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
Ind(D/A)=−18π2∫MTr(F∧F)+dimCE192π2∫MTr(R∧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
Ind(D/A)=∫(F∧F8π2+ˆA(R)),
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?