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This question is about a mathematical subtlety arising in the computation of the partition function of a supersymmetric ensemble of some lower dimensional $D$-branes attached to a stack of higher dimensional ones.

The general prescription to attack this problem is to add chemical potentials for the lower dimensional branes to the action of the worldvolume theory of the stack of higher dimensional branes, after this addition the path integral is performed. Those chemical potentials are defined as Chern characters for some sheaves supported in submanifolds of the worldvolume of the stack of the higher dimensional branes.

Consider a sheaf $\mathcal{F}$ with support on a submanifold $Y$ of the worldvolume $X$ of the higher dimensional branes represented by a sheaf $\mathcal{G}$ over $X$, and let $\iota: Y \hookrightarrow X$ be the inclusion of $Y$ on $X$.

My struggle: My intuition says that the relevant Chern characters in which which I should write the D-brane chemical potentials are computed by Chern characters of $(\mathcal{\iota_{\ast}F})\otimes \iota^{\ast} \mathcal{G}$ on $X$ as $ch_{\bullet}((\mathcal{\iota_{\ast}F}) \otimes \iota^{\ast} \mathcal{G})$. However my (perhaps wrong) impression is that $ch_{\bullet}(\iota_{\ast}\mathcal{F} \otimes \mathcal{G})$ is what is actually computed in the literature.

Take as an example an instanton background term for Yang-Mills theory with field-strenght $F$ on a four-manifold $\mathcal{M}$ written as $\int_{\Gamma} F \wedge F$, with $\Gamma \subset \mathcal{M}$ a two cycle in $\mathcal{M}$.

Question:

1) How should I properly define the relevant Chern characters in which which I should write the D-brane chemical potentials?

Any hint would be helpful.

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The K-theory of the brane receives universal corrections coming from the Todd class of the target manifold. For that reason, the correct way to compute the charge of an embedded brane is

  1. Embed the brane.
  2. Compute the charge.

In other words, computing the charge an performing an embedding are operations that do not commute, in a way measured by the Grothendieck-Riemann-Roch theorem (without denominators). 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).

Reference: A detailed answer to my question can be found in D-branes on Calabi-Yau Manifolds (Page 61 ,equation (151)).

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