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shifted C-field flux quantization

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Context

String theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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 4G_4-flux

For the magnetic G 4G_4-flux, the shifted flux quantization says that the real cohomology class of the flux density (field strength) differential 4-form G 4Ω 4(X)G_4 \in \Omega^4(X) on spacetime XX 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)G˜ 4G 4+14p 1( TX)Ω 4(X) \widetilde G_4 \;\coloneqq\; G_4 + \tfrac{1}{4}p_1(\nabla_{T X}) \;\in\; \Omega^4(X)

(for TX\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 G˜ 4\widetilde G_4 represents an integral cohomology-class:

[G˜ 4]H 4(X,)H 4(X,)H 4(X,). [\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 4G_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 7G_7-flux

In the presence of non-vanishing C-field flux G 4G_4, the electric flux density of M2-branes is not G 7G 4G_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)G˜ 7G 7+12C 3G 4, \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:

dG˜ 7=0. d \widetilde G_7 \;=\; 0 \,.

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

  1. In general G 4=0G_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 4G_4 happens to be intregrally quantizaed (if 14p 1\tfrac{1}{4}p_1 is integral) the appearance of a globally defined C-field potential C 3C_3 in (2),means that the total flux actually does vanish after all.


Charge-quantized G˜ 7\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)G˜ 7h *G 7+12H 3h *G˜ 4 \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

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

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

  2. G˜ 4h *G 4+14h *p 1()\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. dH 3=h *G 414h *p 1()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 7G_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

The suggestion originates in

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

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

Last revised on November 13, 2019 at 15:03:46. See the history of this page for a list of all contributions to it.