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  Physical derivation and applications of the K.H.Mayer`s integrality theorem

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In a previous post   http://www.physicsoverflow.org/24965/supersymmetric-derivation-integrality-differentiable-manifolds  a piece of  the K.H.Mayer`s integrality theorem was considered and its heterotic-supersymmetric derivation was presented.  In this post we consider the full K.H.Mayer`s integrality theorem which reads: (originally in German, please look below the English translation)

English translation:

A very important special case of the K.H.Mayer`s integrality theorem reads (Elliptic Symbols, Christian Bär)

My questions are:

1. How to derive  the K.H.Mayer`s integrality theorem using Heterotic-Supersymmetric quantum mechanics?

2. Do you know physical applications  of the K.H.Mayer`s integrality theorem?

asked Aug 22, 2015 in Theoretical Physics by juancho (1,130 points) [ revision history ]
edited Aug 26, 2015 by juancho

4 Answers

+ 3 like - 0 dislike

Physical application of the K.H.Mayer`s integrality theorem:  Gravitational anomaly for axion strings.

The axion string configuration breaks down the SO(3,1) local Lorentz symmetry to SO(1,1)×SO(2); where  SO(1,1)  acts on the tangent bundle to the string world-sheet denoted TΣ2  and SO(2) acts as gauge group on the normal bundle to the string world-sheet denoted N.

Then,  K.H.Mayer`s integrality theorem for  a normal bundle N with  structure group SO(2) takes the form

2M(N)ˆA(TΣ2)

The total gravitational anomaly of the fermion zero modes living in the world-sheet of the axion string  is given by descent from

(12)[2M(N)ˆA(TΣ2)]4form

In terms of Pontrjagin classes the Mayer class is given by

M=1+18p1+196p2+1384p12+1960p3+13840p1p2+146080p13+...

In terms of Pontrjagin classes the Dirac genus is given by

ˆA=1124p111440p2+75760p12160480p3+11241920p1p231967680p13+...

Then we have

(12)[2M(N)ˆA(TΣ2)]4form=[(1+18p1(N))(1124p1(TΣ2))]4form

which is reduced to

Izeromode=124p1(TΣ2)18p1(N)

and it is the equation (136) on page 40 of http://xxx.lanl.gov/pdf/hep-th/0509097.pdf

Now, we consider that there is a gravitational inflow contribution to the anomaly for the axion string given by the descent from

Iinflow=αp1(TM)

where α is constant to be determined and

TMΣ2=TΣ2N

Then we have that

Iinflow=αp1(TM)=αp1(TΣ2N)=α(p1(TΣ2)+p1(N))=αp1(TΣ2)+αp1(N)

According with all these results we derive that

Iinflow+Izeromode=αp1(TΣ2)+αp1(N)+124p1(TΣ2)18p1(N)

which is reduced to

Iinflow+Izeromode=(α+124)p1(TΣ2)+(α18)p1(N)

In order to cancel  the tangent bundle anomaly we demand that

α+124=0

which is equivalent to α=124.  Using this value of α we obtain

Iinflow+Izeromode=16p1(N)

and it is the equation (137) on page 40 of http://xxx.lanl.gov/pdf/hep-th/0509097.pdf

Also we obtain

Iinflow=124p1(TΣ2)124p1(N)=124p1(TM)

and it is the equation (133) on page 39 of http://xxx.lanl.gov/pdf/hep-th/0509097.pdf

Finally, given that p1(N)=e2(N), where e(N) is the Euler class of the normal bundle; we rewrite the  uncanceled anomaly for the normal bundle as

Iinflow+Izeromode=16e2(N)

answered Aug 22, 2015 by juancho (1,130 points) [ revision history ]
edited Aug 25, 2015 by juancho
+ 2 like - 0 dislike

Physical application of the K.H.Mayer`s integrality theorem:  Gravitational anomaly for fivebranes in M-theory.

It is well known that M theory have two types of BPS extended objects, membranes and fivebranes which are usually  denoted  as M2 and M5 respectively. Since  M2 has an odd-dimensional world-volume it does not have anomalies in continuous symmetries.  M5  is a more interesting and subtle object.

The  M5-brane configuration breaks down the D=11 local Lorentz symmetry SO(10,1) to SO(5,1)×SO(5); where  SO(5,1) acts on the tangent bundle to the fivebrane world-volume denoted TW  and SO(5) acts as gauge group on the normal bundle to the fivebrane world-volume denoted N.

Then,  the K.H.Mayer`s integrality theorem for  a normal bundle N with  structure group SO(5) takes the form

22M(N)ˆA(TW)

The total gravitational anomaly of the fermion zero modes living in the world-volume of the fivebrane  is given by descent from

(12)[22M(N)ˆA(TW)]8form

In terms of Pontrjagin classes the Mayer class is given by

M=1+18p1+196p2+1384p12+1960p3+13840p1p2+146080p13+...

In terms of Pontrjagin classes the Dirac genus is given by

ˆA=1124p111440p2+75760p12160480p3+11241920p1p231967680p13+...

Note that

22M(N)=4+12p1(N)+124p2(N)+196p1(N)2

which coincides with the equation (5.3) on page 31 of http://arxiv.org/pdf/hep-th/9610234v1.pdf    and with the equation (155c)  on page 45 of  http://xxx.lanl.gov/pdf/hep-th/0509097.pdf

Then we have

(12)[22M(N)ˆA(TW)]8form=

2[(1+p1(N)8+p2(N)96+p1(N)2384)(1p1(TW)24p2(TW)1440+7p1(TW)25760)]8

which is reduced to

Izeromode=p2(N)48+p1(N)2192196p1(TW)p1(N)

1720p2(TW)+72880p1(TW)2

From other side, the  K.H.Mayer`s integrality theorem for  the tangent bundle TW with  structure group SO(8) takes the form

24M(TW)ˆA(TW)

The total gravitational anomaly of the chiral two-form living in the world-volume of the fivebrane  is given by descent from

(18)[24M(TW)ˆA(TW)]8form

Then we have

(18)[24M(TW)ˆA(TW)]8form=

2[(1+p1(TW)8+p2(TW)96+p1(TW)2384)(1p1(TW)24p2(TW)1440+7p1(TW)25760)]8

which is reduced to

IA=7360p2(TW)+1360p1(TW)2

and it coincides with the equation (5.4) on page 31 of  http://arxiv.org/pdf/hep-th/9610234v1.pdf      and with the equation (148) on page 44  of  http://xxx.lanl.gov/pdf/hep-th/0509097.pdf

Now, we consider that there is a gravitational inflow contribution to the anomaly for the M5-brane given by the descent from

Iinflow=ap1(TM11)2+bp2(TM11)

where a and b are constants to be determined; and

TM11W=TWN

It is well known that

p1(TM11)=p1(TW)+p1(N)

p2(TM11)=p2(TW)+p2(N)+p1(TW)p1(N)

Then we have

Iinflow=a(p1(TW)+p1(N))2+b(p2(TW)+p2(N)+p1(TW)p1(N))

and it is reduced to

Iinflow=ap1(TW)2+2ap1(TW)p1(N)+ap1(N)2+

bp2(TW)+bp2(N)+bp1(TW)p1(N)

According with all these results we derive that Itotal=Izeromode+IA+Iinflow is given by

Itotal=148p2(N)+1192p1(N)2196p1(TW)p1(N)

148p2(TW)+1192p1(TW)2+ap1(TW)2+

2ap1(TW)p1(N)+ap1(N)2+bp2(TW)+

bp2(N)+bp1(TW)p1(N)

In order to cancel  the tangent bundle anomaly we demand that b=148 and a=1192.  Then with these values we obtain

Iinflow=1192p1(TM11)2+148p2(TM11)

which coincides with the equation (5.5) on page 32 of http://arxiv.org/pdf/hep-th/9610234v1.pdf    and with the equation (152)  on page 44 of  http://xxx.lanl.gov/pdf/hep-th/0509097.pdf

Finally we obtain

Itotal=124p2(N)

which coincides with the equation (5.7) on page 32 of http://arxiv.org/pdf/hep-th/9610234v1.pdf    and with the equation (159)  on page 46 of  http://xxx.lanl.gov/pdf/hep-th/0509097.pdf

Subsequent work gives the following picture

answered Aug 23, 2015 by juancho (1,130 points) [ revision history ]
edited Aug 26, 2022 by juancho
+ 1 like - 0 dislike

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))]8form

Using that

ˆA(TW)=1124p1(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

TQW=TWN

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)21144p1(N)p1(TQ)+1120p1(N)2+

p2(N)90p2(TQ)1440+W4(N)p1(N)48W4(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)]8form

where 2πG is the SU(2) curvature and Tr is the trace in the adjoint representation SU(2).

Using that

ˆA(TW)=1124p1(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)=312Tr(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)]8form

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)=1124p1(TW)11440p2(TW)+75760p1(TW)2

tr(eiG)=212tr(G2)+124tr(G4)

tr(eiF)=3212tr(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)2124W4(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)212W4(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)2148p2(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 .

answered Aug 24, 2015 by juancho (1,130 points) [ revision history ]
edited Aug 27, 2015 by juancho
+ 1 like - 0 dislike

A proof of the K.H.Mayer`s integrality theorem using Heterotic-Supersymmetric quantum mechanics:

The first part of the Mayer integrality theorem is proved using the following effective lagrangian

Leff=12[˙ξμ˙ξμ+iλA˙λA+iRμν˙ξμξν+FABλAλB]

This effective lagrangian can be rewritten as

Leff=12ξμ[2τημν+iRμντ]ξν+12λA[iτηAB+FAB]λB

The Witten index for this heterotic Susy QM is given by

index=MAPBCPBCet0Leff(τ)dτdξdλdM=integer

Then, computing the path integrals we obtain

APBCet012λA[iτηAB+FAB]λBdτdλ=Det(iτηAB+FAB)=si=1(4n=0(1+yi2(2n+1)2π2)2)=2ssi=1cosh(yi2)

and

PBCet012ξμ[2τημν+iRμντ]ξνdτdξ=1Det(2τημν+iRμντ)=1j(n=1(1+xj24π2n2))2=jxj2sinh(xj2)=ˆA(M)

Using these results we derive that

index=M[ˆA(M)2ssi=1cosh(yi2)]topformdM=integer

The second part of the Mayer integrality theorem is proved using the following effective lagrangian

Leff=12[˙ξμ˙ξμ+iλA˙λA+iRμν˙ξμξν+FABλAλB+FABψA0ψB0]

This effective lagrangian can be rewritten as

Leff=12ξμ[2τημν+iRμντ]ξν+12λA[iτηAB+FAB]λB+12FABψA0ψB0

The Witten index for this heterotic Susy QM is given by

index=MPBCPBCet0Leff(τ)dτdξdλdψ0dM=integer

Then, computing the path integrals we obtain

PBCet012λA[iτηAB+FAB]λBdτdλPBCet012FABψA0ψB0dτdψ0=Det(iτηAB+FAB)(si=1y2i)=si=1(y2in=1(1+yi24π2n2)2)=2ssi=1sinh(yi2)

and

PBCet012ξμ[2τημν+iRμντ]ξνdτdξ=1Det(2τημν+iRμντ)=1j(n=1(1+xj24π2n2))2=jxj2sinh(xj2)=ˆA(M)

Using these results we derive that

index=M[ˆA(M)2ssi=1sinh(yi2)]topformdM=integer

answered Aug 28, 2015 by juancho (1,130 points) [ revision history ]
edited Aug 31, 2015 by juancho

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