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I'm studying the Jafferis solution for twisted $N=4$ Yang-Mills theory in four dimensions from the paper Crystals and intersecting branes.

Consider the problem of computing the charges of the allowed lower dimensional branes bound to the worldvolume of a D4-brane. Witten has shown that this problem can be actually solved by computing the $K$-theory of the D4 worldvolume. More precisely, let $\mathcal{E}$ be the Chan-Paton vector bundle of our $D4$ brane; then the spectrum of possible D-branes that can bound to the D4-brane can be read by writing the Chern character of $\mathcal{E}$ in terms of its Chern classes, in the following way

$$ch(\mathcal{E})=1+ch_{1}(\mathcal{E})t+ch_{2}(\mathcal{E})t^{2}+ch_{3}(\mathcal{E})t^{3}$$

$$ =1+c_{1}t+(-c_{2}+ \frac{1}{2}c_{1}^{2})t^{2} + \frac{1}{6}(3c_{3}-3c_{1}c_{2}+c_{1}^{3})t^{3}.$$

From this we learn

$$q_{D4}=ch_{1}=c_{1},$$

$$q_{D2}=ch_{2}=-c_{2}+ \frac{1}{2}c_{1}^{2}.$$

However the $q_{D0}$ charge is reported (see equation (17) in page 9 in Crystals and intersecting branes) to be $$q_{D0}=2ch_{3}=(c_{3}-c_{1}c_{2}+\frac{1}{3}c_{1}^{3}).$$

Question: Why is the $q_{D0}$ charge $2ch_{3}$ and not just $ch_{3}$?

The presence of the factor of two in $q_{D0}$ seems like a pinhead, but it's really odd to me because it imply that we can only have even numbers of D0-branes sticked to a $D4$-brane, or that it is actually not possible to bound a single $D0$-brane to the worldvolume of the D4-brane.

Extra comments:

  1. I think this factor of two, that is currently beyond my understanding, is really important and not a typo. In my previous PSE question D-branes as the atoms of Calabi-Yau crystals I requested help to understand why in the quantum foam description of topological string theory apparently a D2/anti-D2-brane bound state is equivalent to having a single $0$-brane. That seems to point out that $0$-branes are composite objects in this quantum foam story. Is that right?

  2. @AccidentalFourierTransform has made the valuable and important observation that its not obvious at all in my question that the Chern characters o the branes of the theory should be integer valued.The clarification is one of the premises of the quantum foam/crystal melting model of the topological string, the Chern classes of the branes (holomorphic sheaves) are integer valued, and generate the target $\mathbb{Z}$-cohomology. See page 6, equation 2.3 and pages 4 and 24 of Quantum Foam and Topological Strings for further details.

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    $\begingroup$ quick question: is it obvious that $ch_3$ is integral-valued? It generates $\mathbb Q$-homology, but is it also a generator of $\mathbb Z$-homology? $\endgroup$ – AccidentalFourierTransform Dec 31 '20 at 21:53
  • $\begingroup$ @AccidentalFourierTransform Thanks for taking the time to read my question. Your question is a very good, and important one. The answer to your first question is that its is not obvious at all. One of the premises of the quantum foam/crystal melting model of the topological string is that the Chern classes of the branes (holomorphic sheaves) of the theory are integer valued, and generate the target $\mathbb{Z}$-cohomology. See page 6, equation 2.3 and pages 4 and 24 of Quantum Foam and Topological Strings for further details. $\endgroup$ – Ramiro Hum-Sah Dec 31 '20 at 23:59
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    $\begingroup$ Please stop making trivial edits to bump the question into the front page. If someone knows the answer they will post it, eventually. We don't need the extra noise. Thanks! $\endgroup$ – AccidentalFourierTransform Jan 7 at 19:45
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My mistake was that the D0-brane charge is not exactly given as $$q_{D0}^{naive}=ch_{3}(\mathcal{i_{!}E}).$$

The key observation to solve my confusion is that the correct definition must be corrected by contributions from the cotangent bundle of the divisor. To be precise,the correct formula to compute RR-charges of D-branes wrapping subschemes of a scheme X reads: $$q_{D0}^{correct}=ch_{3}(\mathcal{i_{!}E})\sqrt{\hat{A}(\mathcal{T}X)},$$ where $\hat{A}$ is the $\hat{A}$-roof genus. This formula is explained in detail in the paper K-theory and Ramond-Ramond charge (equation 1.1).

To answer my actual question: Is it possible to bound a single D0-brane to a D4-brane (wrapping a toric divisor)? The answer is yes. It simply happens that a single unit of D0-brane is defined as $q_{D0}^{correct}$.

To derive the precise factor of two I was wondering to compute is now just a matter of using the correct definition over the geometry Jafferis is working ($\mathcal{O}(p-2) \oplus \mathcal{O}(-p) \rightarrow \mathbb{P^{1}}$).

If someone is interested in know the origin of the correct definition of the RR-charge can consult the following references:

  1. I-Brane Inflow and Anomalous Couplings on D-Branes (equation (1.7)).
  2. One-loop test of S-duality.
  3. K-theory and Ramond-Ramond charge.
  4. I found extremely useful and readable the original derivation of the Grothendieck-Riemann-Roch theorem (needed to apply the Gysin map and the Chern characters in the correct order when computing D-brane charges).
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As a CW-complex, we can obviously think of $S^4$ as a $4$-cell attached to a single $0$-cell with the usual boundary identifications. So it's certainly possible mathematically. Or rather, insofar as you "can't" bound a single $D0$-brane to a $D4$-brane, there would have to be some extra physical/differential reason why that's not possible in the context of the OP. I'm not familiar enough with string theory to explain that part, unfortunately.

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    $\begingroup$ Thanks for taking the time to read my question. 4-branes cannot wrap an $S^{4}$ because an $S^{4}$ is not a supersymmetric cycle inside a toric Calabi-Yau. My question is nothing about the 0-skeletons of 4-cycles. Mathematically speaking my question is about the $K$-theory of the 4-cylces, more precisely, How is the codimension zero Chern character defined on a 4-cycle? That's the question I'm trying to understand. $\endgroup$ – Ramiro Hum-Sah Jan 2 at 19:24

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