# Physical derivation and applications of the K.H.Mayers 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.Mayers integrality theorem was considered and its heterotic-supersymmetric derivation was presented.  In this post we consider the full K.H.Mayers integrality theorem which reads: (originally in German, please look below the English translation)

English translation:

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

My questions are:

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

2. Do you know physical applications  of the K.H.Mayers integrality theorem?

edited Aug 26, 2015

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Physical application of the K.H.Mayers 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 $T\Sigma^2$  and $SO(2)$ acts as gauge group on the normal bundle to the string world-sheet denoted $N$.

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

$$2M(N)\hat{A}(T\Sigma^2)$$

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

$$(-\frac{1}{2})[2M(N)\hat{A}(T\Sigma^2)]_{4-form}$$

In terms of Pontrjagin classes the Mayer class is given by

$$M=1+\frac{1}{8}\,p_{{1}}+{\frac {1}{96}}\,p_{{2}}+{\frac {1}{384}}\,{p_{{1}}}^ {2}+{\frac {1}{960}}\,p_{{3}}+{\frac {1}{3840}}\,p_{{1}}p_{{2}}+{ \frac {1}{46080}}\,{p_{{1}}}^{3}+...$$

In terms of Pontrjagin classes the Dirac genus is given by

$$\hat{A}=1-\frac{1}{24}\,p_{{1}}-{\frac {1}{1440}}\,p_{{2}}+{\frac {7}{5760}}\,{p_{{1 }}}^{2}-{\frac {1}{60480}}\,p_{{3}}+{\frac {11}{241920}}\,p_{{1}}p_{{2 }}-{\frac {31}{967680}}\,{p_{{1}}}^{3}+...$$

Then we have

$$(-\frac{1}{2})[2M(N)\hat{A}(T\Sigma^2)]_{4-form}=-[(1+\frac{1}{8}\,p_{{1}}(N))(1-\frac{1}{24}\,p_{{1}}(T\Sigma^2))]_{4-form}$$

which is reduced to

$$I^{zeromode} = \frac{1}{24}\,p_{{1}}(T\Sigma^2)-\frac{1}{8}\,p_{{1}}(N)$$

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

$$I^{inflow} =\alpha p_1(TM)$$

where $\alpha$ is constant to be determined and

$$TM \mid _{\Sigma^2} = T\Sigma^2 \oplus N$$

Then we have that

$$I^{inflow} =\alpha p_1(TM)=\alpha p_1(T\Sigma^2 \oplus N)=\alpha(p_1(T\Sigma^2 )+p_1(N))\\=\alpha p_1(T\Sigma^2 )+\alpha p_1(N)$$

According with all these results we derive that

$$I^{inflow}+I^{zeromode}=\alpha p_1(T\Sigma^2 )+\alpha p_1(N)+\frac{1}{24}\,p_{{1}}(T\Sigma^2)-\frac{1}{8}\,p_{{1}}(N)$$

which is reduced to

$$I^{inflow}+I^{zeromode}=(\alpha +\frac{1}{24})p_1(T\Sigma^2 )+(\alpha -\frac{1}{8} )p_1(N)$$

In order to cancel  the tangent bundle anomaly we demand that

$$\alpha +\frac{1}{24} = 0$$

which is equivalent to $\alpha = - \frac{1}{24}$.  Using this value of $\alpha$ we obtain

$$I^{inflow}+I^{zeromode}= -\frac{1}{6}p_1(N)$$

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

Also we obtain

$$I^{inflow} = - \frac{1}{24} p_1(T\Sigma^2 )- \frac{1}{24} p_1(N) = - \frac{1}{24} p_1(TM)$$

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

Finally, given that $p_1(N)= e^2(N)$, where $e(N)$ is the Euler class of the normal bundle; we rewrite the  uncanceled anomaly for the normal bundle as

$$I^{inflow}+I^{zeromode}= -\frac{1}{6}e^2(N)$$

answered Aug 22, 2015 by (1,130 points)
edited Aug 26, 2015 by juancho
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Physical application of the K.H.Mayers 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.Mayers integrality theorem for  a normal bundle $N$ with  structure group $SO(5)$ takes the form

$$2^2M(N)\hat{A}(TW)$$

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

$$(\frac{1}{2})[2^2M(N)\hat{A}(TW)]_{8-form}$$

In terms of Pontrjagin classes the Mayer class is given by

$$M=1+\frac{1}{8}\,p_{{1}}+{\frac {1}{96}}\,p_{{2}}+{\frac {1}{384}}\,{p_{{1}}}^ {2}+{\frac {1}{960}}\,p_{{3}}+{\frac {1}{3840}}\,p_{{1}}p_{{2}}+{ \frac {1}{46080}}\,{p_{{1}}}^{3}+...$$

In terms of Pontrjagin classes the Dirac genus is given by

$$\hat{A}=1-\frac{1}{24}\,p_{{1}}-{\frac {1}{1440}}\,p_{{2}}+{\frac {7}{5760}}\,{p_{{1 }}}^{2}-{\frac {1}{60480}}\,p_{{3}}+{\frac {11}{241920}}\,p_{{1}}p_{{2 }}-{\frac {31}{967680}}\,{p_{{1}}}^{3}+...$$

Note that

$$2^2M \left( N \right) =4+\frac{1}{2}\,p_{{1}} \left( N \right) +\frac{1}{24}\,p_{{2}} \left( N \right) +\frac {1}{96}p_{{1}} \left( N \right)^{2}$$

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

$$(\frac{1}{2})[2^2M(N)\hat{A}(TW)]_{8-form}=$$

$$2[(1+\frac{p_{{1}}(N)}{8}+{\frac {p_{{2}}(N)}{96}}+{\frac {{p_{{1}}}(N)^ {2}}{384}})(1-\frac{p_{{1}}(TW)}{24}-{\frac {p_{{2}}(TW)}{1440}}+{\frac {7{p_{{1 }}}(TW)^{2}}{5760}})]_{8}$$

which is reduced to

$$I^{zeromode} = \frac{p_{{2}} \left( N \right)}{48} +{\frac {p_{{1}} \left( N \right) ^{2}}{192}}-{\frac {1}{96}}\,p_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right) -$$

$${\frac {1}{720}}\,p_{{2}} \left( {\it TW} \right) +\frac {7}{2880}p_{{1}} (TW)^{2}$$

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

$$2^4M(TW)\hat{A}(TW)$$

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

$$(-\frac{1}{8})[2^4M(TW)\hat{A}(TW)]_{8-form}$$

Then we have

$$(-\frac{1}{8})[2^4M(TW)\hat{A}(TW)]_{8-form}=$$

$$-2[(1+\frac{p_{{1}}(TW)}{8}+{\frac {p_{{2}}(TW)}{96}}+{\frac {{p_{{1}}}(TW)^ {2}}{384}})\\(1-\frac{p_{{1}}(TW)}{24}-{\frac {p_{{2}}(TW)}{1440}}+{\frac {7{p_{{1 }}}(TW)^{2}}{5760}})]_{8}$$

which is reduced to

$$I_{A}= -{\frac {7}{360}}\,p_{{2}} \left( {\it TW} \right) +{\frac {1}{360}} p_{{1}} \left( {\it TW} \right) ^{2}$$

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

$$I^{inflow}=a p_{{1}} \left( {{\it TM}}^{11} \right) ^{2}+bp_{{2}} \left( {{\it TM}}^{11} \right)$$

where $a$ and $b$ are constants to be determined; and

$$TM^{11} \mid _{W} = TW \oplus N$$

It is well known that

$$p_{{1}} \left( {{\it TM}}^{11} \right) =p_{{1}} \left( {\it TW}\right) +p_{{1}} \left( N \right)$$

$$p_{{2}} \left( {{\it TM}}^{11} \right) =p_{{2}} \left( {\it TW} \right) +p_{{2}} \left( N \right) +p_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right)$$

Then we have

$$I^{inflow}=a \left( p_{{1}} \left( {\it TW} \right) +p_{{1}} \left( N \right) \right) ^{2}+b \left( p_{{2}} \left( {\it TW} \right) +p_{{2}}\left( N \right) +p_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right) \right)$$

and it is reduced to

$$I^{inflow}=a p_{{1}} \left( {\it TW} \right) ^{2}+2\,ap_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right) +a p_{{1}} \left( N \right) ^{2}+$$

$$bp_{{2}} \left( {\it TW} \right) +bp_{ {2}} \left( N \right) +bp_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right)$$

According with all these results we derive that $I^{total}=I^{zeromode}+I_A+I^{inflow}$ is given by

$$I^{total}=\frac {1}{48}\,p_{{2}} \left( N \right) +{\frac {1}{192}} p_{{1}} \left( N \right) ^{2}-{\frac {1}{96}}\,p_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right) -$$

$$\frac{1}{48}p_{{2}} \left( {\it TW} \right) +{\frac {1}{192}} p_{{1}} \left( {\it TW} \right) ^{2}+a p_{{1}} \left( {\it TW} \right) ^{2}+$$

$$2\,ap_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right) +ap_{{1}} \left( N \right) ^{2}+bp_{{2}} \left( {\it TW} \right) +$$

$$bp_{{2}} \left( N \right) +bp_{{1}} \left( {\it TW} \right)p_{{1}} \left( N \right)$$

In order to cancel  the tangent bundle anomaly we demand that $b=\frac{1}{48}$ and $a= -\frac{1}{192}$.  Then with these values we obtain

$$I^{inflow}=-\frac{1}{192} p_{{1}} \left( {{\it TM}}^{11} \right) ^{2}+\frac{1}{48}p_{{2}} \left( {{\it TM}}^{11} \right)$$

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

$$I^{total}=\frac{1}{24}\,p_{{2}} \left( N \right)$$

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

answered Aug 23, 2015 by (1,130 points)
edited Mar 6 by juancho
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Physical application of the K.H.Mayers 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.Mayers integrality theorem for  a normal bundle $N$ with  structure group $SO(4)$ produces the following two cohomological expressions

$$2^2M(N)\hat{A}(TW)$$

and

$$W_{4}(N)\hat{A}(N)^{-1}\hat{A}(TW)$$

From the original proof of the K.H.Mayers integrality theorem we have that

$$ch(S_{+}(N))+ch(S_{-}(N))= 2^2M(N)$$

and

$$ch(S_{+}(N))-ch(S_{-}(N))= W_{4}(N)\hat{A}(N)^{-1}$$

From these last equations we deduce that

$$ch(S_{+}(N))=\frac{1}{2} [2^2M(N)+ W_{4}(N)\hat{A}(N)^{-1}]$$

and

$$ch(S_{-}(N))=\frac{1}{2} [2^2M(N)-W_{4}(N)\hat{A}(N)^{-1}]$$

which are rewritten as

$$ch(S_{\pm}(N))=\frac{1}{2} [2^2M(N)\pm W_{4}(N)\hat{A}(N)^{-1}]$$

Now, we have that

$$W_{4}(N)\hat{A}(N)^{-1}= W_{{4}} \left( N \right) +\frac{1}{24}\,W_{{4}} \left( N \right) p_{{1}} \left( N \right)$$

$$M(N)=1+\frac{1}{8}\,p_{{1}}(N)+{\frac {1}{96}}\,p_{{2}}(N)+{\frac {1}{384}}\,{p_{{1}}}(N)^ {2}+....$$

Using these last equations we obtain

$$ch(S_{\pm}(N))=\frac{1}{2} [4(1+\frac{1}{8}\,p_{{1}}(N)+{\frac {1}{96}}\,p_{{2}}(N)+{\frac {1}{384}}\,{p_{{1}}}(N)^ {2}) \pm \\ (W_{{4}} \left( N \right) +\frac{1}{24}\,W_{{4}} \left( N \right) p_{{1}} \left( N \right))]$$

which is reduced to

$$ch(S_{\pm}(N))=2+\frac{p_{{1}}(N) \pm W_{{4}} \left( N \right) }{4}+\frac {p_{1}(N)^2+4p_{2}(N)\pm 4 W_{4} ( N )p_{1}(N)}{192}$$

and it is exactly the equation $(6)$ on page 4 of   http://arxiv.org/pdf/hep-th/9709012v1.pdf ; where  $W_4(N) = \chi(N)$, $S_{+}(N)$ 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 $\theta$-fermions and they belong to the $(4_{+},2_{+})$ representation of $SO(5,1) × SO(4)$.

The total gravitational anomaly of the $\theta$-fermion zero modes living in the world-volume of the heterotic $SO(32)$  fivebrane  is given by descent from

$$I_{8}^{\theta}=\frac{1}{2}[\hat{A}(TW)ch(S_{+}(N))]_{8-form}$$

Using that

$$\hat{A}(TW)=1-\frac{1}{24}\,p_{{1}}(TW)-{\frac {1}{1440}}\,p_{{2}}(TW)+{\frac {7}{5760}}\,{p_{{1 }}}(TW)^{2}$$

and

$$ch(S_{+}(N))=2+\frac{p_{{1}}(N) + W_{{4}} \left( N \right) }{4}+\frac {p_{1}(N)^2+4p_{2}(N)+ 4 W_{4} ( N )p_{1}(N)}{192}$$

we obtain

$$I_{8}^{\theta}={\frac {7}{5760}} p_{{1}} \left( {\it TW} \right) ^{ 2}+{\frac {1}{96}}\,p_{{2}} \left( N \right) +{\frac {1}{384}}p_{{1}} \left( N \right) ^{2}-$$

$${\frac {1}{1440}}\,p_{{ 2}} \left( {\it TW} \right) -{\frac {1}{192}}\,p_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right) +$$

$${\frac {1}{96}}\,W_{{4}} \left( N \right) p_{{1}} \left( N \right) -{\frac {1}{96}}\,p_{{1}} \left( { \it TW} \right) W_{{4}} \left( N \right)$$

Now, given that

$$TQ \mid _{W} = TW \oplus N$$

$$p_{{1}} \left( TQ \right) =p_{{1}} \left( {\it TW}\right) +p_{{1}} \left( N \right)$$

$$p_{{2}} \left( TQ \right) =p_{{2}} \left( {\it TW} \right) +p_{{2}} \left( N \right) +p_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right)$$

we deduce that

$$p_{{1}} \left( {\it TW} \right) =p_{{1}} \left( {\it TQ} \right) -p_{{1}} \left( N \right)$$

and

$$p_{{2}} \left( {\it TW} \right) =p_{{2}} \left( {\it TQ} \right) -p_{{ 2}} \left( N \right) -p_{{1}} \left( N \right) p_{{1}} \left( {\it TQ} \right) + p_{{1}} \left( N \right) ^{2}$$

Then using these last equations we obtain

$$I_{8}^{\theta} = {\frac {7}{5760}}p_{{1}} \left( {\it TQ} \right) ^{ 2}-{\frac {1}{144}}\,p_{{1}} \left( N \right) p_{{1}} \left( {\it TQ} \right) +{\frac {1}{120}}p_{{1}} \left( N \right) ^{2}+$$

$${\frac {p_{{2}} \left( N \right) }{90}}-{\frac {p_{ {2}} \left( {\it TQ} \right)}{1440}} +\frac{W_{{4}} \left( N \right) p_{{1}} \left( N \right)}{48} -{\frac {W_{{4}} \left( N \right) p_{{1}}\left( {\it TQ} \right)}{96}}$$

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 $\lambda$-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 $\lambda$-fermion zero modes living in the world-volume of the heterotic $SO(32)$  fivebrane  is given by descent from

$$I_{8}^{\lambda}=-\frac{1}{2}[\hat{A}(TW)ch(S_{-}(N))Tr(e^{iG})]_{8-form}$$

where $2\pi G$ is the $SU(2)$ curvature and $Tr$ is the trace in the adjoint representation $SU(2)$.

Using that

$$\hat{A}(TW)=1-\frac{1}{24}\,p_{{1}}(TW)-{\frac {1}{1440}}\,p_{{2}}(TW)+{\frac {7}{5760}}\,{p_{{1 }}}(TW)^{2}$$

$$ch(S_{-}(N))=2+\frac{p_{{1}}(N) -W_{{4}} \left( N \right) }{4}+\frac {p_{1}(N)^2+4p_{2}(N)- 4 W_{4} ( N )p_{1}(N)}{192}$$

$${\it Tr} \left( {{\rm e}^{{\it iG}}} \right) =3-\frac{1}{2}\,{\it Tr} \left( { G}^{2} \right) +\frac{1}{24}\,{\it Tr} \left( {G}^{4} \right)$$

we obtain

$$I_{8}^{\lambda}=\frac{1}{16}p_{{1}} \left( N \right) {\it Tr} \left( {G}^{2} \right) -\frac{1}{48} p_{{1}} \left( {\it TW} \right) {\it Tr} \left( {G}^{2} \right) -$$

$${\frac {1}{128}}p_{{1}} \left( N \right) ^{2}+{ \frac {1}{480}}\,p_{{2}} \left( {\it TW} \right) -{\frac {7}{1920}}\, p_{{1}} \left( {\it TW} \right) ^{2}-$$

$$\frac{1}{32}p_{{2}} \left( N \right) +\frac{1}{32}W_{{4}} \left( N \right) p_{{1}} \left( N \right) -\frac{1}{32}p_{{1}} \left( {\it TW} \right) W_{{4}} \left( N \right) -$$

$$\frac{1}{24}{\it Tr} \left( {G}^{4} \right) -\frac{1}{8}W_{{4}} \left( N \right) {\it Tr} \left( {G}^{2} \right) +{\frac {1}{64}}\,p_{{1}} \left( {\it TW} \right) p_{{1}} \left( N \right)$$

Using again

$$p_{{1}} \left( {\it TW} \right) =p_{{1}} \left( {\it TQ} \right) -p_{{1}} \left( N \right)$$

and

$$p_{{2}} \left( {\it TW} \right) =p_{{2}} \left( {\it TQ} \right) -p_{{ 2}} \left( N \right) -p_{{1}} \left( N \right) p_{{1}} \left( {\it TQ} \right) + p_{{1}} \left( N \right) ^{2}$$

we obtain

$$I_{8}^{\lambda}={\frac {1}{12}}p_{{1}} \left( N \right) {\it Tr} \left( {G}^{2} \right) -{\frac {1}{48}} {\it Tr} \left( {G}^{2} \right) p_{{1}} \left( {\it TQ} \right) -{\frac {1}{40}} p_{{1}} \left( N \right) ^{2}+$$

$${\frac {1}{480}}\,p_{ {2}} \left( {\it TQ} \right) -{\frac {1}{30}}p_{{2}} \left( N \right) +{\frac {1}{30}}p_ {{1}} \left( N \right) p_{{1}} \left( {\it TQ} \right) -$$

$${\frac {7}{ 1920}}p_{{1}} \left( {\it TQ} \right) ^{2}+{\frac {1}{16}}W_ {{4}} \left( N \right) p_{{1}} \left( N \right) -{\frac {1}{32}}W_{{4}} \left( N \right) p_{{1}} \left( {\it TQ} \right) -$$

$${\frac {1}{24}}{\it Tr} \left( {G}^{ 4} \right) -{\frac {1}{8}}W_{{4}} \left( N \right) {\it Tr} \left( {G}^{2} \right)$$

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 $\psi$-fermions;  and they belong to the $(32,2)$  representation of $SO(32)×SU(2)$.

The total gravitational anomaly of the $\psi$-fermion zero modes living in the world-volume of the heterotic $SO(32)$  fivebrane  is given by descent from

$$I_{8}^{\psi}=\frac{1}{2}[\hat{A}(TW)Tr(e^{iF})Tr(e^{iG})]_{8-form}$$

where $2\pi G$ is the $SU(2)$ curvature,  $2\pi F$ is the $SO(32)$ curvature and $tr$ is the trace in the fundamental representation.

Using that

$$\hat{A}(TW)=1-\frac{1}{24}\,p_{{1}}(TW)-{\frac {1}{1440}}\,p_{{2}}(TW)+{\frac {7}{5760}}\,{p_{{1 }}}(TW)^{2}$$

$${\it tr} \left( {{\rm e}^{{\it iG}}} \right) =2-\frac{1}{2}\,{\it tr} \left( { G}^{2} \right) +\frac{1}{24}\,{\it tr} \left( {G}^{4} \right)$$

$${\it tr} \left( {{\rm e}^{{\it iF}}} \right) =32-\frac{1}{2}\,{\it tr} \left( { F}^{2} \right) +\frac{1}{24}\,{\it tr} \left( {F}^{4} \right)$$

we obtain

$$I_{8}^{\psi}= -\frac{1}{45}\,p_{{2}} \left( {\it TW} \right) +{\frac {7}{180}}p_{{ 1}} \left( {\it TW} \right) ^{2}+\frac{2}{3}\,{\it tr} \left( {G}^{4} \right) +$$

$$\frac{1}{48}\,p_{{1}} \left( {\it TW} \right) {\it tr} \left( {F}^{2 } \right) +\frac{1}{8}\,{\it tr} \left( {G}^{2} \right) {\it tr} \left( {F}^{2 } \right) +\frac{1}{3}\,p_{{1}} \left( {\it TW} \right) {\it tr} \left( {G}^{2 } \right) +\frac{1}{24}\,{\it tr} \left( {F}^{4} \right)$$

Using again

$$p_{{1}} \left( {\it TW} \right) =p_{{1}} \left( {\it TQ} \right) -p_{{1}} \left( N \right)$$

and

$$p_{{2}} \left( {\it TW} \right) =p_{{2}} \left( {\it TQ} \right) -p_{{ 2}} \left( N \right) -p_{{1}} \left( N \right) p_{{1}} \left( {\it TQ} \right) + p_{{1}} \left( N \right) ^{2}$$

we have that

$$I_{8}^{\psi}=-\frac {1}{45}\,p_{{2}} \left( {\it TQ} \right) +\frac {1}{45}\,p_{{2}} \left( N \right) -\frac {1}{18}\,p_{{1}} \left( N \right) p_{{1}} \left( {\it TQ} \right) +{\frac {1}{60}}p_{{1}} \left( N \right) ^{2}+$$

$${\frac {7}{180}}p_{{1}} \left( {\it TQ} \right) ^{2}+\frac {2}{3}\,{\it tr} \left( {G}^{4} \right) +\frac{1}{48}\,{\it tr} \left( {F}^{2} \right) p_{{1}} \left( {\it TQ} \right) -\frac {1}{48}\,{\it tr } \left( {F}^{2} \right) p_{{1}} \left( N \right) +$$

$$\frac {1}{8}\,{\it tr} \left( {G}^{2} \right) {\it tr} \left( {F}^{2} \right) +\frac {1}{3}\,{\it tr} \left( {G}^{2} \right) p_{{1}} \left( {\it TQ} \right) -\frac{1}{3}\,{\it tr} \left( {G}^{2} \right) p_{{1}} \left( N \right) +\frac{1}{24}\,{\it tr} \left( {F}^{4} \right)$$

and it is exactly the equation $(12)$ on page 5 of   http://arxiv.org/pdf/hep-th/9709012v1.pdf .

The total anomaly is $I_8 = I_8^{\theta}+I_8^{\lambda}+I_8^{\psi}$ and then we have

$$I_8=\frac{1}{12}\,W_{{4}} \left( N \right) p_{{1}} \left( N \right) +\frac{1}{48}\,{\it tr } \left( {F}^{2} \right) p_{{1}} \left( {\it TQ} \right) -\frac{1}{48}\,{\it tr} \left( {F}^{2} \right) p_{{1}} \left( N \right) +$$

$$\frac{1}{3}\,{\it tr} \left( {G}^{2} \right) p_{{1}} \left( {\it TQ} \right) -\frac{1}{3}\,{\it tr} \left( {G}^{2} \right) p_{{1}} \left( N \right) -\frac{1}{48}\,p_{{2}} \left( {\it TQ} \right) -\frac{1}{24}\,p_{{1}} \left( N \right) p_{{1}} \left( {\it TQ} \right) +$$

$${\frac {7}{192}}p_{{1}} \left( { \it TQ} \right) ^{2}-\frac{1}{24}\,W_{{4}} \left( N \right) p_{{1}} \left( {\it TQ} \right) -\frac{1}{24}\,{\it Tr} \left( {G}^{4} \right) -\frac{1}{8}\, W_{{4}} \left( N \right) {\it Tr} \left( {G}^{2} \right) +$$

$$\frac{1}{12}\,p_{{1} } \left( N \right) {\it Tr} \left( {G}^{2} \right) -\frac{1}{48}\,{\it Tr} \left( {G}^{2} \right) p_{{1}} \left( {\it TQ} \right) +\frac{2}{3}\,{\it tr} \left( {G}^{4} \right) +$$

$$\frac{1}{24}\,{\it tr} \left( {F}^{4} \right) +\frac{1}{8}\,{ \it tr} \left( {G}^{2} \right) {\it tr} \left( {F}^{2} \right)$$

Now, using $Tr(G^2)=4tr(G^2)$ and $Tr(G^4)=16tr(G^4)$, we obtain

$$I_8=\frac{1}{12}\,W_{{4}} \left( N \right) p_{{1}} \left( N \right) -\frac{1}{24}\,p_{{1}} \left( N \right) p_{{1}} \left( {\it TQ} \right) -\frac{1}{24}\,W_{{4}} \left( N \right) p_{{1}} \left( {\it TQ} \right) +$$

$${\frac {7}{192}}p_{{1}} \left( {\it TQ} \right) ^{2}-\frac{1}{2}\,W_{{4}} \left( N \right) {\it tr} \left( {G}^{2} \right) +\frac{1}{4}\,{\it tr} \left( {G}^{2} \right) p_{{1}} \left( {\it TQ} \right) +$$

$$\frac{1}{8}\,{\it tr}\left( {G}^{2} \right) {\it tr} \left( {F}^{2} \right) +\frac{1}{24}\,{\it tr} \left( {F}^{4} \right) +\frac{1}{48}\,{\it tr} \left( {F}^{2} \right) p_{{1}} \left( {\it TQ} \right) -$$

$$\frac{1}{48}\,{\it tr} \left( {F}^{2} \right) p_{{1 }} \left( N \right) -\frac{1}{48}\,p_{{2}} \left( {\it TQ} \right)$$

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

$$I_8 =[W_{{4}} \left( N \right) -\frac{1}{4}\,{\it tr} \left( {F}^{2} \right) -\frac{1}{2}\,p_{{1}} \left( {\it TQ} \right) ][\frac{1}{12}p_{1}(N)-\frac{1}{2}tr(G^2)-\frac{1}{24}p_{1}(TQ)] +$$

$${\frac {1}{64}}p_{{1}} \left( {\it TQ} \right) ^{2}-\frac{1}{48}\,p_{{2}} \left( {\it TQ} \right) +{\frac {1}{96}}\,{\it tr} \left( {F}^{2} \right) p_{{1}} \left( {\it TQ} \right) +\frac{1}{24}\,{\it tr} \left( {F}^{4} \right)$$

and it is exactly the equation $(15)$ on page 5 of   http://arxiv.org/pdf/hep-th/9709012v1.pdf .

answered Aug 24, 2015 by (1,130 points)
edited Aug 28, 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

$$L_{eff}=\frac{1}{2}[\dot{\xi}_{\mu}\dot{\xi}^{\mu}+i\lambda_{A}\dot{\lambda}^{A}+iR_{\mu \nu}\dot{\xi}^{\mu}\xi^{\nu}+F_{AB}\lambda^A\lambda^B]$$

This effective lagrangian can be rewritten as

$$L_{eff}= -\frac{1}{2}\xi^{\mu}[\partial_{\tau}^2\eta_{\mu \nu}+iR_{\mu \nu}\partial_{\tau}]\xi^{\nu}+\frac{1}{2}\lambda^{A}[i \partial_{\tau} \eta_{A B}+F_{A B}]\lambda^{B}$$

The Witten index for this heterotic Susy QM is given by

$${\it index}=\int_{M}\int_{APBC}\int_{PBC}{\rm e}^{-\int _{0}^{t}\!L_{{{\it eff}}} \left( \tau \right) {d\tau}}d\xi d\lambda dM={\it integer}$$

Then, computing the path integrals we obtain

$$\int_{APBC} e^{-\int _{0}^{t} \frac{1}{2}\lambda^{A}[i \partial_{\tau} \eta_{A B}+F_{A B}]\lambda^{B} {d\tau}}{d\lambda}=\sqrt {{\it Det} \left( i \partial_{\tau} \eta_{A B}+F_{A B} \right) }=\\\sqrt {\prod _{i=1}^{s} \left( 4\,\prod _{n=0}^{ \infty } \left( 1+{\frac {{y_{{i}}}^{2}}{ \left( 2\,n+1 \right) ^{2}{ \pi }^{2}}} \right) ^{2} \right) }={2}^{s}\prod _{i=1}^{s}\cosh \left( \frac{y_{{i}}}{2} \right)$$

and

$$\int_{PBC} e^{-\int _{0}^{t}\frac{1}{2}\xi^{\mu}[\partial_{\tau}^2\eta_{\mu \nu}+iR_{\mu \nu}\partial_{\tau}]\xi^{\nu} {d\tau}} {d\xi}={\frac {1}{\sqrt {{\it Det} \left( \partial_{\tau}^2\eta_{\mu \nu}+iR_{\mu \nu}\partial_{\tau} \right) }}}\\={\frac {1}{\sqrt {\prod _ j \left( \prod _{n=1}^{\infty }(1+{\frac {{x_{{j}}}^{2}}{4{\pi }^{2}{n}^{2}} } )\right) ^{2} }}}=\prod _ j{\frac {\frac{x_{{j}}}{2}}{\sinh \left( \frac{x_{{j}}}{2} \right) }} = \hat{A}(M)$$

Using these results we derive that

$$\mathrm{index}=\int_{M} [\hat{A} \left( M \right) {2}^{s}\prod _{i=1}^{s}\cosh \left( \frac{y_{{i}}}{2} \right)]_{top-form} {dM} =\mathrm{integer}$$

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

$$L_{eff}=\frac{1}{2}[\dot{\xi}_{\mu}\dot{\xi}^{\mu}+i\lambda_{A}\dot{\lambda}^{A}+iR_{\mu \nu}\dot{\xi}^{\mu}\xi^{\nu}+F_{AB}\lambda^A\lambda^B+F_{AB}\psi_{0}^A\psi_{0}^B]$$

This effective lagrangian can be rewritten as

$$L_{eff}= -\frac{1}{2}\xi^{\mu}[\partial_{\tau}^2\eta_{\mu \nu}+iR_{\mu \nu}\partial_{\tau}]\xi^{\nu}+\frac{1}{2}\lambda^{A}[i \partial_{\tau} \eta_{A B}+F_{A B}]\lambda^{B}+\frac{1}{2}F_{AB}\psi_{0}^A\psi_{0}^B$$

The Witten index for this heterotic Susy QM is given by

$${\it index}=\int_{M}\int\int_{PBC}\int_{PBC}{\rm e}^{-\int _{0}^{t}\!L_{{{\it eff}}} \left( \tau \right) {d\tau}}d\xi d\lambda d\psi_{0} dM={\it integer}$$

Then, computing the path integrals we obtain

$$\int_{PBC} e^{-\int _{0}^{t} \frac{1}{2}\lambda^{A}[i \partial_{\tau} \eta_{A B}+F_{A B}]\lambda^{B} {d\tau}}{d\lambda}\int_{PBC} e^{-\int _{0}^{t} \frac{1}{2}F_{AB}\psi_{0}^A\psi_{0}^B {d\tau}}{d\psi_{0}}=\\\sqrt {{\it Det} \left( i \partial_{\tau} \eta_{A B}+F_{A B} \right) }(\prod _{i=1}^{s} y_{i}^2)= \sqrt {\prod _{i=1}^{s} \left( y_{i}^2\,\prod _{n=1}^{ \infty } \left( 1+{\frac {{y_{{i}}}^{2}}{4 { \pi }^{2}n ^{2}}} \right) ^{2} \right) }={2}^{s}\prod _{i=1}^{s}\sinh \left( \frac{y_{{i}}}{2} \right)$$

and

$$\int_{PBC} e^{-\int _{0}^{t}\frac{1}{2}\xi^{\mu}[\partial_{\tau}^2\eta_{\mu \nu}+iR_{\mu \nu}\partial_{\tau}]\xi^{\nu} {d\tau}} {d\xi}={\frac {1}{\sqrt {{\it Det} \left( \partial_{\tau}^2\eta_{\mu \nu}+iR_{\mu \nu}\partial_{\tau} \right) }}}\\={\frac {1}{\sqrt {\prod _ j \left( \prod _{n=1}^{\infty }(1+{\frac {{x_{{j}}}^{2}}{4{\pi }^{2}{n}^{2}} } )\right) ^{2} }}}=\prod _ j{\frac {\frac{x_{{j}}}{2}}{\sinh \left( \frac{x_{{j}}}{2} \right) }} = \hat{A}(M)$$

Using these results we derive that

$$\mathrm{index}=\int_{M} [\hat{A} \left( M \right) {2}^{s}\prod _{i=1}^{s}\sinh \left( \frac{y_{{i}}}{2} \right)]_{top-form} {dM} =\mathrm{integer}$$

answered Aug 28, 2015 by (1,130 points)
edited Aug 31, 2015 by juancho

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