geometry of physics -- fundamental super p-branes

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> this entry is one section of "[[geometry of physics -- supergeometry and superphysics]]"
> which is one chapter of "[[geometry of physics]]"

> previous section: _[[geometry of physics -- supersymmetry]]_

> next section: *[[geometry of physics -- flux quantization]]*

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A remarkable consequence of the general fact that [[supergeometry]] is slightly
[[non-commutative geometry|non-commutative]] is that [[super Minkowski spacetimes]],
regarded as the translational part of [[supersymmetry]] [[super Lie algebras]],
carry non-trivial higher [[spin group|spin]]-[[invariant]] [[cocycles]] in super-[[Lie algebra cohomology]].
Generally, invariant $(p+2)$-cocycles on [[cosets]] $X$ induce interesting
[[action functionals]] for $p+1$-dimensional [[sigma-model]] [[field theories]]
with [[target space]] $X$. These encode the [[dynamics]] of _fundamental $p$-[[branes]]_, in a general sense,
that propagate on (or "in") the space(-time) $X$. For $p = 1$ and $X$ the coset of a [[compact Lie group]],
then this is known as the _[[WZW model]]_ describing a [[string]] that propagates on $X$.
Hence generally we may speak of _[[higher WZW models]]_ here.

Accordingly,  each of the exceptional invariant cocycles on [[super Minkowski spacetime]] defines a [[super p-brane]]
[[sigma model]]. These happen to be the _fundamental [[Green-Schwarz superstring]]_ and the _fundamental [[supermembrane]]_
that appear in, or rather that _define_ [[string theory]] and [[M-theory]] ("M-theory" is a "non-committal" shorthand for "membrane theory", [(Hořava-Witten 95, p. 2](#M-theory#NonCommittal)) -- a fact known as the "old [[brane scan]]" ([Achúcarro-Evans-TownsendWiltshire 87](#AETW87)).

Now in higher generalization of how an invariant 2-cocycle on some [[super Minkowski spacetime]] corresponds to a [[central extension|central]]
super-[[Lie algebra extension]] of it to a higher dimensional spacetime, so every higher cocycle,
i.e. every $p+2$-cocycle for $p \geq 1$, corresponds to a [[super Lie n-algebra|super Lie (p+1)-algebra]] extension
of [[super Minkowski spacetime]], sometimes called an "[[extended super Minkowski spacetime]]".
It turns out that on the [[super Lie n-algebra]] extensions defined by the cocycles for the [[super-string]] and the [[super-membrane]]
make further invariant higher cocycles appear. Interpreting these in turn as [[higher WZW models]] for [[super p-branes]]
it turns out that they correspond to the [[D-branes]] and to the [[M5-brane]] that appear in [[string theory]]/[[M-theory]].
This generalizes the old [[brane scan]] to a tree-like structure of higher invariant extensions that may be called the _[[brane bouquet]]_ of [[string theory]]/[[M-theory]], since it neatly organizes the complete [[super p-brane]] content
purely in terms of [[super Lie n-algebra]] theory ([FSS 13](#FSS13)).

Moreover, it turns out that on [[cocycles]] of [[super Lie n-algebras]] there is a natural [[higher Lie theory|higher Lie theoretic]]
operation of _[[double dimensional reduction]]_: the image in [[rational homotopy theory]] of the [[adjunction unit]]
for the "[[cyclic homology|cyclification]]" [[adjunction]] ([[free loop space]] [[homotopy quotient|homotopy quotiented]] by loop rotation). Applying this to the [[schreiber:brane bouquet]] turns out to yield
relations between the super Lie $n$-algebraic cocycles that embody all the [[dualities in string theory]]
at the level of [[rational homotopy theory]]:
the [[KK-compactification]] from [[M-theory]] to [[type IIA string theory]], the [[T-duality]] of the latter to
[[type IIB string theory]], the [[S-duality]] of the latter, and its relation  to [[F-theory]] ([FSS 16a](#FSS16a), [FSS 16b](#FSS16b)).

Notice that all this concerns _fundamental_ [[super p-branes]] (sometimes referred to as "probe $p$-branes")
-- in generalization of "[[fundamental particles]]" and "fundamental string" --
and in contrast to "solitonic $p$-branes" or "[[black p-branes|black branes]]".
In fact the [[black branes]] (that appear for notably in the [[AdS/CFT correspondence]]) follow from
the fundamental [[super p-branes]]: their [[worldvolumes]] are the [[BPS state]] solutions to the [[supergravity]] [[equations of motion]]
which express but the consistent globalization of the fundamental [[super p-brane]] [[sigma models]]
([Bergshoeff-Sezgin-Townsend 87](#BergshoeffSezginTownsend87)), and moreover their [[worldvolume]] theories
are but the [[perturbation theory]] of the fundamental [[super p-branes]] about these asymptotic singular loci
([Claus-Kallosh-vanProeyen 97](#ClausKalloshProeyen97), [Claus-Kallosh-Kumar-Townsend 98](#ClausKalloshKumarTownsend98)).

Hence a fair bit of the folklore structure of [[string theory]]/[[M-theory]] is (re-)discovered by systematic classification of
the invariant higher extensions of
[[super Minkowski spacetimes]] in [[super Lie n-algebra]] [[homotopy theory]]. But by the
discussion at _[[geometry of physics -- supersymmetry]]_, the relevant [[super-Minkowski spacetimes]]
are themselves already classified as the consecutive ordinary invariant extensions of just the
[[superpoint]] (we review this [below](#SuperMinowskiSpacetimes)).
In conclusion then, a fair bit of the structure of [[string theory]]/[[M-theory]]
is (re-)discovered by a systematic classification of the consecutive invariant higher extension
of the [[superpoint]] in [[super Lie n-algebra]] [[homotopy theory]]. This is what we discuss here.

Computationally, all the these cocycles in the [[schreiber:brane bouquet]] exist
due to special _[[Fierz identities]]_ (which we discuss [below](#FierzIdentities)),
namely due to special spinoral higher [[Clebsch-Gordan coefficients]] that
express the [[tensor product of representations|tensor-product]] operation in the [[real spin representation|real]] [[representation ring]] of the [[spin groups]]
in the given dimensions ([D'Auria-Fré-Maina-Regge 82](#DAuriaFreMainaRegge82), [D'Auria-Fré 82a](#DAuriaFre82a),
[D'Auria-Fré 82b](#DAuriaFre82b))).

Hence there is a tight interplay between spinorial [[representation theory]], [[superalgebra|super-algebraic]] [[higher Lie theory]],
[[spacetime]] that gives rise to a finite system of [[universal exceptionalism|exceptional]] structures: the [[super p-branes]].
This is what we discuss here.

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#Fundamental super p-branes#
* table of contents
{:toc}

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In order to put our discussion in perspective, we start by surveying some

* _[History and background](#HistoryAndBackground)_.

Then in order to lay the basis of all our discussion to follow, which is super [[higher Lie theory]],
we recall [[super Lie algebras]] and discuss their generalization to [[super L-∞ algebras]]
(whose [[formal duals]] are called "[[FDA]]"s in the physics literature):

* _[Super L-infinity cohomology and FDAs](#SuperLInfinityCohomologyAndFDAs)_.

We will be associating a fundamental $p$-brane with each invariant super $L_\infty$-cocycle.
We explain how this is given by a variant of the operation of higher [[Lie integration]] in

* _[From L-infinity-cocycles to higher WZW models](#FromCocyclesToWZW)_

The basic example of [[super Lie algebras]] that induces all phenomena to follow are
the super-translation parts of [[supersymmetry]] algebras, the [[super Minkowski spacetimes]]
introduced in detail in _[[geometry of physics -- supersymmetry]]_. Since here we
frequently need to refer to these structures, we recall their definition again in

* _[Super Minkowski spacetimes](#SuperMinowskiSpacetimes)_.

The key phenomena to be discussed are then the non-trivial invariant [[cocycles]] in the
super-[[Lie algebra cohomology]] of [[super Minkowski spacetimes]] (sometimes called _[[tau cohomology]]_
in the physics literature). Computationally these correspond to certain identities satisfied by
qadrilinear expressions in [[Majorana spinors]]. Such relations are an example of _[[Fierz identities]]_
and so we pause to explain these in

* _[Fierz identities](#FierzIdentities)_.

This then gives rise to the "old [[brane scan]]" of [[Green-Schwarz super-string]] and [[super-membranes]],
which is the classification of the $Spin$-invariant
super-[[Lie algebra cohomology]] of [[super Minkowski spacetimes]] ([[tau cohomology]]):

* _[The super-string and the super-membrane](#TheSuperStringAndTheSuperMembrane)_.

Using our previously established general theory of super-[[L-∞ algebra cohomology]],
we see that the cocycles in the old [[brane scan]] classify
_higher_ central extensions of [[super Minkowski spacetimes]], namely [[super Lie n-algebras]]
called _[[extended super Minkowski spacetimes]]_:

* _[Extended super Minkowski spacetimes](#ExtendedSuperMinkowskiSpacetime)_

A key point then is that these [[super Lie n-algebras]] obtained from [[super Minkowski spacetime]],
turn out to carry _further_ super $L_\infty$-cocycles, not present on the plain
[[super Minkowski spacetimes]]. These correspond to all the [[D-branes]] and to the [[M5-branes]].
This we discuss in

* _[The super D-branes and the M5-brane](#TheSuperDBranes)_

This gives a tree of consecutive invariant [[universal higher central extensions]] of [[super Lie n-algebras]], originating from the [[superpoint]]:
the _[[schreiber:brane bouquet]]_. Next we [[descent|descend]] these iterated central extensions to  single but non-central
higher cocycles. The result turns out to be the image in [[rational homotopy theory]] of the classifying maps of the
[[background fields]] in [[string theory]]/[[M-theory]]: the [[B-field]]-twisted [[RR-fields]] and the M-flux fields.
This we discuss in

* _[Fields](#Fields)_.

There turn out to be special relations among these. In particular passing from  super-[[L-∞ algebra cohomology]]
to the corresponding _[[cyclic cohomology]]_ turns out to be the [[formal dual]] operation of what in
physics is called [[double dimensional reduction]] of [[branes]] (here: of their [[background fields]]). This
crucial operation we discuss in

* _[Double dimensional reduction](#DoubleDimensionalReduction)_ .

By systematically applying this [[super Lie n-algebra|super Lie n-algebraic]] formalization of [[double dimensional reduction]] via [[cyclic cohomology]], we discover all the pertinent [[dualities in string theory]], [[rational homotopy theory|rationally]]:

* _[Dualities](#Dualities)_.

## History and background
 {#HistoryAndBackground}

Below we present fundamental [[super p-branes]] as [[universal exceptionalism|exceptional]]
algebro-geometric structures that are discovered by applying the magnifying glass
(namely a [[Whitehead tower]] construction) of [[super Lie n-algebra]] [[homotopy theory]]
to the atom of [[superspace]]: the [[superpoint]], following ([FSS 13](#FSS13), [FSS 15](#FSS15), [FSS 16a](#FSS16a), [FSS 16b](#FSS16b)).

This is not the perspective from which [[super p-branes]] were arrived at historically. In order
to put our discussion into perspective, here we briefly review some of the historical background.

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[[perturbation theory|Perturbative]] [[string theory]] on geometric backgrounds is defined by the [[Neveu-Schwarz-Ramond model]], namely by [[sigma-model]] [[2d super conformal field theories]] (of [[central charge]] 15) on [[worldsheets]] $\Sigma$ that are [[super Riemann surfaces]], with [[target spaces]] $X$ that are ordinary (i.e. "bosonic") [[spacetime]] [[manifolds]].

These [[worldsheet]] [[field theories]] are induced from _[[action functionals]]_, namely from variants of the standard [[energy functional]] ([[Polyakov action]]) on the [[mapping space]] $[\Sigma,X]$ of smooth functions

$$
  \phi \;\colon\; \Sigma \longrightarrow X
$$

from the [[worldsheet]] $\Sigma$ to target [[spacetime]] $X$.

The central theorem of perturbative superstring theory
(the no ghost theorem with [[GSO projection]]) says that the excitation spectrum of such a [[2d SCFT]] are the quanta of the [[perturbation theory|perturbations]] of a higher dimensional [[effective field theory|effective]] [[supergravity]] [[field theory]] on target spacetime, hence transforms under [[supersymmetry]] on target spacetime.

This is the fundamental prediction of the assumption of fundamental strings:

1. assuming that the [[fundamental particles]] that run in [[Feynman diagrams]] are fundamentally (at high energy) the ground state modes of a fundamental [[string]],

1. demanding that there are [[fermion|fermionic]] particles among these,

implies

1. that the string must be the [[spinning string]] (have fermions in its [[worldsheet]] theory), which in turn implies...

1. that it is the [[superstring]] (worldsheet [[supersymmetry]] mixes the worldsheet bosons and fermions), precisely: the [[Neveu-Schwarz-Ramond superstring]], which then in addition implies...

1. that its [[target space]] [[effective field theory]] is a [[supergravity]] theory, hence that also the effective target space fields exhibit _local_ [[supersymmetry]] (i.e. "high energy supersymmetry", different from "low energy supersymmetry" that the [[LHC]] was looking for).

| main theorem of [[perturbative string theory|perturbative super-string theory]] |
|-------------------------------------------------------------------------------------------------|
| $ \underset{\text{spinning string}}{\underbrace{\text{fermions} \;+\; \text{strings}}} \;=\; \text{superstring} \;\Rightarrow\; \text{supergravity} $ |

The first step in this implication (identifying the [[spinning string]] as the [[superstring]]) is fairly straightforward
(in fact this is how the concept of [[supersymmetry]] was discovered in "the west", in the first place),
but the second step (that the superstring excitations necessarily are quanta of a [[spacetime]] [[supergravity]] theory)
appears as a miracle from the point of view of the [[Neveu-Schwarz-Ramond superstring]].
It comes out this way by non-trivial computation, but is not manifest in the theory.

In order to improve on this situation, [[Michael Green]] and [[John Schwarz]] searched for and found ([Green-Schwarz 81](#GreenSchwarz81), [Green-Schwarz 82](#GreenSchwarz82) [Green-Schwarz 84](#GreenSchwarz84), for the history see [Schwarz 16, slides 24-25](#Schwarz16)) a suitably equivalent string [[action functional]] that would manifestly exhibit [[spacetime]] [[supersymmetry]].
Acordingly, this is now called the _[[Green-Schwarz action functional]]_.

| [[action functional]] for [[superstring]] | manifest [[supersymmetry]] |
|----|---|
| [[Neveu-Ramond-Schwarz super-string]] | on [[worldsheet]]  |
| Green-Schwarz super-string | on [[target space|target]] [[spacetime]] |

The basic idea is to pass to the evident [[supergeometry|supergeometric]] analogue of the bosonic string action:

Let $\Sigma$ be a [[closed manifold]] of [[dimension]] 2 -- representing the abstract _[[worldsheet]]_ of a string. Let $(X,g)$ be a [[pseudo-Riemannian manifold]] -- representing a purely [[gravity|gravitational]] [[spacetime]] background. Then the [[action functional]] governing the [[bosonic string]] propagating in this spacetime is the functional

$$
  \exp(\tfrac{i}{\hbar} S_{bos})
  \;\colon\;
  [\Sigma,X]
   \longrightarrow
  \mathbb{R}/_{\hbar}\mathbb{Z}
$$

on the [[smooth space|smooth]] [[mapping space]] $[\Sigma,X]$ (of [[smooth functions]] $\Sigma \to X$), that simply assigns the proper [[relativity|relativistic]] [[volume]] of the image of the [[worldsheet]] $\Sigma$ in [[spacetime]]:

$$
  (\Sigma\overset{\phi}{\longrightarrow} X)
    \;\mapsto\;
  S_{kin}(\phi)
  \coloneqq
  \int_\Sigma vol_{\phi^\ast g}
  \,.
$$

(This is the _[[Nambu-Goto action]]_. It is classically equivalently to the [[Polyakov action]] which is the genuine starting point for the quantum [[Neveu-Ramond-Schwarz super-string]]. Howver, since, as we discuss below, the Green-Schwarz action naturally generalizes to that of other [[p-branes]] it is more natural to consider the Nambu-Goto form of the action here.)

When here $(X,g)$ is generalized to a [[superspacetime]] [[supermanifold]] with [[orthogonal structure]] encoded by a [[super-vielbein]] $e$, then the same form of the action functional still makes sense and produces a functional on the [[supergeometry|supergeometric]] [[mapping space]] $[\Sigma,X]$. Moreover, by construction this action functional now is invariant under the [[superisometry group]] of $(X,g)$, hence under global spacetime [[supersymmetry]].

$$
\array{
  \text{symmetry of worldsheet theory}
  \\
  \array{
    && \Sigma
    \\
    & {}^{\mathllap{\phi}}\swarrow && \searrow^{\mathrlap{\phi'}}
    \\
    X &&\underset{\simeq}{\longrightarrow}&& X
  }
  \\
  \text{super-isometry of target spacetime}
}
$$

However, [[Michael Green|Green]] and [[John Schwarz|Schwarz]] noticed that this [[kinetic action]] functional $\phi \mapsto \int_\Sigma vol_{\phi^\ast e}$ does _not_ quite yield dynamics that is equivalent to that of the [[Neveu-Schwarz-Ramond super-string]]: when the [[equations of motion]] hold ("on [[shell]]") it has more fermionic degrees of freedom than present in the [[Neveu-Ramond-Schwarz super-string]]. The key insight of Green and Schwarz was that one may add an extra summand $S_{WZW}$ (whose notation we explain in a moment) to the plain super-[[Nambu-Goto action]] $S_{kin}$, such that the resulting [[action functional]] enjoys a further 1-parameter [[symmetry]], called _[[kappa-symmetry]]_. This is the [[Green-Schwarz action functional]]:

$$
  S_{GS} = S_{kin} + S_{WZW}
  \,.
$$

Moreover, they showed that restricting the
[[dynamics]] of the [[Green-Schwarz superstring]] to the $\kappa$-symmetric states, then it does become equivalent, classically
to that of the [[Neveu-Ramond-Schwarz super-string]].

Finally they showed that when [[gauge fixing]] the [[Green-Schwarz action functional]] to [[light-cone gauge]]
(which is possible whenever [[target spacetime]] admits two [[lightlike]] [[Killing vector]]) then the [[Green-Schwarz string]]
may be [[quantization|quantized]] by a standard procedure and the resulting
[[quantum system|quantum]] dynamics is equivalent to that of the [[Neveu-Schwarz-Ramond super-string]].
This provides the desired conceptual [[proof]] for the observed local [[target spacetime]] [[supersymmetry]]
of super-string [[effective field theory]], at least for backgrounds that admit two lightlike Killing vectors.
(The [[quantization]] of the [[Green-Schwarz superstring]] away from [[light cone gauge]] remains an open problem.)

While Green-Schwarz's extra [[kappa-symmetry]] term $S_{WZW}$ this serves a clear purpose as a means to an end, originally its geometric meaning was mysterious. However, in ([Henneaux-Mezincescu 85](#HenneauxMezincescu85)) it was observed (expanded on in ([Rabin 87](#Rabin87), [Azcarraga-Townsend 89](#AzcarragaTownsend89), [Azcarraga-Izquierdo 95,chapter 8](#AzcarragaIzquierdo95))), that the
[[Green-Schwarz action functional]] describing the [[super-string]] in $d+1$-dimensions does have a neat geometrical interpretation: it is simply the ([[parameterized WZW model|parameterized]]) _[[Wess-Zumino-Witten model]]_ for

1. [[target space]] being locally [[super Minkowski spacetime]] $\mathbb{R}^{d-1,1|\mathbf{N}}$ regarded as the [[coset]] [[supergroup]]

   $$
     \mathbb{R}^{d-1,1\vert \mathbf{N}}
       \;\simeq\;
     Iso(\mathbb{R}^{d-1,1\vert \mathbf{N}}) / Spin(d-1,1)
   $$

   for $\mathbf{N}$ a [[real spin representation]] (the "number of supersymmetries"), $Iso(\mathbb{R}^{d-1,1\vert \mathbf{N}})$ the corresponding [[super Poincaré group]] and $Spin(d-1,1)$ its [[Lorentzian manifold|Lorentz-signature]] [[spin group|Spin]] [[subgroup]];

1. [[WZW-term]] being a local potential for the unique (up to rescaling, if it exists) $Spin(d-1,1)$-[[invariant]] [[Lie algebra cohomology|super Lie algebra 3-cocycle]] $\mu_{F1}$ on the [[super Poincaré Lie algebra]] $\mathfrak{iso}(\mathbb{R}^{d-1,1\vert \mathbf{N}})$, with components locally given by the Gamma-matrices of the given [[Clifford algebra]] representation; in terms of the [[super vielbein]] $(e^a, \psi^\alpha)$:

   $$
     \mu_{F1} = \overline{\psi} \wedge \Gamma_A \psi \wedge e_a
   $$

   and so in components the bi-fermionic component of $\mu_{F1}$ is

   $$
     (\mu_{F1})_{a \alpha \beta} = \Gamma_{a \alpha \beta}
   $$

   and all other components vanish.

More in detail, just as ordinary [[Minkowski spacetime]] $\mathbb{R}^{d-1,1}$ may be identified with the [[translation group]]
along itself, with canonical [[linear basis]] of [[left invariant 1-forms]] given by the canonical [[vielbein]] field

$$
  \{e^a \coloneqq \mathbf{d}x^a\}_{a = 0}^{d-1}
  \,,
$$

where $\{x^a\}$ are the canonical [[coordinates]] on $\mathbb{R}^{d-1,1}$, so [[super Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ for some [[real spin representation]] $\mathbf{N}$ is characterized as the [[supergroup]] whose [[left invariant 1-forms]] constitute the $\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$-bigraded differential with generators the [[super-vielbein]]

$$
  \underset{deg = (1,even)}{\underbrace{e^a}}
    \;\coloneqq\;
  \mathbf{d}x^a + \overline{\theta}\Gamma^a \mathbf{d} \theta
  \;\;\;\,,\;\;\;\;\;\;\;\;\;\;
  \underset{deg = (1,odd)}{\underbrace{\psi^\alpha}} \;\coloneqq\; \mathbf{d}\theta^\alpha
  \,,
$$

where $(x^a, \theta^\alpha)$ are the canonical coordinates on $\mathbb{R}^{d-1,1\vert \mathbf{N}}$, with the odd-graded elements $\{\theta^\alpha\}$ spanning the given real [[spin representation|Spin(d-1,1)-representation]] $\mathbf{N}$ with [[Clifford algebra]] generators $\{\Gamma^a\}$.

Now while ordinary [[Minkowski spacetime]] $\mathbb{R}^{d-1,1}$ is an [[abelian group]], reflected by the fact that its [[left-invariant 1-forms]] are all closed

$$
  \mathbf{d}e^a = 0 \;\;\;\;\;\; on \; \mathbb{R}^{d-1,1}
  \,,
$$

the key phenomenon of [[supersymmetry]] (that two [[fermions]] pair to a [[bosons]]) means that $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ is slightly non-abelian, reflected by the fact that the [[super-vielbein]] is not closed

$$
  \mathbf{d} e^a =  \overline{\psi} \wedge \Gamma^a \psi
  \;\,,\;\;\;\;\;\;
  \mathbf{d} \psi^\alpha = 0
  \,.
$$

This elementary effect is the source of all the rich structure seen in the [[Green-Schwarz super-string]]
and generally in all [[super p-brane]] theory.
(The above differential is equivalently that in the [[Chevalley-Eilenberg algebra]] of [[super Minkowski spacetime]],
hence its [[cohomology]] is the super-[[Lie algebra cohomology]] of [[super Minkowski spacetime]]. In parts of the
[[physics]] literature this is referred to a "[[tau cohomology]]".)

In particular, for special combinations of [[spacetime]] [[dimension]] $d$ and number of [[supersymmetries]] $\mathbf{N}$
(i.e. [[real spin representation]] $N$) then the 3-form

$$
  \mu_{F1} =  \overline{\psi} \wedge \Gamma_a \psi \wedge e^a
$$

is a non-trivial [[super Lie algebra|super]] [[Lie algebra cocycle]] on $\mathbb{R}^{d-1,1\vert \mathbf{N}}$, in that
$\mathbf{d}\mu_{F1} = 0$ and so that there is no [[left invariant differential form]] $b$ with $\mathbf{d}b = \mu_{F1}$
(beware here the left-invariance condition: there are of course non-left-invariant potentials for $\mu_{F1}$,
and in fact these are exactly the possible [[Lagrangian densities]] for the WZW action functional $S_{WZW}$).

This happens notably for

1. $d = 10$ and $\mathbf{N} = (1,0) = \mathbf{16}$ ([[heterotic string]])

1. $d = 10$ and $\mathbf{N} = (2,0) = \mathbf{16} + \mathbf{16}$ ([[type IIB superstring]])

1. $d = 10$ and $\mathbf{N} = (1,1) = \mathbf{16} + \mathbf{16}^\ast$ ([[type IIA superstring]]).

(It also happens in some lower dimensions, where however the corresponding [[Neveu-Schwarz-Ramond string]] develops a [[conformal anomaly]] after [8quantization]] ("non-critical strings"). This classification of cocycles is part of what has come to be known as the _[[brane scan]]_ in superstring theory, see below.)

In this equivalent formulation, the [[Green-Schwarz action functional]] for the superstring has the following simple form:

Let $(X,e)$ be a [[superspacetime]], hence a [[supermanifold]] $X$ equipped with a [[super-vielbein]] $e$ (super-[[orthogonal structure]]) which is locally modeled on $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ (technically: a [[torsion of a G-structure|torsion]]-free [[super-Cartan geometry]] modeled on $Spin(d-1,1) \hookrightarrow Iso(\mathbb{R}^{d-1,1\vert \mathbf{N}})$). Write $\mu_{F1}^X \in \Omega^3(X)$ for the [[super differential form]] on $X$ which is the induced [[definite globalization of WZW term|definite globalization]] of the cocycle $\mu_{F1}$ over $X$. For $U \subset X$ any [[contractible topological space|contractible]] [[subspace]], then the restriction of $\mu^X_{F1}|_{U} \in \Omega^3(U)$ of $\mu_{F1}^X$ to $U$ is exact, and hence admits a potential $B_U \in \Omega^2(U)$, i.e. such that $\mathbf{d} B = \mu^X_{F1}|_U$.

Then for $\Sigma$ a 2-dimensional [[closed manifold]], the [[Green-Schwarz action functional]]

$$
  \exp(\tfrac{i}{\hbar} S_{GS})
  \;\colon\;
  [\Sigma,X]_U
    \longrightarrow
  \mathbb{R}/_{\hbar} \mathbb{Z}
$$

is the function on the super-[[smooth mapping space]] $[\Sigma,X]_U$ of morphisms of [[supermanifolds]] $\phi \colon \Sigma \to X$ which factor through $U$, given by

$$
  \phi
    \;\mapsto\;
  \int_\Sigma vol_{\phi^\ast e}
   \;+\;
  \int_\Sigma \phi^\ast B_U
  \;\;\;\,,\;\;\;\;\;\;\;\;\;
  \mathbf{d} B_U = \mu^X_3|_U
  \,.
$$

In order to get rid of the restriction to some [[chart]] $U \subset X$ one needs to add global data. The need for this is at least mentioned briefly in ([Witten 86, p. 261 (17 of 20)](#Witten86)), but seems to have otherwise been ignored in the physics literature. The general solution is to promote the local potentials $B$ to the connection $\hat B$ on a [[Super Gerbes|super gerbe]] ([FSS 13](#FiorenzaSatiSchreiber13)). This is a choice of [[higher prequantization]]

$$
  \array{
    && \mathbf{B}^{2}(\mathbb{R}/_\hbar \mathbb{Z}) & \text{prequantization}
    \\
    & {}^{\mathllap{\hat B}}\nearrow & \downarrow^{\mathrlap{curv}}
    \\
    X
     &\underset{\mu^X_{F1}}{\longrightarrow}&
    \mathbf{\Omega}^3
    &
    \text{3-form curvature}
  }
  \,.
$$

Writing $\int_\Sigma \phi^\ast \hat B$ for the [[volume holonomy]] of a [[circle 2-bundle with connection]] $\hat B$, then the globally defined Green-Schwarz sigma model

$$
  \exp(\tfrac{i}{\hbar} S_{GS})
  \;\colon\;
  [\Sigma, X]
    \longrightarrow
  \mathbb{R}/_\hbar\mathbb{Z}
$$

is given by

$$
  \phi \;\mapsto\; \int_\Sigma vol_{\phi^\ast} + \int_\Sigma \phi^\ast \hat B
  \;\;\,,
  \;\;\;\;\;\;\;
  curv(\hat B) = \mu_{F1}^X
  \,.
$$

This form of the Green-Schwarz action functional for the string has evident generalization to other [[p-branes]]. Whenever there is a [[spin group|Spin(d-1,1)]]-[[invariant]] $(p+2)$-cocycle $\mu_{p+2}$ on $\mathbb{R}^{d-1,1\vert \mathbf{N}}$, then one may ask for a higher gerbe ([[higher prequantum line bundle]]) $\hat C$ with [[curvature]] $\mu^X_{p+2}$ and consider the analogous functional.

The triples $(d,\mathbf{N},p)$ (spacetime dimension, number of supersymmetries, dimension of brane) such that

$$
  \mu_{p+2}
    \;\coloneqq\;
  \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p}
$$

is a nontrivial cocycle, hence for which there is such a Green-Schwarz action functional for $p$-branes on $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ may be classified and form what is called the _[[brane scan]]_ ([Achúcarro-Evans-TownsendWiltshire 87](#AETW87), [Brandt 12-13](#Brandt12-13)):

{#DuffBraneScan}

The graphics on the left is from ([Duff 87](#Duff87)). The diagonal lines indicate [[double dimensional reduction]], taking a $(p+1)$-brane in $(d+1)$ dimensions to a $p$-brane in $d$-dimensions.

For instance for $(d = 11, \; \mathbf{N} = \mathbf{32}, \; p = 2)$ one finds a cocycle, and the corresponding GS-action functional is that of the fundamental [[M2-brane]].

This was a striking confluence of brane physics and classification of [[super Lie algebra|super]] [[Lie algebra cohomology]]. But just as striking as the matching, was what it lacked to match: the [[D-branes]] and the [[M5-brane]] ($d = 11$, $p = 5$) are lacking from the old brane scan. Incidentally, these lacking branes are precisely those branes on which the branes that do appear on the brane scan may end, equivalently those branes that have [[higher gauge fields]] on their [[worldvolume]] (tensor multiplets).

An action functional for the [[M5-brane]] analogous to a Green-Schwarz action functional was found in ([BLNPST 97](#BLNPST97), [APPS 97](#APPS97)). It is again the sum of a kinetic term and a WZW-like term, but the WZW-like term does not come from a cocycle on a (super-)group.

In order to deal with this, it was suggested in ([CAIB 99](#CAIB99), [Sakaguchi 00](#Sakaguchi00), [Azcarraga-Izquierdo 01](#AzcarragaIzquierdo01)) that there is an algebraic structure called "[[extended super-Minkowski spacetimes]]" that generalizes [[super Minkowski spacetime]] and serves to unify the Green-Schwarz-like models for the D-branes and the M5-brane with the original Green-Schwarz models for the string and the M2-brane.

These [[extended super-Minkowski spacetimes]] carry algebraic analogs of super Lie algebra cocycles, such that the relevant terms for the [[D-branes]] and the [[M5-brane]] do appear after all, hence such that all the branes in [[string theory]]/[[M-theory]] are unified. In fact these "[[extended super-Minkowski spacetimes]]" are precisely the "FDA"s that have been introduced before in the [[D'Auria-Fré formulation of supergravity]] and what became identified as the 7-cocycle for the [[M5-brane]] this way had earlier been recognized algebraically as an stepping stone for an elegant re-derivation of [[11-dimensional supergravity]] ([D'Auria-Fré 82](#DAuriaFre82)).

The ([[higher differential geometry|higher]]) geometric meaning of these constructions was found in ([Fiorenza-Sati-Schreiber 13](#FiorezaSatiSchreiber13)): these algebraic structures of "[[extended super-Minkowski spacetimes]]"/FDAs are precisely the [[Chevalley-Eilenberg algebras]] of [[super Lie n-algebra]]-extensions of [[super-Minkowski spacetime]] which are classified by the cocycles that serve as the GS-WZW terms of the $p_1$-branes that may end on those $p_2$-branes whose cocycles are carried by the extended super-Minkowski spacetime.

Hence the missing $p$-branes in the old [[brane scan]] (classifying just cocycles on [[super Lie algebras]]) do appear as one generalizes (super) Lie algebras to (super) [[strong homotopy Lie algebras]] = [[L-infinity algebras]]. Moreover, each brane intersection law (one brane species may end on another) is now matched to a super $L_\infty$-algebra [[extension]] and so the old brane scan is generalized to a tree of branes _[[schreiber:The brane bouquet]]_:

Each item in this bouquet denotes a [[super L-infinity algebra]] and each arrow denotes an [[L-infinity extension]] classified by a cocycle which encodes the GS-WZW term of the brane named by the domain of the arrow. Moreover, arrows pass exactly from one brane species to the brane species that may end on the former.

In ([Fiorenza-Sati-Schreiber 13](#FiorenzaSatiSchreiber13)) it is shown that all these [[super L-infinity algebras]] [[Lie integration|Lie integrate]] to [[smooth super-n-groups]], and all the cocycles [[Lie integration|Lie integrate]] to [[Super Gerbes|super-gerbes]] on these, such that the induced [[volume holonomy]] is the relevant generalized GS-WZW term. For detailed exposition see at _[[schreiber:Structure Theory for Higher WZW Terms]]_.

With this generalized perspective, now the Green-Schwarz-type action functionals describe _all_ the [[p-branes]] in [[string theory]]/[[M-theory]].

Again, in order to make this generally true one needs to apply a [[higher prequantization]] -- a choice of [[line n-bundle with connection|line (p+1)-bundle with connection]] -- in order to globalize the [[WZW-terms]] ([Fiorenza-Sati-Schreiber 13](#FiorenzaSatiSchreiber13))

$$
  \array{
    && \mathbf{B}^{p+1}(\mathbb{R}/_\hbar \mathbb{Z}) & \text{prequantization}
    \\
    & {}^{\mathllap{\hat A_{p+1}}}\nearrow & \downarrow^{\mathrlap{curv}}
    \\
    X
     &\underset{\mu^X_{p+2}}{\longrightarrow}&
    \mathbf{\Omega}^{p+2}
    &
    (p+2)\text{-form curvature}
  }
  \,.
$$

Hence $\hat A_{p+1}$ is the actual [[background field]] that the $p$-brane couples to. There is considerably more information in $\hat A_p$ than in its curvature $curv(\hat A_{p+1}) = \mu_{p+2}$. For instance for the [[M2-brane]] one may find the local super [[moduli space]] for local choices of $\hat A_{p+1}$ for the given $\mu_{4}$ on [[KK-compactifications]] to $d = 4$. It turns out that the bosonic [[body]] of this moduli space is the [[exceptional tangent bundle]] on which the [[U-duality]] group [[E7]] has a canonical action (see at _[[schreiber:From higher to exceptional geometry]]_).

This highlights that Green-Schwarz functionals capture fundamental ("microscopic") aspects of $p$-branes. In contrast, often $p$-branes are discussed in their [[soliton|solitonic]] incarnation as _[[black branes]]_. These solitonic branes sit at asymptotic boundaries of [[anti-de Sitter spacetime]] and carry [[conformal field theories]], related to the ambient [[supergravity]] by [[AdS-CFT duality]].

This phenomenon is indeed a consequence of the fundamental Green-Schwarz branes:

Consider a 1/2-[[BPS state]] solution of [[type II supergravity]] or [[11-dimensional supergravity]], respectively. These solutions locally happen to have the same classification as the Green-Schwarz branes. Hence we may consider a configuration $\phi \colon \Sigma \to X$ of the corresponding fundamental $p$-brane which embeds $\Sigma$ into the asymptotic AdS boundary of the given 1/2 BPS spacetime $X$. Then it turns out that restricting the Green-Schwarz action functional to small fluctuations around this configuration, and applying a [[diffeomorphism]] [[gauge fixing]], then the resulting action functional is that of a [[supersymmetry|supersymmetric]] [[conformal field theory]] on $\Sigma$ as in the [[AdS-CFT]] dictionary:

| fundamental $p$-brane | -fluctuations about asymptotic AdS configuration$\to$ |  solitonic $p$-brane |
|----|-----|---|
| Green-Schwarz action functional |  | [[supersymmetric|super]]-[[conformal field theory]] |

([Claus-Kallosh-Proeyen 97](#ClausKalloshProeyen97), [Claus-Kallosh-Kumar-Townsend 98](#ClausKalloshKumarTownsend98), [AFFFTT 98](#AFFFTT98) [Pasti-Sorokin-Tonin 99](#PastiSorokinTonin99))

In fact the [[BPS-state]] condition itself is neatly encoded in the Green-Schwarz action functionals: by construction they are invariant under the [[spacetime]] [[superisometry group]]. Hence the [[Noether theorem]] implies that there are corresponding [[conserved currents]], whose [[Dickey bracket]] forms a super-[[Lie algebra extension]] of the Lie algebra of supersymmetries.

$$
  \array{
    \left\{
    \array{
      X && \overset{=}{\longrightarrow} && X
      \\
      & {}_{\mathllap{\hat C}}\searrow
        &\swArrow&
      \swarrow_{\mathrlap{\hat C}}
      \\
      && \mathbf{B}^{p+1}(\mathbb{R}/_{\hbar} \mathbb{Z})
    }
    \right\}
      &\longrightarrow&
    \left\{
    \array{
      X && \overset{\simeq}{\longrightarrow} && X
      \\
      & {}_{\mathllap{\hat C}}\searrow
        &\swArrow&
      \swarrow_{\mathrlap{\hat C}}
      \\
      && \mathbf{B}^{p+1}(\mathbb{R}/_{\hbar} \mathbb{Z})
    }
    \right\}
     &\longrightarrow&
    \left\{
      \array{
        X && \overset{\simeq}{\longrightarrow} && X
      }
    \right\}
    \\
    \text{topological currents}
    &&
    \text{Noether currents}
    &&
    \text{symmetries}
  }
$$

Here the "$\swArrow$" filling the triangles is the non-trivial gauge transformation by which the [[WZW term]] (as any WZW term) is preserved under the symmetries (instead of being fixed identically). It is the information in this transformations which makes the currents form an extension of the symmetries.

Here this yields the famous brane charge extensions of the super-isometry super Lie algebra of the schematic form

$$
  \{Q_\alpha, Q_\beta\}
  \;=\;
  (C \Gamma^a_{\alpha \beta}) P_a
  \;+\;
  (C \Gamma^{a_1 \cdots a_p})_{\alpha \beta} Z_{a_1, \cdots, a_p}
$$

(for $Q$ a [[Killing spinor]] and $P$ its corresponding [[Killing vector]]) known as the [[type II supersymmetry algebra]] and the [[M-theory supersymmetry algebra]], respectively ([Azcárraga-Gauntlett-Izquierdo-Townsend 89](#AGIT89)). In fact it yields super-[[L-infinity algebra extensions|Lie n-algebra extensions]] of which the familiar super Lie algebra extensions are the 0-truncation ([Sati-Schreiber 15](#SatiSchreiber15), [[schreiber:Local prequantum field theory|Khavkine-Schreiber 16]]).

In summary, the nature and classification of Green-Schwarz action functionals captures in a mathematically precise way a good deal of the core structure of [[string theory|string]]/[[M-theory]].

In fact, the [[super L-infinity algebra|super Lie-n algebraic]] perspective on the Green-Schwarz functionals via the [[schreiber:brane bouquet]] also solves the following open problem on [[M-branes]]:

it is famously known from [[Freed-Witten anomaly]]-cancellation that the [[D-brane charges]] are not in fact just in [[de Rham cohomology]] in every second degree, but are in [[twisted K-theory]], hence [[rational homotopy theory|rationally]] in [[twisted de Rham cohomology]], with the twist being the [[F1-brane]] charge (from the fundamental). It is an open problem to determine what becomes of these [[twisted K-theory]] charge groups as one lifts F1/D$p$-branes in string theory to M2/M5-branes in [[M-theory]].

|   | intersecting branes | charges in [[generalized cohomology theory]]
|-----|--------------|----------------------|
| [[string theory]] | [[F1-brane|F1]]/[[D-brane|Dp-branes]] | [[twisted K-theory]] |
| [[M-theory]]  | [[M2-brane|M2]]/[[M5-branes]] | ??? |

Notice that there are "microscopic degrees of freedom" of the theory encoded by the choice of [[generalized cohomology theory]] here, generalizing the extra degrees of freedom in the choice of a WZW-term already mentioned above. In general for $E$ a cohomology theory and $E \longrightarrow E \otimes \mathbb{Q}$ its [[Chern character]] map (for instance from [[topological K-theory]] to [[ordinary cohomology]] in every second degree), then a choice of genuine charges is the extra information encoded in a lift

$$
  \array{
    && E
    \\
    & {}^{\mathllap{\text{true charge}}}\nearrow & \downarrow^{\mathrlap{ch}}
    \\
    X &\underset{\text{rational} \atop  \text{charge}}{\longrightarrow}& E \otimes \mathbb{Q}
  }
$$

But [[rational homotopy theory|rationally]] [[schreiber:The brane bouquet]] allows to derive this from first principles:

Above we saw that the naive cocycles of the [[D-branes]] and of the [[M5-brane]] are not defined on the actual [[spacetime]],  but on some "extended" spacetime, which is really a [[smooth super infinity-groupoid]] extension of spacetime. Hence we should ask if these cocycles [[descent|descend]] to the actual super-spacetime while picking up some twists.

One may prove that:

* the F1/D$p$-brane GS-WZW cocycles descend to 10d type II superspacetime to form a single cocycle in rational twisted K-theory, just as the traditional lore reqires ([Fiorenza-Sati-Schreiber 16](#FiorenzaSatiSchreiber16));

* the M2/M5 GS-WZW cocycles descent to 11d superspacetime to form a single cocycle with values in the [[rational n-sphere|rational 4-sphere]] ([Fiorenza-Sati-Schreiber 16](#FiorenzaSatiSchreiber16)).

This has implications on some open conjectures regarding [[M-theory]], for more on this see _[[schreiber:Equivariant cohomology of M2/M5-branes]]_.

$\,$

We now explain all this in detail.

$\,$

## Super $L_\infty$-cohomology and FDAs
 {#SuperLInfinityCohomologyAndFDAs}

We recall [[super Lie algebras]], and amplify their [[formal dual]] description in terms of their
[[Chevalley-Eilenberg algebras]]. This makes it immediate to see what [[super L-∞ algebras]] are,
again they are conveniently expressed (if they are of [[finite type]]) via their [[formal dual]] [[Chevalley-Eilenberg algebras]].

+-- {: .num_defn #SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces}
###### Definition

A _[[super Lie algebra]]_ is a [[Lie algebra]] [[internalization|internal]]
to the [[symmetric monoidal category]] $sVect = (Vect^{\mathbb{Z}/2}, \otimes_k, \tau^{super} )$ of [[super vector spaces]].
Hence this is

1. a [[super vector space]] $\mathfrak{g}$;

1. a homomorphism

   $$
     [-,-] \;\colon\; \mathfrak{g} \otimes_k \mathfrak{g} \longrightarrow \mathfrak{g}
   $$

   of super vector spaces (the _super Lie bracket_)

such that

1. the bracket is skew-symmetric in that the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       \mathfrak{g} \otimes_k \mathfrak{g}
         &
           \overset{\tau^{super}_{\mathfrak{g},\mathfrak{g}}}{\longrightarrow}
         &
       \mathfrak{g} \otimes_k \mathfrak{g}
       \\
       {}^{\mathllap{[-,-]}}\downarrow && \downarrow^{\mathrlap{[-,-]}}
       \\
       \mathfrak{g} &\underset{-1}{\longrightarrow}& \mathfrak{g}
     }
   $$

   (here $\tau^{super}$ is the [[braiding]] [[natural isomorphism]] in the category of [[super vector spaces]])

1. the [[Jacobi identity]] holds in that the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g}
       &&
         \overset{\tau^{super}_{\mathfrak{g}, \mathfrak{g}} \otimes_k id }{\longrightarrow}
       &&
       \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g}
       \\
       & {}_{\mathllap{[-,[-,-]]} - [[-,-],-] }\searrow && \swarrow_{\mathrlap{[-,[-,-]]}}
       \\
       && \mathfrak{g}
     }
     \,.
   $$

=--

Externally this means the following:

+-- {: .num_prop #SuperLieAlgebraTraditional}
###### Proposition

A [[super Lie algebra]] according to def. \ref{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces} is equivalently

1. a $\mathbb{Z}/2$-[[graded vector space]] $\mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}$;

1. equipped with a [[bilinear map]] (the _super Lie bracket_)

   $$
     [-,-] : \mathfrak{g}\otimes_k \mathfrak{g} \to \mathfrak{g}
   $$

   which is _graded_ skew-symmetric: for $x,y \in \mathfrak{g}$ two elements of homogeneous degree $\sigma_x$, $\sigma_y$, respectively, then

   $$
     [x,y] = -(-1)^{\sigma_x \sigma_y} [y,x]
     \,,
   $$

1. that satisfies the $\mathbb{Z}/2$-graded [[Jacobi identity]] in that
for any three elements $x,y,z \in \mathfrak{g}$ of homogeneous super-degree $\sigma_x,\sigma_y,\sigma_z\in \mathbb{Z}_2$ then

   $$
     [x, [y, z]] = [[x,y],z] + (-1)^{\sigma_x \cdot \sigma_y} [y, [x,z]]
     \,.
   $$

A [[homomorphism]] of super Lie algebras is a homomorphisms of the underlying [[super vector spaces]]
which preserves the Lie bracket. We write

$$
  sLieAlg
$$

for the resulting [[category]] of super Lie algebras.

=--

+-- {: .num_defn #CEAlgebraofSuperLieAlgebra}
###### Definition

For $\mathfrak{g}$ a [[super Lie algebra]] of [[finite number|finite]] [[dimension]], then its _[[Chevalley-Eilenberg algebra]]_ $CE(\mathfrak{g})$ is the super-[[Grassmann algebra]] on the [[dual vector space|dual]] super vector space

$$
  \wedge^\bullet  \mathfrak{g}^\ast
$$

equipped with a [[differential]] $d_{\mathfrak{g}}$ that on generators is the linear dual of the super Lie bracket

$$
  d_{\mathfrak{g}} \;\coloneqq\; [-,-]^\ast \;\colon\; \mathfrak{g}^\ast \to \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast
$$

and which is extended to $\wedge^\bullet \mathfrak{g}^\ast$ by the graded Leibniz rule (i.e. as a graded [[derivation]]).

$\,$

Here all elements are $(\mathbb{Z} \times \mathbb{Z}/2)$-bigraded, the first being the _cohomological grading_ $n$ in $\wedge^\n \mathfrak{g}^\ast$, the second being the _super-grading_ $\sigma$ (even/odd).

For $\alpha_i \in CE(\mathfrak{g})$ two elements of homogeneous bi-degree $(n_i, \sigma_i)$, respectively,
the [[signs in supergeometry|sign rule]] is

$$
  \alpha_1 \wedge \alpha_2 = (-1)^{n_1 n_2} (-1)^{\sigma_1 \sigma_2}\; \alpha_2 \wedge \alpha_1
  \,.
$$

(See at _[[signs in supergeometry]]_ for discussion of this sign rule and of an alternative sign rule that is also in use. )

=--

We may think of $CE(\mathfrak{g})$ equivalently as the [[dg-algebra]] of [[invariant differential form|left-invariant]]
[[super differential forms]] on the [[Lie theory|corresponding]] simply connected [[super Lie group]] .

The concept of [[Chevalley-Eilenberg algebras]] is traditionally introduced as a means to define
[[Lie algebra cohomology]]:

+-- {: .num_defn #SuperLieAlgebraCohomology}
###### Definition

Given a [[super Lie algebra]] $\mathfrak{g}$, then

1. an _$n$-cocycle_ on $\mathfrak{g}$ (with [[coefficients]] in $\mathbb{R}$) is an element of degree $(n,even)$ in its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ (def. \ref{CEAlgebraofSuperLieAlgebra}) which is $d_{\mathbb{g}}$ closed.

1. the cocycle is non-trivial if it is not $d_{\mathfrak{g}}$-exact

1. hene the _super-[[Lie algebra cohomology]]_ of $\mathfrak{g}$ (with [[coefficients]] in $\mathbb{R}$) is the [[cochain cohomology]] of its [[Chevalley-Eilenberg algebra]]

   $$
     H^\bullet(\mathfrak{g}, \mathbb{R}) = H^\bullet(CE(\mathfrak{g}))
     \,.
   $$

=--

The following says that the [[Chevalley-Eilenberg algebra]] is an equivalent incarnation of the [[super Lie algebra]]:

+-- {: .num_prop #FormalDualCE}
###### Proposition

The functor

$$
  CE \;\colon\; sLieAlg^{fin}  \hookrightarrow dgAlg^{op}
$$

that sends a finite dimensional [[super Lie algebra]] $\mathfrak{g}$ to its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$
(def. \ref{CEAlgebraofSuperLieAlgebra}) is a [[fully faithful functor]] which hence exibits [[super Lie algebras]]
as a [[full subcategory]] of the [[opposite category]] of [[differential-graded algebras]].

=--

This makes it immediate how to generalize to [[super L-infinity algebras]]:

+-- {: .num_defn #SuperLInfinityAlgebra}
###### Definition

A **[[super L-∞ algebra]]** is an [[L-∞ algebra]] [[internalization|internal to]] the [[symmetric monoidal category]] of [[super vector spaces]] ([def. ](geometry+of+physics+--+superalgebra#CategoryOfSuperVectorSpaces)).

=--

Explicitly this means the following:

+-- {: .num_defn #SuperGradedSignatureOfPermutation}
###### Definition
**(super graded signature of a permutation)**

Let $V$ be a $\mathbb{Z}$-[[graded object|graded]] [[super vector space]], hence a $\mathbb{Z} \times (\mathbb{Z}/2)$-bigraded vector space.

For $n \in \mathbb{N}$ let

$$
  \mathbf{v} = (v_1, v_2, \cdots, v_n)
$$

be an [[n-tuple]] of elements of $V$ of homogeneous degree $(n_i, s_i) \in \mathbb{Z} \times \mathbb{Z}/2$, i.e. such that $v_i \in V_{(n_i,s_i)}$.

For $\sigma$ a [[permutation]] of $n$ elements, write $(-1)^{\vert \sigma \vert}$ for the [[signature of a permutation|signature of the permutation]], which is by definition equal to $(-1)^k$ if $\sigma$ is the composite of $k \in \mathbb{N}$ permutations that each exchange precisely one pair of neighboring elements.

We say that the _super $\mathbf{v}$-graded signature of $\sigma$_

$$
  \chi(\sigma, v_1, \cdots, v_n) \;\in\; \{-1,+1\}
$$

is the product of the [[signature of a permutation|signature of the permutation]] $(-1)^{\vert \sigma \vert}$ with a factor of

$$
  (-1)^{n_i n_j}(-1)^{s_i s_j}
$$

for each interchange of neighbours $(\cdots v_i,v_j, \cdots )$ to $(\cdots v_j,v_i, \cdots )$ involved in the decomposition of the permuation as a sequence of swapping neighbour pairs (see at _[[signs in supergeometry]]_ for discussion of this combination of super-grading and homological grading).

=--

Now def. \ref{SuperLInfinityAlgebra} is equivalent to the following def. \ref{sLInfinityDefinitionViaGeneralizedJacobiIdentity}. This is just the definiton for [L-infinity algebras](#L-infinity-algebra#DefinitionViaHigherBrackets), with the pertinent sign $\chi$ now given by def. \ref{SuperGradedSignatureOfPermutation}.

+-- {: .num_defn #sLInfinityDefinitionViaGeneralizedJacobiIdentity}
###### Definition

An _[[super L-∞ algebra]]_ is

1. a $\mathbb{Z} \times (\mathbb{Z}/2)$-[[graded object|graded]] [[vector space]] $\mathfrak{g}$;

1. for each $n \in \mathbb{N}$ a [[multilinear map]], called the _$n$-ary bracket_, of the form

   $$
     l_n(\cdots)
       \;\coloneqq\;
     [-,-, \cdots, -]_n
     \;\colon\;
       \underset{n \; \text{copies}}{\underbrace{\mathfrak{g} \otimes \cdots \otimes \mathfrak{g}}}
       \longrightarrow
       \mathfrak{g}
   $$

   and of degree $n-2$

such that the following conditions hold:

1. (**super graded skew symmetry**) each $l_n$ is graded antisymmetric, in that for every [[permutation]] $\sigma$ of $n$ elements and for every [[n-tuple]] $(v_1, \cdots, v_n)$ of  homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ then

   $$
     l_n(v_{\sigma(1)}, v_{\sigma(2)},\cdots ,v_{\sigma(n)})
     =
     \chi(\sigma,v_1,\cdots, v_n) \cdot l_n(v_1, v_2, \cdots v_n)
   $$

   where $\chi(\sigma,v_1,\cdots, v_n)$ is the super $(v_1,\cdots,v_n)$-graded signature of the permuation $\sigma$, according to def. \ref{SuperGradedSignatureOfPermutation};

1. (**strong homotopy [[Jacobi identity]]**) for all $n \in \mathbb{N}$, and for all [[n-tuple|(n+1)-tuples]] $(v_1, \cdots, v_{n+1})$ of homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ the followig [[equation]] holds

   \[
     \label{LInfinityJacobiIdentity}
     \sum_{{i,j \in \mathbb{N}} \atop {i+j = n+1}}
     \sum_{\sigma \in UnShuff(i,j)}
     \chi(\sigma,v_1, \cdots, v_{n})
     (-1)^{i(j-1)}
      l_{j} \left(
        l_i \left( v_{\sigma(1)}, \cdots, v_{\sigma(i)} \right),
        v_{\sigma(i+1)} , \cdots , v_{\sigma(n)}
      \right)
     = 0
     \,,
   \]

   where the inner sum runs over all $(i,j)$-[[unshuffles]] $\sigma$ and where $\chi$ is the super graded signature sign from def. \ref{SuperGradedSignatureOfPermutation}.

A _strict [[homomorphism]]_ of super $L_\infty$-algebras

$$is
  \mathfrak{g}_1 \longrightarrow \mathfrak{g}_2
$$

is a [[linear map]] that preserves the bidegree and all the brackets, in an evident sens.

A _strong homotopy homomorphism_ ("sh map") of super $L_\infty$-algebras is something weaker than that, best defined in [[formal duals]], below in def. \ref{SuperLInfinityCEAlgebra}.

=--

+-- {: .num_remark #LInfinityTerminology}
###### Remark

Special cases of the general concept of [[super L-∞ algebras]] def. \ref{sLInfinityDefinitionViaGeneralizedJacobiIdentity} go by special names:

Let $\mathfrak{g}$ be a [[super L-∞ algebra]].

If $\mathfrak{g}$ is concentrated in even $\mathbb{Z}/2$-degree, it is called an _[[L-∞ algebra]]_.

If the only possibly non-vanishing brackets of $\mathfrak{g}$ are the unary one $[-]$ (which induces the structure of a [[chain complex]] on $\mathfrak{g}$) and the binary one, then $\mathfrak{g}$ is equivalently a (super-)[[dg-Lie algebra]].

If $\mathfrak{g}$ is concentrated in $\mathbb{Z}$-degrees 0 to $n-1$ then it is called a _[[super Lie n-algebra]]_.

In particular if $\mathfrak{g}$ is concentrated in degree 0, then it is equivalently a [[super Lie algebra]].

Combining this, if $\mathfrak{g}$ is concentrated in even $\mathbb{Z}/2$-degree and in $\mathbb{Z}$-degree 0 through $n-1$, then it is called a _[[Lie n-algebra]]_.

In particular if $\mathfrak{g}$ is concentrated in $\mathbb{Z}$-degree 0 and in even $\mathbb{Z}/2$-degree, then it is
equivalently a plain [[Lie algebra]].

=--

+-- {: .num_defn #SuperLInfinityCEAlgebra}
###### Definition

A super $L_\infty$ algebra $\mathfrak{g}$ is of _[[finite type]]_ if the underlying $\mathbb{Z} \times (\mathbb{Z}/2)$-[[graded vector space]] is degreewise of [[finite number|finite]] [[dimension]].

If $\mathfrak{g}$ is of finite type, then its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ is the [[dg-algebra]] whose underlying [[graded algebra]] is the super-Grassmann algebra

$$
  \wedge^\bullet \mathfrak{g}^{\ast}
$$

of the graded degreewise [[dual vector space]] $\mathfrak{g}^\ast$, equipped with the [[differential]] which on generators is the sum of the [[dual linear maps]] of the $n$-ary brackets:

$$
  d_{\mathfrak{g}}
   \coloneqq
  [-]^\ast + [-,-]^\ast + [-,-,-]^\ast + \cdots
  \;\colon\;
  \wedge^1 \mathfrak{g}^\ast
    \longrightarrow
  \wedge^\bullet \mathfrak{g}^\ast
$$

and extended to all of $\wedge^\bullet \mathfrak{g}^\ast$ as a super-graded [[derivation]] of degree $(1,even)$.

Notice that here the [[signs in supergeometry]] are such that for $\alpha_i \in \mathfrak{g}^\ast_{(n_i,s_i)}$ elements of homogenous bidegree, then

$$
  \alpha_1 \wedge \alpha_2
  \;=\;
  -(-1)^{n_1 n_2} (-)^{s_1 s_2}
$$

and

$$
  d_{\mathfrak{g}}
  (\alpha_1 \wedge \alpha_2)
  \;=\;
  (d_{\mathfrak{g}} \alpha_1) \wedge \alpha_2
  +
  (-1)^{n_1} \alpha_1 \wedge (d_{\mathfrak{g}} \alpha_2)
  \,.
$$

(see at _[[signs in supergeometry]]_ for more on this).

A _strong homotopy homomorphism_ ("sh-map") between super $L_\infty$-algbras of [[finite type]]

$$
 f \;\colon\; \mathfrak{g}_1 \longrightarrow \mathfrak{g}_2
$$

is defined to be a homomorphism of [[dg-algebras]] between their [[Chevalley-Eilenberg algebras]] going the other way:

$$
  CE(\mathfrak{g}_1) \longleftarrow CE(\mathfrak{g}_2)
    \;\colon\;
  f^\ast
$$

(here $f^\ast$ is the primitive concept, and $f$ is defined as the [[formal dual]] of $f$). Hence the [[category]] of super $L_\infty$-algebras of [[finite type]] is the [[full subcategory]]

$$
  s L_\infty Alg \hookrightarrow dgAlg^{op}
$$

of the [[opposite category]] of [[dg-algebras]] on those that are CE-algebras as above.

Finally, the [[cochain cohomology]] of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ of a super $L_\infty$ algebra of [[finite type]] is its _[[L-∞ algebra cohomology]]_ with [[coefficients]] in $\mathbb{R}$:

$$
  H^\bullet(\mathfrak{g}, \mathbb{R})
   \;=\;
  H^\bullet(CE(\mathfrak{g}))
  \,.
$$

=--

+-- {: .num_remark #SuperLInfintiyAsFDA}
###### Remark
**(history of the concept of (super-)$L_\infty$ algebras)**

The identification of the concept of (super-)$L_\infty$-algebras has a non-linear history:

[[L-∞ algebras]] in the incarnation of higher brackets satisfying a higher Jacobi identity,
def. \ref{sLInfinityDefinitionViaGeneralizedJacobiIdentity}
and remark \ref{LInfinityTerminology}, were introduced in
[Lada-Stasheff 92](https://ncatlab.org/nlab/show/L-infinity-algebra#LadaStasheff92), based on the
example of such a structure on the [[BRST complex]] of the [[bosonic string]] that
was found in the construction of [[closed string field theory]] in [Zwiebach 92](string+field+theory#Zwiebach93).
Some of this history is recalled in [Stasheff 16](#L-infinity-algebra#Stasheff16).

The observation that these systems of higher brackets are fully characterized by their Chevalley-Eilenberg dg-(co-)algebras
(def. \ref{SuperLInfinityCEAlgebra})
is due to [Lada-Markl 94](https://ncatlab.org/nlab/show/L-infinity-algebra#LadaMarkl94).
See [Sati-Schreiber-Stasheff 08, around def. 13](#SatiSchreiberStasheff08).

But in this dual incarnation, [[L-∞ algebras]] and more generally [[super L-∞ algebras]] (of [[finite type]])
had secretly been introduced within the [[supergravity]] literature already in [D'Auria-Fré-Regge 80](#DAuriaFreRegge80) and explicitly in [van Nieuwenhuizen 82](#Nieuwenhuizen82).
The concept was picked up in the [[D'Auria-Fré formulation of supergravity]] ([D'Auria-Fré 82](#DAuriaFre82)) and eventually came to be referred to as "FDA"s (short for "free differential algebra") in the [[supergravity]] literature (but beware that these dg-algebras
are in general [[free construction|free]] only as graded-[[supercommutative superalgebras]], not as differential algebras)
The relation between super $L_\infty$-algebras and the "FDA"s of the [[supergravity]] literature is made explicit in ([FSS 13](#FSS13)).

| [[higher Lie theory]] | [[supergravity]] |
|----------------------------|-----------------------|
| $\,$ [[super Lie n-algebra]] $\mathfrak{g}$ $\,$ | $\,$ "FDA" $CE(\mathfrak{g})$ $\,$ |

The construction in [van Nieuwenhuizen 82](#Nieuwenhuizen82) in turn was motivated by the [[Sullivan algebras]]
in [[rational homotopy theory]] ([Sullivan 77](#rational+homotopy+theory#Sullivan77)). Indeed, their
dual incarnations in rational homotopy theory are [[dg-Lie algebras]] ([Quillen 69](#rational+homotopy+theory#Quillen69)),
hence a special case of $L_\infty$-algebras (remark \ref{LInfinityTerminology})

This close relation between [[rational homotopy theory]] and [[higher Lie theory]] might have remained less of a
secret had it not been for the focus of [[Sullivan minimal models]] on the non-[[simply connected topological space|simply connected]]
case, which excludes the ordinary [[Lie algebras]] from the picture. But the Quillen model of
[[rational homotopy theory]] effectively says that for $X$ a [[rational topological space]] then its [[loop space]]
[[∞-group]] $\Omega X$ is reflected, infinitesimally, by an [[L-∞ algebra]]. This perspective began to
receive more attention when the [[Sullivan construction]] in [[rational homotopy theory]] was
concretely identified as higher [[Lie integration]] in [Henriques 08](#Lie+integration#Henriques).
A modern review that makes this [[L-∞ algebra]]-theoretic nature of [[rational homotopy theory]] manifest is in
[Buijs-Félix-Murillo 12, section 2](rational+homotopy+theory#BuijsFelixMurillo12).

=--

However, what has not been used in the "FDA" literature is that super $L_\infty$-algebras are objects in _[[homotopy theory]]_:

+-- {: .num_prop}
###### Proposition
**([Pridham 10, prop. 4.36](https://ncatlab.org/nlab/show/model+structure+for+L-infinity+algebras#Pridham))**

There exists a [[model category]] such that

1. its [[fibrant objects]] are the (super-)[[L-∞ algebras]]

   with the above [[homomorphisms]] between them;

1. and

   * the weak equivalences between (super-)$L_\infty$-algebras are the [[quasi-isomorphisms]];

   * fibrations between (super-)$L_\infty$-algebras are the surjections

   on the underlying [[chain complex]] (using the unary part of the differential).

For more see at _[[model structure for L-infinity algebras]]_.

=--

$\,$

Concretely, this implies in particular that every [[homomorphisms]] of [[super L-∞ algebras]]

$$
  \array{
    \mathfrak{g}_1
    \\
    & {}_{\mathllap{f}}\searrow
    \\
    && \mathfrak{g}_2
  }
$$

is the composite of a [[quasi-isomorphism]] followed by a surjection

$$
  \array{
    \mathfrak{g}_1
     && \overset{\text{quasi-iso}}{\longrightarrow} &&
    \widetilde \mathfrak{g}_1
    \\
    & {}_{\mathllap{f}}\searrow && \swarrow_{\mathrlap{ {f_{fib}} \atop {\text{surjection}}}}
    \\
    && \mathfrak{g}_2
  }
  \,.
$$

That surjective homomorphism $f_{fib}$

is called a _[[fibrant replacement]]_ of $f$.

$\,$

+-- {: .num_defn }
###### Definition

Given [[homomorphisms]] of [[super L-∞ algebras]]

$$
  \mathfrak{g}_1 \overset{f}{\longrightarrow} \mathfrak{g}_2
$$

then its **[[homotopy fiber]]** $hofib(f)$ is the [[kernel]] of any fibrant replacement

$$
  hofib(f)
   \;\coloneqq\;
  ker(f_{fib})
  \,.
$$

=--

Standard facts in [[homotopy theory]] assert that $hofib(f)$ is well-defined up to [[quasi-isomorphism]].
See at _[Introduction to homotopy theory -- Homotopy fibers](https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+P#HomotopyFibers)_.

The following is the key fact about homotopy fibers in the homotopy theory of super $L_\infty$-algebras
which we will use:

+-- {: .num_prop #HomotopyFibersOfLInfinityCocycles}
###### Proposition
**([Fiorenza-Sati-Schreiber 13, theorem 3.5](#FSS13))**

Write

$$
  b^{p+1}\mathbb{R}
$$

for the **[[line Lie n-algebra|line Lie (p+1)-algebra]]**, given by

$$
  CE(b^{p+1}\mathbb{R})
   \;=\;
  \left(
    \wedge^\bullet \underset{\text{single generator} \atop \text{in deg.} \, (p+2,even)}{\underbrace{\langle c_{p+2} \rangle}}
    \;,\;
    d_{B^{p+1}\mathbb{R}} = 0
  \right)
  \,.
$$

A $(p+2)$-cocycle on an $L_\infty$-algebra is equivalently a homomorphim

$$
  \mu_{p+2} \;\colon\; \mathfrak{g} \longrightarrow B^{p+1}\mathbb{R}
  \,.
$$

The [[homotopy fiber]] of this map

$$
  \array{
    \widehat{\mathfrak{g}}
    \\
    {}^{\mathllap{hofib(\mu_{p+2})}}\downarrow
    \\
    \mathfrak{g}
    &\underset{\mu_{p+2}}{\longrightarrow}& B^{p+1}\mathbb{R}
  }
$$

is given by adjoining to $CE(\mathfrak{g})$ a single generator $b_{p+1}$

forced to be a potential for $\mu_{p+2}$:

$$
  CE(\widehat{\mathfrak{g}})
  \;\simeq\;
  CE(\mathfrak{g})[b_{p+1}]/(d b_{p+1} = \mu_{p+2})
  \,.
$$

=--

As a slogan: _The [[higher central extensions]] classified by higher cocycles are their [[homotopy fibers]]._

+-- {: .num_example}
###### Example

The homotopy fiber of a 2-cocycle on a [[super Lie algebra]] is the classical [[central extension]] that it classifies:

Let $\mathfrak{g}$ be a super Lie algebra of finite dimension, and let $\omega_2 \in \wedge^2 \mathfrak{g}^\ast$
be a 2-cocycle $d_{\mathfrak{g}} \omega_2 = 0$. Then prop. \ref{HomotopyFibersOfLInfinityCocycles} says
that the homotopy fiber $\widehat{\mathfrak{g}} \longrightarrow \mathfrak{g}$
of the corresponding morphism $\mathfrak{g} \overset{\omega_2}{\longrightarrow} b \mathbb{R}$
has [[Chevalley-Eilenberg algebra]] that of $\mathfrak{g}$ with one new generator $c \in \wedge^1 (\widehat {\mathfrak{g}})$
adjoined, with differential given by

$$
  d_{\mathfrak{g}} c = \omega_2
  \,.
$$

Now by def. \ref{CEAlgebraofSuperLieAlgebra} this differential enocodes the linear dual of the
Lie bracket. Hence if $k$ denotes the dual element of $c$ then the Lie brakcet of $\widehat{\mathfrak{g}}$
is modified on elements of $\mathfrak{g}$ to be

$$
  [x,y]_{\widehat{\mathfrak{g}}} = [x,y]_{\mathfrak{g}}+ \omega_2(x,y) k
  \,.
$$

This is exactly the ordinary formula for the central extension of $\mathfrak{g}$ by $\omega_2$.

=--

## From $L_\infty$-Cocycles to higher WZW-type sigma-models
 {#FromCocyclesToWZW}

We have discussed super $L_\infty$-cohomology [above](#SuperLInfinityCohomologyAndFDAs) in generality.
Further [below](#Branes) we consider the exceptional invariant super $L_\infty$-cohomology classes
that emanate out of the [[superpoint]]. There we will see that each of them is to correspond to precisely
one species of [[super p-branes]] as discussed in the [[string theory]] literature. Here, in order
to substantiate this, we discuss in generality how by the mathematics of higher [[Lie integration]]
every super $L_\infty$-cocycle induces a functional on a [[mapping space]] that
may be regarded as the [[action functional]] defining a [[sigma-model]] description for a
fundamental $p$-brane. These are higher order generalizations of the famous _[[Wess-Zumino-Witten model]]_.

### Higher Lie integration
 {#HigherLieIntegration}

We discuss differential refinements of the "path method" of [[nLab:Lie integration]] for [[nLab:L-infinity-algebras]]. The key observation for interpreting the following def. \ref{DefinitionOfLInfinityFormsOnSimplices} is this:

+-- {: .num_remark}
###### Remark

For $\mathfrak{g}$ an [[nLab:L-∞ algebra]], and given a [[nLab:smooth manifold]] $U$, then

1. the _flat_ [[nLab:L-∞ algebra valued differential forms]] on $U$ are equivalently the dg-algebra homomorphisms

   $$
     \Omega_{flat}(U,\mathfrak{g})
     =
     Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet(U)_{\mathrm{dR}})
   $$

1. a finite [[nLab:gauge transformation]] between two such forms is equivalently a homotopy

   $$
    \array{
       & & \Omega_{flat}(U \times \Delta^1,\mathfrak{g})
       \\
       & {}^{(-)|_0}\swarrow && \searrow^{\mathrlap{(-)|_1}}
       \\
       \Omega_{flat}(U ,\mathfrak{g})
       && &&
       \Omega_{flat}(U, \mathfrak{g})
     }
     \,.
   $$

For more details see at _[infinity-Lie algebroid-valued differential form -- Integration of infinitesimal gauge transformations](http://ncatlab.org/nlab/show/infinity-Lie+algebroid-valued+differential+form#InfGaugeTrafo)_.

=--

+-- {: .num_defn #DefinitionOfLInfinityFormsOnSimplices}
###### Definition

For $\mathfrak{g}$ an [[nLab:L-∞ algebra]], write:

* $CE(\mathfrak{g})$ for the [[nLab:Chevalley-Eilenberg algebra]] of an [[nLab:L-∞ algebra]] $\mathfrak{g}$;

* $\Delta^\bullet_{smth} \colon \Delta \to SmoothMfd$ for the [[nLab:cosimplicial object|cosimplicial]] [[nLab:smooth manifold]] [[nLab:manifold with corners|with corners]] which is in degree $k$ the standard $k$-simplex $\Delta^k \hookrightarrow \mathbb{R}^{k+1}$;

* $\Omega^\bullet_{si}(\Delta_{smth}^k)$ for the [[nLab:de Rham complex]] of those [[nLab:differential forms]] on $\Delta_{smth}^k$ which have [[nLab:sitting instants]], in that in an [[nLab:open neighbourhood]] of the [[nLab:boundary]] they are constant perpendicular to any face on their value at that face;

* $\Omega^\bullet_{si}(U \times \Delta_{smth}^k)$ for $U \in SmoothMfd$ for the [[nLab:de Rham complex]] of differential forms on $U \times \Delta^k$ which when restricted to each point of $U$ have sitting instants on $\Delta^k$;

* $\Omega^\bullet_{vert,si}(U \times \Delta_{smth}^k)$ for the subcomplex of forms that in addition are [[nLab:vertical differential forms]] with respect to the projection $U \times \Delta^k \to U$.

=--

+-- {: .num_defn #SimplicialLieIntegrationOfLinfinityAlgebra}
###### Definition

For $\mathfrak{g}$ an [[nLab:L-∞ algebra]], write

* $\exp(\mathfrak{g})_\bullet \in PreSmoothTypes = PSh(CartSp,sSet)$

  for the [[nLab:simplicial presheaf]]

  $$
    \exp(\mathfrak{g})
    \colon
    (U \times k) \mapsto Hom_{dgAlg}( CE(\mathfrak{g}), \Omega^\bullet_{vert,si}(U \times \Delta_{smth}^k) )
    \,.
  $$

  which is the universal _[[nLab:Lie integration]]_ of $\mathfrak{g}$;

* $\flat_{dR}\exp(\mathfrak{g})_\bullet \in PreSmoothTypes = PSh(CartSp,sSet)$

  for the [[nLab:simplicial presheaf]]

  $$
    \flat_{dR}\exp(\mathfrak{g})_\bullet
    \;\colon\;
    (U \times k) \mapsto
    Hom_{dgAlg}( CE(\mathfrak{g}), \Omega^{\bullet\geq 1, \bullet}_{si}(U \times \Delta^k_{smth}) )
  $$

  of those differential forms on $U \times \Delta^\bullet$ with at least one leg along $U$;

* $\Omega^1_{flat}(-,\mathfrak{g})
   \coloneqq
   \flat_{dR}\exp(\mathfrak{g})_0
  \longrightarrow \flat_{dR}\exp(\mathfrak{g})_\bullet$

  for the canonical inclusion of the degree-0 piece, regarded as a simplicial constant simplicial presheaf.

=--

+-- {: .num_example #ExamplesOfLieIntegration}
###### Example

From the discussion at _[[nLab:Lie integration]]_:

1. $\Omega^1_{flat}(-,b^{p+1}\mathbb{R}) = \mathbf{\Omega}^{p+2}_{cl}$;

1. for $\mathfrak{g}$ an ordinary [[nLab:Lie algebra]], then for the [[nLab:simplicial skeleton|2-coskeleton]] (by _[this discussion](http://ncatlab.org/nlab/show/Lie+integration#LieAlgebrasToLieGroups)_)

   $$

     cosk_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G_\bullet
   $$

   for $G$ the simply connected [[nLab:Lie group]] associated to $\mathfrak{g}$ by traditional [[nLab:Lie theory]]. If $\mathfrak{g}$ is furthermore a [[nLab:semisimple Lie algebra]], then also

   $$
     cosk_3 \exp(\mathfrak{g}) \simeq \mathbf{B}G_\bullet
   $$

1. for $\mathfrak{g} = b^{p}\mathbb{R}$ the [[nLab:line Lie n-algebra|line Lie p+1]]-algebra, then (by [this proposition](http://ncatlab.org/nlab/show/Lie+integration#LieIntegrationOfLinenLieAlgebra))

   $$
     \exp(b^p \mathbb{R}) \simeq \mathbf{B}^{p+1}\mathbb{R}
     \,.
   $$

=--

+-- {: .num_remark #LieIntegrationIsFunctorial}
###### Remark

The constructions in def. \ref{SimplicialLieIntegrationOfLinfinityAlgebra} are clearly [[nLab:functor|functorial]]: given a [[nLab:homomorphism]] of [[nLab:L-∞ algebras]]

$$
  \mu \;\colon\; \mathfrak{g} \longrightarrow \mathfrak{h}
$$

it prolongs to a homomorphism of presheaves

$$
  \mu \colon \Omega^1_{flat}(-,\mathfrak{g})
  \longrightarrow \Omega^1(-,\mathfrak{h})
$$

and of [[nLab:simplicial presheaves]]

$$
  \exp(\mu)
  \;\colon\;
  \exp(\mathfrak{g})
  \longrightarrow
  \exp(\mathfrak{h})
$$

etc.

=--

+-- {: .num_example #LInfinityCocyclesAsMorphisms}
###### Example

According to the [above](#LInfinityCohomology), a degree-$(p+2)$-[[nLab:L-∞ cocycle]] $\mu$ on an [[nLab:L-∞ algebra]] $\mathfrak{g}$ is a homomorphism of the form

$$
  \mu \colon \mathfrak{g} \longrightarrow b^{p+1}\mathbb{R}

$$

to the [[nLab:line Lie n-algebra|line Lie (p+2)-algebra]] $b^{p+1}\mathbb{R}$.
The [[nLab:formal dual]] of this is the homomorphism of [[nLab:dg-algebras]]

$$
  CE(\mathfrak{g}) \longleftarrow CE(b^{p+1}\mathbb{R}) \colon \mu^\ast
$$

which manifestly picks a $d_{CE(\mathfrak{g})}$-closed element in $CE^{p+2}(\mathfrak{g})$.

Precomposing this $\mu^\ast$ with a flat [[nLab:L-∞ algebra valued differential form]]

$$
  A \in \Omega^1_{flat}(X,\mathfrak{g}) = Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet(X))
$$

yields, by example \ref{ExamplesOfLieIntegration}, a plain closed $(p+2)$-form

$$
  \mu^\ast A \in \Omega^{p+2}_{cl}(X)
  \,.
$$

=--

+-- {: .num_defn #GroupOfPeriods}
###### Definition

Given an [[nLab:L-∞ cocycle]]

$$
  \mu \colon \mathfrak{g} \longrightarrow b^{p+1}\mathbb{R}
  \,,
$$

as in example \ref{LInfinityCocyclesAsMorphisms},
then its _group of [[nLab:periods]]_ is the [[nLab:discrete group|discrete]] additive [[nLab:subgroup]] $\Gamma \hookrightarrow \mathbb{R}$ of those [[nLab:real numbers]] which are [[nLab:integration of differential forms|integrations]]

$$
  \underset{\partial \Delta^{p+3}_{smth}}{\int} \mu^\ast A \in \mathbb{R}
$$

of the value of $\mu$, as in example \ref{LInfinityCocyclesAsMorphisms},
on [[nLab:L-∞ algebra valued differential forms]]

$$
  A \in \Omega^1_{flat}(\partial \Delta^{p+3}_{smth})
  \,,
$$

over the [[nLab:boundary of a simplex|boundary of the (p+3)-simplex]] (which are forms with sitting instants on the $(p+2)$-dimensional faces that glue together; without restriction of generality we may simply consider forms on the $(p+2)$-[[nLab:sphere]] $S^{p+2}$).

=--

+-- {: .num_prop #TruncatedLieIntegrationOfCocycle}
###### Proposition

Given an [[nLab:L-∞ cocycle]] $\mu \colon \mathfrak{g} \to b^{p+1}\mathbb{R}$,
as in example \ref{LInfinityCocyclesAsMorphisms},
then the universal Lie integration of $\mu$, via def. \ref{SimplicialLieIntegrationOfLinfinityAlgebra} and remark \ref{LieIntegrationIsFunctorial}, descends to the $(p+2)$-[[nLab:coskeleton]]

$$
  \mathbf{B}G \coloneqq cosk_{p+2}\exp(\mathfrak{g})
$$

up to quotienting the coefficients $\mathbb{R}$ by the
group of [[nLab:periods]] $\Gamma$ of $\mu$, def. \ref{GroupOfPeriods}, to yield the bottom morphism in

$$
  \array{
     \exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\longrightarrow}&
     \mathbf{B}^{p+2}\mathbb{R}
     \\
     \downarrow && \downarrow
     \\
     \mathbf{B}G
     &\stackrel{\mathbf{c}}{\longrightarrow}&
     \mathbf{B}^{p+2} (\mathbb{R}/\Gamma)
  }
  \,.
$$

=--

This is due to ([FSS 12](Lie+integration#FSS12)).

Here and in the following we are freely using example \ref{ExamplesOfLieIntegration} to identify $\exp(b^{p+1}\mathbb{R}) \simeq \mathbf{B}^{p+2}\mathbb{R}$. Establishing this is the only real work in prop. \ref{TruncatedLieIntegrationOfCocycle}.

### Higher Maurer-Cartan forms

+-- {: .num_defn #FlatModality}
###### Definition

Write

$$
  \flat \;\colon\; L_{lwhe} PSh(\mathrm{CartSp}, sSet) \longrightarrow L_{lwhe} PSh(\mathrm{CartSp}, sSet)
$$

for the operation that evaluates a [[nLab:simplicial presheaf]] on the point and then extends the result back as a constant presheaf. This comes with a canonical [[nLab:counit of a comonad|counit]] [[nLab:natural transformation]]

$$
  \epsilon^\flat \colon \flat \to Id
  \,.
$$

=--

+-- {: .num_example #Flatbg}
###### Example

For $G$ a [[nLab:Lie group]] and

$$
  \mathbf{B}G  : U \mapsto N C^\infty(U,G) \in L_{lwhe} PSh(\mathrm{CartSp}, sSet)
$$

for its stacky [[nLab:delooping]], which is the universal [[nLab:moduli stack]] of $G$-[[nLab:principal bundles]], then given a $G$-principal bundle $P$ [[nLab:modulating morphism|modulated]] by a map

$$
  X \longrightarrow \mathbf{B}G
$$

then a lift $\nabla$ in the homotopy-commutative diagram

$$
  \array{
     && \flat \mathbf{B}G
     \\
     &{}^{\mathllap{\nabla}}\nearrow& \downarrow
     \\
     X &\longrightarrow& \mathbf{B}G
  }
$$

is equivalently a [[nLab:flat connection]] on $G$. Hence $\flat \mathbf{B}G$ is the universal moduli stack for [[nLab:flat connections]]. Whence the symbol "$\flat$".

=--

+-- {: .num_defn #MaurerCartanForm}
###### Definition

Given $G$ any [[nLab:smooth infinity-group]], denote the double [[nLab:homotopy fiber]] of the [[nLab:counit of a comonad|counit]] $\epsilon^\flat$, def. \ref{FlatModality} as follows:

$$
  \array{
    G
    \\
    \downarrow^{\mathrlap{G}}
    \\
    \flat_{dR} \mathbf{B}G &\longrightarrow& \flat \mathbf{B}G
    \\
    && \downarrow^{\mathrlap{\epsilon^{\flat}}}
    \\
    && \mathbf{B}G
  }
  \,.
$$

We say that

* $\flat_{dR}\mathbf{B}G$ is the _flat de Rham coefficients_ for $G$;

* $\theta_G$ is the _[[nLab:Maurer-Cartan  form]]_ of $G$.

=--

+-- {: .num_example #MaurerCartanOnLieGroup}
###### Example

In the situation of example \ref{Flatbg} where $G$ is an ordinary [[nLab:Lie groups]] and with $\mathfrak{g}$ denoting the Lie algebra of $G$, then we get that

* $\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})$ is the sheaf of flat [[nLab:Lie algebra valued differential forms]];

* $\theta_{G}$ is (under the [[nLab:Yoneda embedding]]) the [[nLab:Maurer-Cartan form]] on $G$ in the traditional sense.

=--

### Higher WZW terms
 {#HigherWZWTerms}

We discuss now how every [[nLab:L-∞ cocycle]] $\mu \;\colon\; \mathfrak{g} \longrightarrow b^{p+1} \mathbb{R}$
induces via differential higher [[nLab:Lie integration]] a [[nLab:higher WZW term]] for a $p$-brane [[nLab:sigma model]]
with [[nLab:target space]] a differential extension $\tilde G$ of a [[nLab:smooth infinity-group]] $G$ that integrates
$\mathfrak{g}$. In the next section [below](#ConsecutiveWZWTermsAndTwists) we characterize these differential
extensions and find that they are given by bundles of [[nLab:moduli stacks]] for [[nLab:higher gauge fields]]
on the $p$-brane [[nLab:worldvolume]]. This means that the higher WZW terms obtained here are in fact higher analogs
of the _[[nLab:gauged WZW model]]_.

(The following construction is from [FSS 13, section 5](#FSS13), streamlined a little.)

$\,$

+-- {: .num_prop #HodgeFiltrationRefinementFromLInfinityCocycles}
###### Proposition

For $\mu \colon \mathfrak{g}\longrightarrow b^{p+1}\mathbb{R}$
an [[nLab:L-∞ cocycle]], then there is the following canonical [[nLab:commuting diagram]] of [[nLab:simplicial presheaves]]

$$
   \array{
    \Omega^1_{flat}(-,G)
    &\stackrel{\mu}{\longrightarrow}&
    \Omega^2_{cl}(-,\mathbb{G})
    \\
    \downarrow && \downarrow
    \\
    \flat_{dR}\mathbf{B}G
    &\stackrel{\flat_{dR} \mathbf{c}}{\longrightarrow}&
    \flat_{dR}\mathbf{B}^2 \mathbb{G}
  }
  \;\;\;
  \coloneqq
  \;\;\;
  \array{
    \Omega^1_{flat}(-,\mathfrak{g})
    &\stackrel{\mu}{\longrightarrow}&
    \mathbf{\Omega}^{p+2}_{cl}
    \\
    \downarrow && \downarrow
    \\
    \flat_{dR}\exp(\mathfrak{g})_\bullet
    &
    \stackrel{\flat_{dR}\exp(\mu)}{\longrightarrow}
    &
    \flat_{dR}\mathbf{B}^{p+2}\mathbb{R}
    \\
    \downarrow && \downarrow
    \\
    \flat_{dR}\mathbf{B}G
    &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}&
    \flat_{dR}\mathbf{B}^{p+2}\mathbb{R}
  }
$$

which is given

* on the top by def. \ref{SimplicialLieIntegrationOfLinfinityAlgebra},
example \ref{ExamplesOfLieIntegration},
remark \ref{LieIntegrationIsFunctorial},

* on the bottom by applying the operation of def. \ref{MaurerCartanForm} to the commuting diagram provided by prop. \ref{TruncatedLieIntegrationOfCocycle},

=--

+-- {: .num_defn #DifferentiallyTiwstedGroup}
###### Definition

Write

$$
  \tilde G
  \coloneqq
  G \underset{\flat_{dR}\mathbf{B}G}{\times} \mathbf{\Omega}^1_{flat}(-,\mathfrak{g})
$$

for the [[nLab:homotopy pullback]] of the left vertical morphism in prop. \ref{HodgeFiltrationRefinementFromLInfinityCocycles} along (the [[nLab:modulating morphism]] for) the [[nLab:Maurer-Cartan form]] $\theta_G$ of $G$, i.e. for the object sitting in a homotopy Cartesian square of the form

$$
  \array{
    \tilde G &\stackrel{\theta_{\tilde G}}{\longrightarrow}&  \Omega^1_{flat}(-,\mathfrak{g})
    \\
    \downarrow && \downarrow
    \\
    G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR}\mathbf{B}G
  }
  \,.
$$

=--

+-- {: .num_example #ExamplesForTildeGInModLayer}
###### Example

For the special case that $G$ is an ordinary [[nLab:Lie group]], then $\flat_{dR}\mathbf{B}G \simeq \Omega^{1}_{flat}(-,\mathfrak{g})$, by example \ref{MaurerCartanOnLieGroup}, hence in this case the morphism being pulled back in def. \ref{DifferentiallyTiwstedGroup} is an [[nLab:equivalence]], and so in this case nothing new happens, we get $\tilde G \simeq G$.

On the other extreme, when $G = \mathbf{B}^{p}U(1)$ is the [[nLab:circle n-group|circle (p+1)-group]], then def. \ref{DifferentiallyTiwstedGroup}  reduces to the [[nLab:homotopy pullback]] that characterizes the [[nLab:Deligne complex]] and hence yields

$$
  \widetilde{\mathbf{B}^p U(1)}
  \simeq

  \mathbf{B}^p U(1)_{conn}
  \,.
$$

This shows that def. \ref{DifferentiallyTiwstedGroup} is a certain non-abelian generalization
of [[nLab:ordinary differential cohomology]]. We find further characterization of this below in corollary \ref{TildeHatGIsDifferentialModuliBundle}, see remark \ref{InterpretationOfTildeHatG}.

=--

+-- {: .num_remark}
###### Remark

From example \ref{ExamplesForTildeGInModLayer}
one reads off the conceptual meaning of def. \ref{DifferentiallyTiwstedGroup}:
For $G$ a [[nLab:Lie group]], then the de Rham coefficients are just globally defined differential forms, $\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})$ (by the discussion [here](smooth+infinity-groupoid+--+structures#deRhamWithCoefficientsInBOfLieGroup)), and in particular therefore the [[nLab:Maurer-Cartan form]] $\theta_G \colon G \to \flat_{dR}\mathbf{B}G$ is a globally defined differential form. This is no longer the case for general [[nLab:smooth ∞-groups]] $G$. In general, the [[nLab:Maurer-Cartan forms]] here is a [[nLab:cocycle]] in [[nLab:hypercohomology]], given only locally by differential forms, that are glued nontrivially, in general, via [[nLab:gauge transformations]] and [[nLab:higher gauge transformations]] given by lower degree forms.

But the WZW terms that we are after are supposed to [[nLab:prequantizations]] of globally defined Maurer-Cartan forms. The homotopy pullback in def. \ref{DifferentiallyTiwstedGroup} is precisely the [[nLab:universal construction]] that enforces the existence of a globally defined Maurer-Cartan form for $G$, namely  $\theta_{\tilde G} \colon \tilde G \to \Omega^1_{flat}(-,\mathfrak{g})$.

=--

+-- {: .num_defn #WZWTermFromLieIntegration}

###### Definition

Given an [[nLab:L-∞ cocycle]] $\mu \colon \mathfrak{g}\longrightarrow b^{p+1}\mathbb{R}$, then via prop. \ref{HodgeFiltrationRefinementFromLInfinityCocycles}, prop. \ref{TruncatedLieIntegrationOfCocycle} and using the [[nLab:natural transformation|naturality]] of the [[nLab:Maurer-Cartan form]], def. \ref{MaurerCartanForm}, we have a morphism of [[nLab:cospan]] [[nLab:diagrams]] of the form

$$
  \array{
    \Omega^1_{flat}(-,\mathfrak{g})
    &\stackrel{\mu}{\longrightarrow}&
    \mathbf{\Omega}^{p+2}_{cl}
    \\
    \downarrow && \downarrow
    \\
    \flat_{dR} \mathbf{B}G
    &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}&
    \flat_{dR}\mathbf{B}^{p+2}\mathbb{R}
    \\
    \uparrow^{\mathrlap{\theta_G}} && \uparrow^{\mathrlap{\theta_{\mathbf{B}^{p+1}(\mathbb{R}/\Gamma)}}}
    \\
    G &\stackrel{\Omega \mathbf{c}}{\longrightarrow}&
    \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)
  }
  \,.
$$

By the [[homotopy fiber product]] characterization of the [[Deligne complex]] ([prop.](Deligne+cohomology#HomotopyFiberProductCharacterization)),
this yields a morphism of the form

$$
  \mathbf{L}_{WZW}^{\mu}
   \;\colon\;
   \tilde G
   \longrightarrow
   \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}
  \,.
$$

which [[nLab:modulating morphisms|modulates]] a [[nLab:circle n-bundle with connection|p+1-connection]]/[[nLab:Deligne cohomology|Deligne cocycle]] on the differentially extended smooth $\infty$-group $\tilde G$ from def. \ref{DifferentiallyTiwstedGroup}.

This we call the _WZW term_ obtained by universal Lie integration from $\mu$.

=--

Essentially this construction originates in ([FSS 13](#FSS13)).

+-- {: .num_remark}
###### Remark

The WZW term of def. \ref{WZWTermFromLieIntegration} is a [[nLab:prequantization]] of  $\omega \coloneqq \mu(\theta_{\tilde G})$, hence a lift $\mathbf{L}_{WZW}^\mu$ in

$$
  \array{
     && \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}
     \\
     & {}^{\mathllap{\mathbf{L}_{WZW}^\mu}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}}
     \\
     \tilde G
     &\stackrel{\mu(\theta_{\tilde G})}{\longrightarrow}& \mathbf{\Omega}^{p+2}
  }
  \,.
$$

=--

### Consecutive WZW terms and twists
 {#ConsecutiveWZWTermsAndTwists}

[Above](#HigherWZWTerms) we discussed how a single [[nLab:L-∞ cocycle]] [[nLab:Lie integration|Lie integrates]] to a [[nLab:higher WZW term]].
More generally, one has a sequence of [[nLab:L-∞ cocycles]], each defined on the extension that is classified by the previous one -- a [[schreiber:The brane bouquet|bouquet]] of cocycles. Here we discuss how in this case the higher WZW terms at each stage
relate to each other. (The following statements are corollaries of [FSS 13, section 5](#FSS13)).

$\,$

In each stage, for $\mu_1 \colon \mathfrak{g}\to b^{p_1+1}\mathbb{R}$ a cocycle and $\hat {\mathfrak{g}} \to \mathfrak{g}$ the extension that it classifies (its [[nLab:homotopy fiber]]), then the next cocycle is of the form $\mu_2 \colon \hat \mathfrak{g} \to b^{p_2+1}\mathbb{R}$

$$
  \array{
    \hat {\mathfrak{g}} &\stackrel{\mu_2}{\longrightarrow}& b^{p_2+1}\mathbb{R}
    \\
    \downarrow
    \\
    \mathfrak{g} &\stackrel{\mu_1}{\longrightarrow}& b^{p_1+1}\mathbb{R}
  }
  \,.
$$

+-- {: .num_lemma #CharacterizationOfLInfinityHomotopyFibers}
###### Lemma

The [[nLab:homotopy fiber]] $\hat \mathfrak{g} \to \mathfrak{g}$ of $\mu_1$ is given by the ordinary [[nLab:pullback]]

$$
  \array{
    \hat \mathfrak{g} &\longrightarrow& e b^{p_1} \mathbb{R}
    \\
    \downarrow && \downarrow
    \\
    \mathfrak{g} &\stackrel{\mu_1}{\longrightarrow}& b^{p_1+1}\mathbb{R}
  }
  \,,
$$

where $e b^{p_1}\mathbb{R}$ is defined by its [[nLab:Chevalley-Eilenberg algebra]]
$CE(e b^{p_1}\mathbb{R})$ being the [[nLab:Weil algebra]] of $b^{p_1}\mathbb{R}$, which is the [[nLab:free construction|free]] differential graded algebra on a generator in degree $p_1$, and where the right vertical map takes that generator to 0 and takes its free image under the differential to the generator of $CE(b^{p_1+1}\mathbb{R})$.

=--

+-- {: .proof}
###### Proof

This follows with the [recognition principle for L-∞ homotopy fibers](model+structure+for+L-infinity+algebras#HomotopyFiberProducts).

=--

+-- {: .num_cor #PullbackOfDifferentialFormCoefficients}
###### Corollary

A [[nLab:homotopy fiber sequence]] of [[nLab:L-∞ algebras]] $\hat \mathfrak{g} \to \mathfrak{g}\stackrel{\mu}{\longrightarrow} b^{p+1}\mathbb{R}$ induces a  [[nLab:homotopy pullback]] diagram of the associated objects of [[nLab:L-∞ algebra valued differential forms]], def. \ref{SimplicialLieIntegrationOfLinfinityAlgebra}, of the form

$$
  \array{
    \mathbf{\Omega}^1_{flat}(-,\hat {\mathfrak{g}})
    &\stackrel{}{\longrightarrow}&
    \mathbf{\Omega}^{p+1}
    \\
    \downarrow && \downarrow^{\mathbf{d}}
    \\
    \mathbf{\Omega}^1_{flat}(-,\mathfrak{g})
    &\stackrel{\mu}{\longrightarrow}&
    \mathbf{\Omega}^{p+2}_{cl}
  }
$$

(hence an ordinary [[nLab:pullback]] of [[nLab:presheaves]], since these are all simplicially constant).

=--

+-- {: .proof}
###### Proof

The construction $\mathfrak{g} \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet(-))$ preserves [[nLab:pullbacks]] ($CE$ is an anti-equivalence onto its image, pullbacks of (pre-)-sheaves are computed objectwise, the [[nLab:hom-functor]] preserves pullbacks in the covariant argument).

Observe then (see the discussion at _[[nLab:L-∞ algebra valued differential forms]]_), that while

$$
  \mathbf{\Omega}^{p+2}_{cl} \simeq Hom_{dgAlg}(CE(b^{p+1}), \Omega^\bullet(-))
$$

we have

$$

  \mathbf{\Omega}^{p+1} \simeq Hom_{dgAlg}(W(b^{p}), \Omega^\bullet(-))
  \,.
$$

With this the statement follows by
lemma \ref{CharacterizationOfLInfinityHomotopyFibers}.

=--

+-- {: .num_defn #ConsecutiveCocycles}
###### Definition

We say that a pair of [[nLab:L-∞ cocycles]] $(\mu_1, \mu_2)$ is _consecutive_ if the domain of the second is the extension ([[nLab:homotopy fiber]]) defined by the first

$$
  \array{
    \hat {\mathfrak{g}} &\stackrel{\mu_2}{\longrightarrow}& b^{p_2+1}
    \\
    \downarrow
    \\
    \mathfrak{g} &\stackrel{\mu_1}{\longrightarrow}& b^{p_1+1}\mathbb{R}
  }
$$

and if the truncated [[nLab:Lie integrations]] of these cocycles via prop. \ref{TruncatedLieIntegrationOfCocycle} preserves the extension property in that also

$$
  \hat G \longrightarrow G \overset{\Omega \mathbf{c}_1}{\longrightarrow} \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)
$$

is a [[nLab:homotopy fiber sequence]] of [[nLab:smooth homotopy types]].

=--

+-- {: .num_remark}
###### Remark

The issue of the second clause in def. \ref{ConsecutiveCocycles} is to do with the truncation degrees: the universal untruncated [[nLab:Lie integration]] $\exp(-)$ preserves homotopy fiber sequences, but if there are non-trivial cocycles on $\mathfrak{g}$ in between $\mu_1$ and $\mu_2$, for $p_2 \gt p_1$, then these will remain as nontrivial homotopy groups in the higher-degree truncation $\mathbf{B}G_{2} \coloneqq \tau_{p_2}\exp(\hat\mathfrak{g})$ (see [Henriques 06, theorem 6.4](Lie+integration#Henriques)) but they will be truncated away in $\mathbf{B}G_1 \coloneqq \tau_{p_1}\exp(\mathfrak{g})$ and will hence spoil the preservation of the homotopy fibers through Lie integration.

Notice that extending along consecutive cocycles is like the extension stages in a [[nLab:Whitehead tower]].

=--

Given two consecutive [[nLab:L-∞ cocycles]] $(\mu_1,\mu_2)$, def. \ref{ConsecutiveCocycles}, let

$$
  \mathbf{L}_1 \colon \tilde G \longrightarrow \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn}
$$

and

$$
  \mathbf{L}_2 \colon \widetilde {\hat G} \longrightarrow \mathbf{B}^{p_2+1}(\mathbb{R}/\Gamma_2)_{conn}
$$

be the WZW terms obtained from the two cocycles via def. \ref{WZWTermFromLieIntegration}.

+-- {: .num_prop #ConsecutiveWZWTermsFromConsecutiveLInfinityCocycles}
###### Proposition

There is a [[nLab:homotopy pullback]] square in [[nLab:smooth homotopy types]] of the form

$$
  \array{
     \widetilde {\hat G}
     &\stackrel{}{\longrightarrow}&
     \mathbf{\Omega}^{p_1+1}
     \\
     \downarrow && \downarrow
     \\
     \tilde G
     &\stackrel{\mathbf{L}_1}{\longrightarrow}&
     \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn}
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof

Consider the following [[nLab:pasting]] composite

$$
  \array{
    \mathbf{\Omega}^{p_1+1}
    &\longrightarrow&
    \ast
    &\longleftarrow&
    \ast
    \\
    {}^{\mathllap{\mathbf{d}}}\downarrow &\swArrow& \downarrow && \downarrow
    \\
    \mathbf{\Omega}^{p_1+2}
    &\longrightarrow&
    \flat_{dR}\mathbf{B}^{p_1+2}\mathbb{R}
    &\stackrel{\theta_{\mathbf{B}^{p_1}\mathbb{R}}}{\longleftarrow}&
    \mathbf{B}^{p_1+1}\mathbb{R}
     \\
     \uparrow^{\mathrlap{\mu_1}} && \uparrow^{\mathrlap{\flat_{dR} \mathbf{c}_1 }} && \uparrow^{\mathrlap{\Omega \mathbf{c}_1}}
     \\
     \mathbf{\Omega}^1_{flat}(-,\mathfrak{g})
     &\stackrel{}{\longrightarrow}&
     \flat_{dR}\mathbf{B}G
     &\stackrel{\theta_G}{\longleftarrow}&
     G
  }
  \,,
$$

where

* the top left square is the evident homotopy;

* the bottom left square is from prop. \ref{HodgeFiltrationRefinementFromLInfinityCocycles};

* the top right square expresses that $\theta$ preserves the basepoint;

* the bottom right square is the naturality of the [[nLab:Maurer-Cartan form]] construction.

Under forming [[nLab:homotopy limits]] over the _horizontal_ cospan diagrams here, this turns into

$$
  \array{
     \mathbf{\Omega}^{p_1+1}
     \\
     \downarrow
     \\
     \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn}
     \\
     \uparrow^{\mathrlap{\mathbf{L}_1}}
     \\
     \tilde G
  }
$$

by def. \ref{WZWTermFromLieIntegration}. On the other hand, forming homotopy limits _vertically_ this turns into

$$
  \array{
    \mathbf{\Omega}^1_{flat}(-,\hat \mathfrak{g})
    &\longrightarrow&
    \flat_{dR}\mathbf{B}G_2
    &\stackrel{\theta_{\hat G}}{\longleftarrow}&
    \hat G
  }
$$

(on the left by corollary \ref{PullbackOfDifferentialFormCoefficients}, on the right by the second clause in def. \ref{ConsecutiveCocycles}).

The homotopy limit over that last [[nLab:cospan]], in turn, is $\widetilde{\hat G}$. This implies the claim by the fact that homotopy limits commute with each other.

=--

+-- {: .num_remark}
###### Remark

Prop. \ref{ConsecutiveWZWTermsFromConsecutiveLInfinityCocycles} says how consecutive pairs of $L_\infty$-cocycles Lie integrate suitably to consecutive pairs of WZW terms.

=--

+-- {: .num_cor #TildeHatGIsDifferentialModuliBundle}
###### Corollary

In the above situation there is a [[nLab:homotopy fiber sequence]] of [[nLab:infinity-group]] objects of the form

$$
  \array{
    \mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)_{conn}
    &\longrightarrow&
    \widetilde {\hat G}
    \\
    && \downarrow
    \\
    && \tilde G &\overset{\mathbf{L}_1}{\longrightarrow}& \mathbf{B}\left( \mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)_{conn} \right)
  }
  \,,
$$

where the bottom horizontal morphism is the [[nLab:higher WZW term]] that Lie integrates $\mu_1$, followed by the
canonical projection

$$
  \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn} \to \mathbf{B}\left( \mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)_{conn} \right)
$$

which removes the top-degree differential form data from a [[nLab:circle n-bundle with connection|higher connection]].

Hence $\widetilde{\hat G}$ is an [[nLab:infinity-group extension]] of $\tilde G$ by the
[[nLab:moduli stack]] of [[nLab:circle n-bundle with connection|higher connections]].

=--

+-- {: .proof}
###### Proof

By prop. \ref{ConsecutiveWZWTermsFromConsecutiveLInfinityCocycles} and the [[nLab:pasting law]], the [[nLab:homotopy fiber]] of $\widetilde {\hat G} \to \tilde G$ is equivalently the homotopy fiber of $\mathbf{\Omega}^{p_1+1}\to \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn}$,
which in turn is equivalently the homotopy fiber of $\ast \to \mathbf{B}\left( \mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)_{conn} \right)$,
which is $\mathbf{B}\left( \mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)_{conn} \right)$:

$$
  \array{
     \mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)_{conn}
       &\longrightarrow&
     \widetilde {\hat G}
       &\overset{}{\longrightarrow}&
     \mathbf{\Omega}^{p_1+1}
       &\overset{}{\longrightarrow}&
     \ast
     \\
     \downarrow &(pb)& \downarrow &(pb)& \downarrow &(pb)& \downarrow
     \\
     \ast
       &\longrightarrow&
     \tilde G
       &\stackrel{\mathbf{L}_1}{\longrightarrow}&
     \mathbf{B}^{p_1+1}(\mathbb{R}/\Gamma_1)_{conn}
       &\overset{}{\longrightarrow}&
     \mathbf{B}\left( \mathbf{B}^{p_1}(\mathbb{R}/\Gamma_1)_{conn} \right)
  }
  \,.
$$

=--

+-- {: .num_remark #InterpretationOfTildeHatG}
###### Remark

Corollary \ref{TildeHatGIsDifferentialModuliBundle} says that $\widetilde {\hat G}$ is a [[nLab:bundle]] of [[nLab:moduli stacks]] for [[nLab:differential cohomology]] over $\tilde G$. This means that maps

$$
  \Sigma \longrightarrow \widetilde{\hat G}
$$

(which are the [[nLab:field (physics)|fields]] of the higher [[nLab:WZW model]] with WZW term $\mathbf{L}_2$) are pairs of plain maps $\phi \colon \Sigma \to \tilde G$ together with a differential cocycle on $\Sigma$, i.e. a $p_1$-form connection on $\Sigma$, which is twisted by $\phi$ in a certain way.

[Below](#TheHigherWZWTermsOfSuperpBranes) we discuss that
this occurs for the (properly globalized) [[nLab:Green-Schwarz super p-brane sigma models]] of all the [[nLab:D-branes]] and of the [[nLab:M5-brane]]. For the D-branes $p_1 = 1$ and so there is a 1-form connection on their [[nLab:worldvolume]], the [[nLab:Chan-Paton gauge field]]. For the [[nLab:M5-brane]] $p_1 = 2$ and so there is a 2-form connection on its worldvolume, the [[nLab:self-dual higher gauge field]] in 6d.

=--

+-- {: .num_example #TargetStackForDBranes}
###### Example

For each [[nLab:D-brane|Dp-brane]] species in [[nLab:type IIA string theory]] there is a pair of consecutive cocycles
(def. \ref{ConsecutiveCocycles}) of the form

$$
  \array{
    \mathfrak{string}_{IIA} &\overset{\mu_{Dp}}{\longrightarrow}& b^{p+1} \mathbb{R}
    \\
    \downarrow
    \\
    \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}
      &\overset{\mu_{F1}^{IIA}}{\longrightarrow}&
    b^2 \mathbb{R}
  }
  \,.
$$

This is by the discussion [below](#TheBraneBouquet). Here

$$
  \mu_{Dp} = (C \wedge \exp(F_2))_{p+2}
$$

reflects the familiar D-brane coupling to the [[nLab:RR-fields]] $C = C_2 + C_4 + \cdots$, given an abelian [[nLab:Chan-Paton gauge field]]
with [[nLab:field strength]] $F_2$, see def. \ref{IIADpbranecocycles} below.

The WZW term induced by $\mu_{F1}^{IIA}$ is the globalization of the original term introduced by
Green and Schwarz in the construction of the [[nLab:Green-Schwarz sigma-model]] for the [[nLab:superstring]].

Now corollary \ref{TildeHatGIsDifferentialModuliBundle} says in this case
that the Dp-brane sigma model has as target space the [[nLab:smooth infinity-group|smooth super 2-group]]
$\widetilde{ String_{IIA} }$
which is an [[nLab:infinity-group extension]] of [[nLab:super Minkowski spacetime]] by
the [[nLab:moduli stack]] $\mathbf{B}U(1)_{conn}$ for [[nLab:complex line bundles]]
with [[nLab:connection on a principal bundle|connection]], sitting in a [[nLab:homotopy fiber sequence]]
of the form

$$
  \array{
    \mathbf{B}U(1)_{conn}
      &\longrightarrow&
    \widetilde{ String_{IIA} }
    \\
    &&
    \downarrow
    \\
    && \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}
    &\overset{\mathbf{L}_{Dp}}{\longrightarrow}&
    \mathbf{B}(\mathbf{B}^{p}U(1)_{conn})
  }
  \,.
$$

It follows that field configurations for the [[nLab:D-brane]] given by morphisms

$$
  \Sigma_{p+1} \longrightarrow \widetilde{ String_{IIA} }
$$

are equivalently pairs, consisting of an ordinary [[nLab:sigma-model]] field

$$
  \phi \;\colon\; \Sigma_{p+1} \longrightarrow \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}
$$

together with a twisted 1-form connection on $\Sigma$, the twist depending on $\phi$.
(In fact here the twist vanishes in bosonic degrees, unless we introduce a nontrivial bosonic component of the [[nLab:B-field]]).
This is just the right datum of the (abelian) [[nLab:Chan-Paton gauge field]] on the D-brane.

=--

### Consecutive WZW terms descending to twisted cocycles

Above we considered consecutive cocycles (def. \ref{ConsecutiveCocycles}) with [[nLab:coefficients]] in [[nLab:line Lie-n algebras]]
$b^{p+1}\mathbb{R}$.  Here we discuss how these may descend to single cocycles with richer [[nLab:coefficients]].

Below we find as examples of this general phenomenon

1. the descent of the separate [[nLab:D-brane]] cocycles to the RR-fields in [[nLab:twisted K-theory]], rationally ([here](#F1DpCocycles))

1. the descent of the [[nLab:M5-brane]] cocycle to a cocycle in degree-4 [[nLab:cohomology]], rationally ([here](#MFluxFields)).

$\,$

Given one stage of consecutive $L_\infty$-cocycles, def. \ref{LInfinityCocycle}
(e.g in the [[schreiber:brane bouquet]] discussed [below](#TheBraneBouquet))

$$
  \array{
    \hat \mathfrak{g}
    &
      \stackrel{\mu_2}{\longrightarrow}
    &
    \mathbf{B}\mathfrak{h}_2
    \\
    {}^{\mathllap{hofib(\mu_1)}}\downarrow
    \\
    \mathfrak{g}
    \\
    & {}_{\mathllap{\mu_1}}\searrow
    \\
    && \mathbf{B}\mathfrak{h}_1
  }
$$

then $\hat \mathfrak{g}$ may be thought of, in a precise sense, as being a $\mathfrak{h}_1$-[[nLab:principal ∞-bundle]] over $\mathfrak{g}$.

This and the following statements all are the general theorems of ([Nikolaus-Schreiber-Stevenson 12](#NSS12)) specified to $L_\infty$-algebras regarded as infinitesimal $\infty$-stacks (aka "[[nLab:formal moduli problems]]") according to [dcct](#dcct). Here we do not not have the space to dwell further on the details of this general theory of higher principal bundles, but the reader familiar with [[nLab:Lie groupoids]] gets an accurate impression by considering the analogous situation in that context (see at _[[nLab:geometry of physics -- principal bundles]]_ for detailed lecture notes that cover the following):

for $H$ a [[nLab:Lie group]] and $\mathbf{B}H$ its one-object [[nLab:delooping]] [[nLab:Lie groupoid]], and for $G$ another Lie group (or just any [[nLab:smooth manifold]]), then a generalized morphism of Lie groupoids

$$
  \array{
     G
     \\
     & \searrow
     \\
     && \mathbf{B}H
  }
$$

(i.e. a morphism between the [[nLab:smooth stacks]] which they represent, or equivalently a [[nLab:bibundle]] of Lie groupoids) classifies a smooth $H$-[[nLab:principal bundle]] over $H$, and the total space $\hat G$ of that bundle is equivalently the [[nLab:homotopy fiber]] of the original map.

$$
  \array{
     \hat G
     \\
     \downarrow
     \\
     G
     \\
     & \searrow
     \\
     && \mathbf{B}H
  }
$$

This is explained in some detail at _[principal bundle -- In a (2,1)-topos](https://ncatlab.org/nlab/show/principal%20bundle#InA2Topos)_.

Back to the abalogous situation of $L_\infty$-algebras instead of Lie groups, it is now natural to ask whether the second cocycle $\mu_2$, defined on the total space (stack) of this bundle is _equivariant_ under the [[nLab:∞-action]] of  $\mathfrak{h}_1$. If $\mu_2$ does not itself already come from the base space, then it can at best be equivariant with respect to an $\mathfrak{h}_1$-[[nLab:∞-action]] on $\mathbf{B}\mathfrak{h}_2$.

A first observation now is that specifying such [[nLab:∞-action]] $\rho$ is equivalent to specifying a second homotopy fiber sequence of the form as on the right of this completed diagram:

$$
  \array{
    \hat \mathfrak{g}
    &&
      \stackrel{\mu_2}{\longrightarrow}
    &&
    \mathbf{B}\mathfrak{h}_2
    \\
    {}^{\mathllap{hofib(\mu_1)}}\downarrow
    && &&
    \downarrow^{\mathrlap{hofib(p_\rho)}}
    \\
    \mathfrak{g}
    && &&
    (\mathbf{B}\mathfrak{h}_2)/\mathfrak{h}_1
    \\
    & {}_{\mathllap{\mu_1}}\searrow  && \swarrow_{\mathrlap{p_\rho}}
    \\
    && \mathbf{B}\mathfrak{h}_1
  }
  \,.
$$

In the simple analogous situation of [[nLab:Lie groupoids]] this comes about as follows (see at _[[nLab:geometry of physics -- representations and associated bundles]]_ for detailed lecture notes on the following):

for $H$ a [[nLab:Lie group]] and $\rho$ a smooth [[nLab:action]] of $H$ on some [[nLab:smooth manifold]] $V$, then there is the [[nLab:action groupoid]] $V/H$. Its [[nLab:objects]] are the points of $V$, but then it has morphisms of the form $v \stackrel{h}{\longrightarrow} \rho(h)(v)$ connecting any two objects that are taken to each other by the Lie group action. For example when $V = \ast$ is the point, then $\ast/H \simeq \mathbf{B}H$ is just the one-object [[nLab:delooping]] Lie groupoid of the Lie group $H$ itself. This also shows that there is canonical map

$$
  \array{
   && V/H
   \\
   & \swarrow
   \\
   \mathbf{B}H
  }
$$

which is given by sending all $v\in V$ to the point, and sending each morphism $v \stackrel{h}{\longrightarrow} \rho(h)(v)$ to $\ast \stackrel{h}{\longrightarrow} \ast$.

This projection is evidently an [[nLab:isofibration]], meaning that if we have a morphism in $\mathbf{B}G$ and a lift of its source object to $V/H$, then there is a compatible lift of the whole morphism. This is a technical condition which ensures that the ordinary [[nLab:fiber]] of this morphism is equivalently already it [[nLab:homotopy fiber]]. But the ordinary fiber of this morphisms, hence the stuff in $V/H$ that gets send to the (identity morphism on) the point, is clearly just $V$ itself again. Hence we conclude that the action of $G$ on $V$ induced a [[nLab:homotopy fiber sequence]]

$$
  \array{
   && V
   \\
   && \downarrow
   \\
   && V/H
   \\
   & \swarrow
   \\
   \mathbf{B}H
  }
  \,.
$$

With a little more work one may show that every homotopy fiber sequence of this form is induced this way by an action, up to equivalence. Hence actions of $H$ are equivalently [[nLab:bundles]] over $\mathbf{B}H$. One way to understand this is to observe that the [[nLab:action groupoid]] $V/H$ is a model for the [[nLab:homotopy quotient]] of the action, and by the [[nLab:Borel construction]] this may equivalently be written as the $\rho$-[[nLab:associated bundle]] to the $H$-[[nLab:universal principal bundle]]:

$$
  V/H \simeq \mathbf{E}H \underset{\rho}{\times} V
  \,.
$$

Hence the statement is that the map that sends $H$-actions $\rho$ the universal $\rho$-[[nLab:associated bundle]] is an equivalence, not just in the context of Lie groups andLie groupoids but much more generally (in every "[[nLab:(infinity,1)-topos]]").

Again back now to the analogous situation with $L_\infty$-algebras instead of Lie groups, a second fact which we are to invoke then is that given $\rho$, then the $\infty$-equivariance of $\mu_2$ is equivalent to it descending down the [[nLab:homotopy fibers]] on both sides to an $L_\infty$-homomorphism of the form

$$
  \mu_2/\mathfrak{h}_1
  \;\colon\;
  \mathfrak{g}
  \longrightarrow
  (\mathbf{B}\mathfrak{h}_2)/\mathfrak{h}_1
$$

making this [[nLab:diagram]] [[nLab:commuting diagram|commute]] in the [[nLab:homotopy category]]:

$$
  \array{
    \hat \mathfrak{g}
    &&
      \stackrel{\mu_2}{\longrightarrow}
    &&
    \mathbf{B}\mathfrak{h}_2
    \\
    {}^{\mathllap{hofib(\mu_1)}}\downarrow
    && &&
    \downarrow^{\mathrlap{hofib(p_\rho)}}
    \\
    \mathfrak{g}
    &&
      \stackrel{\mu_2/\mathfrak{h}_1}{\longrightarrow}
    &&
    (\mathbf{B}\mathfrak{h}_2)/\mathfrak{h}_1
    \\
    & {}_{\mathllap{\mu_1}}\searrow  && \swarrow_{\mathrlap{p_\rho}}
    \\
    && \mathbf{B}\mathfrak{h}_1
  }
  \,.
$$

In our example of Lie group principal bundles this comes down to a classical statement:

one may explicitly check that a morphism of the form

$$
  \array{
     X && \stackrel{\sigma}{\longrightarrow} && V/H
     \\
     & \searrow && \swarrow
     \\
     && \mathbf{B}H
  }
$$

is equivalently a [[nLab:section]] of the $V$-[[nLab:fiber bundle]] which is [[nLab:associated bundle|associated]] via $\rho$ to the $H$-[[nLab:principal bundle]] that is classified by the map on the left. If we pass this to the iterated [[nLab:homotopy fibers]] (the [[nLab:Cech nerve]]) of the vertical maps

$$
  \array{
     P \underset{X}{\times}P \simeq P \times H && \longrightarrow && V \times H
     \\
     \downarrow\downarrow && && \downarrow \downarrow
     \\
     P && \stackrel{\tilde \sigma}{\longrightarrow} && V
     \\
     \downarrow && && \downarrow
     \\
     X && \stackrel{\sigma}{\longrightarrow} && V/H
     \\
     & \searrow && \swarrow
     \\
     && \mathbf{B}H
  }
$$

then this $\sigma$ induces a $V$-valued function on the total space $P$ of the principal bundle with the property that this is $G$-equivariant. It is a classical fact that such equivariant $V$-valued functions on total spaces of principal bundles are equivalent to sections of the associated $V$-fiber bundles. What we are claiming and using here is that this fact again holds in vastly more generality, namely in an [[nLab:(infinity,1)-topos]].

In conclusion:

+-- {: .num_remark #TwistedCocycle}
###### Remark

The resulting triangle diagram

$$
  \array{
    \mathfrak{g}
    &&
      \stackrel{\mu_2/\mathfrak{h}_1}{\longrightarrow}
    &&
    (\mathbf{B}\mathfrak{h}_2)/\mathfrak{h}_1
    \\
    & {}_{\mathllap{\mu_1}}\searrow  && \swarrow_{\mathrlap{p_\rho}}
    \\
    && \mathbf{B}\mathfrak{h}_1
  }
$$

regarded as a morphism

$$
  \mu_2/\mathfrak{h}_1 \;\colon\; \mu_{1} \longrightarrow p_rho
$$

in the [[nLab:slice (infinity,1)-topos|slice]] over $\mathbf{B}\mathfrak{h}_1$
exhibits $\mu_2/\mathfrak{h}_1$ as a [[nLab:cocycle]] in ([[nLab:rational homotopy theory|rational]]) _$\mu_1$-[[nLab:twisted cohomology]]_ with respect to the [[nLab:local coefficient bundle]] $p_\rho$.

=--

([Nikolaus-Schreiber-Stevenson 12](#NSS12))

Notice that a priori this is (twisted) [[nLab:nonabelian cohomology]], though it may happen to land in abelian-, i.e. stable-cohomology.

Such descent is what one needs to find for the [[schreiber:brane bouquet]] [above](#pBranes), in order to interpret each of its branches as encoding $p$-brane model on spacetime itself. This is a purely algebraic problem which has been solved ([Fiorenza-Sati-Schreiber 15](#FSS15)). We discuss the solution in a moment.

## Super Minkowski spacetimes
 {#SuperMinowskiSpacetimes}

In order set the scene, and for reference, we recall the nature of [[super Minkowski spacetimes]] from _[[geometry of physics -- supersymmetry]]_.

In all of the following it is most convenient to regard [[super Lie algebras]] dually
via their [[Chevalley-Eilenberg algebras]] ("[[FDA]]"s), via def. \ref{CEAlgebraofSuperLieAlgebra} and prop. \ref{FormalDualCE}:

+-- {: .num_defn #SuperMinkowskiSpacetime}
###### Definition

Let

$$
  d \in \mathbb{N}
$$

be a [[spacetime]] [[dimension]] and let

$$
  N \in Rep_{\mathbb{R}}(Spin(d-1,1))
$$

be a [[real spin representation]] of the [[spin group]] cover $Spin(d-1,1)$ of the [[Lorentz group]] $O(d-1,1)$
in this dimension. Then the _$d$-dimensional $N$-supersymmetric [[super-Minkowski spacetime]]_ $\mathbb{R}^{d-1,1|N}$
is the [[super Lie algebra]] that is characterized by the fact that its [[Chevalley-Eilenberg algebra]]
$CE(\mathbb{R}^{d-1,1})$ (def. \ref{CEAlgebraofSuperLieAlgebra}) is as follows:

The algebra has generators (as an [[associative algebra]] over $\mathbb{R}$)

$$
   \underset{deg = (1,even)}{\underbrace{e^a}}
   \;\;\;\;
   \text{and}
   \;\;\;\;
   \underset{deg = (1,odd)}{\underbrace{\psi^\alpha}}
$$

for $a \in \{0,1,2, \cdots, 9\}$ and $\alpha \in \{1, 2, \cdots dim_{\mathbb{R}}(N)\}$ subjects to the [[generators and relations|relations]]

$$
  \begin{aligned}
    e^a \wedge e^b = - e^b \wedge e^a
    \\
    \psi^\alpha \wedge \psi^\beta = + \psi^\beta \wedge \psi^\alpha
    \\
    e^a \wedge \psi^\alpha = - \psi^\alpha \wedge e^a
  \end{aligned}
$$

(see at _[[signs in supergeometry]]_), and the [[differential]] $d_{CE}$ acts on the generators as follows:

$$
  \begin{aligned}
    d_{\mathbb{R}^{d-1,1\vert N}} \; \psi^\alpha &
      \coloneqq
      0
    \\
    d_{\mathbb{R}^{d-1,1\vert N}} \; e^a
      & \coloneqq \overline{\psi} \wedge \Gamma^a \psi
    \\
    & \coloneqq   \left(C_{\alpha \alpha'} {\Gamma^a}^{\alpha'}{}_\beta\right)  \psi^\alpha \wedge \psi^\beta
  \end{aligned}
    \,,
$$

where

1. $\overline{\psi} \wedge \Gamma^a \psi$ denotes the $a$-component of the $Spin(d-1,1)$-invariant spinor bilinear pairing $N \otime N \to \mathbb{R}^d$ that comes with every [[real spin representation]] applied to $\psi \wedge \psi$ regarded as an $N \otimes N$-valued form;

1. hence in components (if $N$ is a [[Majorana spinor]] representation, by [this prop.](geometry+of+physics+--+supersymmetry#SpinorToVectorBilinearPairing)):

   1. $C = (C_{\alpha \alpha'})$ is the [[charge conjugation matrix]] (as discussed at _[[Majorana spinor]]_);

   1. $\Gamma^a = ((\Gamma^a)^{\alpha}{}_\beta)$ are the [[matrices]] representing the [[Clifford algebra]] action on $N$ in the [[linear basis]] $\{\psi^\alpha\}_{\alpha = 1}^{dim_{\mathbb{R}}(N)}$

1. [[Einstein summation convention|summation over paired indices]] is understood.

That this indeed yields a [[super Lie algebra]] follows by the symmetry and equivariance of the bilinear spinor pairing (via [this prop.](geometry+of+physics+--+supersymmetry#SpinorToVectorPairing)).

There is a canonical [[Lie algebra action]] of the [[special orthogonal Lie algebra]]

$$
  Lie(Spin(d-1,1)) \simeq \mathfrak{so}(d-1,1)
$$

on $\mathbb{R}^{d-1,1\vert 1}$. The _$N$-supersymmetric [[super Poincaré Lie algebra]] $\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})$ in dimension $d$_ is the [[super Lie algebra]] which is the [[semidirect product Lie algebra]] of this [[Lie algebra action]]

$$
  \mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})
   =
  \mathbb{R}^{d-1,1\vert N} \rtimes \mathfrak{so}(d-1,1)
  \,.
$$

This is characterized by the fact that its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N}))$
is as follows:

it is [[generators and relations|generated]] from elements

$$
   \underset{deg = (1,even)}{\underbrace{e^a}}
   \;\;\;\;
   and
   \;\;\;\;
   \underset{deg = (1,odd)}{\underbrace{\psi^\alpha}}
   \;\;\;\;
   and
   \;\;\;\;
   \underset{deg = (1,even)}{\underbrace{\omega^{a b} = - \omega^{b a}}}
$$

with the [[super vielbein]] $(e^a, \psi^\alpha)$ as before, and with $\omega^{a b}$ the
[[dual basis]] of the induced [[linear basis]] for vectro space of skew-symmetric matricces
underlying the [[special orthogonal Lie algebra]]. The commutation relations are as before,
together with the relation that the generators $\omega^{a b}$ anti-commute with every generator.
Finally the [[differential]] $d_{\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})}$ acts on
these generators as follows:

$$
  \begin{aligned}
    d_{\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})} \; \psi^\alpha
      & \coloneqq
      \left(\tfrac{1}{4}\omega^{a b} \Gamma_{a b} \psi \right)^\alpha
      \\
      & \coloneqq
      \left(\tfrac{1}{4} (\Gamma_{a b})^\alpha{}_{\beta} \right) \omega^{a b} \wedge \psi^\beta
    \\
    d_{\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})} \; e^a
      & \coloneqq \overline{\psi} \wedge \Gamma^a \psi - \omega^a{}_b \wedge e^b
    \\
    & \coloneqq
       \left(
           C_{\alpha \alpha'} {\Gamma^a}^{\alpha'}{}_\beta
       \right)
       \psi^\alpha \wedge \psi^\beta - \omega^a{}_b \wedge e^b
    \\
  \end{aligned}
    \,,
$$

where we are shifting spacetime indicices with the Lorentz metric

$$
  (\eta_{a b}) \coloneqq diag(-1,1,1,\cdots, 1)
  \,.
$$

The canonical maps between these [[super Lie algebras]], dually between their [[Chevalley-Eilenberg algebras]], that send each generator to itself, if present, or to zero if not, constitute the [[diagram]]

$$
  \array{
    \mathbb{R}^{d-1,1\vert N}
      &\hookrightarrow&
    \mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})
    \\
    && \downarrow
    \\
    && \mathfrak{so}(d-1,1)
  }
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

If we think of [[super Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert N}$  as the [[supermanifold]] with

* even [[coordinates]] $\{x^a\}_{a = 0}^{d-1}$;

* odd [[coordinates]] $\{\theta_\alpha\}_{\alpha = 1}^{dim_{\mathbb{R}}(N) }$

then the algebra generators $e^a$ and $\psi^\alpha$ in def. \ref{SuperMinkowskiSpacetime}  correspond to these [[super differential forms]]:

$$
  \begin{aligned}
    e^a & = d_{dR} x^a + \underset{\text{correction term}}{\underbrace{\overline{\theta} \Gamma^a d_{dR} \theta}}
    \\
    \psi^\alpha & = d_{dR} \theta^\alpha
  \end{aligned}
$$

the **super-[[vielbein]]**.

Notice that $d_{dR} x^a$ alone fails to be a left [[invariant differential form]], in that it is not annihilated by the [[supersymmetry]] vector fields

$$
  D_\alpha
    \;\coloneqq\;
  \partial_{\theta^\alpha} - \overline{\theta}_{\alpha'} \Gamma^a{}^{\alpha'}{}_\alpha \partial_{x^a}\,.
$$

Therefore the all-important correction term  above.

=--

+-- {: .num_remark #SuperMinkowskiInvariantCocycles}
###### Remark

By def. \ref{SuperLieAlgebraCohomology} the super-[[Lie algebra cohomology]]
of [[super Minkowski spacetimes]] (def. \ref{SuperMinkowskiSpacetime}) is the cohomology of the [[differential]]

$$
  d_{CE} \, e^a = \overline{\psi} \wedge \Gamma^a \psi
  \;\;\;\,,\;\;\;
  d_{CE} \, \psi^\alpha = 0
  \,.
$$

(also called "[[tau-cohomology]]").

This may superficially look fairly trivial, but in fact it turns out to me very rich.
Specifically we are interested in the $Spin(d-1,1)$-[[invariant]] cohomology. A $Spin$-[[invariant]]
[[cochain]] in the [[Chevalley-Eilenberg algebra]] $CE(\mathbb{R}^{d-1,1\vert M})$ is easily seen to
be equivalently a sum of wedge products of the above generators, such that all indices are
contracted with $\Gamma$-matrices. Hence these are the monomials of the form

$$
  \mu_{p+2}
  \;\coloneqq\;
  \overline{\psi} \Gamma_{a_1 \cdots a_p} \psi
  \,.
$$

We immediately find that the differential takes these to the following expressions (using the fact that the differential is a graded [[derivation]]):

$$
  d_{CE} \mu_{p+2}
    \;=\;
  p
  \;
  \left(
    \overline{\psi} \wedge \Gamma_{a_1 \cdots a_{p-1} b} \psi
  \right)
  \wedge
  \left(
    \overline{\psi} \wedge \Gamma^b \psi
  \right)
  \wedge
  e^{a_1}\wedge \cdots \wedge e^{p_{p-1}}
  \,.
$$

For the these expressions to vanish, all their components have to vanish, hence all the following [[multilinear map|quadrilinear]]
expressions in the [[spinors]] have to vanish identicaly:

$$
  d_{CE} \mu_{p+2} = 0
  \;\;\;\; \Leftrightarrow
  \;\;\;\;
  \left(
    \overline{\psi} \wedge \Gamma_{a_1 \cdots a_{p-1} b} \psi
  \right)
  \wedge
  \left(
    \overline{\psi} \wedge \Gamma^b \psi
  \right)
  = 0
  \,.
$$

=--

Such relation among spinors are known as [[Fierz identities]]. These we discuss now in
_[Fierz identities](#FierzIdentities)_.

## Fierz identities
 {#FierzIdentities}

What are called _[[Fierz identities]]_ in [[physics]] are the relations that hold between [[multilinear map|multilinear]]  expression in [[spinors]]. For example for all [[Majorana spinors]] $\psi$ in Lorentian spacetime dimension 4,5,7, 11, then the following identity holds (example \ref{TheM2andM5CocyclesAsFierzIdentities} below):

$$
  \left(\overline{\psi} \wedge \Gamma_{a b} \psi\right)
  \wedge
  \left(\overline{\psi} \wedge \Gamma^b \psi\right)

  \;=\;
  0
  \,.
$$

(Here $\overline{(-)}$ denotes the Majorana conjugate, $\Gamma_a$ are a Clifford representations, the "$\wedge$"-signs denotes symmetrization in the spinor components and summation over repeated indices is understood. The details of this are discussed [below](#BilinearFierzIdentities).)

In [D'Auria-Fré-Maina-Regge 82](#DAuriaFreMainaRegge82) it was pointed out that all Fierz identities may be understood as expressing the product operation in the [[representation ring]] of the [[spin group]] (in some given dimension): for $\{S_i\}_{i \in I}$ denoting [[isomorphism classes]] of [[irreducible representations|irreducible]] [[spin representations]], then, by definition of [[irreps]], their [[tensor product of representations]] decomposes again as a [[direct sum]] of [[irreducible representations]]

$$
  S_i \otimes S_j  = \underset{k}{\oplus} C_{i j}{}^k S_k
$$

with "[[Clebsch-Gordan coefficients]]" $C_{i j}{}^k$. These coefficients are effectively the Fierz identities.

For example for Lorentzian dimension 11 with $(\tfrac{1}{2})^5$ denoting the unique irreducible [[Majorana spinor]] representation, then one finds ([D'Auria-Fré 82b, section 3](#DAuriaFre82b)) that the symmetric part in the quadruple tensor product of this representation with itself decomposes as a direct sum of irreps as follows

$$
  \left\{
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
  \right\}_{sym}
   \;\simeq\;
  (0)^5
    \;\oplus\;
  (1)^3 (0)^2
    \;\oplus\;
  (1)^4 (0)
    \;\oplus\;
  (1)^5
    \;\oplus\;
  (2) (0)^4
    \;\oplus\;
  (2)(1)(0)^3
    \;\oplus\;
  (2)^2 (0)^3
    \;\oplus\;
  (2)^2 (1)^3
    \;\oplus\;
  (2)^5
$$

where the symbols refer to [[Young diagrams]] canonically labeling representations (details are in example \ref{11dQuadrilinearCGCoefficients} below).

The point is that the expression $ \left(\overline{\psi} \wedge \Gamma_{a b} \psi\right) \wedge \left(\overline{\psi} \wedge \Gamma^b \psi\right)$ from above is a spinor quadrilinear which transforms in the vector representation $(1)(0)^4$ (due to its one free spacetime index). But that vector representation $(1)(0)^4$ is missing from the [[direct sum]] above, meaning that the spinor quadrilinear has vanishing components in this vector representation, hence that this expression vanishes identically.

We discuss Fierz identities as identities among multispinorial elements of the [[Chevalley-Eilenberg algebra]] $CE(\mathbb{R}^{d-1,1\vert N})$ of [[super-Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert N}$, regarded as the super-translation [[supersymmetry]] [[super Lie algebra]]. In this form Fierz identities encode [[cocycles]] in the [[supersymmetry]] super-[[Lie algebra cohomology]], such as those which serve as [[higher WZW terms]] characterizing [[super p-branes]]. We follow [Castellani-D'Auria-Fré 82, section II.8](#CDF).

### Bilinear Fierz identities
  {#BilinearFierzIdentities}

Given a fixed [[real spin representation]] $N$, then the
odd [[coordinates]] $\{\theta^\alpha\}_{\alpha = 1}^{dim_{\mathbb{R}}(N) }$ of the [[super Minkowski spacetime]] [[supermanifold]]
$\mathbb{R}^{d-1,1\vert N}$ span, by construction, precisely that representation space, and hence so do the
spinorial components of the [[super vielbein]] form

$$
  \psi^\alpha = \mathbf{d}\theta^\alpha
   \;\;\;
   \in \Omega^{\bullet}_{li}(\mathbb{R}^{d-1,1\vert N})
   \simeq
    CE(\mathbb{R}^{d-1,1\vert N})
  \,,
$$

since in the construction of [[super differential forms]] on $\mathbb{R}^{d-1,1\vert N}$, the de Rham operator
$\mathbf{d}$ acts on the odd coordinates just formally, by sending the generator $\theta^\alpha$ to the new generator
named $\mathbf{d} \theta^\alpha$.

Therefore we may identify the [[spin representation]] $N$ with the [[linear span]] (over $\mathbb{R}$) of these elements

$$
  N  \simeq \langle \mathbf{d}\theta^\alpha \rangle_{\alpha = 1}^{dim_{\mathbb{R}}(N) }
  \,,
$$

were the [[spin group]] acts on the elements on the right in the defining way (see at _[[geometry of physics -- supersymmetry]]_): a spinorial rotation in a plane $\omega = \{\omega^{a b}\}$ by an angle $\alpha$ acts by

$$
  R_\omega(\psi) \coloneqq \exp(\tfrac{\alpha}{4} \omega^{a b} \Gamma_{a b} ) \psi
  \,.
$$

We may build new [[spin representations]] from this one by forming multilinear expressions in the [[super vielbein]]. For example the elements in $CE(\mathbb{R}^{d-1,1\vert N})$ of the form

$$
  \begin{aligned}
    \overline{\psi} \wedge \Gamma_a \psi
    &=
    \left(C_{\alpha \alpha'} \Gamma_a{}^{\alpha'}_{\beta}\right) \, \psi^\alpha \wedge \psi^{\beta}
    \\
    & =
    \left(C_{\alpha \alpha'} \Gamma_a{}^{\alpha'}_{\beta}\right) \, \mathbf{d}\theta^\alpha \wedge \mathbf{d}\theta^\beta
  \end{aligned}
$$

span, as the spacetime index $a$ ranges in $\{0, 1, \cdots, d-1\}$, a $d$-dimensional [[real vector space]]

$$
  \left\langle
     \,\overline{\psi} \wedge \Gamma_a \psi\,
  \right\rangle_{a = 0}^{d-1}
$$

which still carries a linear [[action]] of the [[spin group]], induced from the spin action on the $\psi$-s:

$$
  \begin{aligned}
    R_\omega(\overline{\psi} \wedge \Gamma_a \psi)
    & =
    \overline{\left( \exp(\tfrac{\alpha}{4}\omega^{a b}\Gamma_{a b} ) \psi \right)}
      \wedge \Gamma_a
    \left( \exp(\tfrac{\alpha}{4}\omega^{a b}\Gamma_{a b} \psi ) \right)
    \\
    & =
    \overline{\psi} \wedge \exp(-\tfrac{\alpha}{4} \omega^{a b} \Gamma_{a b}) \Gamma_a \exp(\tfrac{\alpha}{2}\omega^{a b} \Gamma_{a b})
     \psi
    \\
    & =
    \overline{\psi}
      \wedge
      (R_\omega(\Gamma_a))
    \psi
    \end{aligned}
  \,.
$$

Of course similarly we obtain elements

$$
  \overline{\psi} \Gamma_{a_1 \cdots a_p} \psi
$$

which, if they are non-vanishing at all, span the representation

$$
  \wedge^p \mathbb{R}^d
$$

Now observe that we may say all this more abstractly as follows:

1. the elements $(\psi \wedge \overline{\psi})^{\alpha \beta}$ span the symmetrized [[tensor product of representations]]

   $$
     \{N \otimes N\}_{sym}
       \;\simeq\;
     \langle
       \,  (\psi \wedge \overline{\psi})^\alpha{}_\beta \,
     \rangle_{\alpha,\beta = 1}^{dim_{\mathbb{R}}(N)}
   $$

1. for given $p \in \mathbb{N}$, then the elements of the form $\overline{\psi} \wedge \Gamma_{a_1 \cdots a_p} \psi$ form a [[subrepresentation]] thereof, equivalent to the vector representation $\wedge^p\mathbb{R}^{d}$

1. hence there is a [[direct sum]] decomposition

   $$
     \left\{N \otimes N\right\}_{sym}
       \;\simeq\;
      \underset{p \in \mathbb{N}}{\bigoplus}
      c_p \left(\wedge^p \mathbb{R}^d\right)
   $$

   in the [[category of representations]] of the [[spin group]], which expresses the (symmetrized) [[tensor product of representations]] of the [[Majorana spinor]] representation as a [[direct sum]] of skew-symmetrized tensor products of the vector representation.

Indeed this direct sum decomposition is exhaustive:

+-- {: .num_prop #BilinearFierzDecomposition}
###### Proposition

For $d \in \mathbb{N}$ and $N$ a [[Majorana spinor]] representation of $Spin(d-1,1)$, then the following identity holds:

$$
  (\psi \wedge \overline{\psi})^\alpha{}_\beta
   \;=\;
  \tfrac{1}{dim_{\mathbb{R}}(N)}
  \left(
     \left(
       \overline{\psi}\psi
     \right)
     +
     \left(
       \overline{\psi} \Gamma_a \psi
     \right)
     (\Gamma^a)^\alpha{}_\beta
     +
     \tfrac{1}{2!}
     \left(
       \overline{\psi} \Gamma_{a_1 a_2} \psi
     \right)
     (\Gamma^{a_1 a_2})^\alpha{}_\beta
      +
     \cdots
  \right)
  \,.
$$

=--

+-- {: .proof}
###### Proof

By the discussion there, the [[Majorana spinor]] representation is a real sub-representation of a complex _Dirac representation_ $\mathbb{C}^{(2^\nu)}$. The latter has the special property that

1. the [[Clifford algebra]] contains the full [[matrix algebra]];

1. for $p \geq 1$ the Clifford elements $\Gamma_{a_1 \cdots a_p}$ have vanishing [[trace]].

The first point implies that there exists coefficients $X^{a_1 \cdots a_p} \in \mathbb{C}$ for $p \in \mathbb{N}$ such that

$$
  \psi \wedge \overline{\psi}
    =
  \tfrac{1}{dim_{\mathbb{R}}(N)}
  \left(
    X
    +
    X^a \Gamma_a
    +
    X^{a b} \Gamma_{a b}
    +
    \cdots
  \right)
  \,.
$$

The second condition then implies that multiplying this expression with $\Gamma^{a_1 \cdots a_p}$ and taking the trace projects out the coefficient $X^{a_1 \cdots a_p}$:

$$
  \begin{aligned}
    X^{a_1 \cdots a_p}
      & =
    \frac{1}{p! dim_{\mathbb{R}}(N)}
    tr_N
    \left(
      \left(
        X
        +
        X^a \Gamma_a
        +
        X^{a b} \Gamma_{a b}
        +
        \cdots
      \right)
      \Gamma^{a_1 \cdots a_p}
   \right)
   \\
   & =
   \tfrac{1}{p!}
   tr_N
   \left(
     \psi \wedge \overline{\psi} \, \Gamma^{a_1 \cdots a_p}
   \right)
   \\
   & =
   \tfrac{1}{p!}
   \left(
     \overline \psi \wedge \Gamma^{a_1 \cdots a_p} \psi
   \right)
  \end{aligned}
  \,.
$$

Notice that it is the last step, identifying the trace over $\psi \wedge \overline{\psi} \Gamma^{a_1 \cdots a_p}$ with the $\psi$-$\psi$ component of the matrix $\Gamma^{a_1 \cdots a_p}$, where we use the _symmetrization_ of the spinor tensor product, namely the identity $\psi^\alpha \wedge \overline{\psi}_\beta = \overline{\psi}_\beta \wedge \psi^\alpha$.

=--

Some of the coefficients in prop. \ref{BilinearFierzDecomposition} may vanish identically. These are the _bilinear Fierz identities_, of the form

$$
  \overline{\psi} \Gamma_{a_1 \cdots a_p} \psi = 0
  \,.
$$

+-- {: .num_example}
###### Example

Let $d = 11$. Write $\mathbf{32}$ or $(\tfrac{1}{2})^5$ for the [[Majorana spinor]] representation of $Spin(d-1,1)$. Then

$$
  \left\{
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
  \right\}_{sym}
     \;\simeq\;
  \underset{\simeq \mathbb{R}^d}{\underbrace{(1)^1 (0)^4}}
    \;\oplus\;
  \underset{\simeq \wedge^2 \mathbb{R}^d}{\underbrace{(1)^2 (0)^3}}
    \;\oplus\;
  \underset{\wedge^5 \mathbb{R}^d}{\underbrace{(1)^5}}
  \,.
$$

=--

([D'Auria-Fré 82b (3.1)](#DAuriaFre82b))

+-- {: .proof}
###### Proof

Since we know from prop. \ref{BilinearFierzDecomposition} that the right hand side has to be some direct sum of representations of the form $\wedge^p \mathbb{R}^d$, it is sufficient to check that there is only one choice of sum such that [[dimensions]] match on both sides of the equation.

Now the dimension of $\{N \otimes N\}_{sym}$ is that of the space of symmetric $32 \times 32$ matrices:

$$
  dim_{\mathbb{R}}
  \left(
     \{\mathbf{32} \otimes \mathbf{32}\}_{sym}
  \right)
   \;=\;
  \frac{1}{2}
  \left(
    32 \times 33
  \right)
   =
  528
$$

while the dimension of $\wedge^p \mathbb{R}^d$ is the [[binomial coefficient]]

$$
  dim_{\mathbb{R}}(\wedge^p \mathbb{R}^d)
   \;=\;
  \left(
    11 \atop p
  \right)
  \,.
$$

Hence the claim follows from the fact that

$$
  \begin{aligned}
    582
     & = 11 + 55 + 462
    \\
    & =
  \left(11 \atop 1\right)
   +
  \left(11 \atop 2\right)
   +
  \left(11 \atop 5\right)
  \end{aligned}
  \,.
$$

=--

### Quadrilinear Fierz identities
  {#QuadraticFierzIdentities}

Now we consider the [[direct sum]] decomposition of the [[tensor product of representations]] of _four_ copies of a [[spin representation]]. This yields the quadrilinear Fierz identities.

+-- {: .num_example #11dQuadrilinearCGCoefficients}
###### Example

The group $Spin(10,1)$ has [[rank of a Lie group|rank]] 5, and hene its [[irreducible representation|irreducible]] vector representations are labeled by [[Young diagrams]] consisting of five rows. For instance

$$
  (2)^2 (1)^2 (0)
$$

denotes the representation whose elements may be identified with tensors of the form

$$
  X_{\array{ a_1 & a_2 \\ a_3 & a_4 \\ a_5 }}
$$

which are

1. skew-symmetric in indices in the same column;

1. symmetric and trace-less in indices in the same row.

Write again $(\tfrac{1}{2})^5$ for the [[Majorana spinor]] representation. Then the following identity holds in the [[representation ring]]:

$$
  \left\{
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
  \right\}_{sym}
   \;\simeq\;
  \left.
    \array{
    (0)^5
    \\
    \oplus
    \\
    (2) (0)^4
    \\
    \oplus
    \\
    (1)^3 (0)^2 \oplus  (2)(1)(0)^3
    \\
    \oplus
    \\
    (1)^4 (0) \oplus (2)^2 (0)^3
    \\
    \oplus
     \\
    (1)^5
     \\
     \oplus
     \\
    (2)^2 (1)^3
    \\
    \oplus
    \\
    (2)^5
   }
  \right.
$$

=--

([D'Auria-Fré 82b (3.3) ](#DAuriaFre82b))

+-- {: .proof}
###### Proof

As before, this is supposed to follow already by matching total dimensions on both sides

$$
  \frac{32 \times 33 \times 34 \times 35}{4 \times 3 \times 2}
   \;=\;
  \left.
    \array{
    1
    \\
    +
    \\
    65
    \\
    +
    \\
    165 +  429
    \\
    +
    \\
    330 + 1144
    \\
    +
     \\
    462
     \\
     +
     \\
    17160
    \\
    +
    \\
    32604
   }
  \right.
$$

=--

More in detail we have the following decompositions, in the notation from [above](#QuadraticFierzIdentities).

\[
  \label{Fierz11dA}
  \left(\overline{\psi} \wedge \Gamma_{a_1} \psi\right)
   \wedge
  \left( \overline{\psi} \wedge \Gamma_{a_2} \psi \right)
   \;=\;
  X^{(\mathbf{65})}_{\array{a_1 \\ a_2}}
   +
  \frac{1}{11} \delta_{\array{a_1 a_2}}X^{(\mathbf{1})}
\]

Here for instance the symbol $X^{(\mathbf{65})}_{\array{a_1 \\ a_2}}$ denotes the projection of the term on the left into the direct summand given by the [[representation]] $(2)(0)^4$ of dimension $65$. Similarly:

\[
  \label{Fierz11dB}
  \left(\overline{\psi} \wedge \Gamma_{a_1 a_2} \psi\right)
    \wedge
  \left(\overline{\psi} \wedge \Gamma_{a_3}\right)
    \;=\;
  X^{(\mathbf{429})}_{\array{ a_1 & a_2 \\ a_3}}
   +
  X^{(\mathbf{165})}_{\array{a_1 a_2 a_3}}
\]

\[
  \label{Fierz11dC}
  \left(
    \overline{\psi}\Gamma_{a_1 a_2} \psi
  \right)
  \left(
    \overline{\psi} \Gamma_{a_3 a_4}
  \right)
   \;=\;
  X^{(\mathbf{1144})}_{\array{a_1 a_2 \\ a_3 a_4}}
   +
  X^{(\mathbf{330})}_{\array{a_1 a_2 a_3 a_4}}
   +
  \tfrac{4}{9}\delta_{\array{ [a_1 \\ [a_3} } X^{(\mathbf{65})}_{\array{a_2] \\ a_4] } }
  -
  \tfrac{2}{11} \delta_{\array{a_1 & a_2 \\ a_3 & a_4}} X^{(\mathbf{1})}
\]

\[
  \label{Fierz11dD}
  \left(
    \overline{\psi}
      \wedge
      \Gamma_{a_1 \cdots a_5}
    \psi
  \right)
  \wedge
  \left(
    \overline{\psi}
      \wedge
    \Gamma_{a_6}
   \psi
  \right)
  \;=\;
  \epsilon_{a_1 \cdots a_6}{}^{b_1 \cdots b_5}
  X^{(\mathbf{462})}_{b_1 \cdots b_5}
  +
  X^{(\mathbf{4290})}_{\array{a_1 & \cdots & a_5 \\ a_6}}
  +
  \frac{15}{7}
  \delta_{a_6 [ a_1} X^{(\mathbf{330})}_{\array{a_2 & \cdots & a_5 ] }}
\]

and some more.

([D'Auria-Fré 82b table 2 ](#DAuriaFre82b))

As a corollary:

+-- {: .num_example #TheM2andM5CocyclesAsFierzIdentities}
###### Example

For $d = 11$ we have that

1. the following Fierz identity holds:

   $$
     \left(
       \overline{\psi} \wedge \Gamma_{a b} \psi
     \right)
     \wedge
     \left(
        \overline{\psi} \wedge \Gamma^b \psi
     \right)
     \;= \; 0
     \,.
   $$

   (we will see below that this is the cocycle condition for the [[higher WZW term]] of the [[M2-brane]] ([Bergshoeff-Sezgin-Townsend 87](#BergshoeffSezginTownsend87)), [AETW 87](#AETW87))

1. the following Fierz identity holds:

   $$
     \left(
       \overline{\psi}
         \wedge
         \Gamma_{a_1 \cdots a_4 b}
       \psi
     \right)
     \wedge
     \left(
       \overline{\psi}
        \wedge
        \Gamma^{b}
        \psi
     \right)
     \;=\;
     3
     \left(
     \overline{\psi}
        \Gamma_{[a_1 a_2}
     \psi
     \right)
     \wedge
     \left(
       \overline{\psi}
          \Gamma_{a_3 a_4]}
       \psi
     \right)
   $$

   (we will see below that this is the cocycle condition for the [[higher WZW term]] of the [[M5-brane]] ([BLNPST 97](#BLNPST97), [FSS 15](#FSS15))).

=--

This is due to ([D'Auria-Fré 82b (3.13) and (3.28) ](#DAuriaFre82b))

+-- {: .proof}
###### Proof

The first identity is the result of equation (eq:Fierz11dB) after tracing over the indices $a_2$ and $a_3$. Under this trace both summands on the right of (eq:Fierz11dB) vanish: $X^{(\mathbf{429})}_{\array{ a_1 & a_2 \\ a_3}}$ because it is trace-free in indices in a column, and $X^{(\mathbf{165})}_{\array{a_1 a_2 a_3}}$ because it is skew-symmetric in all indices.

The second identity follows from taking the trace over the indices $a_5 and a_6$ in (eq:Fierz11dD) and of skew-symmetrizing over all indices in (eq:Fierz11dC). By the symmetry properties of the tensors on the right of both equations, in both cases all tensors vanish except, in both cases, the contribution proportional to $X^{(\mathbf{330})}_{[a_1 \cdots a_3]}$, which both identities share. So it only remains to check that the proportionality factor is 3, as claimed. By writing out the skew-symmetrization in the last term in (eq:Fierz11dD) one finds:

{#ComputationOfRelativeCoefficientInM5Cocycle}
$$
  \begin{aligned}
    \frac{15}{7}
    \delta^{a_1 a_6}
    \delta_{a_6 [a_1}
    X^{(\mathbf{330})}_{a_2 \cdots a_5]}
    & =
    \frac{15}{7}
    \delta^{a_1}{}_{[a_1} X^{(\mathbf{330})}_{a_2 \cdots a_5]}
    \\
    & =
    \frac{15}{7}
    \frac{1}{5!}
    \sum_{ \left\{\sigma \atop { {\text{permutation of}} \atop {\{1,\cdots , 5\}} } \right\}}
    (-1)^{\vert \sigma\vert }
    \delta^{a_1}{}_{a_{\sigma(1)}}
    X_{a_{\sigma(2)} \cdots a_{(\sigma(5))}}
    \\
    & =
    \frac{15}{7}
    \frac{1}{5!}
    \sum_{\left\{\sigma \atop { {\text{permutation of}} \atop {\{1,\cdots , 4\}} } \right\} }
    (-1)^{\vert \sigma\vert }
    \left(
      \underset{= 11}{\underbrace{\delta^{a_1}_{a_1}}}
      X^{(\mathbf{330})}_{a_{\sigma(1)}\cdots a_{\sigma(4)}}
      -
      4
      \delta^{a_1}{}_{a_{\sigma(1)}}
      X_{a_1 a_{\sigma(2)} \cdots a_{\sigma(4)}}
    \right)
   \\
   & =
    \frac{15}{7}
    (11-4)
    \frac{1}{5}
    \;
    \underset{X^{(\mathbf{330})}_{a_1\cdots a_4}}{
    \underbrace{
    \frac{1}{4!}
    \sum_{ \left\{ \sigma \atop { {\text{permutation of}} \atop {\{1,\cdots , 4\}} } \right\} }
    (-1)^{\vert \sigma\vert}
    X^{(\mathbf{330})}_{a_{\sigma(1)}\cdots a_{\sigma(4)}}
    }
    }
    \\
    & =
    3 \; X^{(\mathbf{330})}_{a_{\sigma(1)} \cdots a_{\sigma(4)}}
  \end{aligned}
$$

where we used that $X^{(\mathbf{330})}_{a_1 \cdots a_4}$ is already skew-symmetric in all indices.

=--

$\,$

## Super $p$-Branes
 {#Branes}

We now discuss how the [[super p-branes]] arise in the guise of consecutive invariant higher [[super Lie n-algebra]] extensions of
[[super Minkowski spacetimes]]. To put this in perspective, we first
recall in

* _[The super 0-branes and Super-Minowski spacetimes](#SuperMinkowskiSpacetimes)_

(from the end of _[[geometry of physics -- supersymmetry]]_) how the relevant [[super Minkowski spacetimes]] in dimensions 3,4,6,10 and 11 themselves emerge from the [[superpoint]] in a progression of
invariant odinary [[central extensions]] of [[super Lie algebras]].
The last step in this progression (from 10d type IIA spacetime to 11d) is classified by the
cocycle for the [[D0-brane]]. Similarly we may also think of the previous steps as being related to
0-branes.

On all the super-Minkowski spacetimes that appear in this progression in
dimension 3,4,6 and 10 there exist non-trivial invariant 3-cocycles.
These are the [[WZW terms]] of the [[Green-Schwarz superstring]]
in these dimensions. Finally in $d = 11$ there is instead a nontrivial invariant 4-cocycle,
corresponding to the [[super-membrane]] in 11d (the [[M2-brane]]).
The are classical facts from the "old [[brane scan]]" which we review in

* _[The superstring and the super membrane](#TheSuperStringAndTheSuperMembrane)_

While up to this point the progression happens in [[super Lie algebra|super]] [[Lie algebra cohomology]],
now we turn to the [[homotopy theory]] of [[super L-infinity algebras]]: Just like
2-cocycles on [[super Lie algebras]] classify ordinary [[central extensions]], higher
cocycles such as the 3-cocycles and the 4-cocycles of the [[superstring]] and of the
[[supermembrane]] classify [[super Lie n-algebra]] extension of [[super Minkowski spacetime]].
This we discuss in

* _[Extended super Minkowski spacetimes](#ExtendedSuperMinkowskiSpacetime)_

Once this etension into the larger realm of [[super Lie n-algebra]] is made,
the progression continues: On the [[extended super Minkowski spacetime]]
[[super Lie n-algebras]] there appear now furhter invariant cocycles. These
correspond to the [[D-branes]] and the [[M5-brane]]. This we discuss in

* _[The super D-branes and the super  M5-brane](#SuperMinkowskiSpacetimes)_

This yields a complete account of the brane species of [[string theory]]/[[M-theory]]
separately. Next we discuss how these separate cocycles interact (by twisting each other)
and then unify to single but [[non-abelian cohomology|non-abelian]] cocycles. This
is the topic of the next section _[Fields](#Fields)_.

### The super 0-branes and Super Minkowski spacetimes
 {#SuperMinkowskiSpacetimes}

+-- {: .num_prop}
###### Proposition

Consider the [[superpoint]]

$$
  \;\;\;\;\;\; \mathbb{R}^{0\vert 1}
$$

regarded as an abelian [[super Lie algebra]] (def. \ref{SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces}, prop. \ref{SuperLieAlgebraTraditional}). Its maximal [[central extension]] is the $N = 1$ super-[[worldline]] of the [[superparticle]]:

$$
  \array{
    \mathbb{R}^{0,1\vert \mathbf{1}}
    \\
    \downarrow
    \\
    \mathbb{R}^{0\vert 1}
  }
  \,.
$$

* whose even part is spanned by one generator $H$

* whose odd part is spanned by one generator $Q$

* the only non-trivial bracket is

  $$
    \{Q, Q\} = H
  $$

Then consider the [[superpoint]] with two odd dimensions

$$
  \;\;\;\;\;\; \mathbb{R}^{0\vert 2}
  \,,
$$

which is the [[coproduct]] of the atomic 2-cocycle over its
bosonic part $\overset{\rightsquigarrow}{\mathbb{R}^{0 \vert 1}} \simeq \mathbb{R}^0$.

Its maximal [[central extension]] is
the $d = 3$, $N = 1$ [[super Minkowski spacetime]] (def. \ref{SuperMinkowskiSpacetime})

$$
  \array{
    \mathbb{R}^{2,1\vert \mathbf{2}}
    \\
    \downarrow
    \\
    \mathbb{R}^{0\vert 2}
  }
  \,.
$$

* whose even part is $\mathbb{R}^3$, spanned by generators $P_0, P_1, P_2$

* whose odd part is $\mathbb{R}^2$, regarded as

  the [[Majorana spinor]] representation $\mathbf{2}$

  of $Spin(2,1) \simeq SL(2,\mathbb{R})$

* the only non-trivial bracket is the spinor bilinear pairing

  $$
    \{Q_\alpha, Q'_\beta\}
    =
     C_{\alpha \alpha'} \Gamma_a{}^{\alpha'}{}_\beta
    \,P^a
  $$

where $C_{\alpha \beta}$ is the [[charge conjugation matrix]].

=--

This phenomenon continues:

+-- {: .num_theorem}
###### Theorem
**([Huerta-Schreiber 17](#MTheoryFromTheSuperpoint))

The [[diagram]] of [[super Lie algebras]] shown on the right

is obtained by consecutively forming maximal central extensions invariant with respect to the maximal subgroup of automorphisms for which there are invariant cocycles at all. Here $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ is the $d$, $\mathbf{N}$ super-translation [[supersymmetry]] algebra. And these subgroups are the [[spin group]] covers $Spin(d-1,1)$ of the [[Lorentz groups]] $O(d-1,1)$.

=--

+-- {: .num_remark}
###### Remark

That every [[super Minkowski spacetime]] is _some_ [[central extension]] of some [[superpoint]]  is elementary.
This was highlighted in ([Chryssomalakos-Azcárraga-Izquierdo-Bueno 99,
2.1](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#CAIB99)).
But most central extensions of superpoints
are nothing like super-Minkowksi spacetimes.
The point of the above proposition
is to restrict attention to _iterated invariant_ central extensions
and to find that these single out the super-Minkowski spacetimes.

=--

So from studying iterated invariant [[central extensions]]
of [[super Lie algebras]],
starting with the [[superpoint]],
we  (re-)discover

1. [[pseudo-Riemannian geometry|Lorentzian geometry]],

1. [[spin geometry]].

1. [[super spacetimes]].

+-- {: .num_example #D0Cocycle}
###### Example

$\,$

The 2-cocycle that classifies the extension

$$
  \array{
    \mathbb{R}^{10,1\vert \mathbf{32}} && 11d, N = 1
    \\
    \downarrow
    \\
    \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} && 10d, \text{type IIA}
  }
$$

is

$$
  \overline{\psi} \wedge \Gamma_{10} \psi
    \;\in\;
  CE(\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}})
$$

This happens to also be the [[curvature]] of the [[WZW-term]]
in the [[Green-Schwarz sigma-model]] for the [[D0-brane]]. We come back to this
below in remark \ref{TheNatureOfTheD0Cocycle}.

=--

### The super-string and the super-membrane
 {#TheSuperStringAndTheSuperMembrane}

+-- {: .num_prop #CocyclesForSuperStringAndSupermembrane}
###### Proposition
**([Achúcarro-Evans-Townsend-Wiltshire 87](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#AETW87), [Azcárraga-Townsend 89](Green-Schwarz+action+functional#AzcarragaTownsend89), [Brandt 12-13](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#Brandt12-13))**

The maximal invariant 3-cocycle on 10d [[super Minkowski spacetime]] (according to remark \ref{SuperMinkowskiInvariantCocycles}) is

$$
  \mu_{F1} = \left(\overline{\psi} \wedge \Gamma_a \psi\right) \wedge e^a
$$

This is the [[WZW term]] for the [[Green-Schwarz superstring]] ([Green-Schwarz 84](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#GreenSchwarz84)).

The maximal invariant 4-cocycle on 11d [[super Minkowski spacetime]] is

$$
  \mu_{M2} = \tfrac{i}{2} \left(\overline{\psi} \wedge \Gamma_{a b} \psi \right) \wedge e^a \wedge e^b
$$

This is the [[higher WZW term]] for the [[supermembrane]]
([Bergshoeff-Sezgin-Townsend 87](https://ncatlab.org/nlab/show/M2-brane#BergshoeffSezginTownsend87)).

This classification is also known as the **old [[brane scan]]**.

=--

The 4-cocycle here reflects the first [[Fierz identity]] in prop. \ref{TheM2andM5CocyclesAsFierzIdentities}.

| $\stackrel{d}{=}$ |  $p =$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |  8 |  9 |
|--|--------|---|---|---|---|---|---|---|----|----|
| 11 | |  |  [[M2-brane|M2]] |  |   |  |   |   |   |   |
| 10 |  | [[F1-brane|F1]]   |   |  |   | [[NS5-brane|NS5]] |   |   |   |   |
| 9 | |   |   |  | $\;\;\ast\;\;$  |  |   |   |   |   |
| 8 | |   |   | $\ast$ |   |  |   |   |   |   |
| 7 | |   | $\ast$  |  |   |  |   |   |   |   |
| 6 | |  $\ast$  |  | [[3-brane in 6d|S3]] |   |  |   |   |   |   |
| 5 | |  | $\ast$   | |   |  |   |   |   |   |
| 4 | | $\ast$  | [[super 2-brane in 4d|M2]]$_{cmp}$   | |   |  |   |   |   |   |
| 3 | | [[super 1-brane in 3d|F1]]$_{cmp}$  |  | |   |  |   |   |   |   |

Notice that the entries of this table lie on four diagonal lines.
It turns out that each cocycle below and to the left of another cocycle
arises from it by an super Lie alghebraic incarnation of [[double dimensional reduction]].
This we discuss below in _[Double dimensional reduction](#DoubleDimensionalReduction)_

+-- {: .num_remark #TheWZWTermsForStringAndMembrane}
###### Remark

Here "**[[higher WZW term]]**" means the following:

Regard $\mu_{F1} = \left(\overline{\psi} \wedge \Gamma_a \psi\right) \wedge e^a$
as a left [[invariant differential form]]
on [[super-Minkowski spacetime]].
Choose any differential form potential $B_{F1}$, i.e. such that

$$
  d_{dR} B_{F1} = \left(\overline{\psi} \wedge \Gamma_a \psi\right) \wedge e^a
  \,.
$$

(This $B_{F1}$ will _not be left-invariant_.)

Then the [[Green-Schwarz action functional]] for the [[superstring]]
is the function on the space of [[sigma-model]] [[field (physics)|fields]]

$$
  \phi \;\colon\; \underset{\text{worldsheet}}{\underbrace{\Sigma_2}}
    \longrightarrow
    \underset{\text{super-spacetime}}{\underbrace{\mathbb{R}^{9,1\vert \mathbf{N}}}}
$$

(morphisms of [[supermanifolds]]) given by

$$
  \phi
    \;\mapsto\;
  \underset{\text{kinetic action}}{
  \underbrace{
  \int_{\Sigma_2}
  \sqrt{
    -det(\partial_{\sigma^i} e^a(\phi) \partial_{\sigma_j} e_b(\phi))
  }
  \,
  d \sigma^1 \wedge d\sigma^2
  }}
  +
  \underset{\text{WZW term}}{
  \underbrace{
    \int_{\Sigma^2} \phi^\ast B_{F1}
  }
  }
  \,.
$$

The first term is the [[Nambu-Goto action]] the second is a [[WZW term]].

$\,$

Originally [Green-Schwarz 84](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#GreenSchwarz84) introduced $B_{F1}$
to ensure an additional fermionic symmetry: "[[kappa-symmetry]]".

Notice that $B_{F1}$ looks somewhat complicated
and is not unique. That it is simply a [[WZW-term]]
for the [[supersymmetry]] [[supergroup]]

$$
  \mathbb{R}^{9,1\vert \mathbf{N}}
   =
  Iso(\mathbb{R}^{9,1\vert \mathbf{N}}) / Spin(9,1)
$$

was observed in [Henneaux-Mezincescu 85](https://ncatlab.org/nlab/source/Green-Schwarz+action+functional#HenneauxMezincescu85).

$\,$

Similarly, choose any differential form potential $C_{M2}$ such that

$$
  d_{dR} C_{M2} = \left(\overline{\psi} \wedge \Gamma_{a b} \psi\right) \wedge e^a \wedge e^b
  \,.
$$

(This $C_{M2}$ will _not be left-invariant_.)

Then the [[Green-Schwarz action functional|Green-Schwarz type action functional]]
for the [[supermembrane]] is the function on [[sigma-model]] [[field (physics)|fields]]

$$
  \phi \;\colon\; \underset{\text{worldvolume}}{\underbrace{\Sigma_3}} \longrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}
$$

given by

$$
  \phi
    \;\mapsto\;
  \underset{\text{kinetic action}}{
  \underbrace{
  \int_{\Sigma_2}
  \sqrt{
    -det(\partial_{\sigma^i} e^a(\phi)\partial_{\sigma_j} e_b(\phi))
  }
  \,
  d \sigma^1 \wedge d\sigma^2 \wedge d\sigma^3
  }}
  +
  \underset{\text{WZW term}}{
  \underbrace{
    \int_{\Sigma^3} \phi^\ast C_{M2}
  }
  }
  \,.
$$

On the right this is the [[higher WZW term]].

=--

### Extended super Minkowski spacetimes
 {#ExtendedSuperMinkowskiSpacetime}

This way we may finally continue the progression of invariant [[central extensions]] to [[higher central extensions]]:

Recall from prop. \ref{HomotopyFibersOfLInfinityCocycles} that and how higher cocycles classify [[higher central extensions]].

+-- {: .num_defn #TheMembraneAndStringCocycles}
###### Definition

According to remark \ref{TheWZWTermsForStringAndMembrane}
we name the [[homotopy fibers]] (prop. \ref{HomotopyFibersOfLInfinityCocycles}) of the cocycles from prop. \ref{CocyclesForSuperStringAndSupermembrane}, which are the [[higher WZW terms]] of the [[superstring]] and the [[supermembrane]],
as follows

$$
  \array{
    \mathfrak{m}2\mathfrak{brane}
    \\
    {}^{\mathllap{hofib}(\mu_{M2})}\downarrow
    \\
    \mathbb{R}^{10,1\vert \mathbf{32}}
    &\underset{\mu_{M2}}{\longrightarrow}&
    B^3 \mathbb{R}
  }
$$

$\,$

$\,$

$$
  \array{
    \mathfrak{string}_{IIB}
    \\
    {}^{\mathllap{hofib}(\mu_{F1}^B)}\downarrow
    \\
    \mathbb{R}^{9,1\vert \mathbf{16}+ \mathbf{16}}
    &\underset{\mu_{F1}^B}{\longrightarrow}&
    B^2 \mathbb{R}
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    \mathfrak{string}_{het}
    \\
    {}^{\mathllap{hofib}(\mu_{F1}^{het})}\downarrow
    \\
    \mathbb{R}^{9,1\vert \mathbf{16}}
    &\underset{\mu_{F1}^{het}}{\longrightarrow}&
    B^2 \mathbb{R}
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    \mathfrak{string}_{IIA}
    \\
    {}^{\mathllap{hofib}(\mu_{F1}^A)}\downarrow
    \\
    \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}
    &\underset{\mu_{F1}^A}{\longrightarrow}&
    B^2 \mathbb{R}
  }
$$

=--

+-- {: .num_example}
###### Example

By prop. \ref{HomotopyFibersOfLInfinityCocycles} the super Lie 2-algebra $\mathfrak{string}_{het}$ is given by

$$
  CE(\mathfrak{string}_{het})
  =
  \left\{
    \array{
      d e^a = \overline{\psi} \Gamma^a \psi,
      \; d \psi^\alpha = 0
      \\
      d b_2 = \mu_{F1}^{het} = (\overline{\psi} \wedge \Gamma_a \psi)\wedge e^a
    }
  \right\}
$$

This is a super-version of the _[[string Lie 2-algebra]]_ ([Baez-Crans-Schreiber-Stevenson 05](#BCSS05)
which controls [[Green-Schwarz anomaly cancellation]] ([Sati-Schreiber-Stasheff 12](#SatiSchreiberStasheff12))
and the topology of the [[supergravity C-field]] ([Fiorenza-Sati-Schreiber 12a](#FiorenzaSatiSchreiber12a), [12b](#FiorenzaSatiSchreiber12b)).

$\,$

The [[supergravity Lie 3-algebra|membrane super Lie 3-algebra]] $\mathfrak{m}2\mathfrak{brane}$ is given by

$$
  CE(\mathfrak{m}2\mathfrak{brane})
  =
  \left\{
    \array{
      d e^a = \overline{\psi} \wedge \Gamma^a \psi, \; d \psi^\alpha = 0
      \\
      d b_3 = i (\overline{\psi} \wedge \Gamma_{a b} \psi) \wedge e^a \wedge e^b
    }
  \right\}
$$

This dg-algebra was first considered  in [D'Auria-Fré 82 (3.15)](https://ncatlab.org/nlab/show/D'Auria-Fre+formulation+of+supergravity#DAuriaFre)
as a tool for constructing [[11-dimensional supergravity]].
For exposition from the point of view of [[Lie 3-algebras]] see also [Baez-Huerta 10](https://arxiv.org/abs/1003.3436).

=--

$\,$

Hence the progression of maximal invariant extensions of the superpoint continues as a [[diagram]] of [[super L-∞ algebras]] like so:

$\,$

$\,$

(While every extension displayed is an invariant [[universal higher central extension]], not all invariant [[universal higher central extensions]] are displayed. For instance there are string and membrane GS-WZW-terms / cocycles also on the lower dimensional super-Minkowski spacetimes ("non-critical"), e.g. the [[super 1-brane in 3d]] and the [[super 2-brane in 4d]].)

$\,$

But how are we to think of the [[extended super Minkowski spacetimes]] geometrically?

This is clarified by the following result:

$\,$

+-- {: .num_prop #GaugeStackInsideExtendedSpacetime}
###### Proposition
**([Fiorenza-Sati-Schreiber 13, section 5](#FSS13))**

Write $\widetilde {String_{IIA}}$ for the [[super formal smooth infinity-groupoid|super 2-group]]
that [[Lie integration|Lie integrates]] the super [[Lie 2-algebra]] $\mathfrak{string}_{IIA}$
subject to the condition that it carries a globally defined [[Maurer-Cartan form]].
Then for $\Sigma_{p+1}$ a [[worldvolume]] [[smooth manifold]]
there is a [[natural equivalence]]

$$
  \left\{
     \Sigma_{p+1}
       \stackrel{\Phi}{\longrightarrow}
     \widetilde{String_{IIA}}
  \right\}
  \;\;\;
    \leftrightarrow
  \;\;\;
  \left\{
    \array{
      \Sigma_{p+1} \stackrel{\phi}{\longrightarrow} \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}} },
      \\
      \nabla \in Conn(\Sigma_{p+1}, \phi^\ast  \mu_{string_{IIA}} )
    }
  \right\}
$$

between "higher Sigma-model fields" $\Phi$
and pairs, consisting of
an ordinary sigma-model field $\phi$
and a [[gauge field]] $\nabla$ on the [[worldvolume]] of the D-brane
twisted by the [[Kalb-Ramond field]].

This is the _[[Chan-Paton gauge field]]_ on the D-brane.

$\,$

Similarly:

Write $\widetilde {M2Brane}$ for the [[super formal smooth infinity-groupoid|super 3-group]]
that [[Lie integration|Lie integrates]] the super [[Lie 3-algebra]] $\mathfrak{m}2\mathfrak{brane}$
subject to the condition that it carries a globally defined [[Maurer-Cartan form]].
Then for $\Sigma_{5+1}$ a [[worldvolume]] [[smooth manifold]]
there is a [[natural equivalence]]

$$
  \left\{
     \Sigma_{5+1}
       \stackrel{\Phi}{\longrightarrow}
     \widetilde{M2Brane}
  \right\}
  \;\;\;
    \leftrightarrow
  \;\;\;
  \left\{
    \array{
      \Sigma_{5+1} \stackrel{\phi}{\longrightarrow} \mathbb{R}^{10,1\vert \mathbf{32}  },
      \\
      \nabla \in 2Conn(\Sigma_{p+1}, \phi^\ast  \mu_{M2} )
    }
  \right\}
$$

between "higher Sigma-model fields" $\Phi$
and pairs, consisting of
an ordinary sigma-model field $\phi$
and a [[higher gauge field]] $\nabla$ on the [[worldvolume]] of the [[M5-brane]]
and twisted by the [[supergravity C-field]].

=--

$\,$

(See also at _[[schreiber:Structure Theory for Higher WZW Terms]]_, session II).

### The M5-brane and the D-branes
 {#TheSuperDBranes}

The "old [[brane scan]]" classifying the [[superstring]] and the [[supermembrane]]
from [above](#TheSuperStringAndTheSuperMembrane) ran into a conundrum::

Given that  [[superstrings]] and [[supermembranes]] are nicely classified by [[super Lie algebra]] [[Lie algebra cohomology|cohomology]]
(prop. \ref{CocyclesForSuperStringAndSupermembrane})
why do the other [[super p-branes]] known in [[string theory]] not show up similarly? Where are the [[D-branes]] and the [[M5-brane]]?

But from the discussion [above](#ExtendedSuperMinkowskiSpacetime) we see that we should look for
further higher cocycles not on [[super Lie algebras]]
but the [[super L-∞ algebras]]: on the [[extended super Minkowski spacetimes]] of def. \ref{TheMembraneAndStringCocycles}.

+-- {: .num_defn #TheM5BraneCocycle}
###### Definition

In $CE(\mathfrak{m}2\mathfrak{brane})$ from def. \ref{TheM2andM5CocyclesAsFierzIdentities}, define the
element

$$
  \begin{aligned}
    \mu_{M5}
      & \coloneqq
     \tfrac{1}{5!}
     \left(
       \overline{\psi}
       \wedge
        \Gamma_{a_1 \cdots a_5}
       \psi
     \right)
     \wedge
     e^{a_1}
       \wedge
       \cdots
       \wedge
     e^{a_5}
      \;+\;
      h_3 \wedge \mu_{M2}
      \\
      & =
     \tfrac{1}{5!}
     \left(
       \overline{\psi}
       \wedge
        \Gamma_{a_1 \cdots a_5}
       \psi
     \right)
     \wedge
     e^{a_1}
       \wedge
       \cdots
       \wedge
     e^{a_5}
      \;+\;
      h_3 \, \wedge\,
      \tfrac{i}{2}
      \left(
        \overline{\psi}
          \Gamma_{a_1 a_2}
        \psi
      \right)
      \wedge e^{a_1} \wedge e^{a_2}
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop #TheM5BraneCocycleAsACocycle}
###### Proposition
**([D'Auria-Fré 82, (3.27, 3.28)](#DAuriaFre82))**

The element $\mu_{M5} \in \CE(\mathfrak{m}2\mathfrak{brane})$ from def. \ref{TheM5BraneCocycle}
is closed

$$
  d_{CE} \, \mu_{M5} = 0
  \,.
$$

=--

+-- {: .proof}
###### Proof

Recall that  in $CE(\mathfrak{m}2\mathfrak{brane})$ there is the relatioon

$$
  d_{CE} \, h_3 = \mu_{M2}
  \;\;\;
  with
  \;\;\;
  \mu_{M2}
    \; \coloneqq \;
  \tfrac{i}{2}
  \left(
    \overline{\psi}
      \wedge
      \Gamma_{a b}
    \psi
  \right)
  \wedge
    e^a
  \wedge
    e^b
  \,.
$$

Now we compute:

$$
  \begin{aligned}
    d_{CE} \,\underset{\text{M5-brane} \atop {\kappa\text{-symmetry flux}  }}{\underbrace{\mu_{M5}}}
      & =
    \tfrac{1}{5!}
      d_{CE}\left(
         \left(
           \overline{\psi}
           \wedge
           \Gamma_{a_1 \cdots a_5}
           \psi
         \right)
         \wedge
          e_{a_1}
            \wedge
            \cdots
            \wedge
          e_{a_5}
      \right)
      \;+\;
      d_{CE} h_3 \wedge \mu_{M2}
      \\
      & =
      \tfrac{1}{4!}
      \underset{
        \underset{\text{Fierz identity} \atop \text{D'Auria-Fre '82}}{=}
        3
        \left(
           \overline{\psi}
             \wedge
             \Gamma_{[a_1 a_2}
           \psi
        \right)
        \wedge
        \left(
          \overline{\psi}
          \wedge
            \Gamma_{a_3 a_4]}
          \psi
        \right)
      }{
      \underbrace{
      \left(
        \overline{\psi}
          \wedge
          \Gamma_{[a_1 \cdot a_4 a]}
          \psi
      \right)
      \wedge
      \left(
        \overline{\psi}
         \wedge
         \Gamma^a
         \psi
      \right)
     }
     }
     \wedge
     e^{a_1} \wedge \cdots \wedge e^{a_4}
     \;+\;
     \tfrac{1}{2}\mu_{M2} \wedge \mu_{M2}
     \\
     & =
     -
     \tfrac{1}{2}
     \mu_{M2} \wedge \mu_{M2}
     +
     \tfrac{1}{2}
     \mu_{M2} \wedge \mu_{M2}
     \\
     & = 0
  \end{aligned}
$$

Here the identity under the brace is the [[Fierz identity]] from prop. \ref{TheM2andM5CocyclesAsFierzIdentities}.

=--

Notice how it is the presence of the extra higher generator $h_3$ of degree 3 in $CE(\mathfrak{m}2\mathfrak{brane})$ that
makes prop. \ref{TheM5BraneCocycle} work.

That the element $\mu_{M5}$ in def. \ref{TheM5BraneCocycleAsACocycle} is the curvature of the [[higher dimensional WZW model|higher WZW-term]]
of the [[M5-brane]] was argued in [BLNPST97 ](M5-brane#BLNPST97)

It turns out that from just prop. \ref{TheM5BraneCocycleAsACocycle} one obtains the [[D-brane]] cocycles in 10d by applying
[[super Lie n-algebra]] constructions corresponding to
[[double dimensional reduction]] and [[topological T-duality]].
This we discuss below in _[Double dimensional reduction](#DoubleDimensionalReduction)_
and in _[T-duality](#TDuality)_.

Here for the moment we just state the resulting cocycle conditions, below as def. \ref{TheDBraneCocycles},
prop. \ref{TheIIADBraneCocycles} and prop. \ref{TheCocyclesForTheIIBTypeDBranes}.
In order to state this conveniently, we first recall in def. \ref{ConvenientSpinorBasis} the
well adapted bases for type IIA/IIB spinors from _[[geometry of physics -- supersymmetry]]_
_[Example: Spinors in dimension 11, 10 and 9](geometry+of+physics+--+supersymmetry#InDimensions11And10And9)_

+-- {: .num_defn #ConvenientSpinorBasis}
###### Definition
**(basis for IIA/IIB spinors)**

Let $\{\gamma_a\}_{a = 0}^{d-1}$ be a [[Dirac representation]] on $\mathbb{C}^{16}$ of the Lorentzian $d= 9$
  Clifford algebra $Cl(8,1)$. We obtain a Dirac representation of the $d = 10$ and $d = 11$ Clifford algebra by taking the following block matrices acting on $\mathbb{C}^{16} \oplus \mathbb{C}^{16}$
  $$
  \Gamma_{a \leq 8}
  \coloneqq
  \left(
   \array{
       0 & \gamma^a
       \\
       \gamma^a & 0
    }
  \right)
  \;, \qquad
  \Gamma_9
  \coloneqq
  \left(
    \array{
      0 & \mathrm{I}
      \\
      -\mathrm{I} & 0
    }
  \right)
  \;, \qquad
  \Gamma_{10}
  \coloneqq
  \left(
    \array{
      i \mathrm{I} & 0
      \\
      0 & -i \mathrm{I}
    }
  \right)
  \,,
$$
where $I$ is the identity matrix.

 The unique irreducible [[Majorana spinor]] representation of $\mathrm{Spin}(10,1)$ is of real dimension 32.
  Under the inclusions
  $$
    \mathrm{Spin}(8,1) \hookrightarrow \mathrm{Spin}(9,1) \hookrightarrow \mathrm{Spin}(10,1)
  $$
  this representation branches as
  $$
    \mathbf{32} \mapsto \mathbf{16}\oplus \overline{\mathbf{16}} \mapsto \mathbf{16} \oplus \mathbf{16}
    \,,
  $$
  where in the middle $\mathbf{16}$ and $\overline{\mathbf{16}}$ are the left and right chiral Majorana-Weyl
  representations in 10d, while on the right the $\mathbf{16}$ is again the unique irreducible real
  representation in 9d.
 Under this branching we decompose a Majorana spinor $\psi \in \mathbf{32}$ as
  $$
    \psi
    =
    \left(
      \array{
        \psi_1
        \\
        \psi_2
      }
    \right)
  $$
  with $\psi_1 \in \mathbf{16}$ and $\psi_2 \in \overline{\mathbf{16}}$ or $\mathbf{16}$.

  Define another set of matrices $\{\Gamma_a^B\}_{a = 0}^{9}$
  by
  $$
    \Gamma_a^{B}
     :=
    \left\{
      \array{
         \Gamma_a & \vert \; a \leq 8\;,
         \\
         \begin{pmatrix}
              0 & \mathrm{I}
              \\
              \mathrm{I} & 0
           \end{pmatrix}
         &
         \vert \; a = 9\;.
      }
    \right.
  $$
  For emphasis we write the original matrices also as $\Gamma_a^A := \Gamma_a$, for $a \leq 9$.

  Moreover we also write
  $$
    \sigma_1 := \Gamma_9
    \;\,,\;\;\;\;\;
    \sigma_2 := -\Gamma_{9} \Gamma_{10}
    \;\,,\;\;\;\;\;
    \sigma_3 := \Gamma_{10}
    \,.
  $$

  Noice that the matrices $\{\Gamma^B_a\}_{a = 0}^9$ in  do not represent
  a Clifford algebra, but the product of any even number of them represents the correct such
  product acting on $\mathbf{16} \oplus \mathbf{16}$. For instance $\exp(\omega^{a b}\Gamma^B_{a b})$
  are the elements of the $\mathrm{Spin}(d-1,1)$-representation on $\mathbf{16}\oplus \mathbf{16}$.
  Also, for \emph{odd} $p = 2k+1$, each of the pairings
  $$
    \overline{\psi}\Gamma^B_{a_1 \cdots a_p}\psi
      =
    \psi^\dagger \Gamma^B_0 \Gamma^B_{a_1 \cdots a_p} \psi
  $$
  is the sum of the corresponding pairings on two copies of $\mathbf{16}$.

By these definition we have
  $$
    \Gamma_9^B = i \, \Gamma_9 \Gamma_{10} = \Gamma_9 \Gamma_{11}
    \,.
  $$

This relation also makes it manifest that $\Gamma_9^B$ commutes not only with all $\Gamma^B_{a b}$ for $a,b \leq 8$, but also with all $\Gamma_a^B\Gamma_9^B$. Consequently, $\Gamma_{10}$ as well as $\Gamma_9$ are invariant under the IIB Spin-action, in that

  $$
    \exp(-\omega^{a b}\Gamma^B_{a b}) \; \sigma_i \; \exp(\omega^{a b} \Gamma^B_{a b})
      =
    \sigma_i
  $$

for $i \in \{1,2,3\}$.

Conversely, rotation in the $(9,10)$-plane leaves all the $\Gamma_a^B$ invariant, in that

  $$
    \exp(- \tfrac{\alpha}{4} \Gamma_9 \Gamma_{10}) \,\Gamma_a^B\, \exp(\tfrac{\alpha}{4} \Gamma_9 \Gamma_{10})
      \;=\;
    \Gamma_a^B
    \,.
  $$

=--

+-- {: .num_defn #TheDBraneCocycles}
###### Definition
**(the D-brane cocycles)**

Define the following elements in $CE(\mathfrak{string}_{IIA})$: (def. \ref{TheMembraneAndStringCocycles})

$$
  \begin{aligned}
    C_2
      & \coloneqq
    \left(\overline{\psi} \wedge \Gamma^{10} \psi\right)
    \\
    C_4
      & \coloneqq
    \tfrac{i}{2}
    \left(
       \overline{\psi}
         \Gamma_{a_1 a_2}
       \psi
    \right)
    \wedge
     e^{a_1}
     \wedge
     e^{a_2}
     \\
    C_6
      & \coloneqq
    \tfrac{1}{4!}
    \left(
       \overline{\psi}
         \Gamma_{a_1 \cdots a_4}
         \Gamma_{10}
       \psi
    \right)
    \wedge
     e^{a_1}
     \wedge
     \cdots
     \wedge
     e^{a_4}
     \\
    C_8
      & \coloneqq
    \tfrac{i}{6!}
    \left(
       \overline{\psi}
         \Gamma_{a_1 \cdots a_6}
       \psi
    \right)
    \wedge
     e^{a_1}
     \wedge
     \cdots
     \wedge
     e^{a_6}
     \\
    C_{10}
      & \coloneqq
    \tfrac{1}{8!}
    \left(
       \overline{\psi}
         \Gamma_{a_1 \cdots a_8}
         \Gamma_{10}
       \psi
    \right)
    \wedge
     e^{a_1}
     \wedge
       \cdots
     \wedge
     e^{a_8}
     \\
    C_{12}
      & \coloneqq
    \tfrac{i}{10!}
    \left(
       \overline{\psi}
         \Gamma_{a_1 \cdots a_{10}}
       \psi
    \right)
    \wedge
     e^{a_1}
     \wedge
       \cdots
     \wedge
     e^{a_{10}}
  \end{aligned}
$$

Then set

$$
  C_{IIA}
  \;\coloneqq\;
  \underset{i}{\sum}
  C_{2i}
$$

and for $p \in \{2,4,6, \cdots, 10\}$

$$
  \mu_{D p}
    \;\coloneqq\;
  [\exp(f_2) \wedge C_{IIA} ]_{p+2}
$$

where

1. $f_2$ is the extra generator in $CE(\mathfrak{string}_{IIA})$;

1. $\exp(f_2) \coloneqq \underset{k}{\sum} \tfrac{1}{k!} \underset{k \text{ factors}}{\underbrace{f_1 \wedge \cdots \wedge f_2}} $

1. $[-]_{p+2}$ denotes the summand of homogeneous degree $p+2$.

Similarly, define the following elements in $CE(\mathfrak{string}_{IIB})$ (def. \ref{TheMembraneAndStringCocycles})
(with the Clifford elements as in def. \ref{ConvenientSpinorBasis}):

$$
  \begin{aligned}
    C_3
      & \coloneqq
    i
    \left(
      \overline{\psi}
        \wedge
        \Gamma^B_{a} \Gamma_{10}
      \psi
    \right)
    \wedge
    e^a
    \\
    C_5
      & \coloneqq
    \tfrac{1}{3!}
    \left(
      \overline{\psi}
        \wedge
        \Gamma^B_{a_1 \cdots a_3} \Gamma_{9} \Gamma_{10}
      \psi
    \right)
    \wedge
    e^{a_1}
      \wedge
      \cdots
    e^{a_3}
    \\
    C_7
      & \coloneqq
    \tfrac{i}{5!}
    \left(
      \overline{\psi}
        \wedge
        \Gamma^B_{a_1 \cdots a_5} \Gamma_{9}
      \psi
    \right)
    \wedge
    e^{a_1}
      \wedge
      \cdots
    e^{a_5}
    \\
    C_9
      & \coloneqq
    \tfrac{1}{7!}
    \left(
      \overline{\psi}
        \wedge
        \Gamma^B_{a_1 \cdots a_7} \Gamma_{9} \Gamma_{10}
      \psi
    \right)
    \wedge
    e^{a_1}
      \wedge
      \cdots
    e^{a_7}
    \\
    C_{11}
      & \coloneqq
    \tfrac{i}{9!}
    \left(
      \overline{\psi}
        \wedge
        \Gamma^B_{a_1 \cdots a_9} \Gamma_{9}
      \psi
    \right)
    \wedge
    e^{a_1}
      \wedge
      \cdots
    e^{a_9}
  \end{aligned}
$$

Then set

$$
  C_{IIB}
  \;\coloneqq\;
  \underset{i}{\sum}
  C_{2i+1}
$$

and for $p \in \{1,3,5, \cdots 9\}$

$$
  \mu_{D p}
    \;\coloneqq\;
  [\exp(f_2) \wedge C_{IIB} ]_{p+2}
  \,.
$$

=--

The following statements may be obtained from the existence of the [[M5-brane]] cocycle (prop. \ref{TheM5BraneCocycleAsACocycle})
and applying [[double dimensional reduction]] and [[T-duality]]. We discuss this in detail below in
_[Double dimensional reduction](#DoubleDimensionalReduction)_ and _[T-duality](#TDuality)_, for the moment
we are content with stating it as a fact:

+-- {: .num_prop #TheIIADBraneCocycles}
###### Proposition
**([Chryssomalakos-Azcárraga-Izquierdo-Bueno 99](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#CAIB99),

The elements $\mu_{D p} \in CE(\mathfrak{string}_{IIA})$ from def. \ref{TheDBraneCocycles} are closed

$$
  d_{CE}\, \mu_{D p} = 0
$$

and non-exact.

=--

+-- {: .num_prop #TheCocyclesForTheIIBTypeDBranes}
###### Proposition
**([Sakaguchi 99](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#Sakaguchi00))**

The elements $\mu_{D p} \in CE(\mathfrak{string}_{IIB})$ from def. \ref{TheDBraneCocycles} are closed

$$
  d_{CE} \, \mu_{D p} = 0
$$

and non-exact.

=--

+-- {: .num_defn }
###### Definition

By prop. \ref{HomotopyFibersOfLInfinityCocycles} the higher cocycles for the [[M5-brane]] (prop. \ref{TheM5BraneCocycleAsACocycle})
and for the [[D-branes]] (prop. \ref{TheIIADBraneCocycles} and prop. \ref{TheCocyclesForTheIIBTypeDBranes})
classify
further higher super $L_\infty$-algebra extensions. These we name again by the branes that the correspond
to. So the following diagrams denote [[homotopy fiber sequences]]

$$
  \array{
    \mathfrak{m}5\mathfrak{brane}
    \\
    {}^{\mathllap{hofib(\mu_{M5})}}\downarrow
    \\
    \mathfrak{m}2\mathfrak{brane}
      &\underset{\mu_{M5}}{\longrightarrow}&
    B^6 \mathbb{R}
  }
$$

and

$$
  \array{
    \mathfrak{d}p\mathfrak{brane}
    \\
    {}^{\mathllap{hofib(\mu_{D p})}}\downarrow
    \\
   \mathfrak{string}_{IIA/B}
     &\underset{\mu_{D p}}{\longrightarrow}&
   B^{p+1}\mathbb{R}
  }
$$

=--

So in conclusion, by forming iterated invariant [[universal higher central extensions]] of the [[superpoint]], there emerges first [[spacetime]] and then the fundamental [[p-branes]] that propagate in spacetime.

$\,$

$\,$

> Perhaps we need to understand the nature of time itself better. $[...]$ One natural way to approach that question would be to understand in what sense time itself is an emergent concept, and one natural way to make sense of such a notion is to understand how pseudo-Riemannian geometry can emerge from more fundamental and abstract notions such as categories of branes. ([[Greg Moore|G. Moore]], p.41 of "[[Physical Mathematics and the Future]]", talk at [Strings 2014](http://physics.princeton.edu/strings2014/))

$\,$

It serves to have a closer look at the cocycle for the [[D0-brane]]:

+-- {: .num_remark #TheNatureOfTheD0Cocycle}
###### Remark

Notice that all the [[D-brane]] cocycle $\mu_{D p}$ in def. \ref{TheDBraneCocycles} are _higher_ cocycles
except for that of the [[D0-brane]], which is just an ordinary 2-cocycle.
The ordinary central extension that this classifies
is just that which grows the 11th M-theory dimension by the above example \ref{D0Cocycle}.

$$
  \array{
    \mathbb{R}^{10,1\vert \mathbf{32}}
    \\
    {}^{\mathllap{hofib(\mu_{D0})}} \downarrow
    \\
    \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}
       &\underset{\mu_{D0} =  \overline{\psi} \Gamma_{10} \psi }{\longrightarrow}&
    B \mathbb{R}
  }
$$

This may be thought of
as a super $L_\infty$-theoretic incarnation
of D0-brane [[condensate|condensation]]
([Polchinski 99, around p. 8](https://ncatlab.org/nlab/show/D0-brane#Polchinski99)).

More in detail, if we distinguish $\overline{\psi} \wedge \Gamma_{10} \psi$
as an element of $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}^\ast}}$
or as the element $\mu_{D0}$ of $\mathfrak{string}_{IIA}$ (which in components are the same,
just regarded in different contexts), then the relation between the D0-brane and the M-theory spacetime
extension may be stated as follows: The following diagram is [[homotopy pullback|homotopy Cartesian]]
(a [[homotopy pullback]]) square:

$$
  \array{
    \mathbb{R}^{10,1\vert \mathbf{32}}
    &\longleftarrow&
    \mathfrak{d}0\mathfrak{brane}
    \\
    \downarrow &(pb)& \downarrow
    \\
    \mathbb{R}^{9,1\vert \mathbf{16}+ \mathbf{16}^\ast }
    &\longleftarrow&
    \mathfrak{string}_{IIA}
  }
  \,.
$$

=--

+-- {: .num_remark}
###### Remark
**(brane intersection laws)**

In addition to reflecting all the brane speciees, the above [[schreiber:brane bouquet]] knows the _brane intersection laws_: there is a morphism $p_2\mathfrak{brane} \longrightarrow p_1 \mathfrak{brane}$
precisely if the given species of $p_1$-branes may end on the given species of $p_2$-branes
(more discussion of this is in [Fiorenza-Sati-Schreiber 13, section 3](#FSS13)).

But recall from prop. \ref{GaugeStackInsideExtendedSpacetime} the interpretation of the [[extended super Minkowski spacetimes]]
as containing a condensate of branes sourcing the gauge field on the worldvolume of the
branes that they may end on.

In conclusion this shows that
given a cocycle $\mu_{p_1+2}$ for some super $p_1$-brane species
inducing an [[extended super Minkowski spacetime]] via its [[homotopy fiber]]
and then given a consecutive cocycle $\mu_{p_2+2}$ for a $p_2$-brane species on that homotopy fiber
then $p_1$-branes may end on $p_2$-branes
and the $p_2$-branes propagating in the extended spacetime $p_1 \mathfrak{brane}$
see a higher gauge field on their worldvolume
of the kind sourced by boundaries of $p_1$-branes.

=--

$\,$

$$
  \array{
    {
      \text{spacetime}
      \atop
      \text{with}\,p_1\text{-brane condensate}
    }
    &&& p_1 \mathfrak{brane} &\overset{\mu_{p_2+2}}{\longrightarrow}& B^{p_2+2}
    \\
    &&& {}^{\mathllap{hofib(\mu_{p_1+2})}}\downarrow
    \\
    \text{spacetime}
    &&& \mathbb{R}^{d-1,1\vert \mathbf{N}}
      &\underset{\mu_{p_1+2}}{\longrightarrow}&
    B^{p_1+1}\mathbb{R}
  }
$$

$\,$

Hence the [[extended super Minkowski spacetime]] $p_1 \mathfrak{brane}$
is like the original super spacetime $\mathbb{R}^{d-1,1\vert \mathbf{N}}$
but filled with a [[condensation|condensate]] of $p_1$-branes
whose boundaries source a [[higher gauge field]].

$\,$

While this is good,
it means that at each stage of the [[schreiber:brane bouquet]]
we are describing $p_2$-brane dynamics
on a fixed $p_1$-brane [[background field]].
But more generally
we would like to describe the joint dynamics
of all brane species at once.

$\,$

This we turn to now.

$\,$

## Fields
 {#Fields}

In the discussion [above](#Branes) we discovered all the [[p-brane]] species of [[string theory]]/[[M-theory]],
but each separately as a $b^{p+1}\mathbb{R}$-valued super $L_\infty$-cocycle on some
[[extended super Minkowski spacetime]] classified by a previous cocycle.

Here we discuss that all these cocycles unify into single cocycles.
However, these unified cocycles are no longer in [[ordinary cohomology]], meaning that they
no longer have
[[coefficients]] in the simple [[line Lie-n algebras]] $b^{p+1}\mathbb{R}$ as in proposition \ref{HomotopyFibersOfLInfinityCocycles}.

Instead they have coefficients in richer and in general non-abelian [[L-∞ algebras]].
One says they are [[cocycles]] in [[non-abelian cohomology]] theory.
By a phenomenon that has been called the [[Whitehead principle of non-abelian cohomology]],
this means more concretely that these unified cocycles are in a
_[[twisted cohomology|twisted]] [[generalized (Eilenberg-Steenrod) cohomology]] theory_.
Command of the general concept of this name is not strictly necessary for the
analysis of the brane cocycles below, but a rough idea of its basic ingredients
will make the meaning of the constructions and results very transparent. Therefore we
start below with a lightning overview of the idea

* _[Twisted generalized cohomology](#TwistedGeneralizedCohomology)_

Since [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]] is generally exhibited
by classifying [[spaces]], specifically [[topological spaces]], to make use of these
concepts we need a way to regard [[L-∞ algebras]] as certain spaces (or rather as [[homotopy types]] of certain spaces).
This connection is provided by a theory called _[[rational homotopy theory]]_. Again, this is
not strictly necessary for the derivation of the unified brane cocycles below, but it serves to
greatly clarify the resulting structures. Therefore we next give a lightning survey of

* _[Rational homotopy theory](#RationalHomotopyTheory)_.

After these conceptual preliminaries, we then turn to the concrete example of the [[schreiber:brane bouquet]]
established [above](#Branes)
and ask for homotopy descent of consecutive ordinary cocycles on [[extended super Minkowski spacetimes]]
to single but twisted generalized cocycles down on plain [[super Minkowski spacetimes]].

First we consider this for the [[D-branes]] and we derive that the unified F1/Dp-brane cocycles
exist with coefficients the [[rationalization]] of [[twisted K-theory]]. This is the content of the section

* _[RR-Fields](#RRFields)_.

In direct analogy, we then discover the corresponding rationalized coefficients for the [[background fields]]
for the [[M-branes]]. This turns out to be rationalized [[cohomotopy]] in degree 4. This we discuss in

* _[M-Flux Fields](#MFluxFields)_

More technically, what we show now is that
there is _homotopy [[descent]]_ of $p$-brane cocycles
from [[extended super Minkowski spacetime]]
down to ordinary [[super Minkowski spacetime]]
which yields cocycles in [[twisted cohomology]]
for the [[RR-field]] and the M-flux fields ([Fiorenza-Sati-Schreiber 15](#FSS15), [16a](#FSS16a)).

$\,$

### Twisted generalized cohomology
 {#TwistedGeneralizedCohomology}

There is a traditional physics story for how [[twisted cohomology|twisted]] [[generalized (Eilenberg-Steenrod) cohomology]]
appears on [[D-branes]]. Eventually we are after a precise derivation of this phenomenon from "first principles",
but as motivation and for intuition, it serves to recall the folklore:

It is often stated that a
[[Chan-Paton gauge field]] on $n$ coincident [[D-branes]]
is an [[special unitary group|SU(n)]]-[[vector bundle]] $V$,
hence a cocycle in [[nonabelian cohomology]] in degree 1.

But this is not quite true.
In general there are $n$ [[D-branes]] and $n'$ [[anti-D-branes]] coinciding,
carrying [[Chan-Paton gauge fields]]
$V_{brane}$ (of [[rank]] $n$) and $V_{\text{anti-brane}}$ (of rank $n'$), respectively,
yielding a pair of [[vector bundles]]
$$
  (V_{\text{brane}}, V_{\text{anti-brane}})
  \,.
$$
Such pairs are also called _[[virtual vector bundles]]_.

But [[D-branes]] annihilate with [[anti-D-branes]] ([Sen 98](https://ncatlab.org/nlab/show/anti+D-brane#Sen98))
when they are coincident and have exact opposite [[D-brane charge]],
which here means that they carry the same [[Chan-Paton gauge field|Chan-Paton]] vector bundle.
In other words, pairs as above of the special form
$(W,W)$ are equivalent to pairs of the form $(0,0)$.

$$
  (W,W)
    \;\sim\;
  0
  \,.
$$

Hence the net Chan-Paton charge of coincident branes and anti-branes
is really the [[equivalence class]] of $(V_{\text{brane}}, V_{\text{anti-brane}})$
under the [[equivalence relation]] which is generated by the [[relation]]

$$
  (V_{\text{brane}} \oplus W\,,\; V_{\text{anti-brane}} \oplus W)
    \;\sim\;
  (V_{brane}\,,\; V_{anti-brane})
$$

for all [[complex vector bundles]] $W$ ([Witten 98](https://ncatlab.org/nlab/show/anti+D-brane#Witten98),
[Witten 00](https://ncatlab.org/nlab/show/D-brane#Witten00)).

$\,$

The additive [[abelian group]] of such equivalence classes of [[virtual vector bundles]]
is called _[[topological K-theory]].
This behaves in many ways as [[ordinary cohomology]] does,  but is richer.
One says that it is a _[[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theory]]_.

Hence by this story, [[D-brane charge]] should take values in [[twisted K-theory]].
Therefore, by the analog of the [[Maxwell equations]] for [[RR-fields]], where [[D-brane charge]] plays
the role of [[electric charge]] and [[magnetic charge]] in [[electromagnetism]]
also the [[RR-fields]] must be [[cocycles]] in [[twisted K-theory]]
([Moore-Witten 00](https://ncatlab.org/nlab/show/anti+D-brane#MooreWitten99)).

$\,$

A famous fact about [[ordinary cohomology]] $H^n(X,\mathbb{Z})$ is that it is _represented_
by [[topological spaces]] denoted $K(\mathbb{Z},n)$ or $B^n \mathbb{Z}$ and called
_[[Eilenberg-MacLane spaces]]_: For $X$ a [[paracompact topological space]] then there is a [[natural bijection]]

$$
  H^n(X,\mathbb{Z})
    \;\simeq\;
  \left\{
    X \;\longrightarrow\; B^n \mathbb{Z}
  \right\}_{/homotopy}
$$

between [[cohomology]] classes of $X$  and [[homotopy classes]] of [[continuous maps]] from $X$ to the
[[Eilenberg-MacLane space]]. (This turns out to uniquely characterize the spaces $B^n \mathbb{Z}$.)

The collection of all these [[Eilenberg-MacLane spaces]] $B^n \mathbb{Z}$, as $n$ ranges, has the property
that each is the [[based loop space]] of the previous one, up to [[weak homotopy equivalence]]

$$
  B^n \mathbb{Z}
  \underoverset{\text{weak hom. equivalence}}{\simeq}{\longrightarrow}
  \Omega_\ast \left( B^{n+1}\mathbb{Z} \right)
  \,.
$$

More generally, in [[algebraic topology]] any sequence of [[pointed topological spaces]] $E_n$ indexed by the [[natural numbers]] and
equipped with such [[weak homotopy equivalences]]

$$
  E_n \overset{\simeq}{\longrightarrow} \Omega_\ast \left(  E_{n+1} \right)
$$

is called a _[[spectrum]]_, or specifically an _[[Omega-spectrum]]_.
See at _[[geometry of physics -- stable homotopy types]]_ for more on this.

Here for $X$ any [[pointed topological space]],
$\Omega X$ denotes the operation of constructing the [[based loop space|space of continuous loops]]
in $X$, starting and ending at the given basepoint. For later use we mention that a fancy-looking but
most useful alternative way to think of a such as loop is as a _[[homotopy]]_ from the basepoint to itself.
Thought of this way, then the based loop space is equivalently the [[homotopy fiber product]] of the
base point inclusion with itself, denoted as follows:

$$
  \Omega_\ast \left( X  \right)
    \;\simeq\;
  \underset{
    \text{homotopy} \atop \text{fiber product}
  }{
  \underbrace{
    \ast \underoverset{X}{h}{\times} \ast
  }
  }
  \,.
$$

In direct analogy to the above situation, one says for $E_\bullet$ any such
[[Omega-spectrum]] that the [[homotopy classes]] of [[continuous maps]]
into its component space in degree $n$

$$
  E^n(X)
   \;\coloneqq\;
  \left\{
    X \longrightarrow E_n
  \right\}_{/\text{homotopy}}
$$

are the $E$-[[generalized cohomology]] classes of $X$ i degree $n$.

One also says that the cohomology theory $E^\bullet(-)$ is
_[[generalized cohomology theory]] is [[Brown representability theorem|represented]]_ by the _[[spectrum]]_ $E$.

The example of interest for D-brane charge, [[topological K-theory]], is the [[generalized cohomology theory]] denoted by

$$
  E = KU
$$

with

$$
  KU_{2n} \simeq (B U) \times \mathbb{Z}
  \;\;\,,\;\;
  KU_{2n+1} \simeq U
$$

where

1. $U$ denotes the [[stable unitary group]],

1. $B U$ denotes the [[classifying space]] for [[complex vector bundles]].

So in this case a [[cocycle]] is represented by the [[homotopy class]] of a map of the form

$$
  \left(V_{brane}, n_{\text{anti-brane}} \right)
   \;\colon\;
  X \overset{}{\longrightarrow} (B U) \times \mathbb{Z}
$$

which gives a [[complex vector bundle]] $V_{\text{brane}}$ and the trivial vector bundle $\mathbb{C}^{n_{\text{anti-brane}}}$
of [[rank]] $n_{\text{anti-brane}}$, hence a [[virtual vector bundle]] of the form

$$
  (V_{\text{brane}}, \mathbb{C}^{n_{\text{anti-brane}}})
  \,.
$$

A basic fact proven in [[topological K-theory]] is that the K-theory class of every [[virtual vector bundle]]
is represented by one of this form, and it is in this way that $KU_0 = (B U) \times \mathbb{Z}$
is the [[coefficient]] space for [[topological K-theory]].

But this is not yet the full story:
Above we saw
that the [[Chan-Paton gauge field]] on a D-brane
is actually a _[[twisted vector bundle]]_
with twist given by the [[Kalb-Ramond field|Kalb-Ramond]] [[B-field]]
sourced by a string condensate.
([[Freed-Witten anomaly cancellation]]).

Such _[[twisted cohomology|twisted]]_ [[generalized cohomology]] is given by
generalizing the above concept of [[Omega-spectra]] by allowing the base point $\ast$ to
become a _classifying space of twists_. The result is called a _[[parameterized spectrum]]_.
Such consists of:

1. a classifying space of twists $B G$

1. a _[[spectrum object]] in the [[slice category]] $Top_{/B G}$_, namely a sequence of spaces, denoted $E_n/G$, equipped with maps

   $$
     id \;\colon\; B G \to E_n/G \to B G
   $$

   and [[weak homotopy equivalences]] from the $n$th space to the [[homotopy fiber product]] of space inclusion of the space of twists with itself:

   $$
     E_n/G
       \overset{\phantom{AA}\simeq\phantom{AA} }{\longrightarrow}
    \Omega_{B G} (E_{n+1}/G)
      \coloneqq
    \underset{\text{homotopy} \atop {\text{fiber} \atop \text{product} } }{\underbrace{
     B G
       \underoverset{E_{n+1}/G}{h}{\times}
     B G
     }}
   $$

To get a feeling for this definition, consider two extremal cases of [[parameterized spectra]]:

1. An ordinary spectrum $E$ is a parameterized spectrum over the point (i.e. no twists);

   $$
     \array{
       E
       \\
       \downarrow
       \\
       \ast
     }
   $$

1. An ordinary [[topological space]] $X$ is identified with the [[zero-spectrum]] parameterized over $X$, which is just

   $$
     \array{
       X
       \\
       \downarrow^{\mathrlap{id}}
       \\
       X
     }
   $$

Hence a general [[parameterized spectrum]] interpolates between these two extremes, it combines the
[[non-abelian cohomology]] represented by [[topological spaces]] such as $B U$ with the abelian cohomology
[[Brown representability theorem|represented]] by [[spectra]]:

More in detail, given a [[parameterized spectrum]] $E$ over $B G$, then
we have the following elegant picture of twisted $E$-cohomology ([NSS 12, section 4.1](#NSS12)):

1. A _twist_ $\tau$ for the $E$-cohomology of some [[topological space]] $X$ is a map

   $$
     \array{
       X
       \\
       & {}_{\mathllap{\tau}}\searrow
       \\
       && B G
     }
   $$

   from $X$ to the spectrum's classifying space of twists.

1. The _$\tau$-twisted $E$-cohomology_ of $X$ in degree $n$ is the set of [[homotopy classes]]
   of maps $X \overset{\phi}{\longrightarrow} E_n/G$  together with a [[homotopy]] $p_n \circ \phi \simeq \tau$:

   $$
    E^{n+\tau}(X)
      \;\coloneqq\;
    \left\{
     \array{
       X && \longrightarrow && E_n/G
       \\
       & {}_{\mathllap{\tau}} \searrow && \swarrow_{\mathrlap{p_n}}
       \\
       && B G
     }
     \right\}_{/\text{homotopy}}
   $$

1. There is a [[homotopy fiber sequence]] (in parameterized spectra)

   $$
     \array{
       E &\longrightarrow& E/G
       \\
       && \downarrow
       \\
       && B G
     }
   $$

   and this equivalently exhibits $E/G$ as the [[homotopy quotient]] of an ordinary spectrum $E$ by a
[[infinity-action|coherent homotopy action]] of $G$.

We now translate this situation from topological spaces to [[super L-∞ algebras]]
via the central theorem of [[rational homotopy theory]], which we now recall.

$\,$

### Rational homotopy theory
 {#RationalHomotopyTheory}

Recall from the discussion [above](#SuperLInfinityCohomologyAndFDAs)
that an [[L-∞ algebra]] $\mathfrak{g}$ may be thought of as a [[Lie algebra]] "up to [[coherence|coherent]] higher [[homotopy]]":
with the unary bracket $[-]$ regarded as a differential $\partial \coloneqq [-]$, then for instance the trinary bracket is a
[[chain homotopy]] up to which the [[Jacobi identity]] holds

$$
  [x,[y,z]]
    \underoverset{\simeq}{\partial [x,y,z]}{\longrightarrow}
  [[x,y],z] \pm [y,[x,z]]
  \,.
$$

In direct analogy, for $X$ any [[pointed topological space]], then the
[[based loop space]] $\Omega_\ast(X)$ (the [[topological space]] whose elements are
continuous paths from the basepoint to itself) naturally is a [[group]] "up to [[coherence|coherent]] higher homotopy".

Namely

* the operation of concatenating two loops $\alpha,\beta \colon [0,1] \to X$ to a new loop

  $$
    [0,1] \overset{2 \cdot(-)}{\longrightarrow} [0,2]
     \overset{}{\longrightarrow}
    X
  $$

  gives it the structure of a [[semi-group]] whose [[associativity]] law holds up to [[homotopy]];

* the constant loop gives it the structure of a [[monoid]] up to coherent homotopy, called an _[[A-∞ space]]_

* the reversal of loops

  $$
    [0,1] \overset{1-(-)}{\longrightarrow} [0,1] \overset{\alpha}{\longrightarrow} X
  $$

  finally gives it [[inverses]] up to homotopy and makes it what is called a  a _grouplike [[A-∞ space]]_ or _[[∞-group]]_ for short.

Conversely, for $G$ any [[∞-group]]
then there is an essentially unique [[connected topological space|connected]] space $B G$
with $G \;\simeq\; \Omega B G$ (the _[[May recognition theorem]]_).

Now in a similar manner, every double loop space $\Omega_\ast(\Omega_\ast(X))$
becomes a "first order abelian" [[∞-group]]
by exchanging loop directons. This may be called a _[[braided ∞-group]]_,

$\,$

Hence for $G$ a [[braided ∞-group]] then
$B G$ is itself an [[∞-group]]
and so there exists an essentially unique [[simply connected topological space|simply connected]] space

$$
  B^2 G \coloneqq B (B G)
$$

with

$$
  G \;\simeq\; \Omega^2 B^2 G
  \,.
$$

$\,$

And so forth:

Every triple loop space $\Omega^3 X$

becomes a "second order abelian" [[∞-group]]

by exchanging loop directons

called a _[[sylleptic ∞-group]]_.

etc.

$\,$

In a [[spectrum]] $E$,

the maps $E_n \stackrel{\simeq}{\to} \Omega E_{n+1}$

exhbit $E_0$ as an [[infinite loop space]]

hence as a fully [[abelian ∞-group]].

$\,$

It turns out that by a [[homotopy theory|homotopy theoretic]] version of [[Lie theory]],

there is an [[L-∞ algebra]] $\mathfrak{g}$ associated with any [[∞-group]]

$$
  \mathfrak{g} \simeq  \mathfrak{l} B G
  \,.
$$

or

$$
  B\mathfrak{g} \simeq  \mathfrak{l} B^2 G
$$

etc.

$\,$

Its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{B g})$

is called a _[[Sullivan model]]_ for $B^2 G$.

$\,$

For **example** the $L_\infty$-algebra associated with an [[Eilenberg-MacLane space]]

$$
  K(\mathbb{Z},n+1) \simeq B^{n+1}\mathbb{Z}
$$

is the [[line Lie-n algebra]] from above:

$$
  \mathfrak{l}(B^{n+1} \mathbb{Z})
    \;\simeq\;
  B^n \mathbb{R}
  \,.
$$

$\,$

The **main theorem of [[rational homotopy theory]]**  ([Quillen 69](https://ncatlab.org/nlab/show/rational+homotopy+theory#Quillen69),
[Sullivan 77](https://ncatlab.org/nlab/show/rational+homotopy+theory#Sullivan77))

says that the [[L-∞ algebra]] $\mathfrak{l}(B^2 G)$ equivalently reflects the _[[rationalization]]_ of $B^2 G$

(in fact the real-ification, since we are considering $L_\infty$-algebras over the real numbers).

This means that weak equivalence between $L_\infty$-algebras correspond to maps between spaces

that induce [[isomorphism]] on real-ified [[homotopy groups]]

$$
  \left\{
    \;\;\;\;\;
    \array{
      B^2 G_1
        \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow}
      B^2 G_2
      \\
      \text{such that:}
      \\
      \pi_\bullet(B^2 G_1)\otimes_{\mathbb{Z}} \mathbb{R}
         \underoverset{\simeq}{\pi_\bullet(f) \otimes_{\mathbb{Z}} \mathbb{R}   }{\longrightarrow}
      \pi_\bullet(B^2 G_2) \otimes_{\mathbb{Z}} \mathbb{R}
    }
    \;\;\;\;\;
  \right\}
  \;\;\;\leftrightarrow\;\;\;
  \left\{
  \;
  \mathfrak{l}(B^2 G_1)
    \stackrel{\simeq}{\longrightarrow}
  \mathfrak{l}(B^2 G_2)
  \;
  \right\}
  \,.
$$

For concise review in the language that we use here see [Buijs-Félix-Murillo 12, section 2](https://ncatlab.org/nlab/show/rational+homotopy+theory#BuijsFelixMurillo12).

$\,$

We apply this [[rational homotopy theory]] functor

$$
  \mathfrak{l}(-)
    \;\colon\;
  \text{Spaces}
    \longrightarrow
  L_\infty\text{-Algebras}
$$

to find the $L_\infty$-algebraic version of [[parameterized spectra]]

hence of [[twisted cohomology]]:

$\,$

$$
  \array{
  {
  \text{parameterized}
    \atop
  \text{spectrum}
  }
  \;\;\;&\;\;\;\;
  \left\{
  \array{
    id : B^2 G \to E_n/ B G \to B^2 G
    \\
    E_n/ B G \stackrel{\simeq}{\longrightarrow} \Omega_{B^2 G} (E_{n+1}/ B G)
  }
  \right\}
  \;&\;\leftrightarrow\;&\;
  \left(
  \array{
    E &\longrightarrow& E/B G
    \\
    && \downarrow
    \\
    && B^2 G
  }
  \right)
  \\
  & \mathfrak{l}(-)\downarrow
  \\
  L_\infty\text{Algebra}
  \;\;\;&\;\;\;\;
  \left\{
  \array{
    id : \mathfrak{l}(B^2 G) \to \mathfrak{l}(E_n/ B G) \to \mathfrak{l}(B^2 G)
    \\
    \mathfrak{l}(E_n/ B G) \stackrel{\simeq}{\longrightarrow} \Omega_{\mathfrak{l}(B^2 G)} \mathfrak{l}(E_{n+1}/ B G)
  }
  \right\}
  \;&\;\leftrightarrow\;&\;
  \left(
  \array{
    V[1] &\longrightarrow& V[1]/\mathfrak{g}
    \\
    && \downarrow
    \\
    && B \mathfrak{g}
  }
  \right)
  }
$$

$\,$

Here $V \simeq E \otimes \mathbb{R}$ is a chain complex

underlying the real-ification of the spectrum $E$

([[stable Dold-Kan correspondence]]).

$\,$

So for $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ some [[super Minkowski spacetime]],
a cocycle in $\mathfrak{g}$-twisted $V$-cohomology is a diagram of the form

$$
  \array{
    \mathbb{R}^{d-1,1\vert \mathbf{N}}
      && \overset{}{\longrightarrow} &&
    V/\mathfrak{g}
    \\
    & \searrow && \swarrow
    \\
    && B \mathfrak{g}
  }
$$

$\,$

Now given one stage in the [[schreiber:brane bouquet]]

$$
  \array{
    \widehat{\widehat{\mathfrak{g}}}
    \\
    {}^{\mathllap{hofib(\mu_{p_2})}}\downarrow
    \\
    \hat \mathfrak{g}
    &
      \stackrel{\mu_{p_2}}{\longrightarrow}
    &
    B\mathfrak{h}_2
    \\
    {}^{\mathllap{hofib(\mu_{p_1})}}\downarrow
    \\
    \mathfrak{g}
    &\overset{\mu_{p_1}}{\longrightarrow}&
    B\mathfrak{h}_1
  }
$$

we want to descent $\mu_{p_2}$ to $\mathfrak{g}$.

$\,$

By the general theory of [[principal ∞-bundles]] ([Nikolaus-Schreiber-Stevenson 12](#NSS12)):

1. $\widehat{\mathfrak{g}}$ has a $\mathfrak{h}_1$-[[∞-action]]

1. equipping $B \mathfrak{h}_2$ with an $\mathfrak{h}_1$-[[∞-action]]

   is equivalent to finding a [[homotopy fiber sequence]] as on the right here:

   $$
     \array{
       \hat \mathfrak{g}
       &&
         \stackrel{\mu_2}{\longrightarrow}
       &&
       \mathbf{B}\mathfrak{h}_2
       \\
       {}^{\mathllap{hofib(\mu_1)}}\downarrow
       && &&
       \downarrow^{\mathrlap{hofib(p_\rho)}}
       \\
       \mathfrak{g}
       && &&
       (\mathbf{B}\mathfrak{h}_2)/\mathfrak{h}_1
       \\
       & {}_{\mathllap{\mu_1}}\searrow  && \swarrow_{\mathrlap{p_\rho}}
       \\
       && \mathbf{B}\mathfrak{h}_1
     }
     \,.
   $$

1. $\mu_2$ is $\mathfrak{h}_1$-equivariant precisely if it descends to a morphism

   $$
     \mu_2/\mathfrak{h}_1
     \;\colon\;
     \mathfrak{g}
     \longrightarrow
     (\mathbf{B}\mathfrak{h}_2)/\mathfrak{h}_1
   $$

   such that this [[diagram]] [[commuting diagram|commute]] up to homotopy:

   $$
     \array{
       \hat \mathfrak{g}
       &&
         \stackrel{\mu_2}{\longrightarrow}
       &&
       \mathbf{B}\mathfrak{h}_2
       \\
       {}^{\mathllap{hofib(\mu_1)}}\downarrow
       && &&
       \downarrow^{\mathrlap{hofib(p_\rho)}}
       \\
       \mathfrak{g}
       &&
         \stackrel{\mu_2/\mathfrak{h}_1}{\longrightarrow}
       &&
       (\mathbf{B}\mathfrak{h}_2)/\mathfrak{h}_1
       \\
       & {}_{\mathllap{\mu_1}}\searrow  && \swarrow_{\mathrlap{p_\rho}}
       \\
       && \mathbf{B}\mathfrak{h}_1
     }
     \,.
   $$

1. if so, then resulting triangle diagram

   $$
     \array{
       \mathfrak{g}
       &&
         \stackrel{\mu_2/\mathfrak{h}_1}{\longrightarrow}
       &&
       (\mathbf{B}\mathfrak{h}_2)/\mathfrak{h}_1
       \\
       & {}_{\mathllap{\mu_1}}\searrow  && \swarrow_{\mathrlap{p_\rho}}
       \\
       && \mathbf{B}\mathfrak{h}_1
     }
   $$

   exhibits $\mu_2/\mathfrak{h}_1$  as a [[cocycle]] in ([[rational homotopy theory|rational]]) _$\mu_1$-[[twisted cohomology]]_

   with respect to the [[local coefficient bundle]] $p_\rho$.

$\,$

We now work out this general prescription

for the cocycles in the [[schreiber:brane bouquet]].

$\,$

### RR-fields
 {#RRFields}

$\,$

By the [[schreiber:brane bouquet]] above

the type IIA [[D-branes]]

are given by super $L_\infty$ cocycles of the form

$$
  \mathfrak{string}_{IIA }
    \overset{\mu_{Dp}}{\longrightarrow}
  B^{p+1}\mathbb{R}
$$

for $p \in \{0,2,4,6,8,10\}$.

$\,$

Notice that

$$
  H^\bullet(B U, \mathbb{Z})
$$

has one generator in each even degree, the [[universal Chern classes]].

Hence the $L_\infty$-algebra

$$
  \mathfrak{l}(KU)
$$

is given by

$$
  CE(\mathfrak{l}(KU))
    \;\simeq\;
  \left\{
    d \omega_{2p+2} = 0
    \;\vert\;
    p \in \mathbb{Z}
  \right\}
  \,.
$$

This allows to unify the D-brane cocycles

into a single morphism of super $L_\infty$-algebras of the form

$$
  \array{
     \mathfrak{string}_{IIA}
     &&
     \stackrel{\phantom{AA}\mu_D\phantom{AA}}{\longrightarrow}
     &&
     \mathfrak{l}(KU)
     \\
     {}^{\mathllap{hofib(\mu_{F1})}}\downarrow
     \\
     \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}
     \\
     & {}_{\mathllap{\mu_{F1}}}\searrow
     \\
     &&
     \mathbf{B}^2 \mathbb{R}
  }
  \,.
$$

$\,$

By the above prescription, descending $\mu_D$ is equivalent

to finding a [[commuting diagram]] in the [[homotopy category]] of super $L_\infty$-algebras

of the form

$$
  \array{
     \mathfrak{string}_{IIA}
     &&
     \stackrel{\phantom{AA}\mu_D\phantom{AA}}{\longrightarrow}
     &&
     \mathfrak{l}(KU)
     \\
     {}^{\mathllap{hofib(\mu_{F1})}}\downarrow
       && &&
     \downarrow^{\mathrlap{hofib(\phi)}}
     \\
     \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}
       &&\overset{\phantom{AA}\mu_{D}/B \mathbb{R}\phantom{AA} }{\longrightarrow}&&
     \text{something}
     \\
     & {}_{\mathllap{\mu_{F1}}}\searrow && \swarrow_{\mathrlap{\phi}}
     \\
     &&
     \mathbf{B}^2 \mathbb{R}
  }
  \,.
$$

$\,$

This turns out to exist as follows ([Fiorenza-Sati-Schreiber 16a, section 5](#FSS16a)):

Define the $L_\infty$-algebra

$$
  \mathfrak{l}(KU / BU(1))
$$

by

$$
  CE\left(\mathfrak{l}(KU / BU(1))\right)
  \;=\;
  \left\{
    \array{
      d h_3 = 0\;,\;
      \\
      d \omega_{2p+2} = h_3 \wedge \omega_{2p}
    }
  \right\}
  \,.
$$

Moreover write

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

for the super $L_\infty$-algebra whose [[Chevalley-Eilenberg algebra]] is

$$
  CE\left(
    \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}_{res}
  \right)[f_2,h_3]/(d f_2 = \mu_{F1} + h_3)
$$

$\,$

+-- {: .num_prop #DescentOfIIACocycles}
###### Proposition
**([Fiorenza-Sati-Schreiber 16a, theorem 4.16](#FSS16a))**

The super $L_\infty$-algebra $\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}_{res}$

is a [[resolution]] of type IIA [[super-Minkowski spacetime]].

in that there is a weak equivalence

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

This fits into a [[commuting diagram]] of the form

$$
  \array{
     \left\{ {{d e^a = \overline{\psi}\Gamma^a \wedge \psi } \atop {d \psi^\alpha = 0}} \atop { d f_2 = \mu_{F1}} \right\}
     &&
     \mathfrak{string}_{IIA}
     && \stackrel{ \mu_D  }{\longrightarrow} &&
     \mathfrak{l}(KU)
     &&
     \left\{
       d \omega_{2 p+2} = 0
     \right\}
     \\
     && \downarrow^{\mathrlap{hofib(\mu_{F1})}} && && \downarrow^{\mathrlap{\phi}}
     \\
     \left\{ {{d e^a = \overline{\psi}\Gamma^a \wedge \psi } \atop {d \psi^\alpha = 0}} \atop { d f_2 = \mu_{F1} + h_3 } \right\}
     &
     &
     \mathbb{R}_{res}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}
     &&
      \stackrel{ \mu_{F1/D} }{\longrightarrow}
     &&
      \mathfrak{l}(KU / B U(1))
     &&
     \left\{
        d\omega_{2p+2} = h_3\wedge \omega_{2p}
     \right\}
     \\
     && & {}_{\mathllap{\mu_{F 1}}}\searrow && \swarrow_{\mathrlap{\phi}}
     \\
     && && \mathbf{B}^2 \mathbb{R}
     \\
     && && \left\{ d h_3 = 0 \right\}
  }
  \,.
$$

=--

$\,$

In **conclusion**

$\;\;$_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 rationalized [[twisted K-theory]]_.

$\,$

### M-flux fields
 {#MFluxFields}

$\,$

The part of the [[schreiber:brane bouquet]] giving the [[M-branes]] is

$$
  \array{
     \mathfrak{m}2\mathfrak{brane}
     &\stackrel{\mu_{M5}}{\longrightarrow}&
     \mathbf{B}^6 \mathbb{R}
     \\
     {}^{\mathllap{hofib(\mu_{M2})}}\downarrow
     \\
     \mathbb{R}^{10,1\vert \mathbf{32}}
     & \stackrel{\mu_{M2}}{\longrightarrow} &
     \mathbf{B}^3 \mathbb{R}
     \\
     {}^{\mathllap{hofib(\mu_{D 0})}}\downarrow
     \\
     \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}
  }
$$

$\,$

In order to descend this, consider the $L_\infty$-algebra corresponding to the [[4-sphere]]

$$
  \mathfrak{l}(S^4)
  \,.
$$

By standard results on [[rational n-spheres]], this is given by

$$
  CE(\mathfrak{l}S^4)
   \;\simeq\;
  \left\{
    \array{
      d \, g_4 = 0\,,\,
      \\
      d\, g_7 + \tfrac{1}{2} g_4 \wedge _4 = 0
    }
  \right\}
  \,.
$$

(...)

+-- {: .num_defn #RationalSphereAlgebra}
###### Definition

For $p \in \mathbb{N}_{even}$, write

$b^{2p+2} \mathbb{R}/b^p \mathbb{R}$ for the [[L-∞ algebra]] given by the [[Chevalley-Eilenberg algebra]]

$$

  CE(b^{2p+2} \mathbb{R}/b^p \mathbb{R})
  =
  \left(\left\langle g_{p+2}, g_{2p+3}\right\rangle), {{d g_{p+2} = 0} \atop {d g_{2p+3} = g_{p+2} \wedge g_{p+2}}}\right)
  \,.
$$

=--

+-- {: .num_remark #RationalSpheres}
###### Remark

Regarded as a [[Sullivan model]] in [[rational homotopy theory]], then the [[dg-algebra]] of def. \ref{RationalSphereAlgebra} is a minimal model for the [[rationalization]] of the $(p+2)$-[[sphere]].

=--

By the [recognition theorem for L-∞ extensions](model+structure+for+L-infinity+algebras#HomotopyFiberProducts) we get:

+-- {: .num_prop #b2RactionOnb2p2R}
###### Proposition

For $p \in \mathbb{N}_{even}$, there is a [[homotopy fiber sequence]] in the [[homotopy theory of L-∞ algebras]] of the form

$$
  \array{
    b^{2p+2} \mathbb{R}
    \\
    \downarrow
    \\
    b^{2p+2} \mathbb{R}/b^{p}\mathbb{R}
    &\stackrel{}{\longrightarrow}&
    b^{p+1}\mathbb{R}
  }
$$

where on [[formal dual]] [[Chevalley-Eilenberg algebras]] in terms of our defining generating elements the horizontal map is given by $g_{p+4}\mapsto g_{p+4}$ and the vertical map by $g_{p+4}\mapsto 0$ and $g_{2p+3}\mapsto g_{2p+3}$.

By the discussion at _[[∞-action]]_ this exhibits a $b^{p}\mathbb{R}$-action on $b^{2p+2}\mathbb{R}$, for which $b^{2p+2} \mathbb{R}/b^{p}\mathbb{R}$ is the [[homotopy quotient]], whence the notation.

=--

+-- {: .num_example}
###### Example

For $p=2$ and regarded as a statement in [[rational homotopy theory]] via remark \ref{RationalSpheres}, then the extension in prop. \ref{b2RactionOnb2p2R} is a [[Sullivan minimal model]] for the rationalized [[Hopf fibration]]

$$
  \array{
    S^3 &\longrightarrow& S^7
    \\
    && \downarrow
    \\
    && S^4
  }
$$

=--

But since $\mathfrak{m}2\mathfrak{brane}$ is a $b^2 \mathbb{R}$-[[principal ∞-bundle]], it is natural to ask whether $h_3 \wedge \mu_4 + \frac{1}{15} \mu_7$ is $b^2 \mathbb{R}$-[[equivariant cohomology|equivariant]] with respect to some natural $b^2\mathbb{R}$-[[∞-action]] on $b^6 \mathbb{R}$. Such a natural action is given by prop. \ref{b2RactionOnb2p2R}. To exhibit in components the equivariance of the 7-cocycle in corollary \ref{The7CocycleOnm2brane} with respect to this action we need a [[resolution]] of [[super Minkowski spacetime]]:

+-- {: .num_defn}
###### Definition

Write $\mathbb{R}_{res}^{10,1\vert \mathbf{32}}$ for the [[super L-∞ algebra]] whose [[Chevalley-Eilenberg algebra]] is obtained from that of [[super Minkowski spacetime]] by

$$
  CE\left(
   \mathbb{R}_{res}^{10,1\vert \mathbf{32}}
  \right)
  \coloneqq
  \left(
   CE\left(
     \mathbb{R}^{10,1\vert \mathbf{32}}\right)\otimes \left\langle h_3, g_4\right\rangle ,
     {{d h_3 = g_4 - \mu_4} \atop {d g_4 = 0}}
  \right)
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

The canonical morphism

$$
  \mathbb{R}_{res}^{10,1\vert \mathbf{32}}
  \stackrel{\simeq}{\longrightarrow}
  \mathbb{R}^{10,1\vert \mathbf{32}}
$$

given dually by $\psi^\alpha \mapsto \psi^\alpha$, $e^a \mapsto e^a$, is an equivalence of $L_\infty$-algebras. It factors the morphism $\mathfrak{m}2\mathfrak{brane} \longrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}$ from def. \ref{m2braneLie3Algebra} through a morphism $\mathfrak{m}2\mathfrak{brane}  \longrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}_{res}$ which on [[formal dual]] CE-elements is given by $h_3 \mapsto 0$, $g_4 \mapsto 0$ and by being the identity on all other generators.

=--

+-- {: .num_prop #7CocycleOnM2BraneDescends}
###### Proposition

There is a [[diagram]] of [[L-∞ algebras]] of the form

$$
  \array{
    && \vdots && && \vdots
    \\
    && \downarrow \downarrow && && \downarrow \downarrow
    \\
    &&
    \mathfrak{m}2\mathfrak{brane}
    &&
      \stackrel{h_3 \wedge \mu_4  + \frac{1}{15}\mu_7 }{\longrightarrow}
    &&
    b^6 \mathbb{R}
    \\
    &&
    \downarrow && && \downarrow
    \\
    \mathbb{R}^{10,1\vert\mathbf{32}}
    &\stackrel{\simeq}{\longleftarrow}&
    \mathbb{R}_{res}^{10,1\vert\mathbf{32}}
    &&
      \stackrel{h_3 \wedge (g_4 + \mu_4) + \frac{1}{15}\mu_7 }{\longrightarrow}
    &&
    b^6 \mathbb{R}/b^2 \mathbb{R}
    \\
    &&
    & {}_{\mathllap{}}\searrow && \swarrow
    \\
    &&
    && b^3 \mathbb{R}
  }
$$

=--

+-- {: .proof}
###### Proof

That the diagram exists and commutes at the level of the underlying graded algebras of the [[formal dual]] CE-algebras is immediate in terms of the defining generators: each generator is mapped to the generator of the same name, if present, in the codomain, or to zero otherwise, except for $g_7 \in CE(b^6 \mathbb{R})$ which is sent to $h_3 \wedge \mu_4  + \frac{1}{15}\mu_7$ and $g_7 \in CE(b^6 \mathbb{R}/b^2 \mathbb{R})$, which is sent to $h_3 \wedge (g_4 + \mu_4) + \frac{1}{15}\mu_7 $, as indicated.

It remains to check that the middle horizontal map respects the CE-differentials: by prop. \ref{CocyclesOn11dSuperMinkowski} we have

$$
  \begin{aligned}
    d(h_3 \wedge (g_4 + \mu_4) + \frac{1}{15}\mu_7)
    &=
    (g_4 - \mu_4) \wedge (g_4 + \mu_4) + \mu_4 \wedge \mu_4
    \\
    & = g_4 \wedge g_4
  \end{aligned}
$$

and by def. \ref{RationalSphereAlgebra} this says indeed that $g_7 \mapsto h_3 \wedge (g_4 + \mu_4) + \frac{1}{15}\mu_7$ respects the CE-differentials.

=--

+-- {: .num_remark}
###### Remark

The form of the equivariant cocycle in prop. \ref{7CocycleOnM2BraneDescends} is that of the [[curvature]] of the [[geometry of physics -- WZW terms|WZW term]] of the [[sigma model]] describing the [[M5-brane]] as considered in ([BLNPST 97, (6),(8)](#BLNPST97)).

=--

+-- {: .num_remark}
###### Remark

In view of remark \ref{RationalSpheres}, prop. \ref{7CocycleOnM2BraneDescends} says that the CE-elements $\mu_4$ and $\mu_7$ of prop. \ref{CocyclesOn11dSuperMinkowski} define a cocycle with values in the rational 4-sphere. In the discussions in _[[geometry of physics -- WZW terms]]_ and _[[geometry of physics -- BPS charges]]_ we see that under [[Lie integration]] and globalization in [[higher Cartan geometry]], these elements encode the [[supergravity C-field]] and its [[electric-magnetic duality|magnetic dual]]. That these fields should take values in the 4-sphere was first suggested in ([Sati 13, section 2.5](Hisham+Sati#Sati13)).

=--

(...)

+-- {: .num_prop #MBraneDescent}
###### Proposition
[Fiorenza-Sati-Schreiber 15, section 3](FSS15)

There is a [[homotopy fiber sequence]] of $L_\infty$-algebras as on the right

$$
  \left(
  \array{
    && S^7
    \\
    && \downarrow
    \\
    && S^4
    \\
    & \swarrow
    \\
    B SU(2)
    \\
    \downarrow^{\mathrlap{c_2}}
    \\
    B^3 U(1)
  }
  \right)
    \;\;
    \stackrel{\phantom{AA}\mathfrak{l}(-)\phantom{AA} }{\mapsto}
    \;\;
  \left(
  \array{
    && B^6 \mathbb{R}
    \\
    && \downarrow^{\mathrlap{ hofib(\mathfrak{l}(c_2)) } }
    \\
    &&\mathfrak{l}(S^4)
    \\
    & \swarrow_{\mathfrak{l}(c_2)}
    \\
    B^3 \mathbb{R}
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \right)
$$

which is the image under $\mathfrak{l}(-)$ of the [[quaternionic Hopf fibration]].

This makes a [[commuting diagram]]

in the [[homotopy category]] of super $L_\infty$-algebas

of the form

$$
  \array{
    \left\{
      { { d e^a = \overline{\psi}\wedge \Gamma^a \wedge \psi }
       \atop
      { d \psi^\alpha = 0 } }
      \atop
      d h_3 = - \mu_{M2}
    \right\}
    &&
    \mathfrak{m}2\mathfrak{brane}
    &&
      \stackrel{ \mu_{M5} }{\longrightarrow}
    &&
    B^6 \mathbb{R}
    &&
    \left\{
      d g_7 = 0
    \right\}
    \\
    &&
    \downarrow^{\mathrlap{hofib(\mu_{M2})}} && && \downarrow^{\mathrlap{hofib(\mathfrak{l}(c_2))}}
    \\
    \left\{
      { { d e^a = \overline{\psi}\wedge \Gamma^a \wedge \psi }
       \atop
      { d \psi^\alpha = 0 } }
      \atop
      d h_3 = g_4 - \mu_{M2}
    \right\}
    &&
    \mathbb{R}_{res}^{10,1\vert\mathbf{32}}
    &&
      \stackrel{ \mu_{M2/M5} }{\longrightarrow}
    &&
    \mathfrak{l} S^4
    &&
    \left\{
      {d g_4 = 0} \atop {d g_7 + \tfrac{1}{2} g_4 \wedge g_4 = 0}
    \right\}
    \\
    &&
    & {}_{\mathllap{\mu_{M2}}}\searrow && \swarrow_{\mathrlap{\mathfrak{l}(c_2)}}
    \\
    &&
    && \mathbf{B}^3 \mathbb{R}
    \\
    && && \left\{ d g_4 = 0\right\}
  }
$$

=--

In **conclusion**

$\;\;$_this says 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 conjecture due to

([Sati 10, section 6.3](https://ncatlab.org/schreiber/show/Equivariant+cohomology+of+M2/M5-branes#Sati10), [Sati 13, section 2.5](https://ncatlab.org/schreiber/show/Equivariant+cohomology+of+M2/M5-branes#Sati13)).

based on the observation that

the [[equations of motion]] of [[11-dimensional supergravity]]

for the [[supergravity C-field]] [[field strength]] $G_4$  say that

$$
  d G_7 + \tfrac{1}{2} G_4 \wedge G_4 = 0
$$

with $G_7 = \ast G_4$ the [[Hodge dual]]

and this is just the algebraic relation for the Sullivan model of the [[rational n-sphere|rational 4-sphere]].

$\,$

Notice that, unstably, the [[4-sphere]]

is just the space whose [[torsion subgroup|non-torsion]] [[homotopy groups]]

hence those that are visible in [[rational homotopy theory]]

are in degrees 2+2 and 5+2:

| k | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| $\phantom{AA}\pi_k(S^4)\phantom{AA}$ | $\phantom{AA}0\phantom{AA}$ | $\phantom{AA}0\phantom{AA}$ | $\phantom{AA}0\phantom{AA}$ | $\phantom{AA}\mathbb{Z}\phantom{AA}$ | $\phantom{AA}\mathbb{Z}/2\phantom{AA}$ | $\phantom{AA}\mathbb{Z}/2\phantom{AA}$ | $\phantom{AA}\mathbb{Z} \oplus \mathbb{Z}/{12}\phantom{AA}$ |

$\,$

But the correct non-rational lift of the $\mathfrak{l}(S^4)$-coefficients

will also have to be such that it somehow gives rise to [[twisted K-theory]]

under (double) dimensional reduction. This is still an **open problem**.

For further comments see the talk

_[[schreiber:Equivariant cohomology of M2/M5-branes]]_

$\,$

$\,$

Now that we have found

the descended $L_\infty$-cocycles

for all super $p$-branes

in twisted cohomology, rationally,

we may analyze their behaviour under [[double dimensional reduction]]

and discover the super $L_\infty$-algebraic incarnation

of various [[dualities in string theory]].

$\,$

## Double dimensional reduction
 {#DoubleDimensionalReduction}

Underlying most of the [[dualities in string theory]] is the phenomenon of  "[[double dimensional reduction]]"
([Duff-Howe-Inami-Stelle 87](double+dimensional+reduction#DuffHoweInamiStelle87)), so called because:

1. the [[dimension]] of [[spacetimes]] is reduced by [[Kaluza-Klein compactification]] on a [[fiber]] $F$;

1. in parallel the [[dimension]] of [[branes]] is reduced if they _[[wrapped brane|wrap]]_ $F$.

We now explain a mathematical formulation of [[double dimensional reduction]] of brane charges, following ([FSS 16b, section 3](#FSS16b), [BMSS 18, section 2.2](#BMSS18)):

First we give an exposition of the

* _[Idea](#DoubleDimensionalReductionIdea)_.

Then we discuss a first approximation to a mathematical formalization:

* _[Via free looping (no 0-brane effect)](#DoubleDimensionalReductionViaFreeLooping)_

and then the accurate and fully general formalization in

* _[Via cyclification (with 0-brane effects)](#DoubleDimensionalReductionViaCyclification)_

We then specialize this general procedure to [[super L-infinity algebras]] so that it applies
to the super $p$-brane cocycles in

* _[On super p-brane cocycles](#DoubleDimensionalReductionLInfinity)_

$\,$

### Idea
  {#DoubleDimensionalReductionIdea}

The original example of [[double dimensional reduction]]
([Duff-Howe-Inami-Stelle 87](double+dimensional+reduction#DuffHoweInamiStelle87)) is supposed to underly the [[duality in string theory|duality]] between [[M-theory]] and [[type IIA string theory]]. In this case

* [[spacetime]] $X_{11}$ is an 11d [[circle]]-[[fiber bundle]] locally of the form $X_{11} =  X_{10} \times S^1$ over a 10d base spacetime;

* an [[M-theory]] [[membrane]] ([[M2-brane]]) with [[cylinder|cyclindrical]] [[worldvolume]]

  $$
    \Sigma_3 = \Sigma_2 \times S^1
  $$

  [[wrapped brane|wraps]] the circle fiber if its [[trajectory]]

  $$
    \phi_{M2} \;\colon\; \Sigma_3 \longrightarrow X_{11}
  $$

  is of the form

  $$
    \phi_{F1} \times \mathrm{id}_{S^1}
    \;\colon\;
    \Sigma_2 \times S^1 \longrightarrow X_{10} \times S^1
    \,.
  $$

As the [[pseudo-Riemannian metric|Riemannian]] circumference of the circle fiber $S^1$ tends towards zero
this effectively looks like the 2-dimensional [[worldsheet]] $\Sigma_2$ of a [[string]] tracing out a [[trajectory]] in 10-dimensional spacetime:

$$
  \phi_{F1} \;\colon\; \Sigma_2 \longrightarrow X_{10}
$$

$\,$

But there is also "single dimensional reduction" when the [[membrane]] does not [[wrapped brane|wrap]] the fiber space:

$$
  \array{
    \Sigma_3 &&  \overset{\phi_{M2}}{\longrightarrow} && X_{11}
    \\
    & {}_{\phi_{D2}}\searrow && \nearrow
    \\
    && X_{10}
  }
$$

In this case it looks like a [[membrane]] in 10d [[spacetime]], now called the _[[D2-brane]]_.

Similarly the [[M5-brane]] in [[M-theory]]

$$
  \phi_{M5} \;\colon\; \Sigma_{6} \longrightarrow X_{11}
$$

may [[wrapped brane|wrap]] the [[circle]] [[fiber]] to yield a 4-brane in 10d, called the [[D4-brane]] or it may not [[wrapped brane|wrap]] the [[circle]] [[fiber]] to yield a 5-brane in 10d, called the [[NS5-brane]].

$\,$

Beware the naive treatment of branes in this traditional argument. And even naively, this is not the full story yet:
The $S^1$-fibration itself is supposed to re-incarnate in 10d as the [[D0-brane]] and the [[D6-brane]].

Hence [[double dimensional reduction]] from [[M-theory]] to [[type IIA string theory]]
is meant to, schematically, involve decompositions as follows

$$
  \underset{spacetime}{
  \underbrace{
  \array{
    X_{11}
    \\
    \downarrow^{\pi}
    \\
    X_{10}
  }
  }
  }
  \;\;
  \underset{branes}{
  \underbrace{
  \array{
    && && \text{M2-brane} && && && \text{M5-brane}
    \\
    &&
    & {}^{\mathllap{wrapped}}\swarrow && \searrow^{\mathrlap{not \atop wrapped}} &
    &
    &&
    {}^{\mathllap{wrapped}}\swarrow && \searrow^{\mathrlap{not \atop wrapped}}
    \\
    \text{D0-brane} && \text{F1-brane} && &&  \text{D2-brane} && \text{D4-brane} && && \text{NS5-brane}
  }
  }
  }
$$

Above we saw that all [[super p-branes]] are
characterized by the [[flux]] fields $H_{p+2}$ that they are charged under,
more precisely by the bispinorial component of $H_{p+2}$
which is constrained to be super-tangent-space-wise the form

$$
  H_{p+2}^{fermionic}
    \;=\;
  \tfrac{i^{p(p-1)/2}}{p!} \, \left( \overline{\Psi} \wedge \Gamma_{a_1 \cdots a_p}\Psi \right) \wedge E^{a_1} \wedge \cdots \wedge E^{a_p}
$$

where $(E^a, \Psi^\alpha)$ is the [[super vielbein]] ([[graviton]] and [[gravitino]]).

$\,$

Hence we will formalize [[double dimensional reduction]] in terms of these [[fields]].

$\,$

Again there is a naive picture to help the intuition:
Let $G_4 \in \Omega^4_{cl}(X_{11})$ be the [[differential form|differential 4-form]] flux [[field strength]]  of the [[supergravity C-field]].

Under the [[Gysin sequence]] for the [[spherical fibration]]

$$
  \array{
    S^1 &\hookrightarrow& X_{11}
    \\
    && \downarrow^{\mathrlap{\pi}}
    \\
    && X_{10}
  }
$$

this decomposes in cohomology as

$$
  G_4 = (d x^{10}) \wedge H_3 + \pi^\ast F_4
$$

thus giving rise in 10d to

1. a 3-form $H_3$, the [[Kalb-Ramond field|Kalb-Ramond]] [[B-field]] [[field strength]] that the [[string]] couples to;

1. a 4-form $F_4$, the [[RR-field]] [[field strength]] in degree 4, that the [[D2-brane]] couples to.

Similary the 7-form field strength $G_7$ decomposes as

$$
  G_7 = (d x^{10}) \wedge F_6 + \pi^\ast H_7
$$

thus giving rise in 10d to

1. a 6-form $F_6$, the [[RR-field]] [[field strength]] in degree 6, that the [[D4-brane]] couples to

1. a 7-form $H_7$, the dual NS-NS field strength that the [[NS5-brane]] couples to.

$$
  \underset{spacetime}{
  \underbrace{
  \array{
    X_{11}
    \\
    \downarrow^{\pi}
    \\
    X_{10}
  }
  }
  }
  \;\;
  \underset{fluxes}{
  \underbrace{
  \array{
    && && G_4\text{-flux} && && && G_7\text{-flux}
    \\
    &&
    & {}^{\mathllap{wrapped}}\swarrow && \searrow^{\mathrlap{not \atop wrapped}} &
    &
    &&
    {}^{\mathllap{wrapped}}\swarrow && \searrow^{\mathrlap{not \atop wrapped}}
    \\
    F_2\text{-flux} && H_3\text{-flux} && &&  F_4\text{-flux} && F_6\text{-flux} && && H_7\text{-flux}
  }
  }
  }
$$

$\,$

The advantage of this perspective on [[double dimensional reduction]]
from the point of view of the [[background field|background flux fields]]
is that powerful tools from [[cohomology]] theory apply.

$\,$

To first approximation
background fluxes represent classes in [[ordinary cohomology]] (their [[charges]]).

There is a [[classifying space]] $B^n \mathbb{Z}$ for [[ordinary cohomology]]

$$
   H^n(X,\mathbb{Z})
     \;\;\;
     \simeq
     \;\;\;
  \left\{
    \array{
      \text{continuous functions}
      \\
      X \longrightarrow B^n \mathbb{Z}
    }
  \right\}_{/homotopy}
$$

(called an _[[Eilenberg-MacLane space]]_, often denoted $K(\mathbb{Z},n)$).
Hence the charge of $G_4$/$G_7$-flux, to first approximation, is represented by a classifying map

$$
  ([G_4], [G_7])
    \;\colon\;
  X_{11}
    \longrightarrow
  B^4 \mathbb{Z} \,\times\, B^7 \mathbb{Z}
  \,.
$$

and we saw that under [[double dimensional reduction]]
this is supposed to transmute into a map of the form

$$
  ([F_2] , [H_3], [F_4], [F_6], [H_7])
    \;\colon\;
  X_{10}
    \longrightarrow
  B^2 \mathbb{Z} \;\times\; B^3 \mathbb{Z} \;\times\; B^4 \mathbb{Z}  \;\times\; B^6 \mathbb{Z} \;\times\; B^7 \mathbb{Z}
  \,.
$$

$\,$

We ask:

_Which mathematical operation could cause such a transmutation?_

We will now find such an operation
and then use it to give an improved definition of [[double dimensional reduction]],
one that knows about all the fine print of brane charges.

$\,$

### Via free looping (no 0-brane effect)
 {#DoubleDimensionalReductionViaFreeLooping}

$\,$

Let's first record formally
what was going on in the above story.

In the above [[double dimensional reduction]]
of the naive M-fluxes on a trivial 11d circle bundle
we used

1. the [[Cartesian product]] with the circle

1. [[functions]] out of the circle.

Let's have a closer look at these two operations:

$\,$

It is a classical fact about [[locally compact topological spaces]]
(which includes all [[topological spaces]] that one cares about in [[physics]])
that
given topological spaces $\Sigma$, $X$ and $F$, then there is a [[natural bijection]]

$$
  \left\{
    \array{
      \text{continuous functions}
      \\
      \Sigma \times F \longrightarrow X
    }
  \right\}
   \;\;
     \underoverset
       {\text{"forming adjuncts"}}
       {\simeq}
       {\leftrightarrow}
   \;\;
  \left\{
    \array{
      \text{continous functions}
      \\
      \Sigma \longrightarrow Maps(F,X)
    }
  \right\}
$$

where

* $F \times X$ is the [[product topological space]] of $F$ with $X$

  (the set of pairs of points euipped with the [[product topology]])

* $Maps(F,Y)$ is the [[mapping space]] from $F$ to $X$,

  (the set of [[continuous functions]]) $F \to X$ equipped with the [[compact-open topology]])

Except for the subtlety with the topology
this bijection is just rewriting
a [[function]] of two [[variables]] as a function with values in a second function

$$
  (\tilde f(a))(b) = f(a,b)
  \,.
$$

One says that the two [[functors]]

$$
  Top_{cg}
  \;
    \underoverset
      {\underset{Maps(F,-)}{\longrightarrow}}
      {\overset{F \times (-)}{\longleftarrow}}
      {\bot}
  \;
  Top_{cg}
$$

form a _[[adjoint functor|adjoint pair]]_ or an _[[adjunction]]_.

$\,$

A remarkable amount of structure comes with every [[adjunction]]:

* the [[adjunct]] of the [[identity]] $F \times X \overset{id}{\to} F \times X$
  generally called the _[[unit of an adjunction|unit of the adjunction]]_,
  here is the **wrapping operation**

  $$
    X \overset{}{\longrightarrow} Maps(F, F \times X)
  $$

* the [[adjunct]] of the [[identity]] $Maps(F,X) \overset{id}{\to} Maps(F,X)$
  generally called the _[[counit of an adjunction|counit of the adjunction]]_,
  here is the [[evaluation map]]

  $$
    F \times Maps(F, X) \overset{ev}{\longrightarrow} X
  $$

  that evaluates a function on an argument

$\,$

We will see now that the following general fact
about [[adjoint functors]]
serves to implement the above physics story
of [[wrapped branes]]:

$\,$

**Fact.**

The [[adjunct]] of a map of the form

$$
  G \;\colon\; F \times X \overset{}{\longrightarrow} A
$$

is the composite of its image under $Maps(F,-)$ with the adjunction unit $\eta_X$:

$$
  \tilde G
    \;\colon\;
  X \overset{\eta_X}{\longrightarrow} Maps(F,F \times X)
    \overset{Maps(F,G)}{\longrightarrow}
  Maps(F,A)
$$

Moreover, we will see that the following generall fact in [[homotopy theory]]
accurately implements the idea of dimensional reduction of the brane dimensions:

For $F = S^1$ the [[circle]], then

$$
  \mathcal{L} X
  \;\coloneqq\;
  Maps(S^1, X)
$$

is also called the [[free loop space]] of $X$.

$\,$

+-- {: .num_prop #FreeLoopSpacesOfDeloopedAbelianInfinityGroups}
###### Proposition

For $G$ a general [[topological group]], then its [[free loop space]]

$$
  Maps(S^1, B G)
   \;\simeq\;
  G/_{ad}G
$$

is [[weak homotopy equivalence|weakly homotopy equivalent]] to the

[[homotopy quotient]] of $G$ by its [[adjoint action]].

In the special case that $G$ is an [[abelian group|abelian]] [[topological group]].

then this becomes a [[weak homotopy equivalence]] of following simple form

$$
  Maps(S^1 , B G)
    \; \simeq \;
  \underset{\text{wrapped} \atop \text{coefficient}}{\underbrace{G}}
    \; \times \;
  \underset{\text{plain} \atop \text{coefficient}  }{\underbrace{ B G }}
  \,.
$$

*This captures the required reduction on brane dimension!*

In particular if $G = B^n \mathbb{Z}$ then

$$
  Maps(S^1, B^{n+1} \mathbb{Z}) \;\;  \simeq B^n \mathbb{Z} \;\times\; B^{n+1} \mathbb{Z}
  \,.
$$

=--

$\,$

+-- {: .num_example #ReductionOfGFluxChargesOnTrivialCircleFibration}
###### Example

Consider naive $M$-flux fields $G_4$ and $G_7$
on an 11d spacetime that is a trivial circle bundle $X_{11} = X_{10} \times S^1$.
Its charges is represented by a map of the form

$$
  ([G_4], [G_7])
    \;\colon\;
  X_{10} \times S^1 \longrightarrow B^4 \mathbb{Z} \times B^7 \mathbb{Z}
  \,.
$$

By [[adjunction]] this is identified with a map of the form

$$
  \left([H_3], [F_4], [F_6], [H_7]\right)
  \,\coloneqq\,
  \widetilde{([G_4], [G_7])}
    \;\colon\;
  X_{10}
    \longrightarrow
  Maps\left(
    S^1,
    \; B^4 \mathbb{Z} \times B^7 \mathbb{Z}  \;
  \right)
   \;\;\simeq\;\;
  B^3 \mathbb{Z}
    \;\times\;
  B^4 \mathbb{Z}
    \;\times\;
  B^6 \mathbb{Z}
    \;\times\;
  B^7 \mathbb{Z}
  \,.
$$

where on the right we have the transmuted [[coefficients]] by prop. \ref{FreeLoopSpacesOfDeloopedAbelianInfinityGroups}

=--

This is exactly the result we were after.

$\,$

Better yet, the [[adjunction]] yoga
accurately reflects the physics story:
For consider a $p$-brane propagating in 10d spacetimes along a [[trajectory]]

$$
  \phi_p \;\colon\; \Sigma_p \longrightarrow X_{10}
$$

and coupled to these dimensionally reduced background fields

$$
 \Sigma_p
   \overset{\phi_p}{\longrightarrow}
 X_{10}
   \overset{([H_3], [F_4], [F_6], [H_7])}{\longrightarrow}
 B^3 \mathbb{Z} \;\times\; B^4 \mathbb{Z} \;\times\; B^6 \mathbb{Z} \;\times\; B^7 \mathbb{Z}
 \;\;\;\simeq\;\;\;
 Maps(S^1, B^4 \mathbb{Z} \,\times\, B^7 \mathbb{Z})
 \,.
$$

By [[adjunction]] this is identified with a map of the form

$$
  \Sigma_p \times S^1
    \overset{\phi_p \times S^1}{\longrightarrow}
  X_{10} \times S^1 = X_{11}
    \overset{([G_4], [G_7])}{\longrightarrow}
  B^4 \mathbb{Z} \;\times\; B^7 \mathbb{Z}
$$

and this is exactly the coupling we saw in the story of double dimensional reduction.

$\,$

So this works well as far as it goes, but
so far it only applies to trivial circle fibrations
and it does not see the D0-charge.

$\,$

We now disucss the improvement to the full formulation.

$\,$

### Via cyclification (with 0-brane effects)
 {#DoubleDimensionalReductionViaCyclification}

In general the M-theory circle bundle

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

is only locally a product with of $X_{10}$ with $S^1$.

$\,$

For example
the complement of the locus of a **[[KK-monopole]]** [[spacetime]]
is a [[circle]] [[principal bundle]] with [[first Chern class]]
equal to the [[charge]] carried by the [[KK-monopole]].
(which is the corresponding number of coincident [[D6-branes]] in type IIA).

$\,$

Hence in general the above formulation of [[double dimensional reduction]]
via the pair of [[adjoint functors]]

$$
  S^1 \times (-) \;\;\dashv\;\; Maps(S^1, -)
$$

works only locally.

$\,$

But the problem to be solved is easily identified:
Essentially by definition, in a [[circle]] [[principal bundle]]
the [[fibers]] may all be identified with a fixed abstract [[circle]] $S^1$
only _up to rigid rotation_.

$\,$

Hence while in general the above wrapping-map

$$
  X_{10} \overset{}{\longrightarrow} Maps(S^1, X_{11})
$$

given by sending each point of $X_{10}$ to its fiber "wrapping around itself"
does not exist, it does exist up to forgetting at which point in $S^1$ we start the wrapping,
hence the map that always exists lands in the [[quotient space]]

$$
  Maps(S^1, X_{11})/S^1
  \;=\;
  \frac{
    \left\{
      \array{
        \text{continuous functions}
        \\
        S^1 \longrightarrow X_{11}
      }
    \right\}
  }{
    \left\{
      \array{
        \text{rigid loop rotations}
        \\
        S^1 \overset{t \mapsto (t + t_0)}{\longrightarrow} S^1
      }
    \right\}
  }
$$

In general we take this to be the [[homotopy quotient]] space.

$\,$

There is then the following generalization of proposition \ref{FreeLoopSpacesOfDeloopedAbelianInfinityGroups} on
transmutation of [[coefficients]] under [[double dimensional reduction]]

+-- {: .num_prop #FreeLoopSpacesModS1OfDeloopedAbelianInfinityGroups}
###### Proposition

Let $G$ be an [[abelian group|abelian]] [[topological group]].

Then there is a [[weak homotopy equivalence]] of the form

$$
  Maps(S^1 , B G)/S^1
    \; \simeq \;
  \left(
    \underset{wrapped \atop coefficient}{\underbrace{G}}
      \times
    \underset{plain \atop coefficient}{\underbrace{B G}}
  \right)
  \underset{twist}{
  \underbrace{
      \times_{S^1}
  }
  }
  \underset{\text{D0-brane} \atop coeff.}{\underbrace{E S^1}}
  \,.
$$

=--

Notice that a twisting appears. This is a general phenomenon.
We will see [below](#KKReductionToIIA) that for the example of reduction of M-flux
the twist that appears is that in the twisted de Rham cohomology  $d F_4 = H_3 \wedge F_2$
which connects RR-fields $F_{2p}$ with the H-flux $H_3$.

$\,$

Indeed this dimensional reduction is again an _equivalent_ way of regarding the higher dimensional
situation:

$\,$

+-- {: .num_prop #CorrectReductionOnTopologicalSpaces}
###### Proposition
**(double dimensional reduction on topological flux fields)**

There is a pair of [[adjoint functors]] ([[adjoint (∞,1)-functors]] really)

$$
  \array{
    \left\{
      spaces
    \right\}
    &
      \underoverset
        {\underset{Maps(S^1,-)/S^1}{\longrightarrow}}
        {\overset{hofib}{\longleftarrow}}
        {\bot}
    &
    \left\{
      \text{spaces over}\, B S^1
    \right\}
  }
$$

(a proof in more generality is below after prop. \ref{GeneralReduction}).
Equivalently (by [Nikolaus-Schreiber-Stevenson 12](#NSS12)):
There is a pair of [[adjoint functors]] ([[adjoint (∞,1)-functors]] really)

$$
  \array{
    \left\{
      spaces
    \right\}
    &
      \underoverset
        {\underset{[Maps(S^1,-) \to Maps(S^1,-)/S^1]}{\longrightarrow}}
        {\overset{\text{total space}}{\longleftarrow}}
        {\bot}
    &
    \left\{
      S^1\text{-principal}\;\infty\text{-bundles}
    \right\}
  }
$$

Hence for

$$
  \array{
    S^1 &\hookrightarrow& X_{d+1}
    \\
    && \downarrow
    \\
    && X_{d}
  }
$$

an $S^1$-principal bundle and $A$ some [[coefficients]],
then there is a [[natural equivalence]]

$$
  \underset{
    \text{original} \atop \text{fluxes}
  }{
  \underbrace{
    Hom(X_{d+1}\;,\; A)
  }
  }
  \;\;\;
   \underoverset
     {\underset{\text{reduction}}{\longrightarrow}}
     {\overset{\text{oxidation}}{\longleftarrow}}
     {\simeq}
  \;\;\;
  \underset{
    \text{doubly} \atop { \text{dimensionally  reduced} \atop \text{fluxes} }
  }{
  \underbrace{
    Hom_{/B S^1}( X_{d} \; ,\; (\mathcal{L} A)/S^1 )
  }
  }
$$

=--

$\,$

Accordingly we have the following generalization of example \ref{ReductionOfGFluxChargesOnTrivialCircleFibration} to the case with possibly  non-trivial circle-fibration and non-trivial D0-flux:

+-- {: .num_example #ReductionOfGFluxChargesOnTrivialCircleFibration}
###### Example

Consider naive $M$-flux fields $G_4$ and $G_7$ on an 11d [[spacetime]] that
is an $S^1$-[[principal bundle]]

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

Its charges is represented by a map of the form

$$
  ([G_4], [G_7])
    \;\colon\;
  X_{11} \longrightarrow B^4 \mathbb{Z} \;\times\; B^7 \mathbb{Z}
  \,.
$$

By [[adjunction]] this is identified with a map of the form

$$
  \left([F_2], [H_3], [F_4], [F_6], [H_7]\right)
  \,\coloneqq\,
  \widetilde{([G_4], [G_7])}
    \;\colon\;
  X_{10}
    \longrightarrow
  Maps\left(
    S^1,
    \; B^4 \mathbb{Z} \times B^7 \mathbb{Z}  \;
  \right)
   \;\simeq\;
  E S^1
    \;\times_{S^1}\;
  \left(
    B^3 \mathbb{Z}
      \;\times\;
    B^4 \mathbb{Z}
      \;\times\;
    B^6 \mathbb{Z}
      \;\times\;
    B^7 \mathbb{Z}
  \right)
  \,.
$$

where on the right we transmuted the [[coefficients]] by prop. \ref{FreeLoopSpacesModS1OfDeloopedAbelianInfinityGroups}

$\,$

Hence the [[D0-brane]] charge appears! It is the [[first Chern class]]
of the M-theory circle bundle.

=--

$\,$

**Conclusion**

The double dimensional reduction of any flux field

$$
  X_{d+1}
    \overset{G}{\longrightarrow}
  A
$$

is

$$
  \array{
    X_{d}
      && \overset{\tilde G}{\longrightarrow} &&
    Maps(S^1, A)/S^1
    \\
    & \searrow && \swarrow
    \\
    && B S^1
  }
  \,.
$$

$\,$

The operation

$$
  \mathcal{L}(-)/S^1
  \;\coloneqq\;
  Maps(S^1, -)/S^1
$$

may be called **[[cyclic loop space|cyclification]]**
because the [[cohomology]] of this quotient of the [[free loop space]]
is **[[cyclic cohomology]]**.

$\,$

Shadows of this construction appear prominently also at other places in [[string theory]]
notably in discussion of the **[[Witten genus]]**.
A closely related concept in mathematics involving this is the **[[transchromatic character]] map**.

$\,$

In fact this formalization of [[double dimensional reduction]]
works with loads of further data taken into account, such as the
[[differential geometry]] of [[spacetimes]] and the [[differential cohomology]] of
flux fields.

$\,$

For the [[homotopy theory]] cognoscenti, here is the fully general statement:

+-- {: .num_prop #GeneralReduction}
###### Proposition

Let $\mathbf{H}$ be any  [[(∞,1)-topos]] such as

* the [[classical model structure on topological spaces|classical homotopy theory on topological spaces]]  as above

* or better: the [[homotopy theory]] of [[super formal smooth infinity-groupoid|supergeometric homotopy types]]

and let $G$ be an [[∞-group]] in $\mathbf{H}$ such as

* a [[topological group]] as above

* or better [[Lie group]] such as the smooth [[circle group]] $S^1 \simeq U(1)$

then there is a pair of [[adjoint ∞-functors]] of the form

$$
  \mathbf{H}
    \underoverset
      {\underset{[G,-]/G}{\longrightarrow}}
      {\overset{hofib}{\longleftarrow}}
      {\bot}
  \mathbf{H}_{/\mathbf{B}G}
  \,,
$$

where

* $[G,-]$ denotes the [[internal hom]] in $\mathbf{H}$,

* $[G,-]/G$ denotes the [[homotopy quotient]] by the [[conjugation action|conjugation]]
[[∞-action]] for $G$ equipped with its canonical [[∞-action]] by left multiplication and the argument
regarded as equipped with its trivial $G$-$\infty$-action.
(Hence the claim is that $[G,-]/G$ is the right [[base change]]/[[dependent product]] along the canonical $\ast \to \mathbf{B}G$.)

Hence for

* $\hat X \to X$ a $G$ [[principal ∞-bundle]]

* $A$ a [[coefficient]] object, such as for some [[differential cohomology|differential]] [[generalized cohomology theory]]

then there is a [[natural equivalence]]

$$
  \underset{
    \text{original} \atop \text{fluxes}
  }{
  \underbrace{
    \mathbf{H}(\hat X\;,\; A)
  }
  }
    \;\;
    \underoverset
      {\underset{oxidation}{\longleftarrow}}
      {\overset{reduction}{\longrightarrow}}
      {\simeq}
    \;\;
  \underset{
     \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} }
  }{
  \underbrace{
    \mathbf{H}(X \;,\; [G,A]/G)
  }
  }
$$

given by

$$
  \left(
     \hat X \longrightarrow A
  \right)
  \;\;\;
    \leftrightarrow
  \;\;\;
  \left(
    \array{
       X && \longrightarrow && [G,A]/G
       \\
       & \searrow && \swarrow
       \\
       && \mathbf{B}G
    }
  \right)
$$

=--

+-- {: .proof #DimensionalReductionAbstractly}
###### Proof

First observe that the [[conjugation action]] on $[G,X]$ is the [[internal hom]] in
the [[(∞,1)-category]] of $G$-[[∞-actions]] $Act_G(\mathbf{H})$.
Under the [[equivalence of (∞,1)-categories]]

$$
  Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}
$$

(from [Nikolaus-Schreiber-Stevenson 12](#NSS12)) then $G$ with its canonical [[∞-action]] is $(\ast \to \mathbf{B}G)$
and $X$ with the trivial action is $(X \times \mathbf{B}G \to \mathbf{B}G)$.

Hence

$$
  [G,X]/G
    \simeq
  [\ast, X \times \mathbf{B}G]_{\mathbf{B}G}
  \;\;\;\;\;
  \in
   \mathbf{H}_{/\mathbf{B}G}
  \,.
$$

Actually, this is the very definition of what $[G,X]/G \in \mathbf{H}_{/\mathbf{B}G}$ is to mean in the first place, abstractly.
But now since the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}G}$ is itself cartesian closed, via

$$
  E \times_{\mathbf{B}G}(-)
  \;\;\;
   \dashv
  \;\;\;
  [E,-]_{\mathbf{B}G}
$$

it is immediate that
there is the following sequence of [[natural equivalences]]

$$
  \begin{aligned}
   \mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G)
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G})
   \\
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(
     Y \times_{\mathbf{B}G} \ast,
     \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }}
  )
  \\
  & \simeq
  \mathbf{H}(
     \underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}},
     X
  )
  \\
  & \simeq
  \mathbf{H}(hofib(Y),X)
  \end{aligned}
$$

Here $p \colon \mathbf{B}G \to \ast$ denotes the terminal morphism and $p_! \dashv p^\ast$
denotes the [[base change]] along it.

=--

$\,$

We now apply this general mechanism to the [[schreiber:brane bouquet]].

$\,$

### On super $p$-brane cocycles
 {#DoubleDimensionalReductionLInfinity}

By the discussion of [[rational homotopy theory]] [above](#RationalHomotopyTheory)
we may think of [[L-∞ algebras]] as [[rational topological spaces]]
and more generally as rational [[parameterized spectra]].
For instance [above](#RRFields) we found that the [[coefficient]] space
for [[RR-fields]] in rational [[twisted K-theory]] is the
[[L-∞ algebra]] $\mathfrak{l}(KU/BU(1))$.

$\,$

Hence in order to apply [[double dimensional reduction]]
to [[super p-branes]]
we now specialize the above general formalization (prop. \ref{GeneralReduction}) to
 **cyclification of [[super L-∞ algebras]]** ([FSS 16b](#FSS16b))

$\,$

+-- {: .num_defn #Cyclification}
###### Definition

For $\mathfrak{g}$ any [[super L-∞ algebra]] of [[finite type]], its **cyclification**

$$
  \mathfrak{L}\mathfrak{g}/\mathbb{R}
  \in
  s L_\infty Alg_{\mathbb{R}}
$$

is defined by having [[Chevalley-Eilenberg algebra]] of the form

$$
  CE(\mathfrak{L}\mathfrak{g}/\mathbb{R})
  \coloneqq
  \left(
    \wedge^\bullet
      \left(
        \underset{\text{original}}{\underbrace{\mathfrak{g}^\ast}}
        \oplus
         \underset{\text{shifted copy}}{\underbrace{s\mathfrak{g}^\ast}}
          \oplus
         \underset{\text{new generator} \atop \text{in degree 2}}{\underbrace{\langle \omega_2 \rangle}}
      \right)
      \;,\;
  d_{\mathfrak{L}\mathfrak{g}/\mathbb{R}}
  \;\colon\;
  \left\{
    \array{
      \omega_2 &\mapsto& 0
      \\
      \alpha &\mapsto& d_{\mathfrak{g}} \alpha + \omega_2 \wedge s \alpha
      \\
      s \alpha &\mapsto&  - s d_{\mathfrak{g}} \alpha
    }
  \right.
  \right)
$$

where

$$
  s \mathfrak{g}^\ast
$$

is a copy of $\mathfrak{g}^\ast$ with cohomological degrees shifted _down_ by one, and where $\omega$ is a new generator in degree 2.
The differential is given for $\alpha \in \wedge^1 \mathfrak{g}^\ast$ by

$$
  d_{\mathfrak{d}\mathfrak{g}/\mathbb{R}}
  \;\colon\;
  \left\{
    \array{
      \omega_2 &\mapsto& 0
      \\
      \alpha &\mapsto& d_{\mathfrak{g}} \alpha \pm \omega_2 \wedge s \alpha
      \\
      s \alpha &\mapsto&  - s d_{\mathfrak{g}} \alpha
    }
  \right.
$$

where on the right we are extendng $s$ as a graded derivation.
Define

$$
  \mathfrak{L}\mathfrak{g}
  \in
  s L_\infty Alg_{\mathbb{R}}
$$

in the same way, but with $\omega_2 \coloneqq 0$.

=--

For every $\mathfrak{g}$ there is a [[homotopy fiber sequence]]

$$
  \array{
     && \mathfrak{L}\mathfrak{g}
     \\
     && \downarrow
     \\
     && \mathfrak{L} \mathfrak{g}/\mathbb{R}
     \\
     & \swarrow_{\mathrlap{\omega_2}}
     \\
     B \mathbb{R}
  }
$$

which hence exhibits $\mathfrak{L} \mathfrak{g}/\mathbb{R}$ as the [[homotopy quotient]]
of $\mathfrak{L}\mathfrak{g}$ by an $\mathbb{R}$-action.

The following says that the $L_\infty$-cyclification from prop.  \ref{Cyclification}
indeed does model the topological cyclification from prop. \ref{CorrectReductionOnTopologicalSpaces}.

+-- {: .num_prop #FreeLoopSpaceInRarionalHomotopyTheory}
###### Proposition
**([Vigué-Sullivan 76](https://ncatlab.org/nlab/show/Sullivan+model+of+free+loop+space#VigueSullivan76),
[Vigué-Burghelea 85](https://ncatlab.org/nlab/show/Sullivan+model+of+free+loop+space#VigueBurghelea85))**

If

$$
  \mathfrak{g} = \mathfrak{l}(X)
$$

is the $L_\infty$-algebra associated by [[rational homotopy theory]]
to a  [[simply connected topological space]] $X$, then

$$
  \mathfrak{L}( \mathfrak{l}(X) )
    \simeq
  \mathfrak{l}( \mathcal{L}X )
$$

corresponds to the [[free loop space]] of $X$ and

$$
  \mathfrak{L}( \;\mathfrak{l}( X )\; )/\mathbb{R}
    \simeq
  \mathfrak{l}( \;\mathcal{L}X/S^1\; )
$$

corresponds to the [[homotopy quotient]] of the [[free loop space]] by the [[circle group]] [[action]] which rotates the loops.
The [[cochain cohomology]] of the [[Chevalley-Eilenberg algebra]]

$$
  CE(\mathfrak{l}( \;\mathcal{L}X/S^1\; ))
$$

computes the [[cyclic cohomology]] of $X$ with [[coefficients]] in $\mathbb{R}$.
(Whence "cyclification".)
Moreover the homotopy fiber sequence of the cyclification corresponds to that of the [[free loop space]]:

$$
  \left(
    \array{
      \mathcal{L}X
      \\
      \downarrow^{\mathrlap{hofib(p)}}
      \\
      \mathcal{L}X/S^1
      \\
      \downarrow^{\mathrlap{p}}
      \\
      B S^1
    }
    \;\;\;\;\;\;\;\;
  \right)
  \;\;\;\;
    \stackrel{\phantom{AA}\mathfrak{l}(-)\phantom{AA}}{\mapsto}
  \;\;\;\;
  \left(
  \array{
   \mathfrak{L} \mathfrak{l}(X)
   \\
   \downarrow^{ \mathrlap{ hofib( \mathfrak{l}(p) ) } }
   \\
   \mathfrak{L}\mathfrak{l}(X)/\mathbb{R}
   \\
   \downarrow^{\mathrlap{\mathfrak{l}(p)}}
   \\
   B \mathbb{R}
  }
    \;\;\;\;\;\;\;\;
  \right)
$$

=--

$\,$

The following gives the super-$L_\infty$-theoretic formalization
of "[[double dimensional reduction]]"
by which both the spacetime dimension is reduced
while at the same time the [[brane]] dimension
reduces (if [[wrapped brane|wrapping]] the reduced dimension).

$\,$

We have the following $L_\infty$-algebraic incarnation
of the general double dimensional reduction isomorphism prop. \ref{CorrectReductionOnTopologicalSpaces},
prop. \ref{GeneralReduction}:

$\,$

+-- {: .num_prop #ddReduction}
###### Proposition
**([Fiorenza-Sati-Schreiber 16b, prop. 3.8](#FSS16b))**

Let

$$
  \array{
     \widehat{\mathfrak{g}}
     \\
     {}^{\mathllap{\pi}}\downarrow
     \\
     \mathfrak{g}
     \\
     & {}_{\mathllap{\mu_2}}\searrow
     \\
     && B \mathbb{R}
  }
$$

be a [[central extension]] of [[super L-∞ algebras]]. According to prop. \ref{HomotopyFibersOfLInfinityCocycles}
we have

$$
  CE(\widehat{\mathfrak{g}})
    \simeq
  CE(\mathfrak{g})[e, d e = \mu_2]
  \,.
$$

and hence every generator $\alpha_p \in CE(\widehat{\mathfrak{g}})$ has a unique decomposition

$$
  \alpha_p = \beta_p - e \wedge \tilde \alpha_{p-1}
$$

where $\beta_p$ and $\tilde \alpha_{p-1}$ do not involve the generator $e$. We may think of this as

$$
  \pi_\ast(\alpha_p) \coloneqq \tilde \alpha_{p-1}
  \;\;\;\;\,\;
  \alpha_p|_{\mathfrak{g}} \coloneqq \beta_p
  \,.
$$

Under this identification any super $L_\infty$-homomorphism

$$
  \phi \;\colon\; \widehat{\mathfrak{g}} \overset{\phi}{\longrightarrow} \mathfrak{h}
$$

hence a [[dg-algebra]] homomorphism

$$
  \phi^\ast
    \;\colon\;
  CE(\mathfrak{h})
     \longrightarrow
  CE(\widehat{\mathfrak{g}})
$$

gives rise to a homomorphism of the form

$$
  \tilde \phi
    \;\colon\;
  \mathfrak{g}
    \longrightarrow
  \mathfrak{L}\mathfrak{g}/\mathbb{R}
$$

which, in the notation of def. \ref{Cyclification}, is given dually by

$$
  \tilde \phi^\ast
  \colon
  \left\{
    \array{
      \alpha & \mapsto (\phi^\ast \alpha)|_{\mathfrak{g}}
      \\
      s \alpha & \mapsto \pi_\ast(\phi^\ast \alpha)
      \\
      \omega_2 & \mapsto \mu_2
    }
  \right.
  \,.
$$

Moreover, this construction constitutes a [[natural bijection]]

$$
  \array{
    \underset{ \text{original} \atop \text{cocycles} }{
    \underbrace{
      Hom( \widehat{\mathfrak{g}}, \mathfrak{h} )
    }}
     \;&\;
      \underoverset
        {\underset{oxidation}{\longleftarrow}}
        {\overset{reduction}{\longrightarrow}}
        {\simeq}
      \;&\;
    \underset{
      \text{doubly} \atop { \text{dimensionally reduced} \atop \text{cocycles} }
    }{
    \underbrace{
      Hom_{/B\mathbb{R}}( \mathfrak{g}, \mathfrak{L}\mathfrak{h}/\mathbb{R} )
    }
    }
    \\
    \\
    \text{given by}
    \\
    \\
    \left(
      \array{
        \widehat{\mathfrak{g}}
           \overset{}{\longrightarrow}
        \mathfrak{h}
      }
    \right)
      \;&\;
      \leftrightarrow
       \;&\;
    \left(
      \array{
         \mathfrak{g}
           &&
             \overset{}{\longrightarrow}
           &&
         \mathfrak{L}\mathfrak{h}/\mathbb{R}
         \\
         & {}_{\mathllap{\mu_2}}\searrow && \swarrow_{\mathrlap{\omega_2}}
         \\
         && B \mathbb{R}
      }
    \right)
  }
$$

between super $L_\infty$-homomorphisms out of the exteded super $L_\infty$-algebra $\widehat{\mathfrak{g}}$
and homomorphism out of the base $\mathfrak{g}$ into the cyclification (def. \ref{Cyclification}) of the original coefficients
with the latter constrained so that
the canonical 2-cocycle on the cyclification is taken to the 2-cocycle classifying the given extension.

=--

+-- {: .num_remark}
###### Remark

If $CE(\mathfrak{h})$ in prop. \ref{DimensionalReduction} has generators in degree 1, then
the operation $\tilde \phi^\ast$ involves sending generators in degree 0 to multiples of the
ground field. This makes $\tilde \phi$ a "[[curved L-∞ algebra|curved]]" $L_\infty$-homomorphism.
Hence for prop. \ref{DimensionalReduction} to give a pair of [[adjoint functors]] we need to regard it
in the category of $L_\infty$-algebras with curved morphisms between them. But in applications
$\mathfrak{h}$ typically contains no generators of degree 1, in which case the above natural bijection
exists on the category of plain $L_\infty$-homomorphisms.

=--

+-- {: .num_example}
###### Example

Let

$$
  \left[
  \array{
    \widehat{\mathfrak{g}}
    \\
    \downarrow
    \\
    \mathfrak{g}
    \\
    & {}_{}\searrow
    \\
    && b \mathbb{R}
  }
  \right]
  \;\coloneqq\;
  \left[
    \array{
      \mathbb{R}^{d,1\vert N_{d+1}}
      \\
      \downarrow
      \\
      \mathbb{R}^{d-1,1\vert N_d}
      \\
      & {}_{\mathllap{\overline{\psi}\wedge \Gamma^{d}\psi}} \searrow
      \\
      && b \mathbb{R}
    }
  \right]
$$

be the extension of a [[super Minkowski spacetime]] from [[dimension]] $d$ to dimension $d+1$.
Let moreover

$$
  \mathfrak{h} \coloneqq b^{(p+1)+1} \mathbb{R}
$$

be the  [[line Lie n-algebra|line Lie (p+3)-algebra]] (prop. \ref{HomotopyFibersOfLInfinityCocycles})
and consider any [[super p-brane|super (p+1)-brane]] cocycle from the old [[brane scan]] in dimension $d+1$

$$
  \mu_{(p+1)+2}
  \;\coloneqq\;
  \underoverset{a_i = 0}{d}{\sum}
  \left(
    \overline{\psi}
      \wedge
      \Gamma_{a_1 \cdots a_{p+1}}
  \psi
  \right)
  \wedge
  e^{a_1} \wedge \cdots \wedge e^{a_{p+1}}
  \;\;\colon\;\;
  \mathbb{R}^{d,1\vert N_{d+1}}
    \longrightarrow
  b^{p+1} \mathbb{R}
  \,.
$$

Then the cyclification $\mathfrak{L}(b^{p+1}\mathbb{R})/\mathbb{R}$ of the coefficients (prop. \ref{DimensionalReduction}) is

$$
  CE\left(
    \,
    \mathfrak{L}(b^{p+2}\mathbb{R})/\mathbb{R}
    \,
  \right)
  \;=\;
  \left\{
    \array{
      d \omega_2 = 0
      \\
      d \omega_{p + 2} = 0
      \\
      d \omega_{(p+1)+2} = \omega_{p+1} \wedge \omega_2
    }
  \right\}
$$

and the dimensionally reduced cocycle

$$
  \array{
    \mathbb{R}^{d-1,1\vert N_d}
      &&
        \overset{}{\longrightarrow}
      &&
    \mathfrak{L}(b^{p+1}\mathbb{R})/\mathbb{R}
    \\
    & \searrow && \swarrow
    \\
   && b \mathbb{R}
  }
$$

has the following components

$$
  \array{
    && &&
    \overset{
      p+1\text{-brane}
    }{
    \overbrace{
    {
      \mu^{d+1}_{(p+1)+2}
      =
    }
    \atop
    {
      \underoverset{d=0}{d}{\sum}
      \left(
        \overline{\psi}
         \wedge
         \Gamma_{a_1 \cdots a_{p+1}}
         \psi
      \right)
      \wedge
      e^{a_1}
       \wedge
       \cdots
      e {a_{p+1}}
      }
      }
    }
    \\
    && & {}^{\mathllap{\text{wrapped}}}\swarrow && \searrow^{\mathrlap{\text{not} \atop \text{wrapped}}}
    \\
    \underset{
      \text{0-brane}
    }{
    \underbrace{
    {
      \mu^{d}_{0+2}
      =
    }
    \atop
    {
      \left(
      \overline{\psi}
        \wedge
        \Gamma^d
      \psi
      \right)
      }
      }
    }
    &&
    \underset{
      p\text{-brane}
    }{
    \underbrace{
      {
        \mu^{d}_{p+2}
        =
      }
      \atop
      {
        \underoverset{a_i = 0}{d-1}{\sum}
        \left(
        \overline{\psi}
        \wedge
          \Gamma_{a_1 \cdots a_{p}}
        \psi
        \right)
        \wedge
        e^{a_1}
         \wedge \cdots
        e^{a_p}
      }
      }
    }
    &&
    &&
    \underset{
      p+1\text{-brane}
    }{
    \underbrace{
      {
        \mu^{d}_{p+2}
        =
      }
      \atop
      {
        \underoverset{a_i = 0}{d-1}{\sum}
        \left(
        \overline{\psi}
        \wedge
          \Gamma_{a_1 \cdots a_{p+1}}
        \psi
        \right)
        \wedge
        e^{a_1}
         \wedge \cdots
        e^{a_{p+1}}
      }
      }
    }
  }
$$

It follows that with

$$
  d \,\mu^{d+1}_{(p+1)+2} = 0
$$

also

$$
  d\, \mu^d_{p+2} = 0
  \,.
$$

This is the dimensional reduction observed in the old [[brane scan]] ([Achúcarro-Evans-Townsend-Wiltshire 87](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#AETW87))

> graphics grabbed from ([Duff 87](Green-Schwarz+action+functional#Duff87))

But there is more: the un-wrapped component of the dimensionally reduced cocycle satisfies the  twisted cocycle condition

$$
  d \, \mu^d_{(p+1)+2}
   \;=\;
  mu^d_{p+2} \wedge \mu^d_{0+2}
  \,.
$$

These relations are not to be ignored.

=--

This we turn to now.

$\,$

## Dualities
 {#Dualities}

We discuss now how
by repeatedly applying
the super $L_\infty$-algebraic dimensional reduction/oxidation isomorphism of prop. \ref{DimensionalReduction}
to the descended cocycles ([above](#Fields)) from the [[schreiber:brane bouquet]]
yields super $L_\infty$-algebraic equivalences
that reflect the pertinent [[dualities in string theory]]:

1. between [[M-theory]] and [[type IIA string theory]] by [[KK-compactification]]

1. between [[type IIA string theory]] and [[type IIB string theory]] ([[T-duality]])

1. between [[type IIB string theory]] and itself ([[S-duality]])

1. between [[type IIB string theory]] and [[F-theory]].

$\,$

$\,$

We discuss now each aspect of this picture.

$\,$

### M/IIA-Duality via Double dimensional reduction via Cyclification
 {#KKReductionToIIA}

$\,$

The [[M2-brane]]/[[M5-brane]] in 11d

is all controled by the following [[Fierz identities]]

for the $\mathbf{32}$ [[Majorana spinor|Majorana]] [[spin representation]] for $Spin(10,1)$

$\,$

   $$
     \left(
       \overline{\psi} \wedge \Gamma_{a b} \psi
     \right)
     \wedge
     \left(
        \overline{\psi} \wedge \Gamma^b \psi
     \right)
     \;= \; 0
     \,.
   $$

$\,$

and

$\,$

   $$
     \left(
       \overline{\psi}
         \wedge
         \Gamma_{a_1 \cdots a_4 b}
       \psi
     \right)
     \wedge
     \left(
       \overline{\psi}
        \wedge
        \Gamma^{b}
        \psi
     \right)
     \;=\;
     3
     \left(
     \overline{\psi}
        \Gamma_{[a_1 a_2}
     \psi
     \right)
     \wedge
     \left(
       \overline{\psi}
          \Gamma_{a_3 a_4]}
       \psi
     \right)
   $$

([D'Auria-Fré 82b (3.13) and (3.28) ](geometry+of+physics+--+supersymmetry#DAuriaFre82b))

$\,$

$\,$

The first [[Fierz identity]] says that

$$
  \mu_{M2}
  \:\coloneqq\;
  \left(
  \tfrac{i}{2}
  \overline{\psi}
     \wedge
     \Gamma_{a b}
  \psi
  \right)
  \wedge
    e^{a}
    \wedge
    e^b
$$

is a 4-cocycle on $\mathbb{R}^{10,1\vert \mathbf{32}}$, classifying the

[[supergravity Lie 3-algebra]] extension

$$
  \array{
    \mathfrak{m}2\mathfrak{brane}
    \\
    \downarrow
    \\
    \mathbb{R}^{10,1\vert \mathbf{32}}
  }
  \;\;\;\;\;\;\;\;\;
  CE(\mathfrak{m}2\mathfrak{brane})
  =
  \left\{
    \array{
      d \, e^a = \overline{\psi} \wedge \Gamma^a \psi
      \\
      d \, \psi^\alpha = 0
      \\
      d \, h_3 = \mu_{M2}
    }
  \right\}
$$

The second says that

$$
  \mu_{M5}
    \;\coloneqq\;
  \tfrac{1}{5!}
  \left(
    \overline{\psi}
      \wedge
      \Gamma_{a_1 \cdots a_5}
    \psi
  \right)
  \wedge
  e^{a_1}
    \wedge
    \cdots
    \wedge
  e^{a_5}
  \;-\;
  \tfrac{1}{2} h_3 \wedge \mu_{M2}
$$

is a 7-cocycle on $\mathfrak{m}2\mathfrak{brane}$

$$
  \mu_{M5}
  \;\colon\;
  \mathfrak{m}2\mathfrak{brane}
    \longrightarrow
  b^6 \mathbb{R}
$$

$\,$

Equivalently, (by the discussion at _[M-Flux fields](#MFluxFields)_ above)

with $\mathfrak{l}(S^4)$ the [[rational n-sphere|rational 4-sphere]]

$$
  CE(\mathfrak{l}(S^4))
   =
  \left\{
    \array{
      d \, g_4 = 0
      \\
      d \, g_7 = -\tfrac{1}{2} g_4 \wedge g_4
    }
  \right\}
$$

then the two [[Fierz identities]] together say that there is a single [[4-sphere]] valued cocycle

$$
  \mu_{M2/M5}
  \;\colon\;
  \mathbb{R}^{10,1\vert \mathbf{32}}
    \longrightarrow
  \mathfrak{l}(S^4)
$$

Hence $S^4$ is the rational coefficient for the unified M2/M5 M-flux fields.

Now we compute what becomes of this under [[double dimensional reduction]] via $L_\infty$-cyclification (prop. \ref{DimensionalReduction})

+-- {: .num_example}
###### Example

Let $X = S^4$ be the [[4-sphere]].

Its free loop algebra $\mathcal{L}S^4$ is given by

$$
 CE( \; \mathfrak{L}S^4 \; )
  \;=\;
 \left\{
  \array{
    d\, h_3 & = 0
    \\
    d\, \omega_4 & = 0
    \\
    d\, \omega_6 & = h_3 \wedge \omega_4
    \\
    d\, h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4
    \\
  }
  \right\}
$$

and its cyclification $\mathcal{L}S^4 / S^1$ is given by

$$
  CE( \; \mathfrak{L}S^4 / \mathbb{R}  \; )
  \;=\;
  \left\{
    \array{
      d \, h_3 & = 0
      \\
      d \, \omega_2 & = 0
      \\
      d \, \omega_4 & = h_3 \wedge \omega_2
      \\
      d \, \omega_6 & = h_3 \wedge \omega_4
      \\
      d \, h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6
  }
  \right.
  \,.
$$

=--

$\,$

$\,$

+-- {: .num_prop}
###### Proposition
([Fiorenza-Sati-Schreiber 16a, section 3](#FSS16a))

The cyclification of the M-brane coefficient $\mathfrak{l}(S^4)$

is the truncated IIA F1/D-brane coefficient $\mathfrak{l}( KU\langle 0,6 \rangle ( B U(1) )$

with an extra twist for the [[NS5-brane]]:

$$
  CE\left(
    \mathfrak{L} S^4 / \mathbb{R}
  \right)
  \;\simeq\;
  CE\left(\,
     \underset{\text{F1/Dp-branes}}{\underbrace{
     \mathfrak{l}(KU\langle 0,6\rangle/B U(1))
     }}
     \,
  \right)
  \underset{\text{NS5-brane}}{\underbrace{
    [h_7]/(d h_7 = \omega_2 \wedge \omega_6 - \omega_4 \wedge \omega_4)
  }}
  \,.
$$

Under this identification

the $L_\infty$-theoretic dimensional reduction according to prop. \ref{DimensionalReduction}

of the unified M-brane cocycle $\mu_{M2/M5}$ of prop. \ref{MBraneDescent}

along the M-theory extension from example \ref{D0Cocycle}

is the unified type IIA F1/D-brane cocycle from prop. \ref{DescentOfIIACocycles}:

$$
  \array{
    11d, N = 1
    &&&&
    \mathbb{R}^{10,1\vert \mathbf{32}}
      &&
        \overset{\mu_{M2/M5}}{\longrightarrow}
      &&
    \mathfrak{l}(S^4)
    \\
    && && \downarrow
    \\
    10d, \text{type IIA}
    &&
    &&
    \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}
     &&
       \overset{
          { \mathfrak{L}(\mu_{M2/M5})/\mathbb{R}  }
          \atop
          {
            =
            \atop
            {
              \mu^{IIA}_{F1/D0/D2/D4/NS5}
            }
          }
       }{\longrightarrow}
     &&
      {
        { \mathfrak{l}(\mathcal{L}S^4/S^1)  }
      }
    \\
    && && &
     {}_{ \mathllap{\mu^{IIA}}_{F1/D0/D2/D4} } \searrow
    & &
    \swarrow_{\mathrlap{\text{project out} \atop \text{NS5-coefficient}}}
    \\
    &&
    &&
    && \mathfrak{l}( KU\langle 0,6\rangle/B U(1) )
  }
$$

=--

$\,$

Observe that (by [[adjunction]]) this [[double dimensional reduction]] operation

is an [[isomorphism]], hence

* the M2/M5-brane cocycle $\mathbb{R}^{10,1\vert \mathbf{32}} \longrightarrow \mathfrak{l}(S^4)$ in 11d

is equivalent to

* the F1/D$p$/NS5-brane cocycle $\mathbb{R}^{9,1 \vert \mathbf{16} + \overline{\mathbf{16}}} \longrightarrow \mathfrak{l}(\mathcal{L}S^4/S^1)$

Hence this is the M/IIA duality

on super $p$-brane super Lie $n$-algebra cocycles.

$\,$

### IIA/IIB-Duality via T-Duality
 {#TDuality}

The archetypical [[duality in string theory]] is _[[T-duality]]_, which relates the [[F1-brane|F1]]/[[D-brane|Dp]]/[[NS5-brane|NS5]]-[[super p-branes]] of [[type IIA string theory]] on a [[superspacetime]] which is a [[circle]] [[fiber bundle]] over a 9d base to those of [[type IIB string theory]] on a dual circle fibration with the fiber size inverted in string length units. The [[F1-brane|F1]]/[[D-brane|Dp]]-[[super p-brane]] charges on both sides of this [[duality in physics|duality]] take values in [[twisted K-theory]], and hence the mathematical statement here is that dual circle fibrations of this form induce an equivalence in [[twisted K-theory]]. This T-duality equivalence of F1/D$p$-brane charges in [[twisted K-theory]] is known in the literature as "[[topological T-duality]]".

$\,$

We consider now the D-brane cocycles

on type IIA and IIB superspacetime

and dimensional reduce _both_ to their joint 9d base space.

There we discover a hidden [[nLab:duality in string theory|duality]]: _[[nLab:T-duality]]_.

$\,$

+-- {: .num_prop #CoefficientsIsoOnTDuality}
###### Proposition
**([Fiorenza-Sati-Schreiber 16b, prop. 5.1](#FSS16b))**

  The cyclification $\mathfrak{L}\mathfrak{l}(\mathrm{KU}/BU(1))/\mathbb{R}$
  (def. \ref{Cyclification})
  of $\mathfrak{l}(\mathrm{KU}/BU(1))$ (def. \ref{RationalTwistedku})
  has CE-algebra
  $$
    \mathrm{CE}(\mathfrak{L}\mathfrak{l}(\mathrm{KU}/BU(1))/\mathbb{R})
    =
    \left\{
       \array{
         d c_2 = 0\,,\;\;\;\;d \tilde c_2 = 0
         \\
         d h_3 = - c_2 \wedge \tilde c_2
         \\
         d \omega_{2p+2} = h_3 \wedge \omega_{2p} + c_2 \wedge \omega_{2p+1}
         \\
         d \omega_{2p+1} = h_3 \wedge \omega_{2p-1} + \tilde c_2 \wedge \omega_{2p}
       }
    \right\}\;.
  $$
  The cyclification $\mathfrak{L}\mathfrak{l}(\Sigma\mathrm{KU}/BU(1))/\mathbb{R}$
  of $\mathfrak{l}(\Sigma\mathrm{KU}/BU(1))$
  has CE-algebra
  $$
    \mathrm{CE}(\mathfrak{L}\mathfrak{l}(\Sigma\mathrm{KU}/BU(1))/\mathbb{R})
    =
    \left\{
       \begin{array}{l}
         d c_2 = 0 \,, \;\;\;\; d \tilde c_2 = 0
         \\
         d h_3 = - c_2 \wedge \tilde c_2
         \\
         d \omega_{2p+2} = h_3 \wedge \omega_{2p} + \tilde c_2 \wedge \omega_{2p+1}
         \\
         d \omega_{2p+1} = h_3 \wedge \omega_{2p-1} + c_2 \wedge \omega_{2p}
       \end{array}
    \right\}\;.
  $$
  Hence there is an $L_\infty$-isomorphism of the form
  $$
    \phi_T
     \;:\;
    \mathfrak{l}( \mathcal{L}(\mathrm{KU}/BU(1))/S^1 )
      \underoverset{\simeq}{\phantom{AA}\phi_T \phantom{AA}}{\longrightarrow}
    \mathfrak{l}( \mathcal{L}(\Sigma\mathrm{KU}/BU(1))/S^1 )
  $$

  relating the cyclifications of the rational twisted KU-coefficients,
  which is given by
  $$
         c_2 \leftrightarrow \tilde c_2\;, \qquad
         h_3 \mapsto h_3\;, \qquad
         \omega_{p} \mapsto \omega_p
             \;.
  $$

=--

+-- {: .proof}
###### Proof
([skip proof](#HencePostcomposition))

By def. \ref{Cyclification}, as a polynomial algebra the CE-algebra of $\mathfrak{L}\mathfrak{l}(\mathrm{KU}/BU(1))$ is obtained from the CE-algebra of $\mathfrak{l}(\mathrm{KU}/BU(1))$ by adding a shifted copy of each generator. We denote by $\omega_{2p-1}$ the shifted copy of $\omega_{2p}$ and by $-\tilde{c}_2$ the shifted copy of $h_3$. The differential is then defined by
$$
d\omega_{2p+2}=h_3 \wedge \omega_{2p}\;, \qquad d\omega_{2p+1}=h_3 \wedge \omega_{2p-1} + \tilde{c}_2\wedge \omega_{2p}\;, \qquad dh_3=0,\qquad d\tilde{c}_2=0\;.
$$
Next, again by def. \ref{Cyclification}, the CE-algebra of $\mathfrak{L}\mathfrak{l}(\mathrm{KU}/BU(1))/\mathbb{R}$
is obtained by adding a further degree 2 generator $c_2$ and defining the differential as
$$
\array{
d\omega_{2p+2}= h_3 \wedge \omega_{2p} + c_2\wedge \omega_{2p+1}\;, & \qquad
 d\omega_{2p+1}= h_3 \wedge \omega_{2p-1} + \tilde{c}_2\wedge \omega_{2p}\;,
\\
 dc_2=0\;,  \qquad d\tilde{c}_2=0\;,  &
 \qquad \quad  dh_3=  - c_2\wedge \tilde{c}_2\;.
}
$$
The proof for $\mathfrak{L}\mathfrak{l}(\Sigma\mathrm{KU}/BU(1))/\mathbb{R}$ is completely analogous.

=--

$\,$

{#HencePostcomposition} Hence postcomposition with $\phi_T$

sends the doubly dimensionally reduced IIA/B D-brane cocycles

to _some_ other super $L_\infty$-cocycles.

The following theorem says that

indeed it takes them into each other:

$\,$

+-- {: .num_theorem #CocycleTDuality}
###### Theorem
**([Fiorenza-Sati-Schreiber 16b, theorem 5.3](#FSS16b))**

The following [[nLab:commuting diagram|diagram commutes]]

In particular this means that

1. $(\pi^{IIA/IIB}_9)_\ast \left( \mu^{IIA/IIB}_{F1} \right) = - c_2^{IIA/IIB}$.

1. $(\pi_9^{IIA})_\ast - e_9^{IIA} \wedge (-)\vert_{8+1} \;\colon\; C^{IIA} \mapsto C^{IIB}$.

=--

$\,$

The first of these two conditions is

the super $L_\infty$-version of the axiom for

_[[nLab:topological T-duality]]_ due to [Bouwknegt-Evslin-Mathai 04](https://ncatlab.org/nlab/show/topological%20T-duality#BouwknegtEvslinMathai04).

$\,$

$\,$

To understand the second condition

pass to the correspondence super $L_\infty$-algebra.

$\,$

+-- {: .num_prop #CorrespondenceForTDuality}
###### Proposition
**([Fiorenza-Sati-Schreiber 16b, prop. 6.2](#FSS16b))**

 We have a diagram of super $L_\infty$-algebras of the following form

such that  on the classifying 3-cocycles

$\nu$ is given by the _Poincaré form_
$$
  \mathcal{P}
    :=
  e_9^{\mathrm{IIA}} \wedge e_9^{\mathrm{IIB}}
$$
as
$$
  p_B^\ast (\mu_{{}_{F1}}^{\mathrm{IIB}} )- p_A^\ast (\mu_{{}_{F1}}^{\mathrm{IIA}})
    =
   d  \mathcal{P}
   \,.
$$

=--

$\,$

This now is the super $L_\infty$-version of the

refined formulation of [[nLab:topological T-duality]]

due to [Bunke-Schick 04](https://ncatlab.org/nlab/show/topological+T-duality#BunkeSchick04):

$\,$

+-- {: .num_prop #PullPushIso}
###### Proposition
**([Fiorenza-Sati-Schreiber 16b, prop. 64](#FSS16b))**

The [[nLab:integral transform]] of pull-push through this [[nLab:correspondence]] is an
[[nLab:isomorphism]]

  $$
    (\pi_9^{\mathrm{IIA}})_\ast \circ \nu^\ast \circ (\pi_9^{\mathrm{IIB}})^\ast
      \;:\;
    H_{\mu_{F1}^{\mathrm{IIA}}} \left( \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}, \mathfrak{l}(\mathrm{KU})  \right)
      \longrightarrow
    H_{\mu_{F1}^{\mathrm{IIB}}} \left( \mathbb{R}^{9,1\vert \mathbf{16} + {\mathbf{16}}}, \mathfrak{l}(\Sigma \mathrm{KU})   \right)
  $$

 Moreover, it identifies the type IIA D-brane cocycles with those
  of type IIB (from def. \ref{TheDBraneCocycles}), as in Theorem \ref{CocycleTDuality}:
  $$
    \begin{aligned}
      \exp(-f_2^{\mathrm{IIB}}) \wedge  C^{\mathrm{IIB}}
      & =
    (\pi_9^{\mathrm{IIA}})_\ast \circ \nu^\ast \circ (\pi_9^{\mathrm{IIB}})^\ast
      \left(
        \exp(-f_2^{\mathrm{IIA}}) \wedge C^{\mathrm{IIA}}
      \right)
      \\
      & =
      (\pi_9^{\mathrm{IIA}})_\ast \left(
      \exp(\mathcal{P}) \wedge (\pi_9^{\mathrm{IIB}})^\ast \left( \exp(-f^{\mathrm{IIA}}_2) \wedge C^{\mathrm{IIA}} \right)
    \right)
    \end{aligned}
    \,,
  $$

=--

+-- {: .proof}
###### Proof

One checks that

$$
  (\pi_9^{\mathrm{IIA}})_\ast \left(
      \exp(\mathcal{P}) \wedge (\pi_9^{\mathrm{IIB}})^\ast \left( - \right)
  \right)
  =
  (\pi_9^{\mathrm{IIA}})_\ast(-) - e_9^{IIA} \wedge (- )\vert|_{8+1}
  \,.
$$

With this the statement follows from theorem \ref{CocycleTDuality}, item 2.

=--

$\,$

$\,$

Prop. \ref{PullPushIso} is the super $L_\infty$-algebraic analog of the [Bunke-Schick 04, theorem 3.13](https://ncatlab.org/nlab/show/topological+T-duality#BunkeSchick04).

In fact on CE-algebras this is the _Buscher rules for RR-fields_

also known as  _Hori's formula_  ([Hori 99](https://ncatlab.org/nlab/show/T-duality#Hori99)).

$\,$

To see why the above [[nLab:correspondence]] as in [Bunke-Schick 04](https://ncatlab.org/nlab/show/topological+T-duality#BunkeSchick04)

follows from the [[nLab:double dimensional reduction]] isomorphism $\phi_T$

it is useful to break up the doubly dimensionally reduced cocycles $ \mathcal{L}( \, \mu_{F1,Dp}^{IIA/B} \, ) $

into the part for the fiber bundle and the part for the K-cocycles:

$\,$

+-- {: .num_defn #Lie2AlgebraTDuality}
###### Definition

The _[[nLab:T-duality 2-group|delooped T-duality Lie 2-algebra]]_ $B \mathcal{T}_1$

is the [[nLab:homotopy fiber]] of the [[nLab:cup product|cup square]] of the [[nLab:first Chern class]]

$$
  B \mathcal{T}_1
    \overset{}{\longrightarrow}
  B \mathbb{R} \oplus B \mathbb{R}
    \overset{\phantom{A}c_1 \cup \tilde c_1 \phantom{A}}{\longrightarrow}
  B^3 \mathbb{R}
$$

hence by prop. \ref{HigherCentralExtension}

$$
  CE( B \mathcal{T}_1 )
    \;=\;
  \left\{
    \array{
       d c_2 = 0, \;\; d \tilde c_2 = 0
       \\
       d h_3 = - c_2 \wedge \tilde c_2
    }
  \right\}
  \,.
$$

=--

$\,$

+-- {: .num_prop}
###### Proposition
**([FSS 16b, prop. 7.3](#FSS16b))**

The dimensionally reduced twistd K-theory coefficients from prop. \ref{CoefficientsIsoOnTDuality}

sit in a [[nLab:homotopy fiber sequence]] of the form

$$
  \array{
    \mathfrak{l}(\, KU \oplus \Sigma KU \,)
    &\longrightarrow&
    \mathfrak{l}(\, \mathcal{L}( KU/BU(1))/S 1 \,)
    \\
    &&
    \downarrow
    \\
    && B \mathcal{T}_1
  }
$$

exhibiting (via [NSS 12](#NSS12)) an [[nLab:∞-action]]

of the [[nLab:T-duality 2-group|T-duality Lie 2-algebra]] from def. \ref{Lie2AlgebraTDuality}

on $\mathfrak{l}(\, KU \oplus \Sigma KU \,)$.

=--

$\,$

+-- {: .num_prop #TdualityCorrespondenceFromHomotopyFiber}
###### Proposition
**([FSS 16b, prop. 7.5](#FSS16b))**

The [[nLab:homotopy fiber]] of the cyclified IIA/B cocycles (theorem \ref{CocycleTDuality})

projected to the [[nLab:T-duality 2-group|T-duality Lie 2-algebra]] $B \mathcal{T}_1$ (def. \ref{Lie2AlgebraTDuality})

are the super $L_\infty$-algeberas $p_{A}^\ast \widehat{ \mathbb{R}^{9,1 \vert \mathbf{16} + \overline{\mathbf{16}}} }$ and $p_B^\ast( \widehat{ \mathbb{R}^{9,1\vert \mathbf{16} + \mathbf{16}} } )$ from prop. \ref{CorrespondenceForTDuality}

hence the [Bunke-Schick 04](#BunkeSchick04)-type correspondence

follows from theorem \ref{CocycleTDuality}

via $\infty$-functoriality of homotopy fibers:

=--

$\,$

Prop. \ref{TdualityCorrespondenceFromHomotopyFiber} implies that we may study T-duality

in terms of [[nLab:principal 2-bundles]] for the [[nLab:T-duality 2-group]]

over 9d super-manifolds locally modeled on $\mathbb{R}^{8,1\vert \mathbf{16} + \mathbf{16}}$.

This is the [[nLab:supergeometry|supergeometric]] refinement

of the 2-bundle perspective on [[nLab:T-folds]] due to ([Nikolaus 14](https://ncatlab.org/nlab/show/T-fold#Nikolaus14)).

$\,$

### IIB/F

(...)

[FSS16b, section 7](#FSS16b)

(...)

$\,$

$\,$

## References

The interpretation of [[Fierz identities]] as relations satisfied by [[Clebsch-Gordan coefficients]] in the [[representation ring]] of the [[spin group]] originates in

* {#DAuriaFreMainaRegge82} [[Riccardo D'Auria]], [[Pietro Fré]],  E. Maina, [[Tullio Regge]]  _A New Group Theoretical Technique for the Analysis of Bianchi Identities and Its Application to the Auxiliary Field Problem of $D=5$ Supergravity_, Annals Phys. 139 (1982) 93 ([doi:10.1016/0003-4916(82)90007-0](http://dx.doi.org/10.1016/0003-4916(82)90007-0), [spire](http://inspirehep.net/record/167640/))

where it was applied to $Spin(4,1)$, relevant in [[5-dimensional supergravity]].

By this method the Fierz identities for $Spin(9,1)$ (relevant in [[heterotic supergravity]] and [[type II supergravity]]) are discussed in

* {#DAuriaFre82a} [[Riccardo D'Auria]], [[Pietro Fré]], _Geometric Structure of $N=1, D=10$ and $N=4, D=4$ Super Yang-Mills Theory_, Nucl. Phys. B196 (1982) 205 ([spire](http://inspirehep.net/record/167639))

see also appendix C of

* [[José Figueroa-O'Farrill]], Emily Hackett-Jones, George Moutsopoulos, _The Killing superalgebra of ten-dimensional supergravity backgrounds_, Class.Quant.Grav.24:3291-3308,2007 ([arXiv:hep-th/0703192](http://arxiv.org/abs/hep-th/0703192))

and the Fierz identities for $Spin(10,1)$ (relevant in [[11-dimensional supergravity]]) were tabulated in

* {#DAuriaFre82b} [[Riccardo D'Auria]], [[Pietro Fré]], pages 12, 13 of _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) ([[GeometricSupergravityErrata.pdf:file]])

see also

* S. Naito, K. Osada, T. Fukui, _Fierz Identities and Invariance of Eleven-dimensional Supergravity Action_, Phys.Rev. D34 (1986) 536-552 ([spire](http://inspirehep.net/record/236376/?ln=de))

A textbook account of the representation ring method and summary of these results is in

* {#CDF} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter II.8 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

See also

* C. C. Nishi, _Simple derivation of general Fierz-type identities_,  	Am. J. Phys. 73 (2005) 1160-1163 ([arXiv:hep-ph/0412245](http://arxiv.org/abs/hep-ph/0412245))

From the point of view of [[division algebras and supersymmetry]]
the Fierz identities that give the vanishing of trilinear and of quadratic terms in spinors in certain dimensions are discussed in

* [[John Huerta]], section 2.4 _Division Algebras, Supersymmetry and Higher Gauge Theory_ ([arXiv:1106.3385](http://arxiv.org/abs/1106.3385))

The recognition of some [[Fierz identities]] as cocycle conditions defining the [[higher WZW terms]] of the [[super p-branes]] is due to

* {#HenneauxMezincescu85} [[Marc Henneaux]], Luca Mezincescu, _A Sigma Model Interpretation of Green-Schwarz Covariant Superstring Action_, Phys.Lett. B152 (1985) 340 ([web](http://inspirehep.net/record/15922?ln=en))

* {#BergshoeffSezginTownsend87}  [[Eric Bergshoeff]], [[Ergin Sezgin]], [[Paul Townsend]], _Supermembranes and eleven dimensional supergravity_, Phys.Lett. B189 (1987) 75-78, In [[Mike Duff]],  (ed.), _[[The World in Eleven Dimensions]]_ 69-72 ([pdf](http://streaming.ictp.trieste.it/preprints/P/87/010.pdf), [spire](http://inspirehep.net/record/248230?ln=en))

* {#AETW87} Anna Achúcarro, [[Jonathan Evans]], [[Paul Townsend]], [[David Wiltshire]], _Super $p$-Branes_, Phys. Lett. B **198** (1987) 441 ([spire](http://inspirehep.net/record/22286?ln=en), [pdf](https://ir.canterbury.ac.nz/xmlui/bitstream/handle/10092/9041/12616571_pbrane.pdf?sequence=1&isAllowed=y))

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], Mario Tonin, _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. 78 (1997) 4332-4334 ([arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149))

The identification of the [[worldvolume]] [[conformal field theories]] on solitonic branes ([[black branes]])
as the [[perturbation theory]] of fundamenal [[super p-branes]] about asymptotic singular loci in [[BPS state]]
[[supergravity]] solutions is due to

* {#ClausKalloshProeyen97} Piet Claus, [[Renata Kallosh]], [[Antoine Van Proeyen]], _M 5-brane and superconformal $(0,2)$ tensor multiplet in 6 dimensions_, Nucl.Phys. B518 (1998) 117-150 ([arXiv:hep-th/9711161](http://arxiv.org/abs/hep-th/9711161))

* {#ClausKalloshKumarTownsend98} Piet Claus, [[Renata Kallosh]], J. Kumar, [[Paul Townsend]], [[Antoine Van Proeyen]], _Conformal Theory of M2, D3, M5 and 'D1+D5' Branes_, JHEP 9806 (1998) 004 ([arXiv:hep-th/9801206](http://arxiv.org/abs/hep-th/9801206))

* {#AFFFTT98} Gianguido Dall'Agata, Davide Fabbri, Christophe Fraser, [[Pietro Fré]], Piet Termonia, Mario Trigiante, _The $Osp(8|4)$ singleton action from the supermembrane_, Nucl.Phys.B542:157-194, 1999 ([arXiv:hep-th/9807115](http://arxiv.org/abs/hep-th/9807115))

* {#PastiSorokinTonin99} [[Paolo Pasti]], [[Dmitri Sorokin]], Mario Tonin, _Branes in Super-AdS Backgrounds and Superconformal Theories_, Talk given by D.S. at the International Workshop ``Supersymmetry and Quantum Symmetries'', JINR, Dubna, Russia, July 26-31, 1999 ([arXiv:hep-th/9912076](http://arxiv.org/abs/hep-th/9912076))

The bosonic [[string Lie 2-algebra]] is discussed in the appendix of

* {#BCSS05} [[John Baez]], Alissa Crans, [[Urs Schreiber]], [[Danny Stevenson]], _From loop groups to 2-groups_, Homology Homotopy Appl. Volume 9, Number 2 (2007), 101-135. ([arXiv:math/0504123](https://arxiv.org/abs/math/0504123))

Its role in [[Green-Schwarz anomaly cancellation]] is discussed in

* {#SatiSchreiberStasheff12} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Twisted Differential String and Fivebrane Structures]]_, Communications in Mathematical Physics, Volume 315, Issue 1 (2012)  ([arXiv:0910.4001](http://arxiv.org/abs/0910.4001))

and its role in the flux quantization of the [[supergravity C-field]] in

* {#FiorenzaSatiSchreiber12a} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The moduli 3-stack of the C-field]]_, Communications in Mathematical Physics, Volume 333, Issue 1 (2015), Page 117-151 ([arXiv:1202.2455](http://arxiv.org/abs/1202.2455))

and

* {#FiorenzaSatiSchreiber12b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane|Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory]]_, Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014)  ([arXiv:1201.5277](http://arxiv.org/abs/1201.5277))

Our discussion of [[L-infinity algebra cohomology]] is due to

* {#SatiSchreiberStasheff08} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _L-infinity algebra connections and applications to String- and Chern-Simons n-transport_, in Quantum Field Theory, Birkhäuser (2009) 303-424 ([arXiv:0801.3480](http://arxiv.org/abs/0801.3480))

The observation of the [[schreiber:brane bouquet]] in super $L_\infty$-algebra and the general construction of [[higher WZW terms]] from higher $L_\infty$-cocycles is due to

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]],   _[[schreiber:The brane bouquet|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](http://arxiv.org/abs/1308.5264))

see also

* {#HSS19} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], *Real ADE-equivariant (co)homotopy and Super M-branes*, Comm .in Math. Phys. **371** (2019) 425 [[doi:10.1007/s00220-019-03442-3](https://doi.org/10.1007/s00220-019-03442-3), [arXiv:1805.05987](https://arxiv.org/abs/1805.05987)]

The homotopy-[[descent]] of the [[M5-brane]] cocycle and of the type IIA [[D-brane]] cocycles is due to

* {#FSS15} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The WZW term of the M5-brane|The WZW term of the M5-brane and differential cohomotopy]]_, J. Math. Phys. 56, 102301 (2015) ([arXiv:1506.07557](https://arxiv.org/abs/1506.07557))

* {#FSS16a} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Rational sphere valued supercocycles in M-theory]]_, Journal of Geometry and Physics, Volume 114 (2017) ([arXiv:1606.03206](http://arxiv.org/abs/1606.03206))

The derivation of supersymmetric [[topological T-duality]], rationally, and of the higher super Cartan geometry for [[super T-folds]] is due to

* {#FSS16b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]_, ATMP **22** 5 (2018) ([arXiv:1611.06536](https://arxiv.org/abs/1611.06536), [doi:10.4310/ATMP.2018.v22.n5.a3](http://dx.doi.org/10.4310/ATMP.2018.v22.n5.a3))

and the mathematical formulation of [[double dimensional reduction]] that this uses is further discussed in

* {#BMSS18} [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], Section 2.2 of:  _[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory]]_, Communications in Mathematical Physics 371: 197 (2019) ([arXiv:1806.01115](https://arxiv.org/abs/1806.01115), [doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4))

The derivation of the process of higher invariant extensions that leads from the [[superpoint]] to [[11-dimensional supergravity]]:

* {#MTheoryFromTheSuperpoint} [[John Huerta]], [[Urs Schreiber]], _[[schreiber:M-Theory from the Superpoint]]_, Letters in Mathematical Physics **108** (2018) 2695–2727   ([arXiv:1702.01774](https://arxiv.org/abs/1702.01774), [doi:10.1007/s11005-018-1110-z](https://doi.org/10.1007/s11005-018-1110-z))

General discussion of [[twisted cohomology]] is in:

* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- General theory]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015) ([arXiv:1207.0248](http://arxiv.org/abs/1207.0248))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:The Character Map in Twisted Non-Abelian Cohomology]]* ([arXiv:2009.11909](https://arxiv.org/abs/2009.11909))

Foundations and parts of the story is in:

* {#dcct} [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_, Thesis

[[!redirects geometry of physics - fundamental super p-branes]]