Majorana dimer code

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#### Quantum systems
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[[!include quantum systems -- contents]]
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#Contents#
* table of contents
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## Idea

A class of [[quantum error correcting codes]] whose [[code subspaces]] are the [[images]] of [[linear maps]] given by [[tensor networks]] of the shape of [[round chord diagrams]] with chords acting as [[Majorana dimer|Majorana dimers]].

The Majorana dimer code subsumes the [[HaPPY code]]:

\begin{imagefromfile}
    "file_name": "HaPPYCodesAsDimerCode.jpg",
    "width": 680,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [JGPE 19](#JGPE19)"
\end{imagefromfile}

and makes the nature of [[holographic entanglement entropy]] particularly manifest (see [there](holographic+entanglement+entropy#ChordDiagramRepresentation) for more).

## Related concepts

* [[chord diagram]]

## References

### General

Due to:

* {#JGPE19} [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], *Majorana dimers and holographic quantum error-correcting code*, Phys. Rev. Research 1, 033079 (2019) ([arXiv:1905.03268](https://arxiv.org/abs/1905.03268))

based on:

* [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], *Holography and criticality in matchgate tensor networks*, Science Advances 5, eaaw0092 (2019) ([arXiv:1711.03109](https://arxiv.org/abs/1711.03109), [doi:10.1126/sciadv.aaw0092](https://advances.sciencemag.org/content/5/8/eaaw0092))

Review in:

* [[Alexander Jahn]], [[Jens Eisert]], _Holographic tensor network models and quantum error correction: A topical review_ ([arXiv:2102.02619](https://arxiv.org/abs/2102.02619))

See also:

* {#KahnZimborasEisert19} [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Central charges of aperiodic holographic tensor network models_, Phys. Rev. A 102, 042407 ([arXiv:1911.03485](https://arxiv.org/abs/1911.03485))

* [[Han Yan]], *Geodesic string condensation from symmetric tensor gauge theory: a unifying framework of holographic toy models*, Phys. Rev. B 102, 161119 (2020) ([arXiv:1911.01007](https://arxiv.org/abs/1911.01007))

[[!include weight systems on chord diagrams in physics]]

[[!redirects Majorana dimer codes]]


Majorana spinor

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### Context
#### Representation theory
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[[!include representation theory - contents]]#### Representation theory
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[[!include representation theory - contents]]
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#### Higher spin geometry
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[[!include higher spin geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _Majorana spin representation_ is essentially a _real_ [[spin representation]] (see at _[spin representation -- Real representations](spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature)_) but regarded as a _complex_ [[spin representation]] equipped with [[real structure]] (recalled as def. \ref{RealStructureOnLinearRepresentation} below).

Accordingly a _Majorana spinor_ or _Majorana fermion_ is a [[spinor]]/[[fermion]] corresponding to such a representation under [[Wigner classification]]. None of the particles in the [[standard model of particle physics]] except possibly the [[neutrinos]] are Majorana fermions (for neutrinos this remains open). The relevance of Majorana representations is that these appear in [[supersymmetry]], constituting for instance the odd-graded components of [[super-Minkowski spacetimes]]. See remark \ref{SuperPoincareOutlook} below.

The terminology _Majorana spinor_ originates in and is standard in the [[physics]] literature, where it usually refers to the explicit expression of the reality condition in terms of chosen basis components. With standard conventions understood (see prop. \ref{CliffordAlgebraRepresentation} below), then a complex [[spinor]] $\psi$ for $Spin(d-1,1)$, regarded as an element of $\mathbb{C}^{2^\nu}$ (with $d = 2 \nu, 2\nu+1$) is a Majorana spinor if it satisfies the condition

$$
  \psi^t C = \psi^\dagger \Gamma_0
  \,,
$$

where $(-)^T$ denotes forming [[transpose matrices]], $(-)^\dagger = \overline{(-)}^T$ denotes forming [[hermitian adjoint]] and where $C$ is the [[charge conjugation matrix]]. This says that the _Majorana conjugate_ (def. \ref{MajoranaConjugation} below) of $\psi$ (the left hand side) coincides with the "Dirac conjugate" (def. \ref{DiracConjugate} below) of $\psi$ (the right hand side). Equivalently this means that (e.g. [Castellani-D'Auria-Fré, (II.7.22)](#CastellaniDAuriaFre))

$$
  \psi = J(\psi) \coloneqq C \Gamma_0^T \psi^\ast
  \,,
$$

where $J(-)$ is the given real structure (prop. \ref{MajoranaConjugationIsRealStructure} below). See prop. \ref{TheMajoranaConditionInComponents} below.

In some dimensions there are no complex spin representations with real structure, but there may be those with [[quaternionic structure]]. The corresponding physics jargon then is _symplectic Majorana spinor_.

## Definition

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

Let $\rho \colon Spin(s,t) \longrightarrow GL_{\mathbb{C}}(V)$ be a [[unitary representation]] of a [[spin group]]. Then $\rho$ is called _Majorana_ if it admits a real structure $J$ (def. \ref{RealStructureOnLinearRepresentation}) and _symplectic Majorana_ if it admits a [[quaternionic structure]] $J$ (def. \ref{RealStructureOnLinearRepresentation}). An element $\psi \in V$ is called a _Majorana spinor_ if $J(\psi) = \psi$.

=--

## In components
 {#InComponents}

We work out in detail what def. \ref{MajoranaSpinorGeneral} comes down to in components (i.e. in terms of choices of [[linear bases]]), using standard notation and conventions from the physics literature (e.g. [Castellani-D'Auria-Fré](#CastellaniDAuriaFre)).

### Conventions and Notation

In the following we use standard notation for operations on [[matrices]] with entries in the [[complex numbers]] (and of course these matrices may in particular be complex row/column vectors, which may in particular be single complex numbers):

* $(-)^\ast$ -- componentwise [[complex conjugation]];

* $(-)^T$ -- [[transpose matrix]]

* $(-)^\dagger \coloneqq ((-)^\ast)^T = ((-)^T)^\ast$

* $A B$ for the [[matrix product]] of two matrices $A$ and $B$.

We will be discussing three different pairing operations on complex column vectors $\psi_1, \psi_2 \in \mathbb{C}^\nu$:

* $\psi_1^\dagger \psi_2$ -- the standard [[hermitian form]] on $\mathbb{C}^\nu$, this will play a purely auxiliary role.

* $\langle \psi_1,\psi_2\rangle \coloneqq \overline{\psi}_1 \psi_2 \coloneqq \psi_1^\dagger \Gamma_0 \psi_2$ -- the _Dirac pairing_, this will be the alternative [[hermitian form]] with respect to which the [[spin representation]] below is a [[unitary representation]];

* $(\psi_1,\psi_2) \coloneqq \psi_1^T C \psi_2$ -- the _Majorana pairing_ (for $C$ the [[charge conjugation matrix]]), this turns out to coincide with the Dirac pairing above _if_ $\psi_1$ is a Majorana spinor.

Then we use the following conventions on spacetime signature and the correspondig [[Clifford algebra]]:

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

We write $\mathbb{R}^{s,t}$ for the real [[vector space]] $\mathbb{R}^{s+t}$ of [[dimension]] $d = s + t$ equipped with the standard [[quadratic form]] $q$ of [[signature of a quadratic form|signature]] $(t,s)$ ("time", "space"), i.e.

$$
  q(\vec x)
    \coloneqq
  (x^1)^2
    +
    \cdots
    +
  (x^s)^2
    -
  (x^{s+1})^2
    -
   \cdots
  -
  (x^{s+t})^2
  \,.
$$

Hence the corresponding [[metric]] is

$$
  \eta = (\eta_{a b}) \coloneqq diag(\underset{t}{\underbrace{+1 , \cdots, +1}}, \underset{s}{\underbrace{-1, \cdots, -1}})
  \,.
$$

The real [[Clifford algebra]] $Cl(s,t)$ associated with this [[inner product space]] is the $\mathbb{R}$-[[associative algebra|algebra]] [[generators and relations|generated]] from elements $\{\Gamma_a\}_{0 = 1}^{s+t-1}$ subject to the [[generators and relations|relation]]

$$
  \Gamma_a \Gamma_b + \Gamma_b \Gamma_a = 2 \eta_{a b}
  \;\;\;\;
  \forall a,b \in \{0,1,\cdots, t+s-1\}
  \,.
$$

For $n$-[[tuples]] $(a_i)_{i = 1}^n$ of indices we write

$$
  \Gamma_{a_1 \cdots a_n}
  \coloneqq
  \Gamma_{[a_1}
    \cdots
  \Gamma_{a_2]}
   \coloneqq
  \frac{1}{n!} \underset{\sigma}{\sum}
   (-1)^{\vert \sigma\vert}
  \Gamma_{a_{\sigma_1}}
   \cdots
  \Gamma_{a_{\sigma_n}}
$$

for the skew-symmetrized product of Clifford generators with these indices. In partcular if all the $a_i$ are pairwise distinct, then this is simply the plain product of generators

$$
  \Gamma_{a_1 \cdots a_n}
  =
  \Gamma_{a_1} \cdots \Gamma_{a_n}
  \;\;\;
  \text{if}
  \;
  \underset{i,j}{\forall} (a_i \neq a_j)
  \,.
$$

=--

Indices are raised with $\eta^{-1} = (\eta^{a b})$ (which of course as a [[matrix]] coincides with $(\eta_{a b})$)

$$
  \Gamma^a \coloneqq \eta^{a b} \Gamma_b
$$

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

The case $t = 1$ is that of **Lorentzian signature**.

In this case the single [[timelike]] Clifford genrator is $\Gamma_0$ and the remaining spatial Clifford generators are $\Gamma_1, \Gamma_2, \cdots, \Gamma_{d-1}$. So then

* $\Gamma^0 = \Gamma_0$ and $\Gamma_0^2 = + 1$;

* $\Gamma^a = - \Gamma_a$ and $\Gamma_a^2 = -1$ for $a \in \{1,\cdots, d-1\}$.

=--

### Dirac and Weyl representations
 {#DiracAndWeylRepresentations}

The following is a standard convention for the complex representation of the Clifford algebra for $\mathbb{R}^{s,1}$ ([Castellani-D'Auria-Fré, (II.7.1)](#CastellaniDAuriaFre)):

+-- {: .num_prop #CliffordAlgebraRepresentation}
###### Proposition
(**Dirac representation**)

Let $t = 1$ (Lorentzian signature, def. \ref{LorentzianSignature}) and let

$$
  d = s + 1 \in \{ 2\nu, 2 \nu + 1 \}
  \;\;\;\;
  \text{for}\, \nu \in \mathbb{N}\,,\; d\geq 4
  \,.
$$

Then there is a choice of complex linear representation of the [[Clifford algebra]] $Cl(s,1)$ (def. \ref{MinkowskiSpacetime}) on the [[complex vector space]]

$$
  V \coloneqq \mathbb{C}^{2^\nu}
$$

such that

1. $\Gamma_{0}$ is [[hermitian]]

1. $\Gamma_{spatial}$ is [[anti-hermitian]].

Moreover, the pairing

$$
  \langle -,-\rangle \coloneqq (-)^\dagger \Gamma_0 (-)
    \;\colon\;
  V \times V
    \longrightarrow
  \mathbb{C}
$$

is a [[hermitian form]] (def. \ref{HermitianForms}) with respect to which the resulting representation of the [[spin group]] $\exp(\omega^{a b} \Gamma_{a b})$ is [[unitary representation|unitary]]:

$$
  \Gamma_0^{-1} \exp(\omega^{a b} \Gamma_{a b})^{\dagger} \Gamma_0
  =
  \exp(\omega^{a b} \Gamma_{a b})^{-1}
  \,.
$$

These representations are called the **[[Dirac representations]]**, their elements are called **Dirac spinors**.

=--

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

In the case $d = 4$ consider the [[Pauli matrices]] $\{\sigma_{a}\}_{a = 0}^3$, defined by

$$
  \sigma_a x^a
  \coloneqq
  \left(
    \array{
       x^0 + x^1 & x^2 + i x^3
       \\
       x^2 - i x^3 & x^0 - x^1
    }
  \right)
  \,.
$$

Then a Clifford representation as claimed is given by setting

$$
  \Gamma_0
     \coloneqq
  \left(
    \array{
      0 & id
      \\
      id & 0
    }
  \right)
$$

$$
  \Gamma_a
    \coloneqq
  \left(
    \array{
      0 & \sigma_a
      \\
      -\sigma_a & 0
    }
  \right)
  \,.
$$

From $d = 4$ we proceed to higher dimension by [[induction]], applying the following two steps:

**odd dimensions**

Suppose a Clifford representation $\{\gamma_a\}$ as claimed has been constructed in even dimension $d = 2 \nu$.

Then a Clifford representation in dimension $d = 2 \nu + 1$ is given by taking

$$
  \Gamma_a
   \coloneqq
  \left\{
    \array{
      \gamma_a & \vert \; a \leq d - 2
      \\
      \epsilon \gamma_0 \gamma_1 \cdots \gamma_{d-2}
       & \vert\; a = d-1
    }
  \right.
$$

where

$$
  \epsilon
     =
   \left\{
     \array{
       1 & \vert \; \nu \, \text{odd}
       \\
       i & \vert \; \nu \, \text{even}
     }
   \right.
   \,.
$$

**even dimensions**

Suppose a Clifford representation $\{\gamma_a\}$ as claimed has been constructed in even dimension $d = 2 \nu$.

Then a corresponding representation in dimension $d+2$ is given by setting

$$
  \Gamma_{a \lt d}
    \coloneqq
  \left(
    \array{
      0 & \gamma_a
      \\
      \gamma_a & 0
    }
  \right)
  \;\;\,,
  \;\;\;
  \Gamma_{d}
    =
  \left(
    \array{
      0 & id
      \\
      -id & 0
    }
  \right)
  \;\;\,,
  \;\;\;
  \Gamma_{d+1}
    =
  \left(
    \array{
       i \mathrm{id} & 0
       \\
       0 & -i \mathrm{id}
    }
  \right)
  \,.
$$

Finally regarding the statement that this gives a [[unitary representation]]:

That $\langle -,-\rangle \coloneqq (-)^\dagger \Gamma_0 (-)$ is a [[hermitian form]] follows since $\Gamma_0$ obtained by the above construction is a [[hermitian matrix]].

Let $a,b \in \{1, \cdots, d-1\}$ be spacelike and distinct indices. Then by the above we have

$$
  \begin{aligned}
    \Gamma_0^{-1} (\Gamma_a \Gamma_b)^\dagger \Gamma_0
    & =
    \Gamma_0^{-1} \Gamma_0 (\Gamma_b^\dagger \Gamma_a^\dagger)
    \\
    & = (-\Gamma_b) (-\Gamma_a)
    \\
    & = \Gamma_b \Gamma_a
    \\
    & = - \Gamma_a \Gamma_b
  \end{aligned}
$$

and

$$
  \begin{aligned}
    \Gamma_0^{-1} (\Gamma_0 \Gamma_a)^\dagger
    & =
    - \Gamma_0^{-1} \Gamma_0 \Gamma_a^\dagger \Gamma_0^\dagger
    \\
    & =
      - (- \Gamma_a) (\Gamma_0)
    \\
    & = \Gamma_a \Gamma_0
    \\
    & = - \Gamma_0 \Gamma_a
  \end{aligned}
  \,.
$$

This means that the exponent of $\exp(\omega^{a b} \Gamma_a \Gamma_b)$ is an [[anti-hermitian matrix]], hence that exponential is a [[unitary operator]].

=--

+-- {: .num_defn #WeylRepresentation}
###### Definition
(**Weyl representation**)

Since by prop. \ref{CliffordAlgebraRepresentation} the Dirac representations in dimensions $d = 2\nu$ and $d+1 = 2\nu+1$ have the same underlying complex vector space, the element

$$
  \Gamma_{d}
    \propto
  \Gamma_0 \Gamma_1 \cdots \Gamma_{d-1}
$$

acts $Spin(d-1,1)$-invariantly on the representation space of
the Dirac $Spin(d-1,1)$-representation for even $d$.

Moreover, since $\Gamma_0 \Gamma_1 \cdots \Gamma_{d-1}$ squares to $\pm 1$, there is a choice of complex prefactor $c$ such that

$$
  \Gamma_{d+1} \coloneqq c \Gamma_0 \Gamma_1 \cdots \Gamma_{d-1}
$$

squares to +1. This is called the **chirality operator**.

(The notation $\Gamma_{d+1}$ for this operator originates from times when only $d = 4$ was considered. Clearly this notation has its pitfalls when various $d$ are considered, but nevertheless it is commonly used this way e.g. [Castellani-D'Auria-Fré, section (II.7.11) and top of p. 523](#CastellaniDAuriaFre)).

Therefore this representation decomposes as a [[direct sum]]

$$
  V = V_+ \oplus V_-
$$

of the [[eigenspaces]] $V_{\pm}$ of the chirality operator, respectively. These $V_{\pm}$ are called the two **[[Weyl representations]]** of $Spin(d-1,1)$. An element of these is called a **chiral spinor** ("left handed", "right handed", respectively).

=--

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

For a Clifford algebra representation on $\mathbb{C}^\nu$ as in prop. \ref{CliffordAlgebraRepresentation}, we write

$$
  \overline{(-)}
  \coloneqq
  (-)^\dagger \Gamma_0
  \;\colon\;
  Mat_{\nu \times 1}(\mathbb{C})
    \longrightarrow
  Mat(1 \times \nu)(\mathbb{C})
$$

for the map from complex column vectors to complex row vectors which is hermitian congugation $(-)^\dagger = ((-)^\ast)^T$ followed by matrix multiplication with $\Gamma_0$ from the right.

This operation is called **Dirac conjugation**.

In terms of this the [[hermitian form]] from prop. \ref{CliffordAlgebraRepresentation} (Dirac pairing) reads

$$
  \langle -,-\rangle = \overline{(-)}(-)
  \,.
$$

=--

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

The operator adjoint $\overline{A}$ of a $\nu \times \nu$-matrix $A$ with respect to the Dirac pairing of def. \ref{DiracConjugate}, characterized by

$$
  \langle A (-), (-)\rangle
  =
  \langle   - , \overline{A} -\rangle
  \;\;\;\text{and} \;\;\;
  \langle  -, A -\rangle
  =
  \langle \overline{A} - ,  -\rangle
$$

is given by

$$
  \overline{A} = \Gamma_0^{-1} A^\dagger \Gamma_0
  \,.
$$

All the Clifford generators from prop. \ref{CliffordAlgebraRepresentation} are Dirac self-conjugate in that

$$
  \overline{\Gamma}_a = \Gamma_a
  \,.
$$

=--

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

For the first claim consider

$$
  \begin{aligned}
    \langle A \psi_1,  \psi_2\rangle
    & =
    \psi_1^\dagger A^\dagger \Gamma_0  \psi_2
    \\
    & =
    \psi_1^\dagger \Gamma_0 (\Gamma_0^{-1} A^\dagger \Gamma_0) \psi_2
    \\
    & =
    \langle \psi_1, (\Gamma_0^{-1} A \Gamma_0)\psi_2\rangle
  \end{aligned}
  \,.
$$

and

$$
  \begin{aligned}
    \langle \psi_1, A \psi_2\rangle
    & =
    \psi_1^\dagger \Gamma_0 A \psi_2
    \\
    & =
    \psi_1^\dagger \Gamma_0 A \Gamma_0^{-1} \Gamma_0 \psi_2
    \\
    & =
    ( (\Gamma_0^{-1})^\dagger A^\dagger (\Gamma_0)^\dagger \psi_1 )^\dagger \Gamma_0 \psi_2
    \\
    & =
    ( \Gamma_0^{-1} A^\dagger \Gamma_0 \psi_1 )^\dagger \Gamma_0 \psi_2
    \\
    &=
    \langle \overline{A} \psi_1, \psi_2\rangle
  \end{aligned}
  \,,
$$

where we used that $\Gamma_0^{-1} = \Gamma_0$ (by def. \ref{LorentzianSignature}) and $\Gamma_0^\dagger = \Gamma_0$ (by prop. \ref{CliffordAlgebraRepresentation}).

Now for the second claim, use def. \ref{LorentzianSignature} and prop. \ref{CliffordAlgebraRepresentation} to find

$$
  \begin{aligned}
     \overline{\Gamma}_0
     & =
     \Gamma_0^{-1}\Gamma_0^\dagger \Gamma_0
     \\
     & = \Gamma_0^{-1} \Gamma_0 \Gamma_0
     \\
     & =
     \Gamma_0
   \end{aligned}
$$

and

$$
  \begin{aligned}
    \overline{\Gamma}_{spatial}
    & =
    \Gamma_0^{-1} \Gamma_{spatial}^\dagger\Gamma_0
    \\
    &=
    - \Gamma_0^{-1} \Gamma_{spatial} \Gamma_0
    \\
    & = + \Gamma_0^{-1} \Gamma_0 \Gamma_{spatial}
    \\
    &= \Gamma_{spatial}
  \end{aligned}
  \,.
$$

=--

### Charge conjugation matrix
 {#ChargeConjugationMatrix}

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

Given the Clifford algebra representation of the form of prop. \ref{CliffordAlgebraRepresentation}, consider the equation

$$
  C_{(\pm)} \Gamma_a  = \pm \Gamma_a^T C_{(\pm)}
$$

for $C_{(\pm)} \in Mat_{\nu \times n}(\mathbb{C})$.

In even dimensions $d = 2 \nu$ then both these equations have a solution, wheras in odd dimensions $d = 2 \nu + 1$ only one of them does (alternatingly, starting with $C_{(+)}$ in dimension 5). Either $C_{(\pm)}$ is called the _[[charge conjugation matrix]]_.

Moreover, all $C_{(\pm)}$ may be chosen to be real matrices

$$
  (C_{(\pm)})^\ast = C_{(\pm)}
$$

and in addition they satisfy the following relations:

| $d$ |   |    |
|-----|---|----|
| 4 | $C_{(+)}^T = -C_{(+)}$; $C_{(+)}^2 = -1$ | $C_{(-)}^T = -C_{(+)}$; $C_{(-)}^2 = -1$ |
| 5 | $C_{(+)}^T = -C_{(+)}$; $C_{(+)}^2 = -1$  |  |
| 6 | $C_{(+)}^T = -C_{(+)}$; $C_{(+)}^2 = -1$  | $C_{(-)}^T = C_{(-)}$; $C_{(-)}^2 = 1$ |
| 7 |   |  $C_{(-)}^T = C_{(-)}$; $C_{(-)}^2 = 1$ |
| 8 | $C_{(+)}^T = C_{(+)}$; $C_{(+)}^2 = 1$ | $C_{(-)}^T = C_{(-)}$; $C_{(-)}^2 = 1$ |
| 9 | $C_{(+)}^T = C_{(+)}$; $C_{(+)}^2 = 1$ |  |
| 10 | $C_{(+)}^T = C_{(+)}$; $C_{(+)}^2 = 1$ |  $C_{(-)}^T = -C_{(-)}$; $C_{(-)}^2 = -1$ |
| 11 |   |  $C_{(-)}^T = -C_{(-)}$; $C_{(-)}^2 = -1$ |

=--

{#beware} (This is for instance in [Castellani-D'Auria-Fré, section (II.7.2), table (II.7.1)](#CastellaniDAuriaFre), but beware that there $C_{(-)}$ in $d = 10, 11$ is claimed to be symmetric, while instead it is anti-symmetric as shown above, see [van Proeyen 99, table 1](#VanProeyen99), [Laenen, table E.3](#GammaMatrices)).

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

Prop. \ref{ChargeConjugationMatrix} implies that for all $C_{(\pm)}$ listed there then

$$
  C^{-1} = C^T
  \,.
$$

This implies in all cases that

$$
  \Gamma_a C_{(\pm)}^T = \pm C_{(\pm)}^T \Gamma_a^T
  \,.
$$

=--

### Majorana representations and Real structure

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

For $d \in \{4,8,9,10,11\}$, let $V = \mathbb{C}^\nu$ as above. Write $\{\Gamma_a\}$ for a Dirac representation according to prop. \ref{CliffordAlgebraRepresentation}, and write

$$
  C
  \coloneqq
  \left\{
    \array{
      C_{(-)} & \text{for}\; d = 4
      \\
      C_{(+)} & \text{for}\; d = 8
      \\
      C_{(+)} & \text{for}\; d = 9
      \\
      C_{(+)} or C_{(-)} & \text{for}\; d = 10
      \\
      C_{(-)} & \text{for}\; d = 11
    }
  \right.
$$

for the choice of [[charge conjugation matrix]] from prop. \ref{ChargeConjugationMatrix} as shown. Then the function

$$
  J \colon V \longrightarrow V
$$

given by

$$
  \psi \mapsto C \Gamma_0^T \psi^\ast
$$

is a [[real structure]] (def. \ref{RealStructureOnLinearRepresentation}) for the corresponding complex [[spin representation]] on $\mathbb{C}^\nu$.

=--

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

The conjugate linearity of $J$ is clear, since $(-)^\ast$ is conjugate linear and [[matrix multiplication]] is complex linear.

To see that $J$ squares to +1 in the given dimensions: Applying it twice yields,

$$
  \begin{aligned}
    J^2 \psi &=
     C \Gamma_0^T (C \Gamma_0^T\psi^\ast)^\ast
     \\
     & =
     C \Gamma_0^T C \Gamma_0^\dagger \psi
     \\
     &=
     C \underset{= \pm C \Gamma_0}{\underbrace{\Gamma_0^T C}} \Gamma_0 \psi
     \\
     & = \pm C_{(\pm)}^2 \Gamma_0^2 \psi
     \\
     & = \pm C_{(\pm)}^2 \psi
  \end{aligned}
  \,,
$$

where we used $\Gamma_0^\dagger = \Gamma_0$ from prop. \ref{CliffordAlgebraRepresentation}, $C^\ast = \ast$ from prop. \ref{ChargeConjugationMatrix} and then the defining equation of the [[charge conjugation matrix]] $\Gamma_a^T C_{(\pm)} = \pm C_{(\pm)} \Gamma_a$ (def. \ref{ChargeConjugationMatrix}), finally the defining relation $\Gamma_0^2 = +1$.

Hence this holds whenever there exists a choice $C_{(\pm)}$ for the charge conjugation matrix with $C_{(\pm)}^2 = \pm 1$. Comparison with the table from prop. \ref{ChargeConjugationMatrix} shows that this is the case in $d = 4,8,9,10,11$.

Finally to see that $J$ is spin-invariant (in [Castellani-D'Auria-Fré](#CastellaniDAuriaFre) this is essentially (II.2.29)), it is sufficient to show for distinct indices $a,b$, that

$$
  J(\Gamma_a \Gamma_b \psi)
  =
  \Gamma_a \Gamma_b J(\psi)
  \,.
$$

First let $a,b$ both be spatial. Then

$$
  \begin{aligned}
    J(\Gamma_a \Gamma_b \psi)
    & =
    C \Gamma_0^T \Gamma_a^\ast \Gamma_b^\ast \psi^\ast
    \\
    & =
    C \Gamma_0^T (-\Gamma_a^T)(-\Gamma_b^T) \psi^\ast
    \\
    & =
    C \Gamma_0^T \Gamma_a^T \Gamma_b^T \psi^\ast
    \\
    & =
    C  \Gamma_a^T \Gamma_b^T \Gamma_0^T \psi^\ast
    \\
    & =
    \Gamma_a \Gamma_b C \Gamma_0^T \psi^\ast
    \\
    & =
    \Gamma_a \Gamma_b J(\psi)
  \end{aligned}
  \,.
$$

Here we first used that $\Gamma_{spatial}^\dagger = -\Gamma_{spatial}$ (prop. \ref{CliffordAlgebraRepresentation}), hence that $\Gamma_{spatial}^\ast = - \Gamma_{spatial}^T$ and then that $\Gamma_0$ anti-commutes with the spatial Clifford matrices, hence that $\Gamma_0^T$ anti-commutes the the transposeso fthe spatial Clifford matrices. Then we used the defining equation for the [[charge conjugation matrix]], which says that passing it through a Gamma-matrix yields a transpose, up to a global sign. That global sign cancels since we pass through two Gamma matrices.

Finally, that the same conclusion holds for $\Gamma_{spatial} \Gamma_{spatial}$ replaced by $\Gamma_0 \Gamma_{spatial}$: The above reasoning applies with two extra signs picked up: one from the fact that $\Gamma_0$ commutes with itself, one from the fact that it is hermitian, by prop. \ref{CliffordAlgebraRepresentation}. These two signs cancel:

$$
  \begin{aligned}
    J(\Gamma_0 \Gamma_a \psi)
    & =
    C \Gamma_0^T \Gamma_0^\ast \Gamma_a^\ast \psi^\ast
    \\
    & =
    C \Gamma_0^T (+\Gamma_0^T)(-\Gamma_a^T) \psi^\ast
    \\
    & =
    - C \Gamma_0^T \Gamma_0^T \Gamma_a^T \psi^\ast
    \\
    & =
    + C \Gamma_0^T \Gamma_a^T \Gamma_0^T \psi^\ast
    \\
    & = \Gamma_0 \Gamma_a \Gamma_0^T \psi^\ast
    \\
    &= \Gamma_0 \Gamma_a J(\psi)
  \end{aligned}
  \,.
$$

=--

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

Prop. \ref{MajoranaConjugationIsRealStructure} implies that given a [[Dirac representation]] (prop. \ref{CliffordAlgebraRepresentation}) $V$, then the real subspace $S \hookrightarrow V$ of real elements, i.e. elements $\psi$ with $J \psi = \psi$ according to prop. \ref{MajoranaConjugationIsRealStructure} is a [[sub-representation]]. This is called the **Majorana representation** inside the Dirac representation (if it exists).

=--

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

If $C = C_{(\pm)}$ is the [[charge conjugation matrix]] according to prop. \ref{ChargeConjugationMatrix}, then the real structure $J$ from prop. \ref{MajoranaConjugationIsRealStructure} commutes or anti-commutes with the action of _single_ Clifford generators, according to the subscript of $C = C_{(\pm)}$:

$$
  J(\Gamma_a(-)) = \pm \Gamma_a J(-)
  \,.
$$

=--

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

This is same kind of computation as in the proof prop. \ref{MajoranaConjugationIsRealStructure}. First let $a$ be a spatial index, then we get

$$
  \begin{aligned}
    J(\Gamma_a \psi)
    & =
    C \Gamma_0^T \Gamma_a^\ast  \psi^\ast
    \\
    & =
    C \Gamma_0^T (-\Gamma_a^T) \psi^\ast
    \\
    & =
    + C \Gamma_a^T \Gamma_0^T \psi^\ast
    \\
    & =
    \epsilon C^T \Gamma_a^T \Gamma_0^T
    \\
    & =
    \pm \epsilon \Gamma_a C^T \Gamma_0^T \psi^\ast
    \\
    & = \pm \epsilon^2 \Gamma_a C \Gamma_0^T \psi^\ast
    \\
    & = \pm \Gamma_a J(\psi)
  \end{aligned}
  \,,
$$

where, by comparison with the table in prop. \ref{ChargeConjugationMatrix}, $\epsilon$ is the sign in $C^T = \epsilon C$, which cancels out, and the remaining $\pm$ is the sign in $C = C_{(\pm)}$, due to remark \ref{TransposeChargeConjugation}.

For the timelike index we similarly get:

$$
  \begin{aligned}
    J(\Gamma_0 \psi)
    & =
    C \Gamma_0^T \Gamma_0^\ast  \psi^\ast
    \\
    & =
    + C \Gamma_0^T \Gamma_0^T \psi^\ast
    \\
    & =
    \epsilon C^T \Gamma_0^T \Gamma_0^T
    \\
    & =
    \pm \epsilon \Gamma_0 C^T \Gamma_0^T \psi^\ast
    \\
    & = \pm \Gamma_0 C \Gamma_0^T \psi^\ast
    \\
    & = \pm \Gamma_0 J(\psi)
  \end{aligned}
  \,.
$$

=--

We record some immediate consequences:

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

The complex [[bilinear form]]

$$
  (-,-)
   \coloneqq
  \langle J(-),(-)\rangle
$$

induced via the [[real structure]] $J$ of prop. \ref{MajoranaConjugationIsRealStructure} from the [[hermitian form]] $\langle -,-\rangle$ of prop. \ref{CliffordAlgebraRepresentation} is that represented by the [[charge conjugation matrix]] of prop. \ref{ChargeConjugationMatrix}

$$
  (-,-)
   =
  (-)^T C (-)
  \,.
$$

=--

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

By direct unwinding of the various definitions and results from above:

$$
  \begin{aligned}
    \langle J(\psi_1),\psi_2 \rangle
     &=
     \langle C \Gamma_0^T\psi_1^\ast, \psi_2\rangle
     \\
     & =
     (C \Gamma_0^T \psi_1^\ast)^\dagger \Gamma_0 \psi_2
     \\
     & = \psi_1^T C^\dagger \Gamma_0^\ast \Gamma_0 \psi_2
     \\
     & = \psi_1^T C \psi_2
  \end{aligned}
  \,.
$$

=--

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

For a Clifford algebra representation on $\mathbb{C}^\nu$ as in prop. \ref{CliffordAlgebraRepresentation}, then the map

$$
  (-)^T C
  \;\colon\;
  Mat_{\nu \times 1}(\mathbb{C})
    \longrightarrow
  Mat_{1 \times \nu}(\mathbb{C})
$$

(from complex column vectors to complex row vectors) which is given by transposition followed by [[matrix multiplication]] from the right by the [[charge conjugation matrix]] according to prop. \ref{MajoranaConjugationIsRealStructure} is called the
**Majorana conjugation**.

=--

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

In dimensions $d = 4,8,9,10,11$ a spinor $\psi \in \mathbb{C}^{2^\nu}$ is Majorana according to def. \ref{MajoranaSpinorGeneral} with respect to the [[real structure]] from prop. \ref{MajoranaConjugationIsRealStructure}, precisely if

$$
  \psi = C \Gamma_0^T \psi^\ast
$$

(as e.g. in [Castellani-D'Auria-Fré, (II.7.22)](#CastellaniDAuriaFre)),

This is equivalent to the condition that the Majorana conjugate (def. \ref{MajoranaConjugation}) coincides with  the Dirac conjugate (def. \ref{DiracConjugate}) on $\psi$:

$$
  \psi^T C = \psi^\dagger \Gamma_0
  \,,
$$

which in turn is equivalent to the condition that

$$
  (\psi,-) = \langle \psi,-\rangle
  \,,
$$

where on the left we have the complex bilinear form of prop. \ref{ComplexBilinearFormInducedFromMajoranaStructure} and on the right the [[hermitian form]] from prop. \ref{CliffordAlgebraRepresentation}.

=--

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

The first statement is immediate. The second follows by applying the transpose to the first equation, and using that $C^{-1} = C^T$ (from prop. \ref{ChargeConjugationMatrix}). Finally the last statement follows from this by prop. \ref{ComplexBilinearFormInducedFromMajoranaStructure}.

=--

### Pseudo-Majorana spinors and Symplectic structure

In $d = 5$, for example, the reality/Majorana condition

$$
  \psi = C \Gamma_0^T \psi^\ast
$$

from prop. \ref{MajoranaConjugationIsRealStructure}
has no solution. But if we consider the [[direct sum]] of two copies of the complex spinor
representation space, with elements denoted $\psi_1$ and $\psi_2$, then the following condition does have a solution

$$
  C \Gamma_0^T \psi_1^\ast = -\psi_2
  \;\;\;\;
  C \Gamma_0^T \psi_2^\ast = +\psi_1
$$

(e.g [Castellani-D'Auria-Fré, II.8.41](#CastellaniDAuriaFre)). Comparison with prop. \ref{MajoranaConjugationIsRealStructure}
and def. \ref{MajoranaSpinorGeneral} shows that this exhibits a quaternionic structure on the
original complex spinor space, and hence a real structure on its direct sum double.

### Majorana-Weyl spinors

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

In the even dimensions among those dimensions $d$ for which the Majorana projection operator (real structure) $J$ exists (prop. \ref{MajoranaConjugationIsRealStructure}) also the chirality projection operator $\Gamma_{d}$ exists (def. \ref{WeylRepresentation}). Then we may ask that a Dirac spinor $\psi$ is both Majorana, $J(\psi) = \psi$, as well as Weyl, $\Gamma_d \psi = \pm i \psi$. If this is the case, it is called a **Majorana-Weyl spinor**, and the [[sub-representation]] these form is a called a _Majorana-Weyl representation_.

=--

+-- {: .num_prop #WeylMajoranaInLorentzian10d}
###### Proposition

In Lorentzian signature (def. \ref{LorentzianSignature}) for $4 \leq d \leq 11$, then Majorana-Weyl spinors (def. \ref{WeylMajorana}) exist precisely only in $d = 10$.

=--

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

According to prop. \ref{MajoranaConjugationIsRealStructure} Majorana spinors in the given range exist for $d \in \{4,8,9,10,11\}$. Hence the even dimensions among these are $d \in \{4,8,10\}$.

Majorana-Weyl spinors clearly exist precisely if the two relevant projection operators in these dimensions commute with each other, i.e. if

$$
  [J, \epsilon \Gamma_0 \cdots \Gamma_{d-1}] = 0
$$

where

$$
  \epsilon
     =
   \left\{
     \array{
       1 & \vert \; \nu \, \text{odd}
       \\
       i & \vert \; \nu \, \text{even}
     }
   \right.
   \,.
$$

with $d = 2\nu$ (from the proof of prop. \ref{CliffordAlgebraRepresentation}).

By prop. \ref{RealStructureAntiCommutesWithSingleCliffordGenerator} all the $\Gamma_a$ commute or all anti-commute with $J$. Since the product $\Gamma_0 \cdots \Gamma_{d-1}$ contains an even number of these, it commutes with $J$. It follows that $J$ commutes with $\Gamma_d$ precisely if it commutes with $\epsilon$. Now since $J$ is conjugate-linear, this is the case precisely if $\epsilon = 1$, hence precisely if $d = 2\nu$ with $\nu$ odd.

This is the case for $d = 10 = 2 \cdot 5$, but not for $d = 8 = 2 \cdot 4$ neither for $d = 4 = 2 \cdot 2$.

=--

## The spinor bilinear pairing to antisymmetric $p$-tensors
 {#TheSpinorPairingToVectors}

We now discuss, in the component expressions established [above](#InComponents), the complex bilinear pairing operations that take a pair of Majorana spinors to a [[vector]], and more generally to an antisymmetric rank $p$-[[tensor]]. These operations are all of the form

$$
  \psi
    \mapsto
  \epsilon \, \overline{\psi} \Gamma^{a_1 \cdots a_p} \psi
  \,,
$$

where $\epsilon \in \mathbb{C}$ is some prefactor, constrained such as to make the whole expression be real, hence such as to make this the components of an element in $\wedge^p \mathbb{R}^{d-1,1}$.

For a $Spin(d-1,1)$ representation $V$ as in prop. \ref{CliffordAlgebraRepresentation}, with real/Majorana structure as in prop. \ref{MajoranaConjugationIsRealStructure}, write

$$
  S \hookrightarrow V
$$

for the subspace of Majorana spinors, regarded as a real vector space.

Recall, by prop. \ref{TheMajoranaConditionInComponents}, that on Majorana spinors the Majorana conjugate $(-)^T C$ coincides with the Dirac conjugate $\overline{(-)} \coloneqq (-)^\dagger \Gamma_0 $. Therefore we write $\overline{(-)}$ in the following for the conjugation of Majorana spinors, unambiguously defined.

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

For a $Spin(d-1,1)$ representation $V$ as in prop. \ref{CliffordAlgebraRepresentation}, with real/Majorana structure as in prop. \ref{MajoranaConjugationIsRealStructure}, let

$$
  \overline{(-)}\Gamma (-)
    \;\colon\;
  S \times S
   \longrightarrow
  \mathbb{R}^{d-1,1}
$$

be the function that takes Majorana spinors $\psi_1$, $\psi_2$ to the vector with components

$$
  \overline{\psi}_1\Gamma^a \psi_2
    \coloneqq
  \psi_1^T C \Gamma^a \psi_2
  \,.
$$

=--

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

The pairing in def. \ref{SpinorToVectorBilinearPairing} is

1. symmetric:

   $$
     \overline{\psi}_1 \Gamma \psi_2
       =
     \overline{\psi}_2 \Gamma \psi_1
   $$

1. component-wise real-valued (i.e. it indeed takes values in $\mathbb{R}^d \subset \mathbb{C}^d$);

1. $Spin(d-1,1)$-equivariant: for $g \in Spin(d-1,1)$ then

   $$
     \overline{g(-)}\Gamma(g(-))
       =
     g(\overline{(-)}\Gamma(-))
     \,.
   $$

=--

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

Regarding the first point, we need to show that for all $a$ then $C \Gamma_a$ is a symmetric matrix. Indeed:

$$
  \begin{aligned}
     (C \Gamma_a)^T
       & =
     \Gamma_a^T C^T
      \\
      & = \pm \Gamma_a^T C
      \\
      & = \pm \pm C \Gamma_a
      \\
      & = C \Gamma_a
  \end{aligned}
  \,,
$$

where the first sign picked up is from $C^T = \pm C$, while the second is from $\Gamma_a^T C = \pm C \Gamma_a$ (according to prop. \ref{ChargeConjugationMatrix}). Imposing the condition in prop. \ref{MajoranaConjugationIsRealStructure} one finds that these signs agree, and hence cancel out.

(In [van Proeyen99](#VanProeyen99) this is part of table 1, in ([Castellani-D'Auria-Fré](#CastellaniDAuriaFre)) this is implicit in equation (II.2.32a).)

With this the second point follows together with the relation $\psi^T C = \psi^\dagger \Gamma_0$ for Majorana spinors $\psi$ (prop. \ref{TheMajoranaConditionInComponents}) and using the conjugate-symmetry of the [[hermitian form]] $\langle -,-\rangle = (-)^\dagger \Gamma_0 (-)$ as well as the hermiticity of $\Gamma_0 \Gamma_a$ (both from prop. \ref{CliffordAlgebraRepresentation}):

$$
  \begin{aligned}
    (\overline{\psi}_1 \Gamma_a \psi_2)^\ast
    &=
    (\psi_1^T C \Gamma_a \psi_2)^\ast
    \\
    & =
    (\psi_1^\dagger \Gamma_0 \Gamma^a \psi_2)^\ast
    \\
    & =
    \psi_2^\dagger (\Gamma_0 \Gamma^a)^\dagger \psi_1
    \\
    & =
    \psi_2^\dagger \Gamma_0 \Gamma^a \psi_1
    \\
    & =
    \overline{\psi}_2 \Gamma_a \psi_1
  \end{aligned}
  \,.
$$

Regarding the third point: By prop. \ref{MajoranaConjugationIsRealStructure} and prop. \ref{ComplexBilinearFormInducedFromMajoranaStructure} we get

$$
  \begin{aligned}
    (g(\psi_1), \Gamma_a g(\psi_2))
    & =
    \langle g(\psi_1),\Gamma_a g(\psi_2)\rangle
    \\
    & =
     \langle \psi_1, (\Gamma_0^{-1}g^\dagger\Gamma_0) \Gamma_a g \psi_2 \rangle
    \\
    & = \langle \psi_1 (g^{-1} \Gamma_a g) \psi_2 \rangle
  \end{aligned}
  \,,
$$

where we used that $\Gamma_0^{-1}(-)^\dagger \Gamma_0$ is the adjoint with respect to the hermitian form $\langle -,-\rangle = (-)^\dagger \Gamma_0 (-)$ (prop. \ref{CliffordRepresentationIsDiracSelfConjugate}) and that $g$ is unitary with respect to this hermitian form by prop. \ref{CliffordAlgebraRepresentation}.

(In ([Castellani-D'Auria-Fré](#CastellaniDAuriaFre)) this third statement implicit in equations (II.2.35)-(II.2.39).)

=--

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

Proposition \ref{SpinorToVectorPairing} implies that adding a copy of $S$ to the [[Poincaré Lie algebra]] in odd degree, then the pairing of def. \ref{SpinorToVectorBilinearPairing} is a consistent extension of the [[Lie bracket]] of the latter to a [[super Lie algebra]]. This is the _[[super Poincaré Lie algebra]]_.

=--

+-- {: .num_defn #SpinorToRank2TensorBilinearPairing}
###### Definition

For a $Spin(d-1,1)$ representation $V$ as in prop. \ref{CliffordAlgebraRepresentation}, with real/Majorana structure as in prop. \ref{MajoranaConjugationIsRealStructure}, let

$$
  \overline{(-)}\Gamma\Gamma (-)
    \;\colon\;
  S \times S
   \longrightarrow
  \wedge^2 \mathbb{C}^d
$$

be the function that takes Majorana spinors $\psi_1$, $\psi_2$ to the skew-symmetric rank 2-tensor with components

$$
  \overline{\psi}_1\Gamma^{a b} \psi_2
    \coloneqq
  i \psi_1^T C \tfrac{1}{2}(\Gamma^a \Gamma^b - \Gamma^b \Gamma^a) \psi_2
  \,,
$$

=--

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

For $\psi_1 = \psi_2 = \psi$ then the pairing in prop. \ref{SpinorToRank2TensorBilinearPairing} is real

$$
  \underset{a,b}{\forall}
  \;\;\;\;
  i \overline{\psi} \Gamma^{a b} \psi \in \mathbb{R} \subset \mathbb{C}
$$

and $Spin(d-1,1)$-equivariant.

=--

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

The equivariance follows exactly as in the proof of prop. \ref{SpinorToVectorPairing}.

The reality is checked by direct computation as follows:

$$
  \begin{aligned}
    (\overline{\psi}_1 \Gamma_a \Gamma_b \psi_2)^\ast
    & =
    (\psi_1^\dagger \Gamma_a \Gamma_b \psi_2)^\ast
    \\
    & =
    \psi_2^\dagger (\Gamma_0 \Gamma_a \Gamma_b)^\dagger \psi_1
    \\
    & =
    -\langle \psi_2^\dagger \Gamma_0 \Gamma_a \Gamma_b \psi_1 \rangle
    \\
    & =
    -\overline{\psi}_2 \Gamma_a \Gamma_b \psi_1
  \end{aligned}
  \,,
$$

where for the identification $(\Gamma_0 \Gamma_a \Gamma_b)^\dagger = - \Gamma_0 \Gamma_a \Gamma_b$ we used the properties in prop. \ref{CliffordAlgebraRepresentation}.

Hence for $\psi_1 = \psi_2$ then

$$
  (\overline{\psi} \Gamma_a \Gamma_b \psi)^\ast
  =
  -
  \overline{\psi} \Gamma_a \Gamma_b \psi
$$

and so this sign cancels against the sign in $i^\ast = -i$.

=--

Generally:

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

The following pairings are real and $Spin(d-1,1)$-equivariant:

$$
  \begin{aligned}
    & \overline{\psi} \Gamma_a \psi
    \\
    i & \overline{\psi}\Gamma_{a_1 a_2} \psi
    \\
    i & \overline{\psi} \Gamma_{a_1 a_2 a_3} \psi
    \\
     & \overline{\psi} \Gamma_{a_1 \cdots a_4} \psi
    \\
     & \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi
    \\
    i  & \overline{\psi} \Gamma_{a_1 \cdots a_6} \psi
    \\
    i  & \overline{\psi} \Gamma_{a_1 \cdots a_7} \psi
    \\
    & \vdots
  \end{aligned}
  \,.
$$

=--

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

The equivariance follows as in the proof of prop. \ref{SpinorToVectorPairing}.

Regarding reality:

Using that all the $\Gamma_a$ are hermitian with respect $\overline{(-)}(-)$ (prop. \ref{CliffordRepresentationIsDiracSelfConjugate}) we have

$$
  \begin{aligned}
    \left(
      \overline{\psi}
       \Gamma_{a_1 \cdots a_p}
      \psi
    \right)^\ast
    & =
    \overline{\psi}
      \Gamma_{a_p \cdots a_1}
    \psi
    \\
    &=
    (-1)^{p(p-1)/2}
    \overline{\psi}
      \Gamma_{a_1 \cdots a_p}
    \psi
  \end{aligned}
  \,.
$$

=--

## Supersymmetry: Super-Poincaré and super-Minkowski
 {#Supersymmetry}

Every real spin representation of $Spin(d-1,1)$ induces a [[super Lie algebra]] [[Lie algebra extension|extension]] of the [[Poincaré Lie algebra]] $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ in that dimension, i.e. of the Lie algebra of the [[isometry group]] of the [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) in that dimension.

Since we may recover a [[Minkowski spacetime]] from its [[Poincaré Lie algebra]] as the (vector space underlying the) [[coset]] of the [[Poincaré Lie algebra]] by the Lie algebra $\mathfrak{so}(d-1,1)$ of the [[spin group]] (the [[orthogonal Lie algebra]] in Lorentian signature)

$$
  \mathbb{R}^{d-1,1}
   \simeq
  \mathfrak{Iso}(\mathbb{R}^{d-1,1})/\mathfrak{so}(d-1,1)
$$

(namely as the Lie algebra of translations along itself), every [[super Lie algebra]] [[extension of Lie algebras|extension]] of the [[Poincaré Lie algebra]] defines a [[super Lie algebra]] [[extension of Lie algebras|extension]] of Minkowski spacetime. These extensions are the [[super Minkowski spacetimes]] $\mathbb{R}^{d-1,1\vert N}$ of the following definition, and this justifies the following notation:

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

Let $d \in \mathbb{N}$ and let $N \in Rep(Spin(d-1,1))$ be a real spin representation, hence a [[direct sum]] of Majorana representations (def. \ref{MajoranaRepresentation}) and/or Majorana-Weyl representations (def. \ref{WeylMajorana}) (or tensor product of two symplectic Majorana representations...).

We define the corresponding **[[super Poincaré Lie algebra]]**

$$
  \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})
$$

equivalently in terms of its [[Chevalley-Eilenberg algebra]]: $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ (a $\mathbb{N} \times \mathbb{Z}/2$-bigraded [[dg-algebra]], see at _[[signs in supergeometry]]_).

This is generated on

* elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$

* and elements $\{\psi^\alpha\}$ of degree $(1,odd)$

where $a \in \{0,1, \cdots, d-1\}$ is a spacetime index, and where $\alpha$ is an index ranging over a basis of the chosen Majorana spinor representation $N$.

The CE-differential is defined as follows

$$
  d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
$$

and

$$
  d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi
  \,.
$$

(which so far is the differential for the [[semidirect product]] of the [[Poincaré Lie algebra]] [[action|acting]] on the given [[Majorana spinor]] representation)

and

$$
  d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \overline{\psi} \Gamma^a \psi
$$

where on the right we have the spinor-to-vector pairing in $N$ (def. \ref{SpinorToVectorBilinearPairing}).

That this is indeed a [[super Lie algebra]] follows from the fact that the [[Poincaré Lie algebra]] is a [[Lie algebra]] and the fact that the spinor-to-vector pairing is symmetric (which makes it qualify as the odd-odd component of a super-Lie algebra) and $Spin(d-1,1)$-equivariant (which is seen to be the super-[[Jacobi identity]] for it), all according to prop. \ref{SpinorToVectorPairing}.

This defines the [[super Poincaré super Lie algebra]]. After discarding the terms involving $\omega$ this becomes the CE algebra of the [[super translation algebra]] underlying [[super Minkowski spacetime]]

$$
  \mathbb{R}^{d-1,1\vert N}
  \,.
$$

=--

## Examples

### In dimensions 11, 10, and 9
 {#InDimensions11And10And9}

We spell out some of the above constructions and properties for Majorana spinors in Lorentzian spacetimes (def. \ref{LorentzianSignature}) of dimensions 11, 10 and 9, and discuss some relations between these. These spinor structures are relevant for spinors in [[11-dimensional supergravity]] and [[type II supergravity]] in 10d and 9d, as well as to the relation between these via [[Kaluza-Klein compactification]] and [[T-duality]].

+-- {: .num_prop #DiracRepInD11FromD9}
###### Proposition

Let $\{\gamma_a\}$ be any [[Dirac representation]] of $Spin(8,1)$ according to prop. \ref{CliffordAlgebraRepresentation}. By the same logic as in the proof of prop. \ref{CliffordAlgebraRepresentation} we get from this the Dirac representations in dimensions 9+1 and 10+1 by setting

$$
  \Gamma_{a \leq 8}
    \coloneqq
  \left(
    \array{
      0 & \gamma_a
      \\
      \gamma_a & 0
    }
  \right)
  \;\,,\;\;
  \Gamma_{9}
   \coloneqq
  \left(
    \array{
       0 & id
       \\
       -id & 0
    }
  \right)
  \;\,,\;\;
  \Gamma_{10}
   \coloneqq
  \left(
    \array{
      i id & 0
      \\
      0 & -i id
    }
  \right)
  \,.
$$

=--

+-- {: .num_remark #SpinRepsFrom11dToTDuality}
###### Remark

By prop. \ref{CliffordAlgebraRepresentation} the Dirac representation in $d = 11$ has complex dimension $2^{10/2} = 2^{5} = 32$. By prop. \ref{MajoranaConjugationIsRealStructure} and prop. \ref{TheMajoranaConditionInComponents} this representation carries a real structure and hence gives a real/Majorana [[spin representation]] $S \hookrightarrow \mathbb{C}^{32}$ of $Spin(10,1)$ of real dimension 32. This representation often just called "$\mathbf{32}$". This way the corresponding [[super-Minkowski spacetime]] (remark \ref{SuperPoincareOutlook}) is neatly written as

$$
  \mathbb{R}^{10,1\vert \mathbf{32}}
$$

which thus serves to express both, the real dimension of the space of odd-graded coordinate functions at every point on it, as well as the way that the $Spin(10,1)$-cover of the [[Lorentz group]] $SO(10,1)$ acts on these. This is the local model space for [[super spacetimes]] in [[11-dimensional supergravity]].

As we regard $\mathbb{C}^{32}$ instead as the Dirac representation of $Spin(9,1)$ via def. \ref{WeylRepresentation}, then it decomposes into to chiral halfs, each of complex dimension 16. This is the direct sum decomposition in terms of which the block decomposition of the above Clifford matrices is given.

Since in 10d the Weyl condition is compatible with the Majorana condition (by prop. \ref{WeylMajoranaInLorentzian10d}), the real Majorana representation $\mathbf{32}$ correspondingly decomposes as a direct sum two real representations of dimension 116 which often are denoted $\mathbf{16}$ and $\overline{\mathbf{16}}$. Hence as real/Majorana $Spin(9,1)$-representations there is a [[direct sum]] decomposition

$$
  \mathbf{32} \simeq \mathbf{16} \oplus \overline{\mathbf{16}}
  \,.
$$

The corresponding [[super Minkowski spacetime]] (remark \ref{SuperPoincareOutlook})

$$
  \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}
$$

is said to be of "type IIA", since this is the local model space for [[superspacetimes]]  in [[type IIA supergravity]]. This is as opposed to $\mathbb{R}^{9,1\vert \mathbf{16}\oplus \mathbf{16}}$, which is "type IIB" and in contrast to $\mathbb{R}^{9,1\vert \mathbf{16}}$ which is "heterotic" (the local model space for [[heterotic supergravity]]).

Now the Dirac-Weyl representation for $Spin(8,1)$ is of complex dimension $d = 2^{8/2} = 2^4 = 16$. By prop. \ref{MajoranaConjugationIsRealStructure} and prop. \ref{TheMajoranaConditionInComponents} this also admits real structure, and hence gives a Majorana representation fro $Spin(8,1)$, accordingly denoted $\mathbf{16}$. Notice that this is Majorana-Weyl.

We want to argue that both the $\mathbf{16}$ and the $\overline{\mathbf{16}}$ of $Spin(9,1)$ become isomorphic to the single $\mathbf{16}$ of $Spin(8,1)$ under forming the [[restricted representation]] along the inclusion $Spin(8,1)\hookrightarrow Spin(9,1)$ (the one fixed by the above choice of components).

For this it is sufficient to see that $\Gamma_9$, which as a complex linear map goes $\Gamma_9 \colon \mathbf{16} \longrightarrow \overline{\mathbf{16}}$ constitutes an [[isomorphism]] when regarded as a morphism in the [[category of representations]] of $Spin(8,1)$.

Clearly it is a linear isomorphism, so it is sufficient that it is a [[homomorphism]] of $Spin(8,1)$-representations at all. But that's clear since by the Clifford algebra relations $\Gamma_9$ commutes with all $\Gamma_a \Gamma_b$ for $a,b \on \{0,\cdots, 8\}$.

Hence the [[branching rule]] for [[restricted representation|restricting]] the Weyl representation in 11d along the sequence of inclusions

$$
  Spin(8,1) \hookrightarrow Spin(9,1) \hookrightarrow Spin(10,1)
$$

is

$$
  \underset{\in Rep(Spin(10,1))}\underbrace{\mathbf{32}}
    \mapsto
  \underset{\in Rep(Spin(9,1))}\underbrace{\mathbf{16} \oplus \overline{\mathbf{16}}}
    \mapsto
  \underset{\in Rep(Spin(8,1))}\underbrace{\mathbf{16} \oplus \mathbf{16}}
  \,.
$$

If we write a Majorana spinor in $\mathbf{32}$ as $\vartheta \in \mathbb{C}^{32}$, decomposed as a $(1 \times 32)$-matrix as

$$
  \vartheta
    =
  \left(
    \array{
      \psi_1
      \\
      \psi_2
    }
  \right)
  \,.
$$

and if we write for short

$$
  \psi_1
   =
  \left(
    \array{
      \psi_1 \\ 0
    }
  \right)
  \,,\;\;\;
  \psi_2
   =
  \left(
    \array{
      0 \\  \psi_2
    }
  \right)
$$

then this says that after restriction to $Spin(9,1)$-action then $\psi_1$ becomes a Majorana spinor in the $\mathbf{16}$, and $\psi_2$ a Majorana spinor in the $\overline{\mathbf{16}}$, and after further restriction to $Spin(8,1)$-action then either comes a Majorana spinor in one copy of $\mathbf{16}$.

=--

The type IIA spinor-to-vector pairing is just that of 11d under this re-interpretation. We find:

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

The type IIA spinor-to-vector pairing is given by

$$
  \begin{aligned}
    \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
     \Gamma_a^{IIA}
    \left(
      \array{\psi_1 \\ \psi_2}
    \right)
    & =
    \left\{
      \array{
        \overline{\psi}_1 \gamma_a \psi_1
        + \overline{\psi}_2 \gamma_a \psi_2
        &
        \vert \; a \leq 8
        \\
        \overline{\psi}_1 \psi_1
        -
        \overline{\psi}_2 \psi_2
        &
        \vert \; a = 9
      }
    \right.
  \end{aligned}
  \,.
$$

=--

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

Using that on Majorana spinors the Majorana conjugate coincides with the Dirac conjugate (prop. \ref{TheMajoranaConditionInComponents}) and applying prop. \ref{DiracRepInD11FromD9} we compute:

$$
  \begin{aligned}
    \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
     \Gamma_a^{IIA}
    \left(
      \array{\psi_1 \\ \psi_2}
    \right)
      &\coloneqq
    \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
     \Gamma_a
    \left(
      \array{\psi_1 \\ \psi_2}
    \right)
    \\ & =
    \left(\array{\psi_1 \\ \psi_2}\right)^\dagger
     \Gamma_0 \Gamma_a
    \left(
      \array{\psi_1 \\ \psi_2}
    \right)
    \\
    & =
    \left\{
      \array{
    \left(\array{\psi_1 \\ \psi_2}\right)^\dagger
     \left(
       \array{
         \gamma_0 \gamma_a & 0
         \\
         0 & \gamma_0 \gamma_a
       }
     \right)
    \left(
      \array{\psi_1 \\ \psi_2}
    \right)
     & \vert \; a\leq 8
     \\
    \left(\array{\psi_1 \\ \psi_2}\right)^\dagger
     \left(
       \array{
         \gamma_0  & 0
         \\
         0 & -\gamma_0
       }
     \right)
    \left(
      \array{\psi_1 \\ \psi_2}
    \right)
     & \vert \; a = 9
     }
    \right.
    \\
    & =
    \left\{
      \array{
        \overline{\psi}_1 \gamma_a \psi_1
        + \overline{\psi}_2 \gamma_a \psi_2
        &
        \vert \; a \leq 8
        \\
        \overline{\psi}_1 \psi_1
        -
        \overline{\psi}_2 \psi_2
        &
        \vert \; a = 9
      }
    \right.
  \end{aligned}
  \,.
$$

=--

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

The type IIB spinor-to-vector pairing is

$$
  \begin{aligned}
    \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
    \Gamma_a^{IIB}
    \left(\array{\psi_1 \\ \psi_2}\right)
    & =
    \left\{
      \array{
        \overline{\psi}_1 \gamma_a \psi_1
        + \overline{\psi}_2 \gamma_a \psi_2
        &
        \vert \; a \leq 8
        \\
        \overline{\psi}_1 \psi_1
        +
        \overline{\psi}_2 \psi_2
        &
        \vert \; a = 9
      }
    \right.
  \end{aligned}
$$

=--

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

The type II pairing spinor-to-vector pairing is obtained from the type IIA pairing of prop. \ref{typeIIASpinorToVectorPairing} by replacing all bottom right matrix entries (those going $\overline{\mathbf{16}}\to \overline{\mathbf{16}}$ by the corresponding top left entries (those going $\mathbf{16} \to \mathbf{16}$ )). Notice that in fact all these block entries are the same, except for the one at $a = 9$, where they simply differ by a sign. This yields the claim.

=--

Notice also the following relation between the different pairing in dimensions 11, 10 and 9:

+-- {: .num_prop #9ComponentOfIIBPairing}
###### Proposition

The $(9,10)$-component of the spinor-to-bivector pairing (def. \ref{SpinorToRank2TensorBilinearPairing}) in 11d equals the 9-component of the type IIB spinor-to-vector pairing

$$
  \begin{aligned}
    i
    \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
     \Gamma_9 \Gamma_{10}
    \left(\array{\psi_1 \\ \psi_2}\right)
    & =
    \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
    \Gamma_9^{IIB}
    \left(\array{\psi_1 \\ \psi_2}\right)
  \end{aligned}
$$

=--

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

Using prop. \ref{DiracRepInD11FromD9} and prop. \ref{typeIIBSpinorToVectorPairing} we compute:

$$
  \begin{aligned}
    i\,
    \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
     \Gamma_9 \Gamma_{10}
    \left(\array{\psi_1 \\ \psi_2}\right)
    & =
    i
    \left(\array{\psi_1 \\ \psi_2}\right)^\dagger
     \Gamma_0\Gamma_9 \Gamma_{10}
    \left(\array{\psi_1 \\ \psi_2}\right)
    \\
    & =
    i \,
    \left(\array{\psi_1 \\ \psi_2}\right)^\dagger
      \left(
        \array{
          -i \gamma_0 & 0
          \\
          0 & -i \gamma_0
        }
      \right)
    \left(\array{\psi_1 \\ \psi_2}\right)
    \\
    & =
      \overline{\psi}_1 \psi_1
      +
      \overline{\psi}_2 \psi_2
    \\
    & =
    \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
    \Gamma_9^{IIB}
    \left(\array{\psi_1 \\ \psi_2}\right)
  \end{aligned}
$$

=--

The following is an evident variant of the extensions considered in ([CAIB 99](#CAIB99), [FSS 13](#FiorenzaSatiSchreiber13)).

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

We have

1. The 11d $N = 1$ [[super-Minkowski spacetime]] $\mathbb{R}^{10,1\vert \mathbf{32}}$ (def. \ref{SuperMinkowskiSpacetime}) is the  [[central extension|central]] [[super Lie algebra|super]] [[Lie algebra extension]] of the 10d type IIA [[super-Minkowski spacetime]] $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ by the 2-cocycle

   $$
     c_2 \coloneqq \overline{\psi} \wedge \Gamma_{10} \psi
     \;\;\;
     \in
     CE(\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}})
   $$

   (from def. \ref{SpinorToVectorBilinearPairing}).

1. The 10d type IIA [[super-Minkowski spacetime]] $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$  is [[central extension|central]] [[super Lie algebra|super]] [[Lie algebra extension]] of th 9d $N = 2$ [[super-Minkowski spacetime]] by the 2-cocycle given by the type IIA spinor-to-vector pairing

   $$
     c_2^{IIA}
      \;\coloneqq\;
     \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
      \wedge \Gamma_9^{IIA}
      \left(
       \array{\psi_1 \\ \psi_2}
      \right)
        \;\;\;\in
      CE(\mathbb{R}^{8,1\vert \mathbf{16}+ \mathbf{16}})
   $$

   (from prop. \ref{typeIIASpinorToVectorPairing}).

1. The 10d type IIB [[super-Minkowski spacetime]] $\mathbb{R}^{9,1\vert \mathbf{16}+ \mathbf{16}}$  is [[central extension|central]] [[super Lie algebra|super]] [[Lie algebra extension]] of th 9d $N = 2$ [[super-Minkowski spacetime]] by the 2-cocycle given by the type IIB spinor-to-vector pairing

   $$
     c_2^{IIB}
       \;\coloneqq\;
     \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
      \wedge
      \Gamma_9^{IIB}
      \left(
       \array{\psi_1 \\ \psi_2}
      \right)
      \;\;\;
      \in CE(\mathbb{R}^{8,1\vert \mathbf{16} + \mathbf{16}})
   $$

   (from prop. \ref{typeIIBSpinorToVectorPairing}).

In summary, we have the following [[diagram]] in the [[category]] of [[super L-infinity algebras]]

$$
  \array{
    && && \mathbb{R}^{10,1\vert \mathbf{32}}
    \\
    && && \downarrow
    \\
    \mathbb{R}^{9,1\vert \mathbf{16} + \mathbf{16}}
      && &&
    \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}
     &\overset{c_2}{\longrightarrow}&
    B \mathbb{R}
    \\
    & \searrow && \swarrow
    \\
    && \mathbb{R}^{8,1\vert \mathbf{16} + \mathbf{16}}
    \\
    & {}^{\mathllap{c_2^{IIB}}}\swarrow && \searrow^{\mathrlap{c_2^{IIA}}}
    \\
    B \mathbb{R} && && B \mathbb{R}
  }
  \,,
$$

where $B\mathbb{R}$ denotes the [[line Lie 2-algebra]], and where each "hook"

$$
  \array{
    \widehat{\mathfrak{g}}
    \\
    \downarrow
    \\
    \mathfrak{g}
      &\overset{\omega_2}{\longrightarrow}&
    B\mathbb{R}
  }
$$

is a [[homotopy fiber sequence]] (because homotopy fibers of super $L_\infty$-algebra cocycles are the corresponding extension that they classify, see at _[[L-infinity algebra cohomology]]_).

=--

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

To see that the given 2-forms are indeed cocycles: they are trivially closed (by def. \ref{SuperMinkowskiSpacetime}), and so all that matters is that we have a well defined super-2-form in the first place. Since the $\psi^\alpha$ are in bidegree $(1,odd)$, they all commutes with each other (see at _[[signs in supergeometry]]_) and hece the condition is that the pairing is symmetric. This is the case by prop. \ref{SpinorToVectorPairing}.

Now to see the extensions. Notice that for $\mathfrak{g}$ any ([[super Lie algebra|super]]) [[Lie algebra]] (of finite dimension, for convenience), and for $\omega \in \wedge^2\mathfrak{g}^\ast$ a [[Lie algebra cohomology|Lie algebra 2-cocycle]] on it, then the [[Lie algebra extension]] $\widehat{\mathfrak{g}}$ that this classifies is neatly characterized in terms of its dual [[Chevalley-Eilenberg algebra]]: that is simply the original CE algebra with one new generator $e$ (in degree $(1,even)$) adjoined, and with the differential of $e$ taking to be $\omega$:

$$
  CE(\widehat{\mathfrak{g}})
   =
  (CE(\mathfrak{g}) \otimes \langle e\rangle), d e = \omega)
  \,.
$$

Hence in the case of $\omega = c_2^{IIA}$ we identify the new generator with $e^9$ and see that the equation $d e^9 = c_2^{IIA}$ is precisely what distinguishes the CE-algebra of $\mathbb{R}^{8,1\vert \mathbf{16}+ \mathbf{16}}$ from that of $\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}$, by prop. \ref{DiracRepInD11FromD9} and the fact that both spin representation have the same underlying space, by remark \ref{SpinRepsFrom11dToTDuality}.

The other two cases are directly analogous.

=--

Recall the following (e.g. from [FSS 16](#FiorenzaSatiSchreiber16) and references given there):

+-- {: .num_defn #M2CoycleAndIIAStringCocycle}
###### Definition

The cocycle for the [[higher WZW term]] of the [[Green-Schwarz sigma-model]] for the [[M2-brane]] is

$$
  \mu_{M2}
    \coloneqq
  i\,\overline{\vartheta} \wedge \Gamma_a \Gamma_b \vartheta \wedge e^a \wedge e^b
  \;\;\;
  \in
  CE(\mathbb{R}^{10,1\vert \mathbf{32}})
$$

obtained from the spinor-to-bivector pairing of def. \ref{SpinorToRank2TensorBilinearPairing}.
(Here and in the following we are using the nation from remark \ref{SpinRepsFrom11dToTDuality}.)

The cocycle for the [[WZW term]] of the [[Green-Schwarz sigma-model]] for the [[type IIA string theory|type IIA]] [[superstring]] is

$$
  \mu_{IIA}
   \coloneqq
  i\,\overline{\vartheta} \wedge \Gamma_a \Gamma_{10} \vartheta \wedge e^a
  =
  i\,
  \overline{\left(
    \array{\psi_1 \\ \psi_2}
  \right)}
    \Gamma_a \Gamma_{10}
  \left(
    \array{\psi_1 \\ \psi_2}
  \right)
  \;\;\;
  \in
  CE(\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}})
  \,,
$$

i.e. this is the the $e^{10}$-component of $\mu_{M2}$ ("[[double dimensional reduction]]" [FSS 16](#FiorenzaSatiSchreiber16)):

$$
  \mu_{IIA} = (\pi_{10})_\ast \mu_{M2}
  \,.
$$

=--

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

The $e^9$-component of the cocycle for the IIA-superstring (def. \ref{M2CoycleAndIIAStringCocycle}), regarded as an element in $CE(\mathbb{R}^{8,1}\vert \mathbf{16} + \mathbf{16})$, equals the 2-cocycle that defines the type IIB extension, according to prop. \ref{TypeIIExtension}:

$$
  (\pi_9)_\ast \mu_{IIA} = c_2^{IIB}
  \,.
$$

=--

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

We have

$$
  \begin{aligned}
    (\pi_9)_\ast \mu_{IIA}
      & =
    i\,
    \overline{\left(
      \array{\psi_1 \\ \psi_2}
    \right)}
      \Gamma_9 \Gamma_{10}
    \left(
      \array{\psi_1 \\ \psi_2}
    \right)
    \\
    & =
    \overline{\left(\array{\psi_1 \\ \psi_2}\right)}
    \Gamma_9^{IIB}
    \left(\array{\psi_1 \\ \psi_2}\right)
    \\
    & =
    c_2^{IIB}
  \end{aligned}
$$

where the first equality is by def. \ref{M2CoycleAndIIAStringCocycle}, the second is the statement of prop. \ref{9ComponentOfIIBPairing}, while the third is from prop. \ref{TypeIIExtension}.

=--

## Appendix

### Review of unitary representations with real structure

For reference, we here collect some basics regarding [[unitary representations]] equipped with [[real structure]].

All [[vector spaces]] in the following are taken to be [[finite dimensional vector spaces]].

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

Let $V$ be a [[complex vector space]]. A _[[real structure]]_ or _[[quaternionic structure]]_ on $V$ is a real-[[linear map]]

$$
  \phi \;\colon\; V \longrightarrow V
$$

such that

1. $\phi$ is conjugate linear ($\phi(\lambda v) = \overline{\lambda} \phi(v)$ for all $\lambda \in \mathbb{C}$, $v \in V$);

1. $\phi^2 = \left\{ \array{ +id & \text{for real structure} \\ -id & \text{for quaternionic structure} }  \right.$

=--

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

A real structure $\phi$, def. \ref{RealStructure}, on a complex vector space $V$ corresponds to a choice of complex linear isomorphism

$$
  V \simeq \mathbb{C} \otimes_{\mathbb{R}} V_+
$$

of $V$ with the [[complexification]] of a real vector space $V_+$, namely the [[eigenspace]] of $\phi$ for [[eigenvalue]] +1, while $V_- \coloneqq i V_+$ is the eigenspace of eigenvalue -1.

A quaternionic structure, def. \ref{RealStructure}, o $V$ gives it the structure of a left [[module]] over the [[quaternions]] extending the underlying structure of a module over the complex numbers. Namely let

1. $I \coloneqq i(-) \colon V \to V$ be the operation of acting with $i \in \mathbb{C}$

1. $J \coloneqq \phi \colon V \to V$ be the given endomorphisms,

then the conjugate complex linearity of $\phi$ means that

$$
  J \circ I = - I \circ J
$$

and hence with $J^2 = -1$ and $I^2 = -1$ this means that $I$, $J$ and $K \coloneqq I \circ J$ act like the imaginary quaternions.

=--

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

Let $G$ be a [[Lie group]], let $V$ be a [[complex vector space]] and let

$$
  \rho \;\colon\; G \longrightarrow GL_{\mathbb{C}}(V)
$$

be a complex [[linear representation]] of $G$ on $V$, hence a [[group homomorphism]] form $G$ to the [[general linear group]] of $V$ over $\mathbb{C}$.

Then a _real structure_ or _quaternionic structure_ on $(V,\rho)$ is a real or complex structure, respectively, $\phi$ on $V$ (def. \ref{RealStructure}) such that $\phi$ is $G$-invariant under $\rho$, i.e. such that for all $g \in G$ then

$$
  \phi \circ \rho(g) = \rho(g) \circ \phi
  \,.
$$

=--

We will be interested in complex [[finite dimensional vector spaces]] equipped with [[hermitian forms]], i.e. finite-dimensional complex [[Hilbert spaces]]:

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

A [[hermitian form]] (or symmetric complex [[sesquilinear form]]) $\langle -,-\rangle$ on a [[complex vector space]] $V$ is a real [[bilinear form]]

$$
  \langle
    -,-
  \rangle
  \;\colon\;
  V \times V
   \longrightarrow
  \mathbb{C}
$$

such that for all $v_1, v_2 \in V$ and $\lambda \in \mathbb{C}$ then

1. (sesquilinearity) $\langle v_1, \lambda v_2 \rangle = \lambda \langle v_1, v_2 \rangle $,

1. (conjugate symmetry) $\langle v_1, v_2\rangle^\ast = \langle v_2, v_1\rangle $.

1. (non-degeneracy) if $\langle v_1,-\rangle = 0$ then $v_1 = 0$.

A complex [[linear function]]  $f \colon V \to V$ is _[[unitary operator|unitary]]_ with respect to this hermitian form if it preserves it, in that

$$
  \langle f(-), f(-)\rangle
  =
  \langle -,-\rangle
  \,.
$$

Write

$$
  U(V) \hookrightarrow GL_{\mathbb{C}}(V)
$$

for the [[subgroup]] of [[unitary operators]] inside the [[general linear group]].

A complex [[linear representation]] $\rho \colon G \longrightarrow GL_{\mathbb{C}}(V)$ of a [[Lie group]] on $V$ is called a _[[unitary representation]]_ if it factors through this subgroup

$$
  \rho \;\colon\;  G \longrightarrow U(V) \hookrightarrow GL_{\mathbb{C}}(V)
  \,.
$$

=--

The following proposition uses assumptions stronger than what we have in the application to Majorana spinors (compact Lie group, positive definite hermitian form) but it nevertheless helps to see the pattern.

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

Let $V$ be a [[complex vector space|complex]] [[finite dimensional vector space]], $\langle -,-\rangle$ some [[positive definite bilinear form|positive definite]] [[hermitian form]] on $V$, def. \ref{HermitianForms}, let $G$ be a [[compact Lie group]], and $\rho \colon G \to U(V)$ a [[unitary representation]] of $G$ on $V$. Then $\rho$ carries a real structure or quaternionc structure $\phi$ on $\rho$ (def. \ref{RealStructureOnLinearRepresentation}) precisely if it carries a symmetric or anti-symmetric, respectively, non-degenerate complex-[[bilinear map]]

$$
  (-,-) \;\colon\; V \otimes_{\mathbb{C}} V \longrightarrow \mathbb{C}
  \,.
$$

Explicitly:

Given a real/quaternionic structure $\phi$, then the corresponding symmetric/anti-symmetric complex bilinear form is

$$
  (-,-) \coloneqq \langle \phi(-), -\rangle
  \,.
$$

Conversely, given $(-,-)$, first define $\tilde \phi$ by

$$
  (-,-) = \langle \tilde\phi(-),-\rangle
  \,,
$$

and then $\phi \coloneqq \frac{1}{\vert \phi\vert} \phi$ is the corresponding real/quaternionic structure.

If $\tilde\phi = \phi$ then $(-,-)$ is called _compatible_ with $\langle-,- \rangle$.

=--

(e.g. [Meinrenken 13, p. 81](#Meinrenken13))

## Related concepts

* [[charge conjugation matrix]]

* vaguely related only: *[[Majorana zero modes]]* in [[solid state physics]]

## References

The traditional component discussion in terms of a [[charge conjugation matrix]] is discussed for instance in

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], section II.7.3 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* {#VanProeyen99} [[Antoine Van Proeyen]], section 3 of _Tools for supersymmetry_ ([arXiv:hep-th/9910030](http://arxiv.org/abs/hep-th/9910030))

* {#Laenen} [[Eric Laenen]] _Classical Field Theory_ ([web](http://www.nikhef.nl/~t45/ftip/)) chapter on _Gamma matrices_ ([pdf](http://www.nikhef.nl/~t45/ftip/AppendixE.pdf))

The relation to the concept of real structures on complex Spin-representations is highlighted in

* {#FigueroaOFarrill} [[José Figueroa-O'Farrill]], _Majorana spinors_ ([pdf](http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/Majorana.pdf))

See also

* {#Meinrenken13} [[Eckhard Meinrenken]], _Clifford algebras and Lie theory_, Springer (2013)

* Theodor Bröcker, [[Tammo tom Dieck]], _Representations of Compact Lie Groups_, Springer (1985)

* Wikipedia, _[Majorana fermion](https://en.wikipedia.org/wiki/Majorana_fermion)_

The above discussion of cocycles on super-Minkowski spacetimes draws from

* {#CAIB99} C. Chryssomalakos, [[José de Azcárraga]], J. M. Izquierdo and C. Pérez Bueno, _The geometry of branes and extended superspaces_, Nuclear Physics B Volume 567, Issues 1–2, 14 February 2000, Pages 293–330 ([arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137))

* {#Sakaguchi00} Makoto Sakaguchi, section 2 of _IIB-Branes and New Spacetime Superalgebras_, JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](https://arxiv.org/abs/hep-th/9909143))

* {#FiorenzaSatiSchreiber13} [[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))

* {#FiorenzaSatiSchreiber16} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Rational sphere valued supercocycles in M-theory]]_, ([arXiv:1606.03206](https://arxiv.org/abs/1606.03206))

[[!redirects Majorana spinors]]
[[!redirects Majorana representation]]
[[!redirects Majorana representations]]

[[!redirects symplectic Majorana spinor]]
[[!redirects symplectic Majorana spinors]]
[[!redirects symplectic Majorana representation]]
[[!redirects symplectic Majorana representations]]

[[!redirects Majorana-Weyl spinor]]
[[!redirects Majorana-Weyl spinors]]
[[!redirects Majorana-Weyl representation]]
[[!redirects Majorana-Weyl representations]]

[[!redirects Majorana fermion]]
[[!redirects Majorana fermions]]

[[!redirects real spin representation]]
[[!redirects real spin representations]]


Majorana zero mode

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#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], the term "Majorana zero mode" (often abbreviated "MZM" or just "Majorana") has come to refer to (hypothetical and so far elusive) [[ground states]] of certain effectively 1-dimensional [[quantum materials]] (quantum/nano-wires) which are acted on by a "Majorana operator" (namely the [[Hermitian operator|Hermitian]] combination $c + c^\dagger$ of [[fermion]] [[canonical anti-commutation relations|annihilation/creation operators]], only vaguely related to [[special relativity|relativistic]] [[Majorana spinors]]) and which have been argued to potentially behave like [[Majorana anyons]], in some sense (beware that these modes, being stuck to wires, would not be mobile and hence would not admit [[adiabatic theorem|adiabatic]] [[braiding]] operations in the usual sense, see [here](https://quantumcomputing.stackexchange.com/q/25638/19363)).

These Majorana zero modes were theoretically introduced in a [[spin chain]] [[model (in theoretical physics)|model]] by [Kitaev 2001](#Kitaev01) ("Kitaev spin chain"), originally as a theoretical toy example for [[mass gap|gapped]] and degenerate [[ground states]] vaguely as expected for [[topological order]], but then argued to be realizable on interfaces of [[superconductors]] with certain [[topological insulators]] by [Fu & Kane 2008](#FuKane08); and their [[experiment|experimental]] realization in [[superconductor|super-]]/[[semiconductor|semi-conducting]] nano-wires has been proposed in [Lutchyn, Say & Das Sarma 2010](#LutchynSayDasSarma10), [Oreg, Refael, von Oppen 2010](#OregRefaelvonOppen10).

{#Following} Following these proposals and especially after [Microsoft Quantum](https://www.microsoft.com/en-us/research/research-area/quantum-computing) (with [QuTech](https://qutech.nl/) at TU Delft) declared ([Nov 2016](https://blogs.microsoft.com/ai/microsoft-doubles-quantum-computing-bet/)) the concrete aim of realizing [[topological quantum computation]] based on topological qbits given by such "Majorana zero modes" (following the plan laid out in [Das Sarma, Freedman & Nayak 15](#DasSarmaFreedmanNayak15)), the topic attracted [enormous attention](https://scholar.google.com/scholar?cites=12195292414684872731&as_sdt=2005&sciodt=0,5&hl=en) in [[solid state physics]].

But prominent claims of experimental detection of (these kinds of) Majorana zero modes had to be retracted:

* [by *Nature* in 2021](https://www.nature.com/articles/d41586-021-00612-z)

  [[doi:10.1038/s41586-021-03373-x](https://doi.org/10.1038/s41586-021-03373-x), [doi:10.5281/zenodo.4587841](https://doi.org/10.5281/zenodo.4587841), [doi:10.5281/zenodo.4545812](https://doi.org/10.5281/zenodo.4545812), [TU Delft press release](https://www.delta.tudelft.nl/article/majorana-not-fraud-confirmation-bias#)]

* [by *Nature* in 2022](http://retractionwatch.com/2022/04/24/authors-retract-second-majorana-paper-from-nature/) [[doi:10.1038/s41586-022-04704-2](https://doi.org/10.1038/s41586-022-04704-2)]

* [by *Science* in 2022](https://retractionwatch.com/2022/11/17/another-majorana-particle-paper-retracted-this-time-from-science/) [[doi:10.1126/science.adf7575](https://www.science.org/doi/10.1126/science.adf7575)], cf. further commentary by [[Sergey M. Frolov|S.M. Frolov]] [here](https://espressospin.org/2022/11/17/the-fallen-angel-particle/);

* and a range of further claims are under similar criticism, see [Frolov & Mourik 2020](https://doi.org/10.5281/zenodo.6344447), [Das Sarma & Pan 2021](#DasSarmaPan21), [Frolov, Mourik & Zuo 2022](https://doi.org/10.5281/zenodo.6325378) and the extensive list [here](https://twitter.com/spinespresso/status/1503352928656138241).

While some (probably most) researchers now dismiss the whole approach of "Majorana zero modes" (e.g. [BSSA21](#BSSA21), [p. 3](https://arxiv.org/pdf/2106.11840v4.pdf#page=3)) there is a new claim of detection by [Nayak 22](#Nayak22) & [MicrosoftQuantum 23](#MicrosoftQuantum23) -- but see cautionary commentary by [Frolov & Mourik 22a](#FrolovMourik22a), [22b](#FrolovMourik22a), [Frolov 22](#Frolov22) and [Das Sarma 22, p. 9](#DasSarma22).

## References

### General

The general strategy of realizing [[Majorana zero modes]] in supercondocuting/semiconducting nanowires is due to

* {#Kitaev01} [[Alexei Kitaev]], *Unpaired Majorana fermions in quantum wires*,  Physics-Uspekhi, **44** 10S (2001) 131-136 [[arXiv:cond-mat/0010440](https://arxiv.org/abs/cond-mat/0010440), [doi:10.1070/1063-7869/44/10S/S29](https://iopscience.iop.org/article/10.1070/1063-7869/44/10S/S29)]

* {#FuKane08} [[Liang Fu]], [[Charles Kane]], *Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator*, Phys. Rev. Lett. **100** (2008) 096407 [[doi:10.1103/PhysRevLett.100.096407](https://doi.org/10.1103/PhysRevLett.100.096407)]

* {#LutchynSayDasSarma10} Roman M. Lutchyn, Jay D. Sau, [[Sankar Das Sarma]], *Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures*, Phys. Rev. Lett. **105** (2010) 077001 [[doi:10.1103/PhysRevLett.105.077001](https://doi.org/10.1103/PhysRevLett.105.077001)]

* {#OregRefaelvonOppen10} Yuval Oreg, Gil Refael, and Felix von Oppen, *Helical Liquids and Majorana Bound States in Quantum Wires*, Phys. Rev. Lett. **105** (2010) 177002 [[doi:10.1103/PhysRevLett.105.177002](https://doi.org/10.1103/PhysRevLett.105.177002)]

* [[Lukasz Fidkowski]], [[Alexei Kitaev]], *The effects of interactions on the topological classification of free fermion systems*, Phys. Rev. B **81** (2010) 134509  [[arXiv:0904.2197](https://arxiv.org/abs/0904.2197), [doi:10.1103/PhysRevB.81.134509](https://doi.org/10.1103/PhysRevB.81.134509)]

  > ([[interaction|interacting]] generalization)

reviewed in:

* [[Pasquale Marra]], *Majorana nanowires for topological quantum computation: A tutorial*, J. of Applied Physics **132** (2022) 231101   [[arXiv:2206.14828](https://arxiv.org/abs/2206.14828), [doi:10.1063/5.0102999](https://doi.org/10.1063/5.0102999)]

Discussion in the context of [[topological quantum computation]]:

* {#DasSarmaFreedmanNayak15} [[Sankar Das Sarma]], [[Michael Freedman]], [[Chetan Nayak]], *Majorana zero modes and topological quantum computation*, npj Quantum Inf **1** 15001 (2015) $[$[doi:10.1038/npjqi.2015.1](https://doi.org/10.1038/npjqi.2015.1)$]$

General review and experimental status:

* [[Sergey M. Frolov]], *How do we discover MAJORANA PARTICLES in NANOWIRES?*, talk via [ Instituto de Física, Universidade de São Paulo](http://portal.if.usp.br/ifusp/) (May 2021) [[video](https://www.youtube.com/watch?v=7btnPqmSh_0)]

  > [41:25](https://youtu.be/7btnPqmSh_0?t=2485): No, we have not discovered Majorana particles in nanowires. Yes, we should be able to do it.

* {#DasSarma22} [[Sankar Das Sarma]], *In search of Majorana*, Nature Physics **19** (2023) 165-170 [[arXiv:2210.17365](https://arxiv.org/abs/2210.17365), [doi:10.1038/s41567-022-01900-9](https://doi.org/10.1038/s41567-022-01900-9)]

See also:

* Wikipedia, *[Majorana bound states](https://en.wikipedia.org/wiki/Majorana_fermion#Majorana_bound_states)*

### Non-Detection
  {#ReferencesNonDetection}

On the general problem of distinguishing the expected effect from noise:

* {#DasSarmaPan21} [[Sankar Das Sarma]], Haining Pan, *Disorder-induced zero-bias peaks in Majorana nanowires*, Phys. Rev. B **103** 195158 (2021) [[arXiv:2103.05628](https://arxiv.org/abs/2103.05628), [doi:10.1103/PhysRevB.103.195158](https://doi.org/10.1103/PhysRevB.103.195158)]

> we believe that similar confirmation bias applies to many other topological discovery claims in the literature during 2000–2020 where a precise knowledge of what one is looking for has been the key factor in the discovery claim, with the experimental quantization results themselves not being sufficiently compelling. [...] Our results certainly apply to most of the Majorana experiments during 2012–2021 in the literature.

* {#BSSA21} Koen Bertels, Tamara Sarac, Aritra Sarkar, Imran Ashraf, *Quantum Computing -- from NISQ to PISQ*, IEEE Micro **41** (2021) 24-32 [[arXiv:2106.11840v4](https://arxiv.org/abs/2106.11840v4), [doi:10.1109/MM.2021.3099195](https://doi.ieeecomputersociety.org/10.1109/MM.2021.3099195)]

> [p. 3](https://arxiv.org/pdf/2106.11840v4.pdf#page=3): The quantum physics community is sufficiently aware that when certain qubit technologies do not produce any reasonable result after several years of effort, they should be gently removed from the list of quantum candidates. After working with the physics colleagues in Delft, we saw that happening with the Majorana qubits that could not be confirmed in any follow-up experiment.

* {#FrolovMourik22a} [[Sergey M. Frolov]], [[Vincent Mourik]], *We cannot believe we overlooked these Majorana discoveries* [[arXiv:2203.17060](https://arxiv.org/abs/2203.17060), [doi:10.5281/zenodo.6364928](https://doi.org/10.5281/zenodo.6364928), conclusion on: [p. 7](https://arxiv.org/ftp/arxiv/papers/2203/2203.17060.pdf#page=7)]

Non-retracted claims of experimental realization of something in the direction of Majorana zero modes:

* {#MMBDRSC19} Gerbold C. Ménard, Andrej Mesaros, Christophe Brun, François Debontridder, Dimitri Roditchev, Pascal Simon, Tristan Cren, *Isolated pairs of Majorana zero modes in a disordered superconducting lead monolayer*,  Nat Commun **10** 2587 (2019) $[$[doi:10.1038/s41467-019-10397-5](https://doi.org/10.1038/s41467-019-10397-5)$]$

* {#Nayak22} [[Chetan Nayak]], *Microsoft has demonstrated the underlying physics required to create a new kind of qubit*, Microsoft Research Blog (March 2022)

* {#MicrosoftQuantum23} M. Aghaee et al. ([Microsoft Quantum](https://www.microsoft.com/en-us/research/research-area/quantum-computing)), *InAs-Al Hybrid Devices Passing the Topological Gap Protocol*, Phys. Rev. B **107** 245423 (2023) [[doi:10.1103/PhysRevB.107.245423](https://doi.org/10.1103/PhysRevB.107.245423), [arXiv:2207.02472](https://arxiv.org/abs/2207.02472), [video presentation](https://www.microsoft.com/en-us/research/video/inas-al-hybrid-devices-passing-the-topological-gap-protocol/)]

  \linebreak

  with an accompanying caveat editorial:

  Randall D. Kamien, Jessica Thomas, Stephen E. Nagler, Anthony M. Begley, and Sarma Kancharla, *Editorial: Transparency in Physical Review Articles*, Phys. Rev. B **107** 210001 (2023) [[doi:10.1103/PhysRevB.107.210001](https://doi.org/10.1103/PhysRevB.107.210001)]

  saying:

  > "In this issue of Physical Review B, Aghaee et al. [[1](#MicrosoftQuantum23)] report on an advancement towards the goal of topological quantum computing. While Physical Review readers are well aware that the many minutiæ of procedures, computations, and synthesis may be omitted in any particular dispatch, in this publication the intellectual property of the authors’ employer has prevented the release of some parameters of the studied devices that may be needed in order to reproduce them. As a reflection of the traditional values of the scholarly community, this is not in accordance with the usual norms of the Physical Review journals."

but see commentary in:

* [[Sergey Frolov]], *[Twitter:1671558089382957056](https://twitter.com/spinespresso/status/1671558089382957056) * (21 June 2023)

* Karmela Padavic-Callaghan, *Microsoft says its weird new particle could improve quantum computers --- Researchers at Microsoft say they have created elusive quasiparticles called Majorana zero modes -- but scientists outside the company are sceptical*, New Scientist (21 June 2023) [[web](https://www.newscientist.com/article/2378782-microsoft-says-its-weird-new-particle-could-improve-quantum-computers)]

and earlier:

* {#FrolovMourik22b} [[Sergey M. Frolov]], [[Vincent Mourik]]: *Majorana Fireside Podcast, Episode 1: The Microsoft TGP paper live review* [[video](https://www.youtube.com/watch?v=RnYghkDaHH0), conclusion at: [1:01:31](https://www.youtube.com/watch?v=RnYghkDaHH0&t=3691s)]

  > [1:01:52](https://youtu.be/RnYghkDaHH0?t=3712) The signal is fully consistent, from what we see, with *not* having discovered any Majorana or topological superconductivity here. At the same time, the amount of data is extremely narrow.

* {#Frolov22} [[Sergey M. Frolov]], *Superconductors and semiconductors, nanowires and majorana, research and integrity* [[video](https://www.youtube.com/watch?v=FR7J4yH67GA), general caution: [55:34](https://youtu.be/FR7J4yH67GA?t=3334), concrete criticism: [1:01:41](https://youtu.be/FR7J4yH67GA?t=3701)]

  > [1:01:50](https://youtu.be/FR7J4yH67GA?t=3710): The claims of the discovery of Majorana have been overblown and are false. Majorana has not been discovered in nanowires. I don't believe in any other system it has been discovered either.

On how this could happen:

* Elizabeth Gibney, *Inside Microsoft’s quest for a topological quantum computer* (Interview with Alex Bocharov), Nature (2016) [[doi:10.1038/nature.2016.20774](https://doi.org/10.1038/nature.2016.20774)]

  > [Bocharov:] We’re people-centric, rather than problem-centric.

See also:

* {#Frolov21} [[Sergey M. Frolov]], *So, You Think You Discovered a New State of Matter?*, Physics **14** 68 (2021) [[physics.aps:v14/68](https://physics.aps.org/articles/v14/68)]

* [[Sergey M. Frolov]], *Quantum computing’s reproducibility crisis: Majorana fermions*, Nature **592** (2021) 350-352 [[doi:10.1038/d41586-021-00954-8](https://doi.org/10.1038/d41586-021-00954-8)]

[[!include anyons in topological superconductors -- references]]

[[!redirects Majorana zero modes]]

[[!redirects Kitaev spin chain]]
[[!redirects Kitaev spin chains]]


Makkai duality

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{: toc}

## Idea

**Makkai duality** is a kind of [[syntax-semantics duality]], due to [[Michael Makkai]], relating [[pretoposes]] (categories having to do with the [[syntax]] of [[first-order logic]]) and [[ultracategories]] (which are a way of capturing the [[semantics]] of first-order logic). This leads to a proof of [[conceptual completeness]] for first-order logic.

## Related entries

* [[Łoś ultraproduct theorem]]

* [[Stone duality]]

## References

  * [[Michael Makkai]], _Stone duality for first order logic_, Proceedings of the Herbrand symposium (Marseilles, 1981), 217–232, Stud. Logic Found. Math. __107__, North-Holland, Amsterdam, 1982, doi;

  * [[Michael Makkai]], _Stone duality for first order logic_, Adv. Math. __65__ (1987) no. 2, 97--170, doi, [MR89h:03067](http://www.ams.org/mathscinet-getitem?mr=900266);

  * [[Michael Makkai]], _Duality and definability in first order logic_, Mem. Amer. Math. Soc. __105__ (1993), no. 503

* [[Jacob Lurie]], _Lecture 29X-Makkai Duality_, ([lecture notes](http://www.math.harvard.edu/~lurie/278xnotes/Lecture29X-ConceptualCompleteness.pdf))

* [[Jacob Lurie]], _Ultracategories_, ([pdf](http://www.math.harvard.edu/~lurie/papers/Conceptual.pdf))

* [[Francisco Marmolejo]], _Ultraproducts and continuous families of models_, ([thesis](https://dalspace.library.dal.ca/handle/10222/55092), poor quality photocopy).

Some more general variants are achieved in

* Marek W. Zawadowski, _Descent and duality_, Annals of Pure and Applied Logic __71__, n.2 (1995), 131–188

* Henrik Forssell, _First-order logical duality_, Ph.D. thesis, Carnegie Mellon U. 2008, [pdf](http://www.andrew.cmu.edu/user/awodey/students/forssell.pdf)


Makoto Natsuume

* [webpage](http://research.kek.jp/people/natsuume/index-e.html)

## selected writings

* {#Natsuume15} [[Makoto Natsuume]], _AdS/CFT Duality User Guide_, Lecture Notes in Physics 903, Springer 2015 ([arXiv:1409.3575](https://arxiv.org/abs/1409.3575))

## related $n$Lab entries

* [[AdS-CFT duality]]

* [[anti-de Sitter spacetime]]

category: people


Malament–Hogarth spacetime

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{:toc}

## Idea

It has been argued that there are [[spacetimes]] which are such that they allow [[trajectories]] on which a [[computation|computing device]] could travel indefinitely together with spacetime points at which an observer could observe the whole infinite history of the computer in finite time. If this indeed were physically realizable it would to some extent physically implement what is called _[[hypercomputation]]_. This would contradict, to some extent, the [[physical Church-Turing thesis]], which asserts that no physical process can realize a computer more powerful than a Turing machine.

## References

Original articles include

* M. Hogarth, _Does General Relativity Allow an Observer to View an Eternity in a Finite Time?_, Foundations of Physics Letters, 5, 173–181 (1992)

Reviews include

* Wikipedia, _[Malament–Hogarth spacetime](http://en.wikipedia.org/wiki/Malament-Hogarth_spacetime)_

See also

* [[Gabor Etesi]], Istvan Nemeti, _Non-Turing computations via Malament-Hogarth space-times_, Int.J.Theor.Phys. 41 (2002) 341-370 ([arXiv:gr-qc/0104023](http://arxiv.org/abs/gr-qc/0104023))

Discussion relating to the [[cosmic censorship hypothesis]] is in

* {#Etesi12} [[Gabor Etesi]], _A proof of the Geroch-Horowitz-Penrose formulation of the strong cosmic censor conjecture motivated by computability theory_ ([arXiv:1205.4550](http://arxiv.org/abs/1205.4550))

[[!redirects Malament–Hogarth spacetimes]]


Malcev category

[[!redirects Mal'cev category]]
[[!redirects Mal'cev category]]

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## Definition

A __Malcev category__ is a [[finitely complete category|left exact category]] (= having [[finite limits]]) in which any [[reflexive relation|reflexive]] [[internal relation]] is an [[congruence|equivalence relation]]. Equivalently, the [[fibers]] of its [[fibration of points]] are [[unital category|unital]] (equivalently the fibers of the fibration of points are strongly unital).

## Examples

\begin{example}\label{GroupsFormAMalcevCategory}
 The category [[Grp]] of all [[groups]] (including [[non-abelian groups]]) is a Malcev category ([Borceux & Bourn 2004, Ex. 2.2.4](#BorceuxBourn04)). More generally, for $\mathcal{C}$ any category with [[finite limits]], the category $Grp(\mathcal{C})$ of [[group objects]] [[internalization|internal]] to $\mathcal{C}$ is a Malcev category.
\end{example}

More generally, the category of $T$-algebras for any [[Lawvere theory]] $T$ which contains a group operation (an $\Omega$-[[Omega-group|group]]). Other examples include the category, $Heyt$, of [[Heyting algebra]]s and the category of left closed [[magma]]s. The dual category to an [[elementary topos]] is a Malcev category. ([Borceux & Bourn 2004, Ex. 2.2.5-7](#BorceuxBourn04))  A [[Malcev variety]] is a [[variety of algebras]] whose category of models is a Malcev category.

## Properties

\begin{proposition}
In any Malcev category, every [[internal category]] is an [[internal groupoid]].
\end{proposition}

\begin{proposition}
The category of [[internal categories]] and functors in a Mal’cev category is Mal’cev category. ([Gran 1999, Theorem 3.2](#Gran99))
\end{proposition}

## Related notions

* [[subtractive category]]

* [[Goursat category]]

## References

* {#BorceuxBourn04} [[Francis Borceux]], [[Dominique Bourn]], _[[Mal'cev, protomodular, homological and semi-abelian categories]]_, Mathematics and Its Applications __566__, Kluwer 2004 ([doi:10.1007/978-1-4020-1962-3](https://link.springer.com/book/10.1007/978-1-4020-1962-3))

* [[Dominique Bourn]], [_From Groups to Categorial Algebra : Introduction to Protomodular and Mal’tsev Categories_](https://doi.org/10.1007/978-3-319-57219-2), Compact Textbooks in Mathematics, Birkhäuser 2017 (textbook)

* {#Gran99} [[Marino Gran]], _Internal categories in Mal’cev categories_, Journal of Pure and Applied Algebra **143** 1-3 (1999) 221-229 [doi:10.1016/S0022-4049(98)00112-1]

[[!redirects Mal'cev categories]]
[[!redirects Mal'cev category]]
[[!redirects Mal'cev category]]
[[!redirects Malʹcev category]]
[[!redirects Malcev category]]
[[!redirects Maltsev category]]
[[!redirects Мальцев category]]


Malcev completion

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## Idea

In a 1949 paper devoted to the study of the coset spaces of nilpotent Lie groups, [[Anatoly Malcev]] exhibited an equivalence between the category of torsion free radicable nilpotent finite rank groups and the category of finite dimensional, nilpotent rational Lie algebras. This involves a completion construction which is used also in the general formulation of [[Hausdorff series]] (cf. Bourbaki) which takes values in the Mal'cev completion of the universal enveloping algebra on two generators.

## Definition

(There are several variants of the definition.)

Mal'cev completion is a left [[adjoint functor]] to the embedding of the category of uniquely [[divisible group|divisible]] [[nilpotent groups]] into the category of nilpotent groups.

## Applications

[[Hausdorff series]], [[rational homotopy theory]], surgery, combinatorial group theory

## Related concepts

* [[completion of a group]]

* [[rationalization]], [[rational homotopy theory]]

## References

* [[A. I. Mal'cev]], _On a class of homogeneous spaces_, Izvestiya Akad. Nauk. SSSR. Ser. Mat. __13__, (1949) 9–32, [MR28842](http://www.ams.org/mathscinet-getitem?mr=28842).
Original reference: [[А. И. Мальцев]], _Об одном классе однородных пространств_, Известия Академии наук СССР. Серия математическая 13:1 (1949), 9–32.  [PDF](http://mi.mathnet.ru/izv3161).

* [[Nicolas Bourbaki]], _Lie groups and Lie algebras_

* Jaume Amorós, _On the Malcev completion of Kähler groups_,
Commentarii Mathematici Helvetici __71__, n. 1, 192-212, [doi](http://dx.doi.org/10.1007/BF02566416)
[alg-geom/9410013](http://arxiv.org/abs/alg-geom/9410013)

* Robert B. Warfield, _The Malcev correspondence_, Ch. 12 in __Nilpotent Groups__, Springer Lec. Notes in Math. __513__, 1976, 104-111, [doi](http://dx.doi.org/10.1007/BFb0080164)

* Bohumil Cenkl, Richard Porter, _Mal'cev's completion of a group and differential forms_, [MR628342](http://www.ams.org/mathscinet-getitem?mr=628342), [euclid](http://projecteuclid.org/euclid.jdg/1214435841)

* [[Benoit Fresse]], _Operads & Grothendieck-Teichmüller groups - draft document_, [pdf](https://hal.archives-ouvertes.fr/hal-00656333)

[[!redirects Malcev completions]]
[[!redirects Mal'cev completion]]
[[!redirects Mal'cev completions]]
[[!redirects Maltsev completion]]
[[!redirects Maltsev completions]]
[[!redirects Mal'tsev completion]]
[[!redirects Mal'tsev completions]]

[[!redirects Mal'cev completion]]
[[!redirects Mal'cev completion]]


Malcev group

## Definition

A __Malcev group__ is a [[filtered group]] $G=F_1 G\supset F_2 G\supset\cdots$
such that

* the [[associated graded]] [[Lie algebra]] $gr(G)$ over [[integers]]
is uniquely divisible, hence a rational [[Lie algebra]];

* the graded rational [[Lie algebra]] $gr(G)$ is generated as a [[Lie algebra]] by its degree 1 component;

* the [[filtered object]] $G$ is complete: the canonical map $G\to lim_n G/F_n G$
is an [[isomorphism]].

Morphisms of Malcev groups are morphisms of [[filtered groups]].

## Properties

The category of Malcev groups is equivalent to the category of rational [[complete Hopf algebras]].  See there for more information.

## Related concepts

* [[Malcev Lie algebra]]

* [[complete Hopf algebra]]

## References

* [[А. И. Мальцев]], _Нильпотентные группы без кручения_, Изв. АН СССР. Сер. матем. 13:3 (1949), 201–212.  [PDF](http://mi.mathnet.ru/izv3184).

* [[Daniel Quillen]], _Rational homotopy theory_, Annals of Mathematics 90:2 (1969), 205.  [doi](http://dx.doi.org/10.2307/1970725).

[[!redirects Malcev groups]]


Malcev Lie algebra

## Definition

We work over a [[field]] of characteristic 0, such as the [[rationals]].

A __Malcev Lie algebra__ is a [[filtered Lie algebra]] $\mathfrak{g}=F_1 \mathfrak{g}\supset F_2 \mathfrak{g}\supset\cdots$
such that

* the graded [[Lie algebra]] $gr(\mathfrak{g})$ is generated as a [[Lie algebra]] by its degree 1 component;

* the [[filtered object]] $\mathfrak{g}$ is complete: the canonical map $\mathfrak{g}\to lim_n \mathfrak{g}/F_n \mathfrak{g}$
is an [[isomorphism]].

Morphisms of Malcev Lie algebras are morphisms of [[filtered Lie algebras]].

## Properties

The category of Malcev Lie algebras is equivalent to the category of [[complete Hopf algebras]].  See there for more information.

## Related concepts

* [[Malcev group]]

* [[Malcev completion]]

* [[complete Hopf algebra]]

## References

* [[А. И. Мальцев]], _Нильпотентные группы без кручения_, Изв. АН СССР. Сер. матем. 13:3 (1949), 201–212.  [PDF](http://mi.mathnet.ru/izv3184).

* [[Daniel Quillen]], _Rational homotopy theory_, Annals of Mathematics 90:2 (1969), 205.  [doi](http://dx.doi.org/10.2307/1970725).

[[!redirects Malcev Lie algebras]]