Contents

Contents

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 linear combination of the cup square of the first fractional Pontryagin class with itself, and the second Pontryagin class:

(1)$I_8 \;\coloneqq\; \tfrac{1}{48} \Big( p_2 \;-\; \big( \tfrac{1}{2} p_1\big)^2 \Big) \;\; \in H^8\big( -, \mathbb{Q}\big) \,.$

In general this is a cohomology class in ordinary cohomology with 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 (1) controls certain quantum anomaly cancellation in M-theory and type IIA string theory (Vafa-Witten 95, Duff-Liu-Minasian 95 (3.10) with (3.14)). Since it was first obtain as a 1-loop-contribution in perturbative quantum supergravity, it is often known as the one-loop anomaly term or the one-loop anomaly polynomial in M-theory/type IIA string theory.

Properties

Inflow to M5-brane anomalies

Consider an 11-dimensional 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

(2)$I^{M5} \;\coloneqq\; I^{M5}_{\psi} + I_{C} \;\in\; H^8(F \times Q_{M5},\mathbb{Z})$

where

1. the first summand is the class of the chiral anomaly of chiral fermions on $Q_{M5}$ (Witten 96, (5.1)),

2. the second term the class of the quantum anomaly of a self-dual higher gauge field (Witten 96, (5.4))

Moreover, there is the restriction of the $I_8$-term (1) 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)

(3)$I_8\vert_{M5} \;\coloneqq\; I_8 \big( T_{Q_{M5}} X \big) \;\in\; H^8(F \times Q_{M5},\mathbb{Z}) \,.$

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

(4)$I^{M5} \;+\; I_8\vert_{M5} \;=\; \tfrac{1}{24} p_2(N_{Q_{M5}})$

This result used to be “somewhat puzzling” (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” (Witten 96, p. 37).

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

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

1. 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 (5) below (Monnier 13, around (3.11));

2. for $G^{M5}_4$ the “restriction” of the class of the supergravity C-field to $Q_{M5}$, the term $I^{M5}_{C}$ in (2) should have a further summand $-\tfrac{1}{2}\big( G_4^{M5} \big)^2$ (Monnier 13, around (3.7), using Monnier 14b, (2.13))

3. for 11d spacetime a 4-sphere-fiber bundle,

$\array{ S^4 &\longrightarrow& X^{(11)} \\ && \big\downarrow^{\mathrlap{\pi}} \\ && X^{(6)} }$

as befits the near horizon geometry of a black M5-brane, the supergravity C-field should be taken to be of the form (Monnier 13, (3.12))

$G_4 =\coloneqq \tfrac{1}{2}\chi + \pi^\ast(G^{M5}_4)$

with $\tfrac{1}{2}\chi$ the degree-4 Euler class, whose integral over the 4-sphere fiber is unity (this Prop.), reflecting the presence of a single M5.

By this proposal, the anomaly inflow from the bulk would not be just $I_8$, as in (3) but would be all of the following fiber integration

(5)$\array{ \pi_\ast \Big( - \tfrac{1}{6} G_4 G_4 G_4 + G_4 I_8 \Big) & = - \tfrac{1}{24} p_2 + \tfrac{1}{2}G^{M5}_4 + I_8 }$

Here we used this Prop to find that

$\pi_\ast\big( \chi^3 \big) \;=\; 2 p_2$

which would cancel against the first term $\tfrac{1}{24} p_2$ in (5). Hence with this proposal, the remaining M5-brane anomaly (4) would be canceled.

References

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

For further discussion see

• part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020),

part II: Twisted $String$ and $String^c$ structures, J. Australian Math. Soc. 90 (2011), 93-108 (arXiv:1007.5419);

part III: Twisted higher structures, Int. J. Geom. Meth. Mod. Phys. 8 (2011), 1097-1116 (arXiv:1008.1755)

In relation to the quantum anomaly of the M5-brane:

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

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

that this term disappears (cancels) when properly taking into account the singularity of the supergravity C-field at the locus of the black M5-brane.

A more transparent version of this argument was offered in

• Samuel Monnier, Global gravitational anomaly cancellation for five-branes, Advances in Theoretical and Mathematical Physics, Volume 19 (2015) 3 (arXiv:1310.2250)

based on a refined discussion of the quantum anomaly of the self-dual higher gauge field on the M5-brane in

• Samuel Monnier, The anomaly line bundle of the self-dual field theory, Comm. Math. Phys. 325 (2014) 41-72 (arXiv:1109.2904)

• Samuel Monnier, The global gravitational anomaly of the self-dual field theory, Comm. Math. Phys. 325 (2014) 73-104 (arXiv:1110.4639, pdf slides)

• Ibrahima Bah, Federico Bonetti, Ruben Minasian, Emily Nardoni, Class $\mathcal{S}$ Anomalies from M-theory Inflow (arXiv:1812.04016)