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\section*{I8}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{InflowToM5BraneAnomaly}{Inflow to M5-brane anomalies}\dotfill \pageref*{InflowToM5BraneAnomaly} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In the [[string theory]]-literature ``$I_8$'' is the standard notation for a certain [[characteristic class]] of [[manifolds]] (of their [[tangent bundles]]): It is a [[rational numbers|rational]] linear combination of the [[cup product|cup square]] of the [[first fractional Pontryagin class]] with itself, and the [[second Pontryagin class]]:

\begin{equation}
I_8
  \;\coloneqq\;
  \tfrac{1}{48}
  \Big(
    p_2 \;-\; \big( \tfrac{1}{2} p_1\big)^2
  \Big)
  \;\; 
  \in
  H^8\big( -, \mathbb{Q}\big)
  \,.
\label{TheTerm}\end{equation}
In general this is a [[cohomology class]] in [[ordinary cohomology]] with [[rational number|rational]] [[coefficients]], though in applications it appears in further rational combination with other classes that in total yield a class in [[integral cohomology]].

The expression \eqref{TheTerm} controls certain [[quantum anomaly cancellation]] in [[M-theory]] and [[type IIA string theory]] (\hyperlink{VafaWitten95}{Vafa-Witten 95}, \hyperlink{DuffLiuMinasian95}{Duff-Liu-Minasian 95 (3.10) with (3.14)}). Since it was first obtain as a [[1-loop]]-contribution in [[perturbative quantum gravity|perturbative quantum]] [[supergravity]], it is often known as \emph{the one-loop anomaly term} or the \emph{one-loop anomaly polynomial} in [[M-theory]]/[[type IIA string theory]].

$\backslash$linebreak

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{InflowToM5BraneAnomaly}{}\subsubsection*{{Inflow to M5-brane anomalies}}\label{InflowToM5BraneAnomaly}

Consider an 11-dimensional [[spin structure|spin]]-[[manifold]] $X^{(11)}$ and a 2-parameter family of 6-dimensional [[submanifolds]] $Q_{M5} \hookrightarrow X^{(11)}$. When regarded as a family of [[worldvolumes]] of an [[M5-brane]], the family of [[normal bundles]] $N_X Q_{M5}$ of this inclusion carries a [[characteristic class]]

\begin{equation}
I^{M5}
  \;\coloneqq\;
  I^{M5}_{\psi} 
  + 
  I_{C}
  \;\in\;
  H^8(F \times Q_{M5},\mathbb{Z})
\label{IM5}\end{equation}
where

\begin{enumerate}%
\item the first summand is the class of the [[chiral anomaly]] of [[chiral fermions]] on $Q_{M5}$ (\hyperlink{Witten96}{Witten 96, (5.1)}),


\item the second term the class of the [[quantum anomaly]] of a [[self-dual higher gauge field]] (\hyperlink{Witten96}{Witten 96, (5.4)})



\end{enumerate}
Moreover, there is the restriction of the $I_8$-term \eqref{TheTerm} to $Q_{M5}$, hence to the [[tangent bundle]] of $X^{11}$ to $Q_{M5}$ (the ``anomaly inflow'' from the [[bulk spacetime]] to the M5-brane)

\begin{equation}
I_8\vert_{M5}
  \;\coloneqq\;
   I_8
   \big(
     T_{Q_{M5}} X
   \big)
  \;\in\;
  H^8(F \times Q_{M5},\mathbb{Z})
  \,.
\label{I8AnomalyInflow}\end{equation}
The sum of these cohomology classes, evaluated on the [[fundamental class]] of $Q_{M5}$ is proportional to the [[second Pontryagin class]] of the [[normal bundle]]

\begin{equation}
I^{M5}
  \;+\;
  I_8\vert_{M5}
  \;=\;
  \tfrac{1}{24} p_2(N_{Q_{M5}})
\label{SumOfM5AndInflowAnomalyIsp2}\end{equation}
(\hyperlink{Witten96}{Witten 96 (5.7)})

This result used to be ``somewhat puzzling'' (\hyperlink{Witten96}{Witten 96, p. 35}) since consisteny of the [[M5-brane]] in [[M-theory]] should require its total [[quantum anomaly]] to vanish. But $p_2(N_{Q_{M5}})$ does not in general vanish, and the right conditions to require under which it does vanish were ``not clear'' (\hyperlink{Witten96}{Witten 96, p. 37}).

(For more details on computations involved this and the following arguments, see also \hyperlink{BilalMetzger03}{Bilal-Metzger 03}).

A resolution was proposed in (\hyperlink{FreedHarveyMinasianMoore98}{Freed-Harvey-Minasian-Moore 98}), further clarified in (\hyperlink{Monnier13}{Monnier 13}), see also (\hyperlink{BahBonettiMinasianNardoni18}{Bah-Bonetti-Minasian-Nardoni 18}). There it is asserted that

\begin{enumerate}%
\item the correct bulk anomaly inflow is not just that from $I_8$ itself, but includes also a contribution from the class $G_4$ of the [[supergravity C-field]], as per \eqref{FiberIntegration} below (\hyperlink{Monnier13}{Monnier 13, around (3.11)});


\item for $G^{M5}_4$ the ``restriction'' of the class of the [[supergravity C-field]] to $Q_{M5}$, the term $I^{M5}_{C}$ in \eqref{IM5} should have a further summand $-\tfrac{1}{2}\big( G_4^{M5} \big)^2$ (\hyperlink{Monnier13}{Monnier 13, around (3.7)}, using \hyperlink{Monnier14b}{Monnier 14b, (2.13)})


\item for 11d spacetime a [[4-sphere]]-[[fiber bundle]],

\begin{displaymath}
\itexarray{
     S^4 &\longrightarrow& X^{(11)} 
     \\
     && \big\downarrow^{\mathrlap{\pi}}
     \\
     && X^{(6)}
  }
\end{displaymath}
as befits the [[near horizon geometry]] of a [[black brane|black]] [[M5-brane]], the [[supergravity C-field]] should be taken to be of the form (\hyperlink{Monnier13}{Monnier 13, (3.12)})

\begin{displaymath}
G_4 =\coloneqq \tfrac{1}{2}\chi + \pi^\ast(G^{M5}_4)
\end{displaymath}
with $\tfrac{1}{2}\chi$ the degree-4 [[Euler class]], whose integral over the 4-sphere fiber is unity (\href{Sullivan+model+of+a+spherical+fibration#SullivanModelForSphericalFibration}{this Prop.}), reflecting the presence of a single M5.



\end{enumerate}
By this proposal (also \hyperlink{BahBonettiMinasianNardoni18}{Bah-Bonetti-Minasian-Nardoni 18 (5)}, \hyperlink{BBMN19}{BBMN 19 (2.9) and appendix A.4, A.5}), the anomaly inflow from the bulk would not be just $I_8$, as in \eqref{I8AnomalyInflow} but would be all of the following [[fiber integration]]

\begin{equation}
\itexarray{
    \pi_\ast
    \Big(
      -
      \tfrac{1}{6}
      G_4  G_4  G_4
      +
      G_4  I_8
    \Big)
    & =
    - \tfrac{1}{24} p_2 + \tfrac{1}{2}(G^{M5}_4)^2 + I_8
  }
\label{FiberIntegration}\end{equation}
Here we used \href{Spin5#FiberIntegrationOfCupPowersOfChiOver4Sphere}{this Prop} to find that

\begin{displaymath}
\pi_\ast\big(
    \chi^3
  \big)
  \;=\;
  2 p_2
\end{displaymath}
which would cancel against the first term $\tfrac{1}{24} p_2$ in \eqref{FiberIntegration}. Hence with this proposal, the remaining M5-brane anomaly \eqref{SumOfM5AndInflowAnomalyIsp2} would be canceled.

$\backslash$linebreak

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Pontryagin classes]], [[Stiefel-Whitney classes]], [[Wu classes]], [[Euler class]]


\item [[C-field tadpole cancellation]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The term showed in [[string theory]]/[[M-theory]] [[anomaly cancellation]] in

\begin{itemize}%
\item [[Cumrun Vafa]], [[Edward Witten]], \emph{A One-Loop Test Of String Duality}, Nucl.Phys.B447:261-270, 1995 (\href{https://arxiv.org/abs/hep-th/9505053}{arXiv:hep-th/9505053})


\item [[Mike Duff]], [[James Liu]], [[Ruben Minasian]], \emph{Eleven Dimensional Origin of String/String Duality: A One Loop Test}, Nucl.Phys. B452 (1995) 261-282 (\href{https://arxiv.org/abs/hep-th/9506126}{arXiv:hep-th/9506126})


\item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory}, J.Geom.Phys.22:103-133, 1997 (\href{https://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234})



\end{itemize}
For further discussion see

\begin{itemize}%
\item [[Hisham Sati]], \emph{[[Geometric and topological structures related to M-branes]]} ,

part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (\href{http://arXiv.org/abs/1001.5020}{arXiv:1001.5020}),

part \emph{II: Twisted $String$ and $String^c$ structures}, J. Australian Math. Soc. 90 (2011), 93-108 (\href{http://arxiv.org/abs/1007.5419}{arXiv:1007.5419});

part \emph{III: Twisted higher structures}, Int. J. Geom. Meth. Mod. Phys. 8 (2011), 1097-1116 (\href{http://arxiv.org/abs/1008.1755}{arXiv:1008.1755})



\end{itemize}
In relation to the [[quantum anomaly]] of the [[M5-brane]]:

The original computation of the total M5-brane anomaly due to

\begin{itemize}%
\item \hyperlink{Witten96}{Witten 96}

\end{itemize}
left a remnant term of $\tfrac{1}{24} p_2$. It was argued in

\begin{itemize}%
\item [[Dan Freed]], [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], \emph{Gravitational Anomaly Cancellation for M-Theory Fivebranes}, Adv.Theor.Math.Phys.2:601-618, 1998 (\href{https://arxiv.org/abs/hep-th/9803205}{arXiv:hep-th/9803205})


\item [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], \emph{Non-abelian Tensor-multiplet Anomalies}, JHEP9809:004, 1998 (\href{https://arxiv.org/abs/hep-th/9808060}{arXiv:hep-th/9808060})


\item [[Adel Bilal]], Steffen Metzger, \emph{Anomaly cancellation in M-theory: a critical review}, Nucl.Phys. B675 (2003) 416-446 (\href{https://arxiv.org/abs/hep-th/0307152}{arXiv:hep-th/0307152})



\end{itemize}
that this term disappears (cancels) when properly taking into account the singularity of the [[supergravity C-field]] at the locus of the [[black brane|black]] [[M5-brane]].

A more transparent version of this argument was offered in

\begin{itemize}%
\item Samuel Monnier, \emph{Global gravitational anomaly cancellation for five-branes}, Advances in Theoretical and Mathematical Physics, Volume 19 (2015) 3 (\href{https://arxiv.org/abs/1310.2250}{arXiv:1310.2250})

\end{itemize}
based on a refined discussion of the quantum anomaly of the [[self-dual higher gauge field]] on the M5-brane in

\begin{itemize}%
\item Samuel Monnier, \emph{The anomaly line bundle of the self-dual field theory}, Comm. Math. Phys. 325 (2014) 41-72 (\href{http://arxiv.org/abs/1109.2904}{arXiv:1109.2904})


\item Samuel Monnier, \emph{The global gravitational anomaly of the self-dual field theory}, Comm. Math. Phys. 325 (2014) 73-104 (\href{http://arxiv.org/abs/1110.4639}{arXiv:1110.4639}, \href{http://www.physics.rutgers.edu/het/video/monnier11b.pdf}{pdf slides})



\end{itemize}
This formulation via an anomaly 12-form is (re-)derived also in

\begin{itemize}%
\item Ibrahima Bah, Federico Bonetti, [[Ruben Minasian]], Emily Nardoni, \emph{Class $\mathcal{S}$ Anomalies from M-theory Inflow}, Phys. Rev. D 99, 086020 (2019) (\href{https://arxiv.org/abs/1812.04016}{arXiv:1812.04016})


\item Ibrahima Bah, Federico Bonetti, [[Ruben Minasian]], Emily Nardoni, \emph{Anomaly Inflow for M5-branes on Punctured Riemann Surfaces} (\href{https://arxiv.org/abs/1904.07250}{arXiv:1904.07250})



\end{itemize}
[[!redirects one-loop anomaly polynomial I8]]



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