Contents

cohomology

# Contents

## Idea

In string theory/M-theory, the shifted C-field flux quantization condition is a charge quantization-condition on the supergravity C-field expected in M-theory.

## For the magnetic $G_4$-flux

For the magnetic $G_4$-flux, the shifted flux quantization says that the real cohomology class of the flux density (field strength) differential 4-form $G_4 \in \Omega^4(X)$ on spacetime $X$ becomes integral after shifted by one quarter of the first Pontryagin class, hence the condition that with the shifted 4-flux density defined as

(1)$\widetilde G_4 \;\coloneqq\; G_4 + \tfrac{1}{4}p_1(\nabla_{T X}) \;\in\; \Omega^4(X)$

(for $\nabla_{T X}$ any affine connection on spacetime, in particular the Levi-Civita connection) we have (using the de Rham theorem to translate from de Rham cohomology to real cohomology) that $\widetilde G_4$ represents an integral cohomology-class:

$[\widetilde G_4] \;\in\; H^4(X, \mathbb{Z}) \overset{H^4(X, \mathbb{Z}\hookrightarrow \mathbb{R})}{\longrightarrow} H^4(X, \mathbb{R}) \,.$

This condition was originally argued for in (Witten 96a, Witten 96b) as a sufficient condition for ensuring that the prequantum line bundle for the 7d Chern-Simons theory on an M5-brane worldvolume is divisible by 2.

Proposals for encoding this condition by a Wu class-shifted variant of stable ordinary differential cohomology were considered in Hopkins-Singer 02, Diaconescu-Freed-Moore 03, FSS 12.

It turns out that the shifted flux quantization condition on the C-field is naturally implied (FSS1 19b, Prop. 4.12) by the requirement that $G_4$ is the differential form datum underlying, via Sullivan's theorem, a cocycle in unstable J- twisted Cohomotopy in degree 4 (Hypothesis H).

## For the electric $G_7$-flux

In the presence of non-vanishing C-field flux $G_4$, the electric flux density of M2-branes is not $G_7 \coloneqq \star G_4$ alone, but receives corrections, first due to the quadratic C-field self-interaction in D=11 supergravity, but then also due to the shifted C-field flux quantization expected in M-theory:

The 11d supergravity literature states the corrected 7-flux to be the following combination, also known as the Page charge (due to Page 83 (8), Duff-Stelle 91 (43), reviewed e.g. in BLMP 13, p. 21):

(2)$\widetilde G'_7 \;\coloneqq\; G_7 + \tfrac{1}{2} C_3 \wedge G_4 \,,$

where the second term subtracts the electric flux induced by the self-intersection of the field, and also ensures that the full expression is a closed differential form if the naive 11d supergravity equations of motion hold:

$d \widetilde G_7 \;=\; 0 \,.$

But in fact (2) does not quite make general sense, for two reasons:

1. In general $G_4 = 0$ is not an admissible condition and is not the actual vanshing of the C-field, due to the shifted C-field flux quantization.

2. Even if $G_4$ happens to be intregrally quantizaed (if $\tfrac{1}{4}p_1$ is integral) the appearance of a globally defined C-field potential $C_3$ in (2),means that the total flux actually does vanish after all.

Charge-quantized $\widetilde G_7$-flux with shifted C-field flux quantization (FSS 19b, Prop. 4.3, FSS 19c, Section 4)

Both of these issues are solved if the C-field is taken to be charge quantized in J-twisted Cohomotopy (Hypothesis H). This gives the corrected formula

(3)$\widetilde G_7 \;\coloneqq\; h^\ast G_7 + \tfrac{1}{2} H_3 \wedge h^\ast \widetilde G_4$

where

1. the expression lives on the homotopy pullback of the Sp(2)-parametrized quaternionic Hopf fibration

$h \coloneqq h_{\mathbb{H}}\sslash Sp(2)$

to spacetime, along the twisted Cohomotopy-cocycle that represents the C-field under Hypothesis H;

2. $\widetilde G_4 \coloneqq h^\ast G_4 + \tfrac{1}{4}h^\ast p_1(\nabla)$ is the integral shifted C-field pulled back to that 3-spherical fibration over spacetime;

3. $d H_3 = h^\ast G_4 - \tfrac{1}{4}h^\ast p_1(\nabla)$ trivializes not the C-field itself, but its pullback, and not absolutely but relative to the background charge implied by shifted C-field flux quantization.

With the corrected 7-flux in twisted Cohomotopy it becomes true that

1. the integral of $G_7$ around the 7-sphere linking a black M2-brane is always integer (FSS 19c, Theorem 4.6);

2. this integer satisfies the C-field tadpole cancellation condition (FSS 19b, Section 4.6).

## References

### General

The suggestion originates in

Proposals to model the condition by a Wu class-shifted variant of ordinary differential cohomology include

Suggestion that an actual E8-principal bundle on 11d spacetime plays a role here:

The observation that the condition is implied by C-field charge quantization in J-twisted Cohomotopy (Hypothesis H) is due to

Discussion of the Page charge in relation to the Myers effect in M-theory for M2-branes polarizing into M5-branes of fuzzy 3-sphere-shape:

### Relation to Freed-Witten anomaly

On relating the Freed-Witten anomaly to the shifted C-field flux quantization:

On D4-branes:

On D6-branes:

Last revised on March 8, 2021 at 13:36:58. See the history of this page for a list of all contributions to it.