote:

> 2) Page 22: An A-brane carrying a line bundle is represented by a
> Lagrangean submanifold of the Calabi-Yau, or, more generally,
> symplectic manifold Y. If transformed along a Hamiltonian flow, one
> gets a physically equivalent A-brane. Now consider an A-brane
> carrying a rank n > 1 bundle. It might be viewed as a collection of
> n line-bundle A-branes with coincident locations. Now, move each
> of those components along a different Hamiltonian flow. The result
> is no longer a higher rank A-brane. Is it still physically
> equivalent? If so, where highes the additional gauge symmetry in
> this representation?

I'm too lazy to download the paper right now, but it sounds like
you're just moving the N coincident branes apart along a flat
direction. This breaks the gauge symmetry U(N) [itex]-> U(1)^N[/itex] just as in
flat space.

Kris

Squark

<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>Kris Kennaway <kkenn@xor.obsecurity.org> wrote in message news:<iv4jm1-ptk1.ln1-100000@xor.obsecurity.org>...\n\n> I\'m too lazy to download the paper right now, but it sounds like\n> you\'re just moving the N coincident branes apart along a flat\n> direction. This breaks the gauge symmetry U(N) -> U(1)^N just as in\n> flat space.\n\nThe whole point is that it probably doesn\'t. The reason is that\nmoving an A-brane along a Hamiltonian direction is deforming\nthe theory by a BRST-exact operator which shouldn\'t have any\neffect. So, unless I\'m missing something, the full gauge\nsymmetry is still hiding somewhere, the question is where.\n\nBest regards,\nSquark\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>Kris Kennaway <kkenn@xor.obsecurity.org> wrote in message news:<iv4jm1-ptk1.ln1-100000@xor.obsecurity.org>...

> I'm too lazy to download the paper right now, but it sounds like
> you're just moving the N coincident branes apart along a flat
> direction. This breaks the gauge symmetry U(N) [itex]-> U(1)^N[/itex] just as in
> flat space.

The whole point is that it probably doesn't. The reason is that
moving an A-brane along a Hamiltonian direction is deforming
the theory by a BRST-exact operator which shouldn't have any
effect. So, unless I'm missing something, the full gauge
symmetry is still hiding somewhere, the question is where.

Best regards,
Squark

Squark

<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>Hello everyone.\n\nSince there was no reply to my previous question, I will...\nPost more questions. Maybe someone finds one of these more\ninteresting.\n\n\n5) On page 5 Aspinwall defines the fermions of N = (2,2)\nnon-linear sigma-model. If phi is the bosonic part of the\nsigma-model, the fermions are sections of the following\nbundles on the worldsheet Sigma:\n\na) psi^i_+ of K^1/2 (x) phi*T_X\nb) psi^jbar_+ of K^1/2 (x) phi*Tbar_X\nc) psi^i_- of Kbar^1/2 (x) phi*T_X\nd) psi^jbar_- of Kbar^1/2 (x) phi*Tbar_X\n\nX is the target space, K is the holomorphic cotangent\nbundle on Sigma. The question is, what are conditions for\nK^1/2 to exist and how unique is it? What is the physical\nmeaning of these conditions?\n\n\n6) On pages 19-20, Aspinwall defines tge Maslov class of a\nLagrangian submanifold L as a map pi_1(L) -> pi_1(S^1) = Z.\nThe definition goes by defining u = z / |z| where\nz = Omega|_L / dV_L. Here, Omega is the unique up to\nnormalization holomorphic 3-form on Y, the ambient space of\nL (a Calabi-Yau), dV_L is the volume form* on L. u is then a\nmap L -> S^1 which induces the Maslov class.\nThe vanishing of the Maslov class is an anomaly vanishing\ncondition for A-branes on Y, and since the A-model only\ndepends on the Kahler form / complexified symplectic\nstructure, there might be a definition which doesn\'t\nexplicitely use the complex structure. Almost certainly\nthere should be since the A-model is not only defined for\nYs coming from a Calabi-Yau. What is this definition?\n\n\n7) In section 6, starting on page 63, Aspinwall talks about\nstability of A and B branes. Is it correct to assume that\nwhen he speaks here of the decay procces of a brane M into\nbranes N1 ... Nk, the decay products may actually contain\nclosed strings as well?\n\n\n8) On page 63 Aspinwall claims BPS branes in the untwisted\ntheory descend to D-branes in A and B models. Why is true?\nIntuitively it appears BPS D-branes should correspond to\n"classical" boundary conditions on the world sheet, which\nare not affected quantum corrections, therefore, they can\nbe painlessly transferred to the twisted case. Can one\nmake this more rigorous? Now, it seems possible that BPS\nD3-branes would become A-branes and BPS D-2k-branes would\nbecome B-branes. What about BPS D1-branes and BPS\nD5-branes? Or can\'t these exist for some reason**? What\nabout NS5-branes?\n\n\n9) On page 67, Aspinwall discusses the formation of a\nbound state of two A-branes given by Lagrangian\nsubmanifolds L_1 and L_2 with a so-called "type 1"\nintersection. The intersection region is then replaced by\na Lawlor neck. It is then claimed the geometrical\noperation described is assymetric with repsect to\nexchange of L_1 and L_2. Why? It might be that the Lawlor\nneck is asymmetrical but then, how do the A-branes know\nin which "order" to bind?\n\n\n10) On page 68, it is claimed the central charge of an\nA-brane L is given by (integral over L) Omega where Omega\nis again the unique-up-to-constant (3,0)-form on the\ntarget space Y. The problem is, Omega is only defined up\nto a constant whereas the central charge should be\nabsolutely defined (rather than its rations only), no?\n\n\n11) On page 74 Aspinwall gives the physical\ninterpretation of the axioms of a triangulated category.\nOne of the axioms (regarding completion of commutative\ndiagrams between distinguished triangles) is interpreted\nas follows: "given open strings between D-branes A and A\'\nand between B and B\' we may construct open strings\nbetween the corresponding bound states". Do I correctly\nunderstand the thus constructed open string is a linear\ncombination (in the sense of quantum superposition) of\nthe A-A\' string and the B-B\' string? Are such\nsuperpositions in 1-1 correspondence with arrows\ncompleting the diagram?\n\n\n12) On page 75, the homological mirror conjecture is\ndiscussed. It is explained the Fukaya category F(Y) is\nnot triangulated and therefore it has to be completed to a\ntriangulated category Tr F(Y) for mirror symmetry to have\na chance (and supposedly to triumph). Firstly, is Tr F(Y)\nsomething like the "free triangulated category" generated\nby F(Y) or, is some a priori information about\ndistringuised triangles assumed? Is the translation\nautomorphism fixed a priori (to be adding 1 to all ghost\nnumbers)? Secondly, has it been fully demonstrated Tr F(Y)\nis "bigger than" F(Y) in some example?\n\n\n13) What exactly happens to the correspondence between\nB-branes (viewed as vector bundles on the target space X)\nand objects of D(X) when a non-zero B-field is on? Then\nthe vector bundles should be replaced by twisted vector\nbundles (at least in the untwisted theory?). Do I\nunderstand correctly that precisely because of that the\ncorrespondence between B-branes and D(X) is non-canonical\nand we may have mondromies of the set of stable objects\nwhen going around a non-contractible loop in moduli space?\n\n\n14) On page 78, the tachyon condensation approach to\nB-brane decay is discussed. Given two objects A and B of\nD(X) and a morphism f:A -> B[1], Cone(f) represents the\nbound state of A with B via condesation of the potentially\ntachyonic string state f. It is claimed that whereas, from\nthe viewpoint of algebraic geometry, Cone(f) is invariant\nunder replacing f by zf for any non-zero number z, in the\nuntwisted theory the scale of f is determined by\nminimizing the quantum potential of the tachyon. Am I to\nunderstand that the "scale of f" is really the scale of\nthe "quantum field" corresponding to f, i.e. the e.v of\na_f + ia*_f where a_f and a*_f are the annihilation and\ncreation operators of the string state f.\n\n\n15) On page 77, Aspinwall defines the parameters ksi\nwhich controll B-brane stability as a function of the\nKahler form on the target space. These are defined as the\narguments of the central charges which are in turn\ncomputed via the so-called Picard-Fuchs equations and the\nsmall alpha\' approximation given in equation 191, page\n76. Is there a more elegant way to define these\nquantities or at least have a grasp of their physical\nmeaning (besides just saying they\'re the given by the\nA-brane ksi-s in the mirror)?\n\n\n16) On page 79, Aspinwall discusses decay into multiple\nB-branes. It is claimed that for certain value of the\nksi-parameters for the A_1, A_2, A_3 the objects E_2 and\nF are stable w.r.t. decay via the triangles appearing in\nthe octahedron diagram 197. It is not clear to me why E_2\nis stable w.r.t. E_2 -> A_3 + E_3 and why F is stable\nw.r.t. F -> A_1 + E_3?\n\n\n17) On page 84, Pi-stability for quintic threefold is\nanalyzed. The periods (integrals over elements of the\nhomology group) of the holomorphic 3-form Omega are\nclaimed to be given by equation 210:\n\nomega_j = (-1/5)(Sum over m = 1 to infinity)\nalpha^(2+j)m Gamma(m/5) z^-m/5 / Gamma(m) Gamma(1 - m/5)^4\n\nwhere the quintic is given by the equation\n\nx_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 +\nz^5 x_0 x_1 x_2 x_3 x_5 = 0\n\nin homogeneous coordinates. What is alpha in this equation?\nAlso, what dzeta in equation 214 on page 85?\n\n\n18) On page 86, Aspinwall claims a 4-brane is always\nmue-stable (i.e. BPS in the large volume limit) since it\nhas no subobjects. How come? What about 2-branes?\n\n\n19) On page 103, the Bridge-Kind-Reid (BKR) theorem is\nstated: "Suppose X is a smooth resolution of the orbifold\nC^d / G with G a finite subgroup of SU(d) and d < 4. Then\nthe derived category D(X) is equivalent to the derived\ncategory of G-equivariant sheaves on C^d". Is it known\nwhat happens for more general orbifold resolutions? Also,\nfootnote 42 says that "the only reason why it should fail\nfor d > 3 is that smooth resolutions need not exist". What\nif a smooth resolution happens to exist after all?\n\n\n20) On page 105 Aspinwall explains the Sardo Infirri view\nof the Douglas-Moore construction (linking D-branes on the\norbifold C^d/G to McKay quivers). I don\'t quite follow.\nBelow Aspinwall says "In this case one studies translation\ninvariant G-equivariant holomorphic bundles on Q = C^d.\nLet the fiber of a vector bundle be given by a\nepresentation V of G". I.e. each fiber is V and this is\nhow the G-equivariant structure works? Then he continues\n"Then, for G-equivariance the connection on this bundle\ntransforms yet again in Hom_G(V, Q (x) V). Why?\n\n\n21) On page 107, Aspinwall talks about Pi-stability in\nthe vicinity of the orbifold point in the moduli space.\nThe stability condition is termed "theta-stability" and it\nis claimed that analogically to mue-stability, one has to\nimpose the condition that the G-equivariant bundle +\nconnection describing the D-brane has is\n"Hermitian-Yang-Mills", i.e. it satisfied equation 201 on\npage 81:\n\ng^(j kbar) F^beta_(alpha j k bar) = mue(E) delta^alpha_beta\n\nHere F is the curvature tensor, E is the vector bundle, g\nis the metric, j and k are target-space-indices, alpha and\nbeta are E-indices, mue(E) is the so-called "slope" of E:\n\nmue(E) = deg(E) / k Vol(X)\n\nwhere k is the rank of E, X is the target space and\n\ndeg(E) = (integral over X) J ^ J ^ c_1(E)\n\nwhere J is the Kahler form. However, on page 81 it is said\nthat "a bundle is mue-stable iff it admits an _irreducibe_\nHermitian-Yang-Mills connection".\n\nDon\'t we have to demand irreducibility for theta-stability\neither?\n\n\n22) On the same page 207, Aspinwall calims the blow-up of\nthe oribifold singularity is produced by "twisted closed\nstring marginal operators". What does that "twisted" part\nmean?\n\n\nThx in advance!\n\nBest regards,\nSquark\n\n\n* Apparently, the definition doesn\'t really depend on the\nchoice of dV_L since rescaling it by a positive function\ndoesn\'t affect u.\n\n** By D-n-branes here I mean D-branes which have an\nn-dimensional compactified-dimensions part. The Calabi-Yau\nD-branes can be cross-multiplied with 3-space D-branes, so,\nfor instance, both the IIA and the IIB type models will\ncontain both even and odd dimensional Calabi-Yau D-branes\n(cross multiplied with 3-space D-branes of corresponding\ndimension parity).\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>Hello everyone.

Since there was no reply to my previous question, I will...
Post more questions. Maybe someone finds one of these more
interesting.

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:a) \psi^i_+[/itex] of [itex]K^1/2[/itex] (x) [itex]\phi*T_Xb) \psi^jbar_+[/itex] of [itex]K^1/2[/itex] (x) [itex]\phi*Tbar_Xc) \psi^i_-[/itex] of [itex]Kbar^1/2 (x) \phi*T_Xd) \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?

6) On pages [itex]19-20,[/itex] Aspinwall defines tge Maslov class of a
Lagrangian submanifold L as a map [itex]\pi_1(L) -> \pi_1(S^1) = Z[/itex].
The definition goes by defining [itex]u = z / |z|[/itex] where
[itex]z = \Omega|_L / dV_L.[/itex] Here, [itex]\Omega[/itex] is the unique up to
normalization holomorphic 3-form on Y, the ambient space of
L (a Calabi-Yau), [itex]dV_L[/itex] is the volume [itex]form*[/itex] on L. u is then a
map [itex]L -> S^1[/itex] which induces the Maslov class.
The vanishing of the Maslov class is an anomaly vanishing
condition for A-branes on Y, and since the A-model only
depends on the Kahler form / complexified symplectic
structure, there might be a definition which doesn't
explicitely use the complex structure. Almost certainly
there should be since the A-model is not only defined for
Ys coming from a Calabi-Yau. What is this definition?

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?

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?

9) On page 67, Aspinwall discusses the formation of a
bound state of two A-branes given by Lagrangian
submanifolds [itex]L_1[/itex] and [itex]L_2[/itex] with a so-called "type 1"
intersection. The intersection region is then replaced by
a Lawlor neck. It is then claimed the geometrical
operation described is assymetric with repsect to
exchange of [itex]L_1[/itex] and [itex]L_2[/itex]. Why[itex]? It[/itex] might be that the Lawlor
neck is asymmetrical but then, how do the A-branes know
in which "order" to bind?

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?

11) On page 74 Aspinwall gives the physical
interpretation of the axioms of a triangulated category.
One of the axioms (regarding completion of commutative
diagrams between distinguished triangles) is interpreted
as follows: "given open strings between D-branes A and A'
and between B and B' we may construct open strings
between the corresponding bound states". Do I correctly
understand the thus constructed open string is a linear
combination (in the sense of quantum superposition) of
the [itex]A-A'[/itex] string and the [itex]B-B'[/itex] string? Are such
superpositions in 1-1 correspondence with arrows
completing the diagram?

12) On page 75, the homological mirror conjecture is
discussed. It is explained the Fukaya category F(Y) is
not triangulated and therefore it has to be completed to a
triangulated category Tr F(Y) for mirror symmetry to have
a chance (and supposedly to triumph). Firstly, is Tr F(Y)
something like the "free triangulated category" generated
by F(Y) or, is some a priori information about
distringuised triangles assumed? Is the translation
automorphism fixed a priori (to be adding 1 to all ghost
numbers)? Secondly, has it been fully demonstrated Tr F(Y)
is "bigger than" F(Y) in some example?

13) What exactly happens to the correspondence between
B-branes (viewed as vector bundles on the target space X)
and objects of D(X) when a non-zero B-field is on? Then
the vector bundles should be replaced by twisted vector
bundles (at least in the untwisted theory?). Do I
understand correctly that precisely because of that the
correspondence between B-branes and D(X) is non-canonical
and we may have mondromies of the set of stable objects
when going around a non-contractible loop in moduli space?

14) On page 78, the tachyon condensation approach to
B-brane decay is discussed. Given two objects A and B of
D(X) and a morphism f:A [itex]-> B[1],[/itex] Cone(f) represents the
bound state of A with B via condesation of the potentially
tachyonic string state f. It is claimed that whereas, from
the viewpoint of algebraic geometry, Cone(f) is invariant
under replacing f by zf for any non-zero number z, in the
untwisted theory the scale of f is determined by
minimizing the quantum potential of the tachyon. Am I to
understand that the "scale of f" is really the scale of
the "quantum field" corresponding to f, i.e. the [itex]e.v ofa_f + ia*_f[/itex] where [itex]a_f[/itex] and [itex]a*_f[/itex] are the annihilation and
creation operators of the string state f.

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)?

16) On page 79, Aspinwall discusses decay into multiple
B-branes. It is claimed that for certain value of the
ksi-parameters for the [itex]A_1, A_2, A_3[/itex] the objects [itex]E_2[/itex] and
F are stable w.r.t. decay via the triangles appearing in
the octahedron diagram 197. It is not clear to me why [itex]E_2[/itex]
is stable w.r.t[itex]. E_2 -> A_3 + E_3[/itex] and why F is stable
w.r.t. [itex]F -> A_1 + E_3[/itex]?

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:

[tex]\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[/tex]

where the quintic is given by the equation

[tex]x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 +z^5 x_0 x_1 x_2 x_3 x_5 =[/tex]

in homogeneous coordinates. What is [itex]\alpha[/itex] in this equation?
Also, what dzeta in equation 214 on page 85?

18) On page 86, Aspinwall claims a 4-brane is always
mue-stable (i.e. BPS in the large volume limit) since it
has no subobjects. How come? What about 2-branes?

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?

20) On page 105 Aspinwall explains the Sardo Infirri view
of the Douglas-Moore construction (linking D-branes on the
orbifold [itex]C^d/G[/itex] to McKay quivers). I don't quite follow.
Below Aspinwall says "In this case one studies translation
invariant G-equivariant holomorphic bundles on [itex]Q = C^d[/itex].
Let the fiber of a vector bundle be given by a
epresentation V of G". I.e. each fiber is V and this is
how the G-equivariant structure works? Then he continues
"Then, for G-equivariance the connection on this bundle
transforms yet again in [itex]Hom_G(V, Q (x) V)[/itex]. Why?

21) On page 107, Aspinwall talks about [itex]\Pi-stability[/itex] in
the vicinity of the orbifold point in the moduli space.
The stability condition is termed "[itex]\theta-stability[/itex]" and it
is claimed that analogically to mue-stability, one has to
impose the condition that the G-equivariant bundle +
connection describing the D-brane has is
"Hermitian-Yang-Mills", i.e. it satisfied equation 201 on
page 81:

[tex]g^(j[/itex] kbar) [itex]F^\beta_(\alpha j k[/itex] bar) = mue(E) [itex]\delta^\alpha_beta[/tex]

Here F is the curvature tensor, E is the vector bundle, g
is the metric, j and k are target-space-indices, [itex]\alpha[/itex] and
[itex]\beta[/itex] are E-indices, mue(E) is the so-called "slope" of E:

mue(E) = deg(E) / k Vol(X)

where k is the rank of E, X is the target space and

deg(E) = (integral over [itex]X) J ^ J ^ c_1(E)[/itex]

where J is the Kahler form. However, on page 81 it is said
that "a bundle is mue-stable iff it admits an _irreducibe_
Hermitian-Yang-Mills connection".

Don't we have to demand irreducibility for [itex]\theta-stability[/itex]
either?

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?

Thx in advance!

Best regards,
Squark

* Apparently, the definition doesn't really depend on the
choice of [itex]dV_L[/itex] since rescaling it by a positive function
doesn't affect u.

** By D-n-branes here I mean D-branes which have an
n-dimensional compactified-dimensions part. The Calabi-Yau
D-branes can be cross-multiplied with 3-space D-branes, so,
for instance, both the IIA and the IIB type models will
contain both even and odd dimensional Calabi-Yau D-branes
(cross multiplied with 3-space D-branes of corresponding
dimension parity).

Ilarion Melnikov

<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.

Squark

<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>Ilarion Melnikov <lmel@kaluza.cgtp.duke.edu> wrote in message news:<Pine.LNX.4.44.0405091600070.11112-100000@kaluza.cgtp.duke.edu>...\n\n> Dear Squark,\n>\n> here are answers to some of your questions from the\n> second batch.\n\nThx a lot for your answers!\n\n> Remember, that the discussion in Aspinwall\'s notes is\n> restricted to zero genus worldsheet. There all is well, and\n> K^(1/2) (or equivalently, a spin structure) exists and is\n> unique. For a Riemann surface of higher genus there is no\n> longer a unique spin structure, and I imagine (I hope the\n> experts will correct me) that K^(1/2) has the same kind of\n> ambiguity.\n\nSo we have to sum over spin structures on the worldsheet in\nstring perturbation theory? In other words, integrate over the\nmoduli space of curves-with-spin-structure rather than just\ncurves?\n\n> I don\'t think so, or at least not directly. Of course, when\n> real branes (i.e. not topological and not at g_s = 0) decay,\n> you are, of course, correct. However,\n> I think that the production of closed strings would occur at\n> higher order in g_s that is considered (namely g_s^0 !) in this\n> framework. The closed strings only show up through the background\n> geometry.\n\nAre you saying that the whole notion of Pi-stability is essentially\na tree level thing? However, isn\'t any D-brane transformation\nnonperturbative, since D-branes are completely static in string\nperturbation theory? Also, if you are right that closed strings\ncan\'t be considered here, a brane-anti-brane pair (of identical\ntypes, i.e. A + antiA) would be stable against decay (into the\n"nothing" brane 0).\n\n> I know of no reason why the boundary conditions are not\n> affected by quantum corrections.\n\nWell, I\'m familiar with some statement I don\'t completely understand\nthat BPS states are not affected by quantum corrections. However,\nin this case it would probably protect them against g-corrections\n(which correspond to turning on the "spacetime Planck constant) as\nopposed to alpha\' corrections (which correspond to turning on the\n"worldsheet Planck constant). Btw this would imply Pi-stability as\ndefined in g = 0 would hold water at finite g (since it is really\nthe BPS condition)!\n\n> In fact, one would assume\n> they are so. The picture of branes as boundary conditions is\n> only valid in the large radius limit, where it clearly emerges\n> from worldsheet or spacetime (Becker^2, Strominger) considerations.\n> However, quantum effects modify this quite a bit. The framework\n> that Aspinwall presents in his notes allows to track these objects\n> away from the large radius limit in the topological field theory.\n\nIndeed you are right. Hence I\'m left with wondering why the BPS\nstates of the topological models are in 1-1 correspondence with\nthose of the full string theory.\n\n> NS5: these are hard to get hold of in a controlled TQFT.\n\nDo they still exist in the TQFT though?\n\n> D5: these can appear as so-called co-isotropic branes. (look\n> at Kapustin\'s papers on the subject).\n> D1: these require a non-trivial H_1 for the target space, which\n> is not oft considered.\n\nFor them (the D1) to be stable against decay into the brane 0\n(closed strings!) since they wouldn\'t carry any conserved\ncharge. Also, according to page 15 the mirror of the quintic\nthreefold is its (orbifold) quotient by a (Z_5)^3 action. It\nseems hard to believe the resulting space has H_1 = 0! For\ninstance, if the quintic threefold is simply connected (is\nit?), the mirror would have H_1 = pi_1 = (Z_5)^3 (for the\ncorrect definition of these concepts for orbifolds). So H_1\nwould be torsion but it wouldn\'t vanish!\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>\n> First, I hope that you now know what the ... in Eq.(36) stands\n> for.\n\nNo, except supposing it is of order z^3. In fact I asked about\nthe ... in equation 35 in the first question batch!\n\n> Second, the Phi_i given are a particular set of periods\n> with the requisite asymptotic properties (namely, those that\n> match (216)).\n\nAsymptotic w.r.t. what? alpha\' -> 0?\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>\n> Hmm, how more general would you like?\n\nI\'d like it to replace C^d by a general algebraic variety X\n(say) over C with the action of a finite group G.\n\n> Bridgeland-King-Reid\n> already tells you that for SUSY orbifolds of C^3 the result\n> is true. Now you can (and people do) consider non-SUSY\n> orbifolds. In principle, this is not so bad. The sigma\n> model still has N=(2,2) SUSY, and the only thing to go wrong\n> is the integrality of the R-charges no longer holds, so that\n> spectral flow is gone. This is almost bearable, but much of\n> the machinery described in the notes does not apply at all!\n\nHowever the theorem is formulated completely mathematically\nwithout any reference to SUSY, R-charges, spectral flow or\nthe sigma model!\n\nBest regards,\nSquark.\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>Ilarion Melnikov <lmel@kaluza.cgtp.duke.edu> wrote in message news:<Pine.LNX.4.44.0405091600070.11...p.duke.edu>...

> Dear Squark,
>
> here are answers to some of your questions from the
> second batch.

Thx a lot for your answers!

> 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.

So we have to sum over spin structures on the worldsheet in
string perturbation theory? In other words, integrate over the
moduli space of curves-with-spin-structure rather than just
curves?

> 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.

Are you saying that the whole notion of [itex]\Pi-stability[/itex] is essentially
a tree level thing? However, isn't any D-brane transformation
nonperturbative, since D-branes are completely static in string
perturbation theory? Also, if you are right that closed strings
can't be considered here, a brane-anti-brane pair (of identical
types, i.e. A + antiA) would be stable against decay (into the
"nothing" brane 0).

> I know of no reason why the boundary conditions are not
> affected by quantum corrections.

Well, I'm familiar with some statement I don't completely understand
that BPS states are not affected by quantum corrections. However,
in this case it would probably protect them against g-corrections
(which correspond to turning on the "spacetime Planck constant) as
opposed [itex]to \alpha'[/itex] corrections (which correspond to turning on the
"worldsheet Planck constant). Btw this would imply [itex]\Pi-stability[/itex] as
defined in g = would hold water at finite g (since it is really
the BPS condition)!

> 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.

Indeed you are right. Hence I'm left with wondering why the BPS
states of the topological models are [itex]in 1-1[/itex] correspondence with
those of the full string theory.

> NS5: these are hard to get hold of in a controlled TQFT.

Do they still exist in the TQFT though?

> 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.

For them (the D1) to be stable against decay into the brane
(closed strings!) since they wouldn't carry any conserved
charge. Also, according to page 15 the mirror of the quintic
threefold is its (orbifold) quotient by [itex]a (Z_5)^3[/itex] action. It
seems hard to believe the resulting space has [itex]H_1 = 0![/itex] For
instance, if the quintic threefold is simply connected (is
it?), the mirror would have [itex]H_1 = \pi_1 = (Z_5)^3[/itex] (for the
correct definition of these concepts for orbifolds)[itex]. So H_1[/itex]
would be torsion but it wouldn't vanish!

> > 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.

No, except supposing it is of order [itex]z^3[/itex]. In fact I asked about
the ... in equation 35 in the first question batch!

> Second, the [itex]\Phi_i[/itex] given are a particular set of periods
> with the requisite asymptotic properties (namely, those that
> match (216)).

Asymptotic w.r.t. what[itex]? \alpha' ->[/itex] ?

> > 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?

I'd like it to replace [itex]C^d[/itex] by a general algebraic variety X
(say) over C with the action of a finite group G.

> 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!

However the theorem is formulated completely mathematically
without any reference to SUSY, R-charges, spectral flow or
the [itex]\sigma[/itex] model!

Best regards,
Squark.

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