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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.