group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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:
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.
If $X^{8}$ has
or
then
is the Euler class (see this Prop. and this Prop., respectively), hence then
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))
where the flux forms themselves appear in their higher order corrected form as power series in the Planck length
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
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
where
the first summand is the class of the chiral anomaly of chiral fermions on $Q_{M5}$ (Witten 96, (5.1)),
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)
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
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
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 (6) below (Monnier 13, around (3.11));
for $G^{M5}_4$ the “restriction” of the class of the supergravity C-field to $Q_{M5}$, the term $I^{M5}_{C}$ in (3) should have a further summand $-\tfrac{1}{2}\big( G_4^{M5} \big)^2$ (Monnier 13, around (3.7), using Monnier 14b, (2.13))
for 11d spacetime a 4-sphere-fiber bundle,
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))
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 (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 $I_8$, as in (4) but would be all of the following fiber integration
Here we used this Prop to find that
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.
The term showed in string theory/M-theory anomaly cancellation in
Cumrun Vafa, Edward Witten, A One-Loop Test Of String Duality, Nucl.Phys.B447:261-270, 1995 (arXiv:hep-th/9505053)
Mike Duff, James Liu, Ruben Minasian, Eleven Dimensional Origin of String/String Duality: A One Loop Test, Nucl.Phys. B452 (1995) 261-282 (arXiv:hep-th/9506126)
Edward Witten, Five-Brane Effective Action In M-Theory, J.Geom.Phys.22:103-133, 1997 (arXiv:hep-th/9610234)
For further discussion see
Hisham Sati, Geometric and topological structures related to M-branes ,
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)
Derivation from classification of higher curvature corrections to D=11 supergravity:
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
Dan Freed, Jeff Harvey, Ruben Minasian, Greg Moore, Gravitational Anomaly Cancellation for M-Theory Fivebranes, Adv.Theor.Math.Phys.2:601-618, 1998 (arXiv:hep-th/9803205)
Jeff Harvey, Ruben Minasian, Greg Moore, Non-abelian Tensor-multiplet Anomalies, JHEP9809:004, 1998 (arXiv:hep-th/9808060)
Adel Bilal, Steffen Metzger, Anomaly cancellation in M-theory: a critical review, Nucl.Phys. B675 (2003) 416-446 (arXiv:hep-th/0307152)
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
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 $\mathcal{S}$ 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.