# Is there a difference of sign conventions of Dirac Index between mathematics and physics?

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 Condensed Matter > Strongly Correlated Electrons 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?