Physical application of the K.H.Mayer`s integrality theorem: Anomaly for Heterotic SO(32) Fivebrane.
The Heterotic SO(32) fivebrane configuration breaks down the D=10 local Lorentz symmetry SO(9,1) to SO(5,1)×SO(4); where SO(5,1) acts on the tangent bundle to the heterotic SO(32) fivebrane world-volume denoted TW; and SO(4) acts as gauge group on the normal bundle to the heterotic SO(32) fivebrane world-volume denoted N.
Then, the K.H.Mayer`s integrality theorem for a normal bundle N with structure group SO(4) produces the following two cohomological expressions
22M(N)ˆA(TW)
and
W4(N)ˆA(N)−1ˆA(TW)
From the original proof of the K.H.Mayer`s integrality theorem we have that
ch(S+(N))+ch(S−(N))=22M(N)
and
ch(S+(N))−ch(S−(N))=W4(N)ˆA(N)−1
From these last equations we deduce that
ch(S+(N))=12[22M(N)+W4(N)ˆA(N)−1]
and
ch(S−(N))=12[22M(N)−W4(N)ˆA(N)−1]
which are rewritten as
ch(S±(N))=12[22M(N)±W4(N)ˆA(N)−1]
Now, we have that
W4(N)ˆA(N)−1=W4(N)+124W4(N)p1(N)
M(N)=1+18p1(N)+196p2(N)+1384p1(N)2+....
Using these last equations we obtain
ch(S±(N))=12[4(1+18p1(N)+196p2(N)+1384p1(N)2)±(W4(N)+124W4(N)p1(N))]
which is reduced to
ch(S±(N))=2+p1(N)±W4(N)4+p1(N)2+4p2(N)±4W4(N)p1(N)192
and it is exactly the equation (6) on page 4 of http://arxiv.org/pdf/hep-th/9709012v1.pdf ; where W4(N)=χ(N), S+(N) is the spin bundle with positive chirality constructed from N by using the spinor representation of SO(4); and S−(N) is the spin bundle with negative chirality constructed from N by using the spinor representation of SO(4).
One first kind of chiral fermions that are living in the worldvolume of the heterotic SO(32) fivebrane are called θ-fermions and they belong to the (4+,2+) representation of SO(5,1)×SO(4).
The total gravitational anomaly of the θ-fermion zero modes living in the world-volume of the heterotic SO(32) fivebrane is given by descent from
Iθ8=12[ˆA(TW)ch(S+(N))]8−form
Using that
ˆA(TW)=1−124p1(TW)−11440p2(TW)+75760p1(TW)2
and
ch(S+(N))=2+p1(N)+W4(N)4+p1(N)2+4p2(N)+4W4(N)p1(N)192
we obtain
Iθ8=75760p1(TW)2+196p2(N)+1384p1(N)2−
11440p2(TW)−1192p1(TW)p1(N)+
196W4(N)p1(N)−196p1(TW)W4(N)
Now, given that
TQ∣W=TW⊕N
p1(TQ)=p1(TW)+p1(N)
p2(TQ)=p2(TW)+p2(N)+p1(TW)p1(N)
we deduce that
p1(TW)=p1(TQ)−p1(N)
and
p2(TW)=p2(TQ)−p2(N)−p1(N)p1(TQ)+p1(N)2
Then using these last equations we obtain
Iθ8=75760p1(TQ)2−1144p1(N)p1(TQ)+1120p1(N)2+
p2(N)90−p2(TQ)1440+W4(N)p1(N)48−W4(N)p1(TQ)96
and it is exactly the equation (8) on page 4 of http://arxiv.org/pdf/hep-th/9709012v1.pdf .
The second kind of chiral fermions that are living in the worldvolume of the heterotic SO(32) fivebrane are called SU(2) gauginos or λ-fermions; and they belong to the (1,3,4−,2−) representation of SO(32)×SU(2)×SO(5,1)×SO(4).
The total gravitational anomaly of the λ-fermion zero modes living in the world-volume of the heterotic SO(32) fivebrane is given by descent from
Iλ8=−12[ˆA(TW)ch(S−(N))Tr(eiG)]8−form
where 2πG is the SU(2) curvature and Tr is the trace in the adjoint representation SU(2).
Using that
ˆA(TW)=1−124p1(TW)−11440p2(TW)+75760p1(TW)2
ch(S−(N))=2+p1(N)−W4(N)4+p1(N)2+4p2(N)−4W4(N)p1(N)192
Tr(eiG)=3−12Tr(G2)+124Tr(G4)
we obtain
Iλ8=116p1(N)Tr(G2)−148p1(TW)Tr(G2)−
1128p1(N)2+1480p2(TW)−71920p1(TW)2−
132p2(N)+132W4(N)p1(N)−132p1(TW)W4(N)−
124Tr(G4)−18W4(N)Tr(G2)+164p1(TW)p1(N)
Using again
p1(TW)=p1(TQ)−p1(N)
and
p2(TW)=p2(TQ)−p2(N)−p1(N)p1(TQ)+p1(N)2
we obtain
Iλ8=112p1(N)Tr(G2)−148Tr(G2)p1(TQ)−140p1(N)2+
1480p2(TQ)−130p2(N)+130p1(N)p1(TQ)−
71920p1(TQ)2+116W4(N)p1(N)−132W4(N)p1(TQ)−
124Tr(G4)−18W4(N)Tr(G2)
and it is exactly the equation (10) on page 4 of http://arxiv.org/pdf/hep-th/9709012v1.pdf .
The third kind of chiral fermions that are living in the worldvolume of the heterotic SO(32) fivebrane are called ψ-fermions; and they belong to the (32,2) representation of SO(32)×SU(2).
The total gravitational anomaly of the ψ-fermion zero modes living in the world-volume of the heterotic SO(32) fivebrane is given by descent from
Iψ8=12[ˆA(TW)Tr(eiF)Tr(eiG)]8−form
where 2πG is the SU(2) curvature, 2πF is the SO(32) curvature and tr is the trace in the fundamental representation.
Using that
ˆA(TW)=1−124p1(TW)−11440p2(TW)+75760p1(TW)2
tr(eiG)=2−12tr(G2)+124tr(G4)
tr(eiF)=32−12tr(F2)+124tr(F4)
we obtain
Iψ8=−145p2(TW)+7180p1(TW)2+23tr(G4)+
148p1(TW)tr(F2)+18tr(G2)tr(F2)+13p1(TW)tr(G2)+124tr(F4)
Using again
p1(TW)=p1(TQ)−p1(N)
and
p2(TW)=p2(TQ)−p2(N)−p1(N)p1(TQ)+p1(N)2
we have that
Iψ8=−145p2(TQ)+145p2(N)−118p1(N)p1(TQ)+160p1(N)2+
7180p1(TQ)2+23tr(G4)+148tr(F2)p1(TQ)−148tr(F2)p1(N)+
18tr(G2)tr(F2)+13tr(G2)p1(TQ)−13tr(G2)p1(N)+124tr(F4)
and it is exactly the equation (12) on page 5 of http://arxiv.org/pdf/hep-th/9709012v1.pdf .
The total anomaly is I8=Iθ8+Iλ8+Iψ8 and then we have
I8=112W4(N)p1(N)+148tr(F2)p1(TQ)−148tr(F2)p1(N)+
13tr(G2)p1(TQ)−13tr(G2)p1(N)−148p2(TQ)−124p1(N)p1(TQ)+
7192p1(TQ)2−124W4(N)p1(TQ)−124Tr(G4)−18W4(N)Tr(G2)+
112p1(N)Tr(G2)−148Tr(G2)p1(TQ)+23tr(G4)+
124tr(F4)+18tr(G2)tr(F2)
Now, using Tr(G2)=4tr(G2) and Tr(G4)=16tr(G4), we obtain
I8=112W4(N)p1(N)−124p1(N)p1(TQ)−124W4(N)p1(TQ)+
7192p1(TQ)2−12W4(N)tr(G2)+14tr(G2)p1(TQ)+
18tr(G2)tr(F2)+124tr(F4)+148tr(F2)p1(TQ)−
148tr(F2)p1(N)−148p2(TQ)
and it is exactly the equation (14) on page 5 of http://arxiv.org/pdf/hep-th/9709012v1.pdf .
Finally the total anomaly is rewritten as
I8=[W4(N)−14tr(F2)−12p1(TQ)][112p1(N)−12tr(G2)−124p1(TQ)]+
164p1(TQ)2−148p2(TQ)+196tr(F2)p1(TQ)+124tr(F4)
and it is exactly the equation (15) on page 5 of http://arxiv.org/pdf/hep-th/9709012v1.pdf .