I poseted this question here, but it seems that no one so far has been interested in this question.

In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by

$$\mathrm{Ind}(D\!\!\!\!/_{A})=\frac{-1}{8\pi^{2}}\int_{M}\mathrm{Tr}(F\wedge F)+\frac{\dim_{\mathbb{C}}E}{192\pi^{2}}\int_{M}\mathrm{Tr}(R\wedge R),$$

where $E$ is a vector bundle over $M$, $D\!\!\!\!/_{A}$ is a Dirac operator twisted by the gauge field $A$, $F$ is the assocciated field strength, and $R$ is the Riemann tensor of $M$.

However, in Gapped Boundary Phases of Topological Insulators via Weak Coupling by

Nathan Seiberg and Edward Witten, their version of index theorem is

$$\mathrm{Ind}(D\!\!\!\!/_{A})=\int\left(\frac{F\wedge F}{8\pi^{2}}+\widehat{A}(R)\right),$$

where the sign of the second chern character differs from that in Nakahara.

Is this related with the different conventions of Lie algebra (Hermitian in physics vs. Anti-Hermitian in mathematics) between the two communities?