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I8

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Cohomology

cohomology

Special and general types

Special notions

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Extra structure

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String theory

Contents

Idea

In the string theory-literature “I 8I_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 8148(p 2(12p 1) 2)H 8(,). 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)Spin(7)- or Sp(1)Sp(2)Sp(1)\cdot Sp(2)-structure

If X 8X^{8} has

or

then

12(p 214(p 1) 2)=χ 8 \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=124χ 8. 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 8I_8 appear as the higher curvature correction at order 6\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 4G_4 and its dual G 7G_7 is (Souères-Tsimpis 17, (4.3))

dG 7()=12G 4()G 4()+ 6I 8(R), 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()=G 4+ 6G 4 (1)+ G_4(\ell) \;=\; G_4 + \ell^6 G_4^{(1)} + \cdots
G 7()=G 7+ 6G 7 (1)+ G_7(\ell) \;=\; G_7 + \ell^6 G_7^{(1)} + \cdots

(Souères-Tsimpis 17, (4.4))

Beware that this is not the lowest order higher curvature correction: there is already one at 𝒪( 3)\mathcal{O}(\ell^3), given by 3G 412p 1(R)\ell^3 G_4 \wedge \tfrac{1}{2}p_1(R) (Souères-Tsimpis 17, Section 3.2). Hence the full correction at 𝒪( 6)\mathcal{O}(\ell^6) should be the further modification of (2) to

(2)dG 7()=12G 4()(G 4()2 312p 1(R))+ 6I 8(R), 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)X^{(11)} and a 2-parameter family of 6-dimensional submanifolds Q M5X (11)Q_{M5} \hookrightarrow X^{(11)}. When regarded as a family of worldvolumes of an M5-brane, the family of normal bundles N XQ M5N_X Q_{M5} of this inclusion carries a characteristic class

(3)I M5I ψ M5+I CH 8(F×Q M5,) 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 M5Q_{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 8I_8-term (1) to Q M5Q_{M5}, hence to the tangent bundle of X 11X^{11} to Q M5Q_{M5} (the “anomaly inflow” from the bulk spacetime to the M5-brane)

(4)I 8| M5I 8(T Q M5X)H 8(F×Q M5,). 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 M5Q_{M5} is proportional to the second Pontryagin class of the normal bundle

(5)I M5+I 8| M5=124p 2(N Q M5) I^{M5} \;+\; I_8\vert_{M5} \;=\; \tfrac{1}{24} p_2(N_{Q_{M5}})

(Witten 96 (5.7))

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)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 8I_8 itself, but includes also a contribution from the class G 4G_4 of the supergravity C-field, as per (6) below (Monnier 13, around (3.11));

  2. for G 4 M5G^{M5}_4 the “restriction” of the class of the supergravity C-field to Q M5Q_{M5}, the term I C M5I^{M5}_{C} in (3) should have a further summand 12(G 4 M5) 2-\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,

    S 4 X (11) π X (6) \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=12χ+π *(G 4 M5) G_4 =\coloneqq \tfrac{1}{2}\chi + \pi^\ast(G^{M5}_4)

    with 12χ\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 (also Bah-Bonetti-Minasian-Nardoni 18 (5), BBMN 19 (2.9) and appendix A.4, A.5), the anomaly inflow from the bulk would not be just I 8I_8, as in (4) but would be all of the following fiber integration

(6)π *(16G 4G 4G 4+G 4I 8) =124p 2+12(G 4 M5) 2+I 8 \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

π *(χ 3)=2p 2 \pi_\ast\big( \chi^3 \big) \;=\; 2 p_2

which would cancel against the first term 124p 2\tfrac{1}{24} p_2 in (6). Hence with this proposal, the remaining M5-brane anomaly (5) would be canceled.


References

General

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

For further discussion see

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 124p 2\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)

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

  • Ibrahima Bah, Federico Bonetti, Ruben Minasian, Emily Nardoni, Class 𝒮\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 November 11, 2019 at 04:58:53. See the history of this page for a list of all contributions to it.