Is there a difference of sign conventions of Dirac Index between mathematics and physics?
$begingroup$
In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by
$$mathrm{Ind}(D!!!!/_{A})=frac{-1}{8pi^{2}}int_{M}mathrm{Tr}(Fwedge F)+frac{dim_{mathbb{C}}E}{192pi^{2}}int_{M}mathrm{Tr}(Rwedge R),$$
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 Strongly Correlated Electrons
Gapped Boundary Phases of Topological Insulators via Weak Coupling by
Nathan Seiberg and Edward Witten, their version of index theorem is
$$mathrm{Ind}(D!!!!/_{A})=intleft(frac{Fwedge F}{8pi^{2}}+widehat{A}(R)right),$$
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?
I also posted this question here.
characteristic-classes convention
asked Dec 19 '18 at 14:57
The Last Knight of Silk RoadThe Last Knight of Silk Road
1856
$endgroup$
add a comment |
$begingroup$
In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by
$$mathrm{Ind}(D!!!!/_{A})=frac{-1}{8pi^{2}}int_{M}mathrm{Tr}(Fwedge F)+frac{dim_{mathbb{C}}E}{192pi^{2}}int_{M}mathrm{Tr}(Rwedge R),$$
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 Strongly Correlated Electrons
Gapped Boundary Phases of Topological Insulators via Weak Coupling by
Nathan Seiberg and Edward Witten, their version of index theorem is
$$mathrm{Ind}(D!!!!/_{A})=intleft(frac{Fwedge F}{8pi^{2}}+widehat{A}(R)right),$$
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?
I also posted this question here.
characteristic-classes convention
asked Dec 19 '18 at 14:57
The Last Knight of Silk RoadThe Last Knight of Silk Road
1856
$endgroup$
add a comment |
$begingroup$
In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by
$$mathrm{Ind}(D!!!!/_{A})=frac{-1}{8pi^{2}}int_{M}mathrm{Tr}(Fwedge F)+frac{dim_{mathbb{C}}E}{192pi^{2}}int_{M}mathrm{Tr}(Rwedge R),$$
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 Strongly Correlated Electrons
Gapped Boundary Phases of Topological Insulators via Weak Coupling by
Nathan Seiberg and Edward Witten, their version of index theorem is
$$mathrm{Ind}(D!!!!/_{A})=intleft(frac{Fwedge F}{8pi^{2}}+widehat{A}(R)right),$$
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?
I also posted this question here.
characteristic-classes convention
asked Dec 19 '18 at 14:57
The Last Knight of Silk RoadThe Last Knight of Silk Road
1856
$endgroup$
In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by
$$mathrm{Ind}(D!!!!/_{A})=frac{-1}{8pi^{2}}int_{M}mathrm{Tr}(Fwedge F)+frac{dim_{mathbb{C}}E}{192pi^{2}}int_{M}mathrm{Tr}(Rwedge R),$$
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 Strongly Correlated Electrons
Gapped Boundary Phases of Topological Insulators via Weak Coupling by
Nathan Seiberg and Edward Witten, their version of index theorem is
$$mathrm{Ind}(D!!!!/_{A})=intleft(frac{Fwedge F}{8pi^{2}}+widehat{A}(R)right),$$
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?
I also posted this question here.
characteristic-classes convention
characteristic-classes convention
asked Dec 19 '18 at 14:57
The Last Knight of Silk RoadThe Last Knight of Silk Road
1856
asked Dec 19 '18 at 14:57
The Last Knight of Silk RoadThe Last Knight of Silk Road
1856
asked Dec 19 '18 at 14:57
The Last Knight of Silk RoadThe Last Knight of Silk Road
1856
asked Dec 19 '18 at 14:57
The Last Knight of Silk RoadThe Last Knight of Silk Road
1856
asked Dec 19 '18 at 14:57
The Last Knight of Silk RoadThe Last Knight of Silk Road
1856
1856
add a comment |
add a comment |
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