The Chern-Simons term of an (abelian brane) is commonly written as $$ \sim\int_{\mathcal M_{p+1}}\sum_iC_{i}[e^{2\pi\alpha'F+B}], $$ where $C_i$ is the background Ramond-Ramond $i$-form, $F$ is the brane's gauge field strength, and $B$ is the background NS-NS antisymmetric field -- there is actually a non-abelian extension to this, but I think this is enough for the purposes of the question.

This Chern-Simons term gets "anomalous curvature coupling" corrections, given by $$ \sim\int_{\mathcal M_{p+1}}\sum_iC_{i}[e^{2\pi\alpha'F+B}] \sqrt{\mathcal A(4\pi^2\alpha'R)}, $$ with (this is the "A-roof") $$ \sqrt{\mathcal A(R)}=1-\frac{p_1(R)}{48}+p_1^2(R)\frac7{11520} -\frac{p_2(R)}{2880}+... $$ with the $p_i(R)$ being the $i$th Pontryagin class. The relevant classes for the previous expression are $$ p_1(R)=-\frac1{8\pi^2}Tr R\wedge R,~p_2(R)=\frac1{(2\pi)^4} (-\frac14TrR^4+\frac18(TrR^2)^2) $$ where $R$ is the curvature two-form.

I am confused about in which regime we can neglect these corrections. Naively I would say that for low curvature backgrounds would be safe to neglect them, but $$ p_1(4\pi^2\alpha'R)\sim\alpha'^2 m,~m\in\mathbb Z, $$ since as far as I understand $p_1(R)$ is an integer, and topological values, thus should not depend on the local values that the curvature might have.

Adding to my confusion is the KPV paper https://arxiv.org/abs/hep-th/0112197, where the (non-abelian anti-D3 and abelian NS5) brane worldvolume actions do not contain these corrections, yet when setting the Klebanov-Strassler background configuration, they use the global tadpole cancellation condition (their equation (2)), which needs quantum corrections that are finite-valued because the Pontryagin classes are not zero for this type IIB background (see Becker-Becker-Schwarz, section "Tadpole-cancellation condition", Chapter 10). For me, this sounds that the background is receiving more $\alpha'$ corrections than the worldvolume theory, which makes the system inconsistent. Is this the case?


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