Schreiber Generalized cohomology of M2/M5-branes

---

$\,$

Urs Schreiber

Generalized cohomology of M2/M5-branes

talk at

Higher Structures in String Theory and Quantum Field Theory

ESI Vienna, Dec 7 - 11, 2015

Abstract. While it is well-known that the charges of F1/Dp-branes in type II string theory need to be refined from de Rham cohomology to certain twisted generalized differential cohomology theories, it is an open problem to determine the generalized cohomology theory for M2-brane/M5-branes in 11 dimensions. I discuss how a careful re-analysis of the old brane scan (arXiv:1308.5264 , arXiv:1506.07557, joint with Fiorenza and Sati) shows that rationally and unstably, the M2/M5 brane charge is in degree-4 cohomotopy. While this does not integrate to the generalized cohomology theory called stable cohomotopy, it does integrate to $G$-equivariant stable cohomotopy, for $G$ a non-cyclic finite group of ADE type. On general grounds, such an equivariant cohomology theory needs to be evaluated on manifolds with ADE orbifold singularities, and picks up contributions from the orbifold fixed points. Both of these statements are key in the hypothesized but open problem of gauge enhancement in M/F-theory.

$\,$

Acknowledgement. The following note profited from discussion with David Barnes, Charles Rezk, David Roberts, Hisham Sati. An adapted version appears also at “Equivariant generalized cohomology of M2/M5-branes”, talk at MPI Bonn, 15 Jan 2016.

$\,$

Motivation and Introduction

As reviewed at the end of last week’s lecture

• Prequantum field theory and the Green-Schwarz WZW terms, lecture series at Higher Structures in String Theory and Quantum Field Theory, ESI Vienna, Nov 30 -Dec 4,

what is known for sure about M-theory is all encoded in the prequantum Green-Schwarz-type sigma-models describing the propagation of M2-branes and M5-branes on super-spacetimes. In particular:

1. The BPS charges of such spacetimes – which are traditionally argued to detect the properties of the full quantum regime of the elusive theory – are identified with the charges of the Noether currents of these sigma-models.

   In fact the Heisenberg Lie n-algebra (Fiorenza-Rogers-Schreiber 13) of these prequantum field theories is The M-Theory BPS charge super Lie 6-algebra, whose 0-truncation is the M-theory super Lie algebra “$\{Q_\alpha, Q_\beta\} = (C \Gamma^a_{\alpha \beta}) P_a + (\Gamma^{a_1 a_2}_{\alpha \beta}) Z_{a_1 a_2} + \Gamma^{a_1 a_2 \cdots a_5}_{\alpha \beta} Z_{a_1 a_2 \cdots a_5}$” (Sati-Schreiber 15, Schreiber-Khavkine 16).

2. The membrane instanton contributions – which are argued to detect further non-perturbative effects – are the volume holonomy, i.e. the magnetic flux, of the complexified higher WZW term of the M2-brane over supersymmetric cycles (Schreiber 15).

3. The definite globalization of the M2-WZW term over a superspacetime implies the equations of motion of 11-dimensional supergravity (hence in particular the Hodge duality between the rational M-brane charges) together with the classical anomaly cancellation that makes the M2-sigma model be globally well defined on this target (Schreiber 15).

Therefore, for making progress with the open question of formulating M-theory proper, a key issue is a precise understanding of the cohomological nature of M-brane charges (Sati 10).

In most of the existing literature, these charges had been regarded simply in de Rham cohomology. But it is well known (see (Distler-Freed-Moore 09) for the state of the art) that in the small coupling limit where the perturbation theory of type II string theory applies, the brane charges are not just in (twisted, self-dual) de Rham cohomology, but instead in a (twisted, self-dual) equivariant generalized cohomology theory, namely in real ($\mathbb{Z}/2$-equivariant) topological K-theory, of which de Rham cohomology is only the rational shadow under the Chern character map. This makes a crucial difference (Maldacena-Moore-Seiberg 01, Evslin-Sati 06): the differentials in the Atiyah-Hirzebruch spectral sequence for K-theory describe how de Rham cohomology classes receive corrections as they are lifted to K-theory: some charges may disappear, others may appear.

But the lift of this situation to M-branes had been missing. The open question is: Which equivariant generalized cohomology theory $E_G$ do M-brane charges take values in?

The answer needs to satisfy (at least) the following two consistency conditions:

1. the rationalization $E_G(X_{11})\otimes \mathbb{Q}$ of the generalized cohomology classes has to reproduce the correct rational brane charges, we analyze these below in The rational cohomology of M2/M5-brane charges;

2. the $G$-equivariant Atiyah-Hirzebruch spectral sequence for $E_G(X_{11})$ along an M-theory circle fibration

   $$\array{ S^1 &\to& X_{11} \\ && \downarrow \\ && X_{10} }$$

   needs to be a suitable higher order correction to the AHSS for topological KR-theory $KU_{\mathbb{Z}/2}(X_{10})$.

Here we are concerned with the first item. By the analysis in (Sati-Varghese 03, section 4), at the rational level the second item is implied by the first, see the conclusion below.

Super $p$-brane sigma-models and super $L_\infty$-cohomology

All the Green-Schwarz-type sigma models describing super p-branes propagating on target superspacetimes are sigma-models that are given by definite globalizations of higher WZW terms, where these higher WZW terms are classified by the Spin(d-1,1)-invariant Lie n-algebra cocycles of the bouquet of super L-∞ algebras that arise by iterated higher extensions from the superpoints (Fiorenza-Sati-Schreiber 13, Schreiber 15).

Here for $\mathfrak{g}$ an super-L-∞ algebra of finite type, it is formally dual to its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$, which is a dg-algebra structure on the Grassmann algebra over the degreewise linear dual of the $\mathbb{N}$-graded super vector space underlying $\mathfrak{g}$ (so in total everything is $(\mathbb{Z},\mathbb{Z}_2)$-bigraded, see at signs in supergeometry).

For $\mathbf{B}^{p+1} \mathbb{R}$ denoting the line Lie (p+2)-algebra (whose Chevalley-Eilenberg algebra is generated in degree $(p+2)$ with vanishing differential) then an $L_\infty$-algebra homomorphism

$$\mathfrak{g} \stackrel{\mu}{\longrightarrow} \mathbf{B}^{p+1}\mathbb{R}$$

is equivalently a $(p+2)$-cocycle in L-infinity algebra cohomology. Its homotopy fiber (with respect to the natural homotopy theory of L-∞ algebras) is the L-∞ algebra extension $\hat \mathfrak{g}$ that it classifies

$$\array{ \hat \mathfrak{g} \\ \downarrow^{\mathrlap{hofib(\mu)}} \\ \mathfrak{g} &\stackrel{\mu}{\longrightarrow}& \mathbf{B}^{p+1}\mathbb{R} } \,.$$

This $L_\infty$-extension will in general carry new cocycles, so that towers and bouquets of higher extensions emanate from any one super $L_\infty$-algebra

$$\array{ \widehat{\hat \mathfrak{g}} \\ \downarrow^{\mathrlap{hofib(\mu_2)}} \\ \hat \mathfrak{g} &\stackrel{\mu_2}{\longrightarrow}& \mathbf{B}^{p_2 + 1} \mathbb{R} \\ \downarrow^{\mathrlap{hofib(\mu_1)}} \\ \mathfrak{g} &\stackrel{\mu_1}{\longrightarrow}& \mathbf{B}^{p_1+1}\mathbb{R} } \,.$$

The super Minkowski spacetimes regarded as supersymmetry super Lie algebras give examples of super L-∞ algebras, and the bouquets of higher extensions induced by them via $Spin(d-1,1)$-invariant cocycles gives the brane bouquet of string/M-theory: each such $(p+2)$-cocycle is the curvature of the higher WZW term of the Green-Schwarz sigma model of the corresponding $p$-brane, hence is the Chern character of the corresponding rational cohomology theory in which this brane’s charges take values. This fact allows an accurate mathematical analysis of the rational brane charges.

We will have to assume a basic familarity with these matters in the following. They may be found in the accompanying lecture

• Prequantum field theory and the Green-Schwarz WZW terms, lecture series at Higher Structures in String Theory and Quantum Field Theory, ESI Program, Vienna, in the week Nov 30 - Dec 4

for which lecture notes are at

• Structure Theory for Higher WZW Terms

• super Cartan geometry.

Warmup: The generalized cohomology of F1/Dp-brane charges

For example the super Minkowski spacetime $\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}$ locally modeling super spacetimes in 10d type IIA supergravity carries super $L_\infty$-extensions of the following form (FSS 13):

$$\array{ \mathbb{R}^{10,1\vert \mathbf{32}} && \widehat{\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}} &\stackrel{\underset{p=0,2,4,6}{\oplus} \mu_{D p}}{\longrightarrow}& \underset{p = 0,2,4,6}{\oplus} \mathbf{B}^{p+1}\mathbb{R} \\ & {}_{\mathllap{hofib(\mu_{D0})}}\searrow & \downarrow^{\mathrlap{hofib(\mu_{F1})}} \\ && \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} &\stackrel{\mu_{F1}}{\longrightarrow}& \mathbf{B}^2 \mathbb{R} } \,.$$

Here the homotopy fiber $\widehat{\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}}$ is the $\mathbf{B}\mathbb{R}$-principal ∞-bundle classified by the WZW term $\mu_{F1}$ for the F1-brane (the type IIA superstring). By (Nikolaus-Schreiber-Stevenson 12), asking whether the cocycles $\mu_{D p}$ for the D-branes are $\mathbf{B}\mathbb{R}$-equivariant and descend as twisted cocycles down to super-Minkowski spacetime is equivalent to asking whether there is a homotopy fiber sequence $\underset{p = 0,2,4,6}{\oplus} \mathbf{B}^{p+1}\mathbb{R} \to something \to \mathbf{B}^2\mathbb{R}$ and a homotopy-commuting diagram of the form

$$\array{ \widehat{\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}} && \stackrel{\underset{p=0,2,4,6}{\oplus} \mu_{D p}}{\longrightarrow} && \underset{p = 0,2,4,6}{\oplus} \mathbf{B}^{2p+1}\mathbb{R} \\ \downarrow^{\mathrlap{hofib(\mu_{F1})}} && && \downarrow^{\mathrlap{hofib(\phi)}} \\ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} && \stackrel{}{\longrightarrow} && something \\ & {}_{\mathllap{\mu_{F 1}}}\searrow && \swarrow_{\mathrlap{\phi}} \\ && \mathbf{B}^2 \mathbb{R} } \,.$$

Inspection shows that this indeed exists: write $\left(\underset{p = 0,2,4,6}{\oplus}\mathbf{B}^{2p+1}\mathbb{R}\right)/\mathbf{B} \mathbb{R}$ for the L-∞ algebra whose Chevalley-Eilenberg algebra has generators $\omega_2, \omega_4, \omega_6, \omega_8$ and $h_3$ in the indicated degrees, with non-trivial differential given by $d(\omega_{2(k+1)}) = h_3 \wedge \omega_{2k}$:

$$CE \left( \left(\underset{p = 0,2,4,6}{\oplus}\mathbf{B}^{2p+1}\mathbb{R}\right)/\mathbf{B} \mathbb{R} \right) \coloneqq \left\{ \{ \omega_{p+2}, h_3\}_{p = 0,2,4,6}, d \omega_{2(k+1)} = h_3 \wedge \omega_{2k} \right\} \,.$$

Moreover write $\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}_{res}$ for the super $L_\infty$-algebra whose Chevalley-Eilenberg algebra is that of $\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}$ with generators $f_2$ and $h_3$ added, subject to $d f_2 = \mu_{F1} + h_3$. This is a resolution

$$\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}_{res} \stackrel{\simeq}{\longrightarrow} \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}$$

of type IIA super-Minkowski spacetime which serves to represent morphisms in the homotopy theory for super L-infinity algebras in the following. Because with this, the above descent problem indeed has a solution as follows:

$$\array{ \left\{ {{d e^a = \overline{\psi}\Gamma^a \wedge \psi } \atop {d \psi^\alpha = 0}} \atop { d f_2 = \mu_{F1}} \right\} && \widehat{ \mathbb{R}^{ 9,1\vert \mathbf{16} + \overline{\mathbf{16}} } } && \stackrel{\underset{p=0,2,4,6}{\oplus} \mu_{D p}}{\longrightarrow} && \underset{p = 0,2,4,6}{\oplus} \mathbf{B}^{2p+1}\mathbb{R} && \left\{ d \omega_{2 k} = 0 \right\} \\ && \downarrow^{\mathrlap{hofib(\mu_{F1})}} && && \downarrow \\ \left\{ {{d e^a = \overline{\psi}\Gamma^a \wedge \psi } \atop {d \psi^\alpha = 0}} \atop { d f_2 = \mu_{F1} + h_3 } \right\} & & \mathbb{R}_{res}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} && \stackrel{ \left( \omega_{p+2} \mapsto \mu_{D p} \right) }{\longrightarrow} && \left( \underset{p = 0,2,4,6}{\oplus} \mathbf{B}^{2p+1}\mathbb{R} \right)/\mathbf{B} \mathbb{R} && \left\{ d\omega_{2(k+1)} = h_3\wedge \omega_{2k} \right\} \\ && & {}_{\mathllap{\mu_{F 1}}}\searrow && \swarrow \\ && && \mathbf{B}^2 \mathbb{R} \\ && && \left\{ d h_3 = 0 \right\} } \,.$$

This says that the type IIA F1-brane and D-brane cocycles with $\mathbb{R}$-coefficients do descent to super-Minkowski spacetime as one single cocycle with coefficients in the homotopy quotient $\left( \underset{p = 0,2,4,6}{\oplus} \mathbf{B}^{2p+1}\mathbb{R} \right)/\mathbf{B} \mathbb{R}$.

But these rational coefficients are precisely the rational image of twisted topological K-theory.

Accordingly, the Lie integration of this rational situation to twisted K-theory, and its globalization over a 10-dimensional IIA super spacetime $X_{10}$, yields a diagram of parameterized spectra in smooth infinity-stacks of the form

$$\array{ X_{10} && \stackrel{RR_{/B}}{\longrightarrow} && KU / \mathbf{B} U(1) \\ & {}_{\mathllap{B}}\searrow && \swarrow \\ && \mathbf{B}^2 U(1) }$$

According to (Sati-Schreiber-Stasheff 09, Nikolaus-Schreiber-Stevenson 12) here the morphism denoted $B$ represents the Kalb-Ramond B-field under which the F1-brane is charged and the morphism denoted $RR_{/B}$ represents the RR-field under which the D-branes are charged.

This is how one may re-discover the familiar cohomological nature of the F1/Dp-brane charges in type II string theory from an analysis of the super $L_\infty$-cohomology embodied in the brane bouquet-refinement of the old brane scan.

The rational cohomology of M2/M5-brane charges

We now consider the analogue of this re-derivation, but up in 11-dimensions, where it provides a previously missing derivation of the rational cohomology of M-brane charges.

For the 11-dimensional super Minkowski spacetime on which 11-dimensional supergravity is locally modeled (via super Cartan geometry) the iterative extension of $L_\infty$-cocycles in the brane bouquet looks like so (Fiorenza-Sati-Schreiber 13):

$$\array{ \widehat{\mathbb{R}^{10,1\vert \mathbf{32}}} &\stackrel{\mu_{M5}}{\longrightarrow}& \mathbf{B}^6 \mathbb{R} \\ \downarrow^{\mathrlap{hofib(\mu_{M2})}} \\ \mathbb{R}^{10,1\vert \mathbf{32}} & \stackrel{\mu_{M2}}{\longrightarrow} & \mathbf{B}^3 \mathbb{R} \\ \downarrow^{\mathrlap{hofib(\mu_{D 0})}} \\ \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}} }$$

Hence the M5-brane WZW term exists on the $\mathbf{B}^2 \mathbb{R}$-principal infinity-bundle classified by the M2-brane WZW term. Again using (Nikolaus-Schreiber-Stevenson 12), we may ask if this is equivariant and descends back to 11-dimensional super-Minkowski spacetime.

And it does (Fiorenza-Sati-Schreiber 15):

write $\mathbf{B}^6 \mathbb{R}/\mathbf{B}^2 \mathbb{R}$ for the L-infinity algebra whose Chevalley-Eilenberg algebra is generated from elements $\omega_4$ and $\omega_7$, in degrees 4 and 7 as indicated, and whose differential is given by $d \omega_4 = 0$ and $d \omega_7 = \omega_4 \wedge \omega_4$. This sits in a homotopy fiber sequence of L-infinity algebras of the form

$$\mathbf{B}^6 \mathbb{R} \longrightarrow \mathbf{B}^6\mathbb{R}/\mathbf{B}^2 \mathbb{R} \longrightarrow \mathbf{B}^3 \mathbb{R} \,.$$

Notice that if we think of the Chevalley-Eilenberg algebras of these $L_\infty$-algebras as being Sullivan models in rational homotopy theory, then this homotopy fiber sequence is the rational image of the quaternionic Hopf fibration

$$S^7 \longrightarrow S^4 \to \mathbf{B}SU(2) \stackrel{\mathbf{c_2}}{\to} \mathbf{B}^3 U(1) \,.$$

Now computation shows (Fiorenza-Sati-Schreiber 15) that indeed the WZW term for the M5-brane does descend back to super-Minkowski spacetime as a cocycle with coefficients in this rational 4-sphere:

$$\array{ \left\{ { { d e^a = \overline{\psi}\wedge \Gamma^a \wedge \psi } \atop { d \psi^\alpha = 0 } } \atop d h_3 = - \mu_{M2} \right\} && \widehat{\mathbb{R}^{10,1\vert \mathbf{32}}} && \stackrel{h_3 \wedge \mu_4 + \frac{1}{15}\mu_{M5} }{\longrightarrow} && \mathbf{B}^6 \mathbb{R} && \left\{ d \omega_7 = 0 \right\} \\ && \downarrow^{\mathrlap{hofib(\mu_{M2})}} && && \downarrow \\ \left\{ { { d e^a = \overline{\psi}\wedge \Gamma^a \wedge \psi } \atop { d \psi^\alpha = 0 } } \atop d h_3 = g_4 - \mu_{M2} \right\} && \mathbb{R}_{res}^{10,1\vert\mathbf{32}} && \stackrel{h_3 \wedge (g_4 + \mu_4) + \frac{1}{15}\mu_7 }{\longrightarrow} && \mathbf{B}^6 \mathbb{R}/\mathbf{B}^2 \mathbb{R} && \left\{ {d g_4 = 0} \atop {d g_7 = g_4 \wedge g_4} \right\} \\ && & {}_{\mathllap{\mu_{M2}}}\searrow && \swarrow \\ && && \mathbf{B}^3 \mathbb{R} \\ && && \left\{ d g_4 = 0\right\} }$$

Hence we read off from this computation that, rationally, M2-brane charge is in degree-4 ordinary cohomology and it twists M5-brane charge which is, rationally, in unstable degree-4 cohomotopy. This confirms a statement made earlier in (Sati 10, section 6.3, Sati 13, section 2.5).

An unstable Lie integration of this situation, in direct analogy to the above situation for twisted K-theory, would be given by maps into the quaternionic Hopf fibration

$$\array{ && && S^7 \\ && && \downarrow \\ X && \stackrel{G_7_{/G_4}}{\longrightarrow} && S^4 \\ & {}_{\mathllap{G_4}} \searrow && \swarrow & \downarrow \\ && \mathbf{B}^3 U(1) &\stackrel{\mathbf{c2}}{\longleftarrow}& \mathbf{B}SU(2) }$$

where the left map $G_4$ would represent the magnetic M2-brane charge and the horizontal map the $G_4$-twisted magnetic M5-brane charge. (Here we are displaying a diagram of smooth infinity-stacks, there is a further refinement of these cocycles to nonabelian differential cohomology (FSS 15)).

Notice that, unstably, the 4-sphere is just the space whose non-torsion homotopy groups (hence those that are visibe rationally) are in degrees 4 and 7

k	1	2	3	4	5	6	7
$\pi_k(S^4)$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z} \oplus \mathbb{Z}_{12}$

Hence, unstably, the 4-sphere $S^4$ may be thought of as the coefficient which is just right for detecting integral M2-brane charge and M5-brane charge. For instance the near-horizon limit of a black M2-brane is the spacetime $X_{11} = AdS_4 \times S^7$ and the degree-4 cohomotopy classes

$$[AdS_4 \times S^7, S^4] \simeq [S^7,S^4] \simeq \mathbb{Z} \oplus \mathbb{Z}_12$$

detect the integral charge of these (the M2 being the magnetic source for M5-brane charge), with the unit of charge being represented by the quaternionic Hopf fibration. Similarly the near-horizon limit of a black M5-brane is $AdS_7 \times S^4$ and again the degree-4 cohomotopy classes are

$$[AdS_7 \times S^4 , S^4] \simeq [S^4, S^4] \simeq \mathbb{Z}$$

detecting the integral charge of these branes.

But unstable cohomotopy – which may be thought of as a very nonabelian cohomology theory – is unlikely to satisfy consistency condition 2 of reproducing topological K-theory in a suitable limit, for that we need an actual “abelian” cohomology theory represented by a spectrum. This we turn to now.

Lift to ADE-equivariant stable cohomotopy theory

It follows that the first of our two consistency conditions is to be solved by finding a (possibly equivariant) generalized cohomology theory whose rational image is 4-shifted cohomotopy.

The immediate guess might be that this is 4-shifted stable cohomotopy, i.e. the generalized cohomology theory which is represented by the suspension spectrum of the 4-sphere, hence by the 4-suspended sphere spectrum $\mathbb{S}$:

$$\Sigma^\infty S^4 = \Sigma^4 \mathbb{S} \,.$$

However, this does not work: the non-torsion element in $\pi_7(S^4) = \mathbb{Z} \oplus \mathbb{Z}/12$, which is the one represented by the quaternionic Hopf fibration, becomes torsion after stabilization – since the third stable homotopy group of spheres is the cyclic group $\pi_3^S = \mathbb{Z}/24$ – and hence disappears rationally.

A natural way to evade this problem is to ask for a finite group $G$ acting on the quaternionic Hopf fibration and to consider $G$-equivariant stable cohomotopy. Since this forces all homotopies to exist $G$-equivariantly, it potentially makes some unstable non-torsion elements remain stably non-torsion.

More concretely, there is “genuine” $G$-equivariant cohomology theory, motivated as follows. The traditional suspension isomorphism

$$H^n(X,E) \simeq H^{n+k}(S^k \wedge X, E)$$

relates the integer grading of cohomology groups with $k$-fold suspensions of base spaces given by smash product with the $k$-sphere. In a context where all spaces and coefficients are equipped with $G$-action, then one may consider not just plain spheres $S^k$ but representation spheres $S^V$ given by one-point compactification of linear $G$-representations $V$. The usual spheres are subsumed by this as coming from the trivial representations: $S^k \simeq S^{\mathbb{R}^k}$. This gives rise to a generalized grading of cohomology groups not just by integers, but by linear $G$-representations – called RO(G)-grading – such that an equivariant suspension isomorphism holds

$$H^W_G(X,E) \simeq H^{W+V}_G(S^V\wedge X, E)$$

for any linear $G$-representations $W$ and $V$.

An equivariant version of the Brown representability theorem states that every RO(G)-graded equivariant cohomology theory is represented by what is called a genuine G-spectrum. Where an ordinary spectrum $E$ is a system of pointed topological spaces $E_n$ indexed by the integers, hence by the ordinary spheres $S^n$, and equipped with compatible comparison maps

$$S^{n-k}\wedge E_k \longrightarrow E_{n} \,,$$

a genuine G-spectrum is a system of pointed topological spaces $E_V$ indexed by representation spheres $S^V$ and equipped with compatible comparison maps of the form

$$S^{V-W} \wedge E_W \longrightarrow E_V \,.$$

In particular for $X$ any pointed topological G-space, there is the corresponding $G$-equivariant suspension spectrum $\Sigma^\infty_G X$ whose value on a representation sphere $S^V$ is the smash product $S^V \wedge X$.

Hence the problem that $\pi_7(\Sigma^\infty S^4)$ is torsion may potentially be fixed by finding a finite group $G$ and a $G$-action on the quaternionic Hopf fibration, $S^7_G \to S^4_G$, such that $S^7_G \simeq S^{\mathbb{R}^7_G}$ is a representation sphere for a 7-dimensional linear $G$-representation, and such that the $G$-equivariant quaternonic Hopf fibration represents a non-torsion element in

$$H^{-\mathbb{R}^7_G}(\ast, \Sigma^\infty_G S^4) = [\Sigma^\infty_G S^7, \Sigma^\infty_G S^4] \,.$$

Now, by the way the quaternionic Hopf fibration is obtained, via the Hopf construction, from the product operation on the quaternions $\mathbb{H}$, it is equivariant under the induced action of the automorphism group of the quaternions. This automorphism group is the special orthogonal group $SO(3)$, acting on the imaginary quaternions by rotation:

$$Aut_{\mathbb{R}}(\mathbb{H}) \simeq SO(3) \,.$$

Since we need equivariance under a finite group, our options are finite subgroups of $SO(3)$

$$G \hookrightarrow SO(3) \,.$$

These finite subgroups have an ADE classification. In the A-series they are the cyclic groups, sitting in the inclusion $SO(2) \hookrightarrow SO(3)$. In the D-series these are the dihedral groups which are those subgroups generated from a cyclic subgroup rotating in some plane and a reflection at that plane. Finally there are three exceptional finite subgroups: the tetrahedral group, the octahedral group and the icosahedral group

Regard both $S^7$ and $S^4$ as pointed topological G-spaces via the $SO(3)$-action induced via automorphisms of the quaternions. Write

$$\Sigma^\infty_G S^7, \Sigma^\infty_G S^4 \in G Spectra$$

for the corresponding equivariant suspension spectra.

Remark

The 4-sphere with this action is manifestly a representation sphere

$$S^4 = \mathbb{H}\mathbb{P}^1 \simeq \mathbb{H} \cup \{\infty\} \simeq S^{\mathbb{H}} \,.$$

We will write $S^{\mathbb{R}^4_G}$ for this representation sphere.

Moreover, the 7-sphere with this action is also a representation sphere, via stereographic projection

$$\begin{aligned} S^7 & =_G S(\mathbb{H} \times \mathbb{H}) \\ & \simeq_G S( \mathbb{R} \oplus (Im(\mathbb{H} \oplus \mathbb{H})) ) \\ & \simeq_G S^{Im(\mathbb{H} \oplus \mathbb{H})} \end{aligned} \,.$$

We will write $S^{\mathbb{R}^7_G}$ for this.

Recall again that if we took trivial $G$, then in the stable homotopy category

$$[\Sigma^\infty S^7, \Sigma^\infty S^4] \simeq \mathbb{Z}_{24}$$

by the above. In contrast:¹

Theorem

Let $G \hookrightarrow SO(3)$ be a non-cyclic finite subgroup. Then in $G$-equivariant homotopy theory the quaternionic Hopf fibration $S^7 \to S^4$ becomes a non-torsion group, i.e.

$$H_G^{-\mathbb{R}^7_G}(\ast, \Sigma^\infty_G S^4) \;\coloneqq\; [\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G \;\simeq\; \mathbb{Z} \oplus \cdots$$

with the quaternionic Hopf fibration, regarded as a $G$-equivariant map, representing a non-torsion element.

Proof

First use the Greenlees-May decomposition which says that for any two $G$-equivariant spectra $X,Y$ and writing $\pi_\bullet(X), \pi_\bullet(Y)$ for their equivariant homotopy groups, organized as Mackey functors $H \mapsto \pi_n^H(X)$ for all subgroups $H \subset G$, then the canonical map

$$[X,Y]_G \longrightarrow \underset{n}{\oplus} Hom_{\mathcal{M}[G]}(\pi_n(X), \pi_n(Y))$$

is rationally an isomorphism.

With this we are reduced to showing that there exists $n \in \mathbb{Z}$ and a morphism of Mackey functors of equivariant homotopy groups $\pi_n(\Sigma^\infty_G S^7) \to \pi_n(\Sigma^\infty_G S^4)$ which is not a torsion element in the abelian hom-group of Mackey functors.

To analyse this, we use the tom Dieck splitting which says that the equivariant homotopy groups of equivariant suspension spectra $\Sigma^\infty_G X$ contain a direct summand which is simply the ordinary stable homotopy groups of the naive fixed point space $X^H$:

$$\pi_n^H(\Sigma^\infty_G X) \;\simeq\; \pi_n(\Sigma^\infty (X^H)) \; \underset{{[J \subset H]} \atop {J \neq H}}{\oplus} \pi_\bullet^{W J}(\Sigma^\infty (E (W J)_+ \wedge X^J))$$

(here $W J \coloneqq (N_H J)/J$ denotes the Weyl group, the quotient group of the normalizer subgroup (of $J$ in $H$) by $J$).

Now observe that the fixed points of the $SO(3)$-action on the quaternionic Hopf fibration that we are considering is just the real Hopf fibration:

$$(p_{\mathbb{H}})^{SO(3)} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1$$

since $SO(3)$ acts transitively on the quaternionic quaternions and fixes the real quaternions. By our assumption that $G \subset SO(3)$ does not come through $SO(2) \hookrightarrow SO(3)$ it follows that this statment is still true for $G$:

$$(p_{\mathbb{H}})^{G} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1 \,.$$

But the real Hopf fibration defines a non-torsion element in $\pi_0^S \simeq \mathbb{Z}$.

In conclusion then, at $n = 1$ and $H = G$ we find that the $G$-equivariant quaternionic Hopf fibration contributes a non-torsion element in

$$Hom_{Ab}(\pi_1^G(\Sigma^\infty_G S^7), \pi_1^G(\Sigma^\infty_G S^4))$$

which appears as a non-torsion element in

$$Hom_{\mathcal{M}[G]}( \pi_1(\Sigma^\infty_G S^7), \pi_1(\Sigma^\infty_G S^4) )$$

and hence in $[\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G$.

Conclusion and outlook

In conclusion, a consistent possibility for the equivariant generalized cohomology theory in which M2/M5-brane charges take value is 4-shifted DE-equivariant stable cohomotopy in RO(G)-degree

$$\mathbb{R}^7 - \mathbb{R}^7_G \;\; \in Rep(G)$$

(in the notation of remark 1), hence the cohomology theory

$$H^{\mathbb{R}^{7}-\mathbb{R}^{7}_G }_G( -,\; \Sigma^\infty_G S^4 ) \simeq H(- , [\Sigma^\infty_G S^7, \Sigma^7 \Sigma^\infty_G S^4]) \,.$$

The ADE-equivariance (or rather: DE-equivariance) which we discovered this way from the mathematics is a pleasant surprise: the key conjecture about the elusive microscopic degrees of freedom of M-theory states (see e.g. Acharya-Gukov 04) that

1. its spacetimes are ADE orbifolds;

2. the theory is non-degenerate – in that it exhibits the nonabelian gauge enhancement – only at the ADE singularities, i.e. at the fixed points of the ADE-action.

Both these statements follow mathematically from the above analysis:

1. a $G$-equivariant cohomology theory is to be evaluated on topological G-spaces;

2. by the tom Dieck splitting principle (already used above) it picks up its contributions from the $G$-fixed points.

Now the conjectured gauge enhancement in M-theory at these fixed points (see e.g. Acharya-Gukov 04) is to be the M-theory lift of the familiar appearance of Chan-Paton gauge fields with nonabelian structure group on coincident D-branes. But mathematically the Chan-Paton gauge fields are just elements in the twisted topological K-theory in which the D-brane charges take values, as above. Hence it seems that in order to mathematically exhibit the conjectured gauge enhancement in M-theory at ADE-singularities, it is now sufficient to show that DE-equivariant stable cohomotopy on an 11-dimensional circle fibration reduces to twisted K-theory on the 10-dimensional base. But that is just the second consistency check already mentioned at the beginning.

To check this, one may run the Serre spectral sequence/Atiyah-Hirzebruch spectral sequence for $G$-equivariant stable cohomotopy theory on circle fibrations

$$\array{ S^1 &\to& X_{11} \\ && \downarrow \\ && X_{10} } \,.$$

We may check this rationally: By Greenlees-May decomposition our equivariant cohomology theory rationally decomposes as a direct sum of equivariant Eilenberg-MacLane spectra on the equivariant homotopy group Mackey functors

$$H^{\mathbb{R}^{7}-\mathbb{R}^{7}_G }_G( -,\; \Sigma^\infty_G S^4 ) \simeq_{\mathbb{Q}} \underset{n}{\prod} \Sigma^n H(\pi_n(\Sigma^\infty_G S^4))$$

and for each of the EM-summands there is a Serre-Atiyah-Hirzebruch spectral sequence (Kronholm 10, theorem 3.1)

$$E_2^{p,q} \;=\; H^p(X_{10}, H^{(\mathbb{R}^{7}-\mathbb{R}^{7}_G) + q}(S^1,\pi_{p+q}(\Sigma^\infty_G S^4))) \;\;\Rightarrow\;\; H^{\mathbb{R}^{7}-\mathbb{R}^{7}_G + \bullet}_G(X_{11} , \pi_\bullet(\Sigma^\infty_G S^4)) \,.$$

But by the above analysis the $E_2$-page here rationally reduces to the Gysin sequence analysis in (Sati-Varghese 03, section 4).

For instance for $p + q = 6 - 5$ there is the following contribution that gives the correct double dimensional reduction of the charge seen by the M5-brane to the charge seen by the D4-brane, rationally:

$$\begin{aligned} H^6(X_{10}, H^{(\mathbb{R}^{7}-\mathbb{R}^{7}_G) -5}( S^1,\pi_{1}(\Sigma^\infty_G S^4) ) ) & = H^6(X_{10}, H^{-\mathbb{R}^{7}_G + 2}( S^1, \pi_{1}(\Sigma^\infty_G S^4) ) ) \\ & \simeq H^6( X_{10}, Hom_{\mathcal{M}[G]}( \pi_2(\Sigma^1 \Sigma^\infty_G S^7),\pi_{1}(\Sigma^\infty_G S^4)) ) \\ & \simeq H^6( X_{10}, Hom_{\mathcal{M}[G]}( \pi_1(\Sigma^\infty_G S^7),\pi_{1}(\Sigma^\infty_G S^4)]_G ) \\ & \simeq H^6( X_{10}, \mathbb{Z} ) \oplus torsion \end{aligned} \,.$$

Here in the second step we unraveled the definition of cohomology with values in a Mackey for the case of the domain being a sphere, in the third step we used stability of the stable equivariant homotopy groups and in the fourth we used the proof of theorem 1.

This shows that the (self-dual) ADE-equivariant cohomotopy charges on $X_{11}$ do give the (self-dual) twisted K-theory data on $X_{10}$, rationally.

$\,$

$\,$

$\,$

$\,$

$\,$

References

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 ? 142 (arXiv:1304.6292)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018 (arXiv:1308.5264)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The WZW term of the M5-brane, J. Math. Phys. 56, 102301 (2015) (arXiv:1506.07557)

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles -- General theory, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801 (arXiv:1207.0248)

• Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)

• Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)

• Hisham Sati, Urs Schreiber, Lie n-algebras of BPS charges (arXiv:1507.08692)

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics, Volume 315, Issue 1, pp 169-213 (arXiv:0910.4001)

• Hisham Sati, Mathai Varghese, Some Relations between Twisted K-theory and $E_8$ Gauge Theory, JHEP0403:016,2004 (arXiv:hep-th/0312033)

• Urs Schreiber, Structure Theory for Higher WZW Terms, lecture course at Hisham Sati (org.) Flavors of Cohomology, Pittsburgh June 3-5, 2015

• Urs Schreiber, Igor Khavkine, Lie n-algebras of higher Noether currents

---

1. The proof of theorem 1 profited crucially from Charles Rezk, who suggested here that the reduction to fixed points will make the real Hopf fibration give a non-torsion contribution; and from David Barnes who amplified the use of the Greenless-May decomposition theorem. ↩