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  Difference of sign conventions of Dirac Index between mathematics and physics

+ 2 like - 0 dislike
62 views

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?

asked Dec 19, 2018 in Theoretical Physics by New Student (220 points) [ no revision ]

1 Answer

+ 2 like - 0 dislike

Yes, Nakahara uses an antihermitian $F$, whereas Seiberg and Witten use an hermitian $F$.

For an antihermitian $F$, the Chern character is given by $Tr(e^{\frac{i}{2\pi}F})$, and so $ch_2=Tr(\frac{1}{2}(\frac{i}{2\pi}F)^2 )=-\frac{1}{8 \pi^2} Tr(F^2)$.

Using an hermitian $F$ is equivalent to replacing $F$ by $\pm i F$, and so flips the sign of $F^2$.

Remark that Seiberg and Witten are only considering a $U(1)$ gauge group, which is why their formula does not include $Tr$ and $dim E$.

answered Dec 20, 2018 by 40227 (5,050 points) [ no revision ]

Thank you very much for answering my question!

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