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

<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 &gt; 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) &lt; 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">&nbsp;&nbsp;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 $mue(\pi_b) < mue(\pi_a) + 1$?

Best regards,
Squark.

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On $2004-05-03,$ Squark 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) $-> U(1)^N$ just as in flat space. Kris



Kris Kennaway wrote in message news:... > 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) $-> U(1)^N$ 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

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

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 $N = (2,2)$
non-linear $\sigma-model$. If $\phi$ is the bosonic part of the
$\sigma-model,$ the fermions are sections of the following
bundles on the worldsheet $\Sigma:a) \psi^i_+$ of $K^1/2$ (x) $\phi*T_Xb) \psi^jbar_+$ of $K^1/2$ (x) $\phi*Tbar_Xc) \psi^i_-$ of $Kbar^1/2 (x) \phi*T_Xd) \psi^jbar_-$ of $Kbar^1/2 (x) \phi*Tbar_X$

X is the target space, K is the holomorphic cotangent
bundle on $\Sigma$. The question is, what are conditions for
$K^1/2$ to exist and how unique is it? What is the physical
meaning of these conditions?

6) On pages $19-20,$ Aspinwall defines tge Maslov class of a
Lagrangian submanifold L as a map $\pi_1(L) -> \pi_1(S^1) = Z$.
The definition goes by defining $u = z / |z|$ where
$z = \Omega|_L / dV_L.$ Here, $\Omega$ is the unique up to
normalization holomorphic 3-form on Y, the ambient space of
L (a Calabi-Yau), $dV_L$ is the volume $form*$ on L. u is then a
map $L -> S^1$ 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

9) On page 67, Aspinwall discusses the formation of a
bound state of two A-branes given by Lagrangian
submanifolds $L_1$ and $L_2$ 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 $L_1$ and $L_2$. Why$? It$ 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 $L) \Omega$ where $\Omega$
is again the unique-up-to-constant (3,0)-form on the
target space Y. The problem is, $\Omega$ 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 $A-A'$ string and the $B-B'$ 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 $-> B[1],$ 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 $e.v ofa_f + ia*_f$ where $a_f$ and $a*_f$ 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 $\alpha'$ 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 $A_1, A_2, A_3$ the objects $E_2$ and
F are stable w.r.t. decay via the triangles appearing in
the octahedron diagram 197. It is not clear to me why $E_2$
is stable w.r.t$. E_2 -> A_3 + E_3$ and why F is stable
w.r.t. $F -> A_1 + E_3$?

17) On page 84, $\Pi-stability$ for quintic threefold is
analyzed. The periods (integrals over elements of the
homology group) of the holomorphic 3-form $\Omega$ are
claimed to be given by equation 210:

$$\omega_j = (-1/5)(Sum[/itex] over $m = 1$ to infinity) $\alpha^(2+j)m \Gamma(m/5) z^-m/5 / \Gamma(m) \Gamma(1 - m/5)^4$$ where the quintic is given by the equation $$x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 +z^5 x_0 x_1 x_2 x_3 x_5 =$$ in homogeneous coordinates. What is [itex]\alpha$ 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
$C^d / G$ 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 $C^d$". 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 $C^d/G$ to McKay quivers). I don't quite follow.
Below Aspinwall says "In this case one studies translation
invariant G-equivariant holomorphic bundles on $Q = C^d$.
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 $Hom_G(V, Q (x) V)$. Why?

21) On page 107, Aspinwall talks about $\Pi-stability$ in
the vicinity of the orbifold point in the moduli space.
The stability condition is termed "$\theta-stability$" 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:

$$g^(j[/itex] kbar) $F^\beta_(\alpha j k$ bar) = mue(E) $\delta^\alpha_beta$$ Here F is the curvature tensor, E is the vector bundle, g is the metric, j and k are target-space-indices, [itex]\alpha$ and
$\beta$ 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 $X) J ^ J ^ c_1(E)$

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 $\theta-stability$
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?

Best regards,
Squark

* Apparently, the definition doesn't really depend on the
choice of $dV_L$ 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).



Dear Squark, here are answers to some of your questions from the second batch. Ilarion > 5) On page 5 Aspinwall defines the fermions of $N = (2,2)$ > non-linear $\sigma-model$. If $\phi$ is the bosonic part of the > $\sigma-model,$ the fermions are sections of the following > bundles on the worldsheet $\Sigma:$ > > $a) \psi^i_+$ of $K^1/2$ (x) $\phi*T_X$ > $b) \psi^jbar_+$ of $K^1/2$ (x) $\phi*Tbar_X$ > $c) \psi^i_-$ of $Kbar^1/2 (x) \phi*T_X$ > $d) \psi^jbar_-$ of $Kbar^1/2 (x) \phi*Tbar_X$ > > X is the target space, K is the holomorphic cotangent > bundle on $\Sigma$. The question is, what are conditions for > $K^1/2$ 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 $K^(1/2) (or$ 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 $K^(1/2)$ 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 $H^1(M, Z_2)$. > 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 $at g_s = 0)$ decay, you are, of course, correct. However, I think that the production of closed strings would occur at higher order in $g_s$ that is considered (namely $g_s^0 !)$ 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 $(Becker^2,$ 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 $H_1$ 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 $L) \Omega$ where $\Omega$ > is again the unique-up-to-constant (3,0)-form on the > target space Y. The problem is, $\Omega$ 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 $\Omega,$ 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 $\alpha'$ 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, $\Pi-stability$ for quintic threefold is > analyzed. The periods (integrals over elements of the > homology group) of the holomorphic 3-form $\Omega$ are > claimed to be given by equation 210: > > $\omega_j = (-1/5)(Sum$ over $m = 1$ to infinity) > $\alpha^(2+j)m \Gamma(m/5) z^-m/5 / \Gamma(m) \Gamma(1 - m/5)^4$ > > where the quintic is given by the equation > > $x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 +$ > $z^5 x_0 x_1 x_2 x_3 x_5 =$ > > in homogeneous coordinates. What is $\alpha$ 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 $\Phi_i$ given are a particular set of periods with the requisite asymptotic properties (namely, those that match (216)). The $\alpha$ is a fifth root of unity, $\zeta$ is the Riemann $\zeta$ function! > 19) On page 103, the Bridge-Kind-Reid (BKR) theorem is > stated: "Suppose X is a smooth resolution of the orbifold > $C^d / G$ 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 $C^d$". 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 $C^3$ the result is true. Now you can (and people do) consider non-SUSY orbifolds. In principle, this is not so bad. The $\sigma$ model still has $N=(2,2)$ 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 $C^4/G$ 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.



Ilarion Melnikov wrote in message news:... > 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 > $K^(1/2) (or$ 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 $K^(1/2)$ 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 $at g_s = 0)$ decay, > you are, of course, correct. However, > I think that the production of closed strings would occur at > higher order in $g_s$ that is considered (namely $g_s^0 !)$ in this > framework. The closed strings only show up through the background > geometry. Are you saying that the whole notion of $\Pi-stability$ 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 $to \alpha'$ corrections (which correspond to turning on the "worldsheet Planck constant). Btw this would imply $\Pi-stability$ 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 $(Becker^2,$ 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 $in 1-1$ 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 $H_1$ 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 $a (Z_5)^3$ action. It seems hard to believe the resulting space has $H_1 = 0!$ For instance, if the quintic threefold is simply connected (is it?), the mirror would have $H_1 = \pi_1 = (Z_5)^3$ (for the correct definition of these concepts for orbifolds)$. So H_1$ would be torsion but it wouldn't vanish! > > 17) On page 84, $\Pi-stability$ for quintic threefold is > > analyzed. The periods (integrals over elements of the > > homology group) of the holomorphic 3-form $\Omega$ are > > claimed to be given by equation 210: > > > > $\omega_j = (-1/5)(Sum$ over $m = 1$ to infinity) > > $\alpha^(2+j)m \Gamma(m/5) z^-m/5 / \Gamma(m) \Gamma(1 - m/5)^4$ > > > > where the quintic is given by the equation > > > > $x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 +$ > > $z^5 x_0 x_1 x_2 x_3 x_5 =$ > > > > in homogeneous coordinates. What is $\alpha$ 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 $z^3$. In fact I asked about the ... in equation 35 in the first question batch! > Second, the $\Phi_i$ given are a particular set of periods > with the requisite asymptotic properties (namely, those that > match (216)). Asymptotic w.r.t. what$? \alpha' ->$ ? > > 19) On page 103, the Bridge-Kind-Reid (BKR) theorem is > > stated: "Suppose X is a smooth resolution of the orbifold > > $C^d / G$ 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 $C^d$". 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 $C^d$ 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 $C^3$ the result > is true. Now you can (and people do) consider non-SUSY > orbifolds. In principle, this is not so bad. The $\sigma$ > model still has $N=(2,2)$ 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 $\sigma$ model! Best regards, Squark.

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