Contents

cohomology

# 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

### For $Spin(7)$- or $Sp(1)\cdot Sp(2)$-structure

If $X^{8}$ has

or

then

$\tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2 \big) \;=\; \chi_8$

is the Euler class (see this Prop. and this Prop., respectively), hence then

$I_8 = \tfrac{1}{24} \chi_8 \,.$

### As a higher curvature correction to 11d Supergravity

A systematic analysis of the possible supersymmetric higher curvature corrections of D=11 supergravity makes the $I_8$ appear as the higher curvature correction at order $\ell^6$, where $\ell$ is the Planck length in 11d (Souères-Tsimpis 17, Section 4).

At this order, the equation of motion for the supergravity C-field flux $G_4$ and its dual $G_7$ is (Souères-Tsimpis 17, (4.3))

$d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge G_4(\ell) + \ell^6 I_8(R) \,,$

where the flux forms themselves appear in their higher order corrected form as power series in the Planck length

$G_4(\ell) \;=\; G_4 + \ell^6 G_4^{(1)} + \cdots$
$G_7(\ell) \;=\; G_7 + \ell^6 G_7^{(1)} + \cdots$

Beware that this is not the lowest order higher curvature correction: there is already one at $\mathcal{O}(\ell^3)$, given by $\ell^3 G_4 \wedge \tfrac{1}{2}p_1(R)$ (Souères-Tsimpis 17, Section 3.2). Hence the full correction at $\mathcal{O}(\ell^6)$ should be the further modification of (2) to

(2)$d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge \big( G_4(\ell) - 2\ell^3 \tfrac{1}{2} p_1(R) \big) + \ell^6 I_8(R) \,,$

### 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

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

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)

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

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

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

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, see also Bah-Bonetti-Minasian-Nardoni 18 (5), BBMN 19 (2.9) and appendix A.4, A.5. By this proposal, the anomaly inflow from the bulk would not be just $I_8$, as in (4) but would be all of the following fiber integration

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

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 (6). Hence with this proposal, the remaining M5-brane anomaly (5) would be canceled – except for a remaining term $\tfrac{1}{2}(G^{M5}_4)^2$ which is ignored by fiat.

## References

### General

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)

### From higher curvature corrections to 11d Supergravity

Derivation from classification of higher curvature corrections to D=11 supergravity:

• Bertrand Souères, Dimitrios Tsimpis, Section 4 of: The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity, Phys. Rev. D 95, 026013 (2017) (arXiv:1612.02021)

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

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.

This formulation via an anomaly 12-form is (re-)derived also in

• Ibrahima Bah, Federico Bonetti, Ruben Minasian, Emily Nardoni, Class $\mathcal{S}$ Anomalies from M-theory Inflow, Phys. Rev. D 99, 086020 (2019) (arXiv:1812.04016)

• Ibrahima Bah, Federico Bonetti, Ruben Minasian, Emily Nardoni, Anomaly Inflow for M5-branes on Punctured Riemann Surfaces (arXiv:1904.07250)

Last revised on January 20, 2020 at 09:17:54. See the history of this page for a list of all contributions to it.