Spin(5)

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### Context
#### Group Theory
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[[!include group theory - contents]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[spin group]] in [[dimension]] 5.

## Properties

### Exceptional isomorphism

+-- {: .num_prop #ExceptionalIsoToSp2}
###### Proposition

There is an [[exceptional isomorphism]]

$$
  Spin(5)
  \;\simeq\;
  Sp(2)
$$

between [[Spin(5)]] and the [[quaternionic unitary group]] $Sp(2) = U(2,\mathbb{H})$.

=--

This is an indirect consequence of [[triality]], see e.g. [Čadek-Vanžura 97](#CadekVanzura97).  Alternatively, it can be shown as follows.

Let $V$ be a 4-dimensional complex vector space with an inner product and a compatible complex volume form.    As explained [here](https://ncatlab.org/nlab/show/Spin%286%29#ExceptionalIsomorphism), this structure can be used to define a conjugate-linear Hodge star operator on $\Lambda^2 V$ whose $+1$ and $-1$ eigenspaces, say $\Lambda_{\pm}^2 V$, are each 6-dimensional *real* inner product spaces.  Thus, the group $\mathrm{SU}(V)$ acts as linear transformations of $\Lambda_{\pm}^2 V$ that preserve the inner product, giving a homomorphism $\rho: \mathrm{SU}(V) \to \mathrm{O}(\Lambda_{\pm}^2 V)$.  In fact $\rho$ maps $\mathrm{SU}(V)$ in a 2-1 and onto way to $\mathrm{SO}(\Lambda_{\pm}^2 V)$.  Taking $V = \mathbb{C}^4$ this shows $\mathrm{SU}(4) \cong \mathrm{Spin}(6)$.

Now suppose $V$ is additionally equipped with an complex symplectic structure, i.e. a nondegenerate skew-symmetric complex-bilinear form
$J \in \Lambda^2 V$. The subgroup $\mathrm{Sp}(V)$ of $\mathrm{SU}(V)$ preserving this extra structure is isomorphic to the [[compact symplectic group]] $\mathrm{Sp}(2)$, which is also the [[quaternionic unitary group]].   This subgroup $\mathrm{Sp}(V)$ acts on $\Lambda_+^2 V$ and $\Lambda_-^2 V$
preserving $J \in \Lambda^2 V = \Lambda_+^2 V \oplus \Lambda_-^2 V$.  Since $\mathrm{Sp}(V)$ is compact, every invariant subspace has an invariant complement, so one or both of the 6-dimensional subspaces $\Lambda_+^2 V$ and $\Lambda_-^2 V$ must have a 5-dimensional subspace invariant under the action of $\mathrm{Sp}(V)$.  This shows that the double cover $\rho: \mathrm{SU}(4) \to \mathrm{SO}(6)$ restricts to a 2-1 homomorphism $\sigma : \mathrm{Sp}(2) \to \mathrm{SO}(5)$.   Since
$$   \dim(\mathrm{Sp}(2)) = 10 = \dim(\mathrm{SO}(5)) $$
the differential $d\sigma$, being injective, must also be surjective.  Thus $\sigma : \mathrm{Sp}(2) \to \mathrm{SO}(5)$ is actually a double cover.  Since $\mathrm{Sp}(2)$ is connected this implies $\mathrm{Sp}(2) \cong \mathrm{Spin}(5)$.

### Action on quaternionic Hopf fibration

+-- {: .num_prop #Spin5EquivarianceOfQuaternionicHopfFibration}
###### Proposition
**([[Spin(5)]]-[[action|equivariance]] of [[quaternionic Hopf fibration]])**

Consider

1. the [[Spin(5)]]-[[action]] on the [[4-sphere]] $S^4$ which is induced by the defining action on $\mathbb{R}^5$ under the identification $S^4 \simeq S(\mathbb{R}^5)$;

1. the [[Spin(5)]]-action on the [[7-sphere]] $S^7$ which is induced under the exceptional isomorphism $Spin(5) \simeq Sp(2) = U(2,\mathbb{H})$ (from Prop. \ref{ExceptionalIsoToSp2}) by the canonical left action of $U(2,\mathbb{H})$ on $\mathbb{H}^2$ via $S^7 \simeq S(\mathbb{H}^2)$.

Then the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ is [[equivariant]] with respect to these [[actions]].

=--

This is almost explicit in [Porteous 95, p. 263](#Porteous95)

\begin{center}
\begin{imagefromfile}
  "file_name": "Spin5OnHopfH.jpg",
  "width": 280
\end{imagefromfile}
\end{center}

### Cohomology

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

The [[integral cohomology|integral]] [[cohomology ring]] of the [[classifying space]] $B Spin(5)$ is spanned by two generators

1. the [[first fractional Pontryagin class]] $\tfrac{1}{2}p_1$

1. the [[linear combination]] $\tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2$ of the half the [[second Pontryagin class]] with half the [[cup product]]-square of the [[first Pontryagin class]]:

$$
  H^\bullet
  \big(
    B Spin(5), \mathbb{Z}
  \big)
  \;\simeq\;
  \mathbb{Z}
  \left[
    \tfrac{1}{2}p_1,
    \;
    \tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2
  \right]
$$

=--

This is a special case of the general statement in [Pittie 91](#Pittie91), see e.g. [Kalkkinen 06, Section 3](#Kalkkinen06)).

\linebreak

+-- {: .num_prop  #FiberIntegrationOfCupPowersOfChiOver4Sphere}
###### Proposition

Let

$$
  \array{
    S^4
    &\longrightarrow&
    B Spin(4)
    \\
    &&
    \big\downarrow^{\mathrlap{\pi}}
    \\
    &&
    B Spin(5)
  }
$$

be the [[spherical fibration]] of [[classifying spaces]] induced from the canonical inclusion of [[Spin(4)]] into [[Spin(5)]] and using that the [[4-sphere]] is equivalently the [[coset space]] $S^4 \simeq Spin(5)/Spin(4)$ ([this Prop.](sphere#nSphereAsCosetSpace)).

Then the [[fiber integration]] of the odd [[cup product|cup powers]] $\chi^{2k+1}$ of the [[Euler class]] $\chi \in H^4\big( B Spin(4), \mathbb{Z}\big)$ (see [this Prop](Spin4#IntegralCohomologyOfClassifyingSpace))  are proportional to [[cup product|cup powers]] of the [[second Pontryagin class]]

$$
  \pi_\ast
  \left(
    \chi^{2k+1}
  \right)
  \;=\;
  2 \big( p_2 \big)^k
  \;\;\in\;\;
  H^4\big( B Spin(5), \mathbb{Z} \big)
  \,,
$$

for instance

$$
  \begin{aligned}
    \pi_\ast
    \big(
      \chi
    \big)
    & =
    2
    \\
    \pi_\ast
    \left(
      \chi^3
    \right)
    & =
    2 p_2
    \\
    \pi_\ast
    \left(
      \chi^5
    \right)
    & =
    2 (p_2)^2
  \end{aligned}
    \;\;\in\;\;
    H^4\big( B Spin(5), \mathbb{Z} \big)
  \,;
$$

while the [[fiber integration]] of the even [[cup product|cup powers]] $\chi^{2k}$ vanishes

$$
  \pi_\ast
  \left(
    \chi^{2k}
  \right)
  \;=\;
  0
  \;\;\in\;\;
  H^4\big( B Spin(5), \mathbb{Z} \big)
  \,.
$$

=--

([Bott-Cattaneo 98, Lemma 2.1](#BottCattaneo98))

### Coset spaces

[[!include coset space structure on n-spheres -- table]]

### $G$-Structure and exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

\linebreak

## Related concepts

* [[Spin(5,1)]]

[[!include low dimensional rotation groups -- table]]

linebreak

## References

* {#Porteous95} [[Ian Porteous]], _Clifford Algebras and the Classical Groups_, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1995)

* {#CadekVanzura97} [[Martin Čadek]], [[Jiří Vanžura]], _On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures  in 8-dimensional vector bundles_, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 ([jstor:43737249](https://www.jstor.org/stable/43737249))

* {#BottCattaneo98} [[Raoul Bott]], [[Alberto Cattaneo]], _Integral Invariants of 3-Manifolds_, J. Diff. Geom. **48** (1998) 91-133 [[arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001), [doi:10.4310/jdg/1214460608](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-48/issue-1/Integral-invariants-of-3-manifolds/10.4310/jdg/1214460608.full)]

* {#Pittie91} [[Harsh Pittie]], _The integral homology and cohomology rings of SO(n) and Spin(n)_, Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E)

* {#Kalkkinen06} [[Jussi Kalkkinen]], _Global Spinors and Orientable Five-Branes_, JHEP 0609: 028, 2006 ([arXiv:hep-th/0604081](https://arxiv.org/abs/hep-th/0604081))

[[!redirects Sp(2)]]

[[!redirects Spin5]]


Spin(5)/SU(2) is the 7-sphere

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#### Geometry
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* table of contents
{:toc}

## Statement

The [[coset space]] of [[Spin(5)]] by its [[subgroup]] [[SU(2)]] is [[diffeomorphism|diffeomorphic]] to the standard [[7-sphere]]:

\[
  \label{Spin5BySU2}
  Spin(5)/SU(2)
  \;\simeq_{diff}\;
  S^7
  \,.
\]

This is however not an [[isometry]] to the standard [[Riemannian manifold]]-[[structure]] ("[[round n-sphere]]"), whence one speaks of a [[squashed n-sphere]].

The identification (eq:Spin5BySU2) follows via the exceptional [[isomorphisms]]

[[Spin(5)]] $\simeq$ [[Sp(2)]]

and

[[SU(2)]] $\simeq$ [[Sp(1)]]

as a special case of the general statement

$$
  S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)
$$

(see [this Prop.](sphere#SphereAsSymplecticUnitaryCoset)).

## Related concepts

[[!include coset space structure on n-spheres -- table]]

## References

* {#GrayGreen70} [[Alfred Gray]], Paul S. Green, p. 2 of _Sphere transitive structures and the triality automorphism_, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 ([euclid:1102976640](https://projecteuclid.org/euclid.pjm/1102976640))

Discussion of this [[squashed n-sphere|squashed]] 7-sphere [[coset space]] as a [[fiber]] for [[KK-compactification]] of [[11-dimensional supergravity]]:

* {#AwadaDuffPope83} M. A. Awada, [[Mike Duff]], [[Christopher Pope]], _$N=8$ Supergravity Breaks Down to $N=1$_, Phys. Rev. Lett. 50, 294 – Published 31 January 1983 ([doi:10.1103/PhysRevLett.50.294](https://doi.org/10.1103/PhysRevLett.50.294))

* {#DuffNilssonPope83} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere_, Phys. Rev. Lett. 50, 2043 – Published 27 June 1983; Erratum Phys. Rev. Lett. 51, 846  ([doi:10.1103/PhysRevLett.50.2043](https://doi.org/10.1103/PhysRevLett.50.2043))


Spin(5,1)

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#Contents#
* table of contents
{:toc}

## Idea

The [[Lorentz group|Lorentzian]] [[spin group]] in [[dimension]] $5 + 1$

## Properties

### Isomorphism with $SL(2,\mathbb{H})$

See at _[[SL(2,H)]]_.

[[!include exceptional spinors and division algebras -- table]]

## Related concepts

[[!include low dimensional rotation groups -- table]]

## References

Via [[quaternion]]-calculus and [[SL(2,H)]]:

* [[Joás Venâncio]], [[Carlos Batista]], _Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes_, Adv. Appl. Clifford Algebras **31** 71 (2021) ([arXiv:2007.04296](https://arxiv.org/abs/2007.04296), [doi:10.1007/s00006-021-01172-1](https://doi.org/10.1007/s00006-021-01172-1))


Spin(6)

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### Context
#### Group Theory
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[[!include group theory - contents]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[spin group]] in [[dimension]] 6.

## Properties

### Exceptional isomorphism
 {#ExceptionalIsomorphism}

+-- {: .num_prop #IsomorphismBetweenSpin6AndSU4}
###### Proposition

There is an [[exceptional isomorphism]]

\begin{tikzpicture}

  \node (Spin6) at (0,1.4)  {$\mathrm{Spin}(6)$};
  \node (SU4) at (3.4,1.4)  {$\mathrm{SU}(4)$};

  \node at (1.7,1.4) {$\simeq$};

  \node (center) at (0,0) {};
  \node (topright) at (30:1) {};
  \node (left) at (180-30:1) {};
  \node (botright) at (0,-1) {};

  \draw[fill=black] (center) circle (.1);
  \draw[fill=black] (topright) circle (.1);
  \draw[draw=lightgray, fill=lightgray] (botright) circle (.1);
  \draw[fill=black] (left) circle (.1);

  \draw (center) to (topright);
  \draw[lightgray] (center) to (botright);
  \draw (center) to (left);

  \begin{scope}[shift={(3.4,0)}]
  \node (center) at (0,0) {};
  \node (left) at (-1,0) {};
  \node (right) at (+1,0) {};

  \draw[fill=black] (center) circle (.1);
  \draw[fill=black] (left) circle (.1);
  \draw[fill=black] (right) circle (.1);

  \draw (center) to (left);
  \draw (center) to (right);
  \end{scope}

\end{tikzpicture}

between [[Spin(6)]] and [[SU(4)]], reflecting, under the [[classification of simple Lie groups]], the coincidence of the [[Dynkin diagrams]] "[[D3]]" and [[A3]].

=--

(e.g. [Figueroa-O'Farrill 10, Lemma 8.1](#FigueroaFarrill10))

One way to see the isomorphism $\mathrm{Spin}(6) \cong \mathrm{SU}(4)$ is as follows.  Let $V$ be a 4-dimensional [[complex vector space]] with an [[inner product]] and a compatible complex [[volume form]], meaning an element of the [[exterior product]] $\Lambda^4 V$ whose [[norm]] is 1 in the norm coming from the inner product on $V$. The inner product defines a conjugate-linear isomorphism $V \cong V^\ast$ (with the complex [[dual vector space]]) that together with the complex volume form can be used to define a conjugate-linear [[Hodge star operator]] on $\Lambda^2 V$.  This Hodge star operator squares to the identity, and its $+1$ and $-1$ eigenspaces, say $\Lambda_{\pm}^2 V$, each become 6-dimensional *real* inner product spaces in a natural way.  Thus, the group $\mathrm{SU}(V)$, consisting of all complex-linear transformations of $V$ that preserve the inner product and complex volume form, acts as linear transformations of $\Lambda_+^2 V$ that preserve the inner product, giving a homomorphism $\rho: \mathrm{SU}(V) \to \mathrm{O}(\Lambda_+^2 V)$.  Since $\mathrm{SU}(V)$ is connected we in fact have $\rho: \mathrm{SU}(V) \to \mathrm{SO}(\Lambda_+^2 V)$.

Specializing to the case $V = \mathbb{C}^4$ we get a [[Lie group]] [[homomorphism]] $\rho: \mathrm{SU}(4) \to \mathrm{SO}(6)$.  Since $d\rho$ is nonzero and $\mathrm{SU}(4)$ is [[simple Lie group|simple]], $d\rho$ must be [[injective function|injective]].   Since

$$ \mathrm{dim}(\mathrm{SU}(4)) = 15 = \mathrm{dim}(\mathrm{SO}(6)), $$

$d\rho$ must also be [[surjective function|surjective]].   Since $\mathrm{SO}(6)$ is [[connected topological space|connected]] and $d\rho$ is a [[bijection]], $\rho$ must be a [[covering map]].  Since $\rho(\pm 1) = 1$, $\rho$ exhibits $\mathrm{SU}(4)$ as a connected cover of $\mathrm{SO}(6)$ that is *at least* a [[double cover]].  But the [[universal cover]] of $\mathrm{SO}(6)$, namely $\mathrm{Spin}(6)$, is only a double cover.  Thus $\mathrm{SU}(4)$ is a double cover of $\mathrm{SO}(6)$, and $\mathrm{SU}(4) \cong \mathrm{Spin}(6)$.

### Coset spaces

[[!include coset space structure on n-spheres -- table]]

### $G$-Structure and exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

\linebreak

## Related concepts

[[!include low dimensional rotation groups -- table]]

* [[U(1)]], [[SU(2)]], [[SU(3)]], [[SU(5)]]

\linebreak

## References

* {#FigueroaFarrill10} [[José Figueroa-O'Farrill]], _[PG course on Spin Geometry](https://empg.maths.ed.ac.uk/Activities/Spin/)_ lecture 8: _Parallel and Killing spinors_, 2010 ([pdf](https://empg.maths.ed.ac.uk/Activities/Spin/Lecture8.pdf))

[[!redirects SU(4)]]


Spin(7)

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#### Group Theory
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#Contents#
* table of contents
{:toc}

## Idea

The [[spin group]] in [[dimension]] $n = 7$.

\linebreak

## Properties

### Subgroups and Supgroups

+-- {: .num_prop #Spin7SubgroupsInSpin8}
###### Proposition
**([[Spin(7)]]-[[subgroups]] in [[Spin(8)]])

There are precisely 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[Spin(7)]]-[[subgroups]] inside [[Spin(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

\begin{center}
  \begin{xymatrix}
      \mathrm{Spin}(7)
      \ar@{^{(}->}[dr]^{\iota'}
      \\
      & \mathrm{Spin}(8)
      &
      \mathrm{Spin}(7)
      \ar@{_{(}->}[l]^-{\iota}
      \\
      \mathrm{Spin}(7)
      \ar@{^{(}->}[ur]_-{ \iota'' }
  \end{xymatrix}
\end{center}

=--

([Varadarajan 01, Theorem 5 on p. 6](#Varadarajan01))

+-- {: .num_prop #G2IsIntersectionOfSpin7SubgroupsInSpin8}
###### Proposition
**([[G2]] is [[intersection]] of [[Spin(7)]]-[[subgroups]] of [[Spin(8)]])**

The [[intersection]] inside [[Spin(8)]] of any two [[Spin(7)]]-subgroups from distinct [[conjugacy classes of subgroups]] (according to Prop. \ref{Spin7SubgroupsInSpin8})
is the [[exceptional Lie group]] [[G2]], hence we have [[pullback squares]] of the form

\begin{center}
\begin{xymatrix}
  G_2
   \ar@{^{(}->}[r]
   \ar@{^{(}->}[d]
  &
  \mathrm{Spin}(7)
  \ar@{^{(}->}[d]^-{\iota^\prime}
  \\
  \mathrm{Spin}(7)
   \ar@{^{(}->}[r]_-{\iota}
  &
  \mathrm{Spin}(8)
 \end{xymatrix}
\end{center}

=--

([Varadarajan 01, Theorem 5 on p. 13](#Varadarajan01))

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

We have the following [[commuting diagram]] of [[subgroup]] inclusions, where each [[square]] exhibits a [[pullback]]/[[fiber product]], hence an [[intersection]] of subgroups:

\begin{center}
\begin{xymatrix}
    \mathrm{SU}(2)
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{SU}(3)
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{G}_2
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{Spin}(7)
    \ar@{^{(}->}[d]^{\iota'}
    \\
    \mathrm{Spin}(5)
    \ar@{^{(}->}[r]
    &
    \mathrm{Spin}(6)
    \ar@{^{(}->}[r]
    &
    \mathrm{Spin}(7)
    \ar@{^{(}->}[r]_-{\iota}
    &
    \mathrm{Spin}(8)
\end{xymatrix}
\end{center}

Here in the bottom row we have the [[Lie groups]]

[[Spin(5)]] $\hookrightarrow$ [[Spin(6)]] $\hookrightarrow$ [[Spin(7)]] $\hookrightarrow$ [[Spin(8)]]

with their canonical [[subgroup]]-inclusions, while in the top row we have

[[SU(2)]] $\hookrightarrow$ [[SU(3)]] $\hookrightarrow$ [[G2]] $\hookrightarrow$ [[Spin(7)]]

and the right vertical inclusion $\iota'$ is the one of the two non-standard inclusions, according to Prop. \ref{Spin7SubgroupsInSpin8}.

=--

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

The square on the right is that from Prop. \ref{G2IsIntersectionOfSpin7SubgroupsInSpin8}.

The square in the middle is [Varadarajan 01, Lemma 9 on p. 10](#Varadarajan01).

The statement also follows with
[Onishchik 93, Table 2, p. 144](#Onishchik93):

\begin{center}
\begin{imagefromfile}
  "file_name": "SO8SubgroupIntersection.jpg",
  "width": 690
\end{imagefromfile}
\end{center}

=--

\linebreak

### Coset spaces

+-- {: .num_prop #QuotientOfSpin7ByG2IsS7}
###### Proposition
**([[coset space]] of [[Spin(7)]] by [[G2]] is [[7-sphere]])**

Consider the canonical [[action]] of [[Spin(7)]] on the [[unit sphere]] in $\mathbb{R}^8$ (the [[7-sphere]]),

1. This action is is [[transitive action|transitive]];

1. the [[stabilizer group]] of any point on $S^7$ is [[G2]];

1. all [[G2]]-subgroups of [[Spin(7)]] arise this way, and are all [[conjugate subgroup|conjugate]] to each other.

Hence the [[coset space]] of [[Spin(7)]] by [[G2]] is the [[7-sphere]]

$$
  Spin(7)/G_2 \;\simeq\; S^7
  \,.
$$

=--

(e.g [Varadarajan 01, Theorem 3](#Varadarajan01))

[[!include coset space structure on n-spheres -- table]]

\linebreak

### $G$-Structure and exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

## Related concepts

[[!include low dimensional rotation groups -- table]]

[[!include special holonomy table]]

## References

* {#Onishchik93} A. L. Onishchik (ed.) _Lie Groups and Lie Algebras_

  * *I.* A. L. Onishchik, E. B.  Vinberg, _Foundations of Lie Theory_,

  * *II.* V. V. Gorbatsevich, A. L. Onishchik, _Lie Transformation Groups_

  Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

* {#Varadarajan01} [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae Volume 19, Issue 2, 2001, Pages 163-177 (doi:10.1016/S0723-0869(01)80027-X, [pdf](https://core.ac.uk/download/pdf/81114499.pdf))

[[!redirects Spin7]]


Spin(7) manifold

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###Context###
#### Riemannian geometry
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[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[8-manifold]] of [[special holonomy]] [[Spin(7)]].

Equivalently: an [[8-manifold]] equipped with a globalization of the [[Cayley 4-form]].

## Properties

### As part of the Berger classification

[[!include special holonomy table]]

### As $\mathbb{O}$-Riemannian manifolds

[[!include normed division algebra Riemannian geometry -- table]]

### As exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

### Relation to J-twisted Cohomotopy
 {#RelationToJTwistedCohomotopy}

On a [[spin-manifold]] [[8-manifold|of dimension 8]] a choice of topological [[Spin(7)]]-[[G-structure|structure]] is equivalently a choice of [[cocycle]] in [[J-homomorphism|J-]][[twisted Cohomotopy cohomology theory]]. This follows ([[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation|FSS 19, 3.4]]) from

1. the standard [[coset space]]-[[structures]] on the [[7-sphere]] (see [here](7-sphere#CosetSpaceRealization))

1. the fact that [[coset spaces]] $G/H$ are the [[homotopy fibers]] of the maps $B H \to B G$ of the corresponding [[classifying spaces]] (see [here](coset#ForInfinityGroups))

\begin{imagefromfile}
  "file_name": "ExceptionalSpheres.jpg",
  "width": 730
\end{imagefromfile}

### Characteristic classes

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

Let $X$ be a  [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] 8 with [[Spin structure]]. If the [[frame bundle]] moreover admits [[G-structure]] for

$$
  G = Spin(7) \hookrightarrow Spin(8)
$$

then the [[Euler class]] $\chi$, the [[second Pontryagin class]] $p_2$ and the [[cup product]]-square $(p_1)^2$ of the [[first Pontryagin class]] (the combination proportional to the [[I8]]-term) of the [[frame bundle]]/[[tangent bundle]] are related by

\[
  \label{EulerClassInTermsOfPontryagin}
  8 \chi
  \;=\;
  4 p_2 - (p_1)^2
  \,.
\]

=--

([Isham-Pope 88 (3.36)](#IshamPope88))

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

The same conclusion (eq:EulerClassInTermsOfPontryagin) also holds for [[Spin(5).Spin(3)]]-structure, see [there](quaternion-Kähler+manifold#CharacteristicClassesForSpin5Spin3Structure).

=--

See also at _[[C-field tadpole cancellation]]_.

\linebreak

## Related concepts

* [[Spin(7)-instanton]]

* [[M-theory on Spin(7)-manifolds]]

* [[F-theory on Spin(7)-manifolds]]

* [[8-manifold]]

## References

### General

The concept goes back to:

* E. Bonan, (1966), _Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)_, C. R. Acad. Sci. Paris 262: 127–129.

Characterization in terms of the [[Euler class|Euler 8-class]] and the [[I8]]-[[invariant polynomial]] of the [[tangent bundle]]:

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], Thm. 10.7 in: _Spin geometry_, Princeton University Press (1989) [[ISBN 9780691085425](https://press.princeton.edu/books/hardcover/9780691085425/spin-geometry-pms-38-volume-38)]

Construction of [[compact topological space|compact]] [[Spin(7)-manifolds]]:

* Christine Taylor, _Compact Manifolds with Holonomy Spin(7)_ (1996) ([pdf](https://people.maths.ox.ac.uk/joyce/theses/TaylorMSc.pdf))

* [[Dominic Joyce]], _A new construction of compact 8-manifolds with holonomy $Spin(7)$_, J. Differential Geom. Volume 53, Number 1 (1999), 89-130 ([euclid:jdg/1214425448](https://projecteuclid.org/euclid.jdg/1214425448))

* [[Dominic Joyce]], _Compact Manifolds with Special Holonomy_, Oxford Mathematical Monographs, Oxford University Press (2000) ([ISBN:9780198506010](https://global.oup.com/academic/product/compact-manifolds-with-special-holonomy-9780198506010?cc=de&lang=en&))

In terms of [[G-structure]]:

* [[Robert Bryant]], _Metrics with Exceptional Holonomy_, Annals of Mathematics Second Series, Vol. 126, No. 3 (Nov., 1987), pp. 525-576 ([jstor:1971360](https://www.jstor.org/stable/1971360))

and motivated from special [[supersymmetry]] (such as in [[M-theory on Spin(7)-manifolds]]):

* {#IshamPope88} [[Chris Isham]], [[Christopher Pope]], _Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings_, Class. Quant. Grav. 5 (1988) 257 ([spire:251240](http://inspirehep.net/record/251240), [doi:10.1088/0264-9381/5/2/006](https://doi.org/10.1088/0264-9381/5/2/006))

* {#IshamPopeWarner88} [[Chris Isham]], [[Christopher Pope]], [[Nicholas Warner]], _Nowhere-vanishing spinors and triality rotations in 8-manifolds_,  Classical and Quantum Gravity, Volume 5, Number 10, 1988 ([cds:185144](http://cds.cern.ch/record/185144), [doi:10.1088/0264-9381/5/10/009](https://iopscience.iop.org/article/10.1088/0264-9381/5/10/009))

  (focus on [[Spin(7)-structure]])

On [[Spin(7)-orbifolds]]:

* Ya. V. Bazaikin, _On the new examples of complete noncompact Spin(7)-holonomy metrics_,  Sib Math J **48**, 8–25 (2007) ([doi:10.1007/s11202-007-0003-7](https://doi.org/10.1007/s11202-007-0003-7))

### Relation to Higgs bundles

Relating [[M-theory on Spin(7)-manifolds]] with [[F-theory on Spin(7)-manifolds]] via [[Higgs bundles]]:

* {#CHRTZ20} [[Mirjam Cvetic]], [[Jonathan Heckman]], Thomas B. Rochais, Ethan Torres, [[Gianluca Zoccarato]], _Geometric Unification of Higgs Bundle Vacua_ ([arXiv:2003.13682](https://arxiv.org/abs/2003.13682))

[[!redirects Spin(7) manifolds]]

[[!redirects Spin(7)-manifold]]

[[!redirects Spin(7)-manifolds]]

[[!redirects Spin7-manifold]]
[[!redirects Spin7-manifolds]]

[[!redirects Spin(7)-structure]]
[[!redirects Spin(7)-structures]]

[[!redirects Spin(7)-orbifold]]
[[!redirects Spin(7)-orbifolds]]

[[!redirects Spin(7) structure]]
[[!redirects Spin(7) structures]]

[[!redirects Spin7-structure]]
[[!redirects Spin7-structures]]

[[!redirects Spin7 structure]]
[[!redirects Spin7 structures]]


Spin(7)-instanton

#Contents#
* table of contents
{:toc}

## Idea

The concept of _$Spin(7)$-instantons_ ([Lewis 98](#Lewis98)) is an analogue of that of [[Yang-Mills instantons]] for [[special unitary group|SU(2)]]-[[principal connections]] on 4-dimensional manifolds, to (8-dimensional) [[Spin(7) manifolds]].

## Definition

On a [[Spin(7) manifold]] the [[differential n-form|differential 2-forms]] decompose into two [[irreducible representations]], $\Lambda_7$ and $\Lambda_{21}$ under the [[action]] of [[Spin(7)]] (e.g. [Lewis 98, p. 13](#Lewis98)).

A connection on the $Spin(7)$-manifold is called a _$Spin(7)$-instanton_ it its [[curvature]] 2-form has vanishing component in the representation $\Lambda_7$ ([Lewis 98, def.3.1](#Lewis98)).

## References

* {#Lewis98} C. Lewis, _$Spin(7)$-Instantons_, 1998 ([pdf](https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf))

* [[Simon Donaldson]],[[Richard Thomas]], _Gauge theory in higher dimensions_ ([pdf](http://wwwf.imperial.ac.uk/~rpwt/skd.pdf))

* {#Leung02} [[Naichung Conan Leung]], _Riemannian Geometry Over Different Normed Division Algebras_, J. Differential Geom. Volume 61, Number 2 (2002), 289-333. ([publisher page](http://projecteuclid.org/euclid.jdg/1090351387))

[[!redirects Spin(7)-instantons]]

[[!redirects Spin(7) instanton]]
[[!redirects Spin(7) instantons]]


Spin(7)/G2 is the 7-sphere

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## Statement

The [[coset space]] of [[Spin(7)]] by the [[exceptional Lie group]] [[G2]] is [[homeomorphism|homeomorphic]] to the [[7-sphere]]:

$$
  Spin(7)/G_2
  \;\simeq\;
  S^7
  \,.
$$

## Related concepts

[[!include coset space structure on n-spheres -- table]]

## References

* {#GrayGreen70} [[Alfred Gray]], Paul S. Green, p. 2 of _Sphere transitive structures and the triality automorphism_, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 ([euclid:1102976640](https://projecteuclid.org/euclid.pjm/1102976640))

* {#Varadarajan01} [[Veeravalli Varadarajan]], Theorem 3 in _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae, 19 (2001): 163-177 ([pdf](https://core.ac.uk/download/pdf/81114499.pdf))


Spin(8)-subgroups and reductions -- table

**[[Spin(8)]]-[[subgroups]] and [[reduction of structure group|reductions]] to [[exceptional geometry]]**

| [[reduction of structure group|reduction]] | from [[spin group]] | to [[maximal subgroup|maximal]] [[subgroup]] |
|---|----|---|
| [[Spin(7)-structure]] | [[Spin(8)]] | [[Spin(7)]] |
| [[G2-structure]] | [[Spin(7)]] | [[G2]] |
| [[Calabi-Yau 3-fold|CY3-structure]] | [[Spin(6)]] | [[SU(3)]] |
| [[SU(2)-structure]] | [[Spin(5)]] | [[SU(2)]] |
|  |  |  |
| **[[generalized tangent bundle|generalized]] [[reduction of structure group|reduction]]** | **from [[Narain group]]** | **to  [[direct product group]]** |
|  |  |  |
| [[generalized Spin(7)-structure]] | $Spin(8,8)$ | $Spin(7) \times Spin(7)$ |
| [[generalized G2-structure]] | $Spin(7,7)$ | $G_2 \times G_2$ |
| [[generalized Calabi-Yau manifold|generalized CY3]] | $Spin(6,6)$ | $SU(3) \times SU(3)$ |

see also: _[[coset space structure on n-spheres -- table|coset space structure on n-spheres]]_


Spin(9)

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## Idea

The [[spin group]] in [[dimension]] 9.

## Properties

### Relation to octonionic Hopf fibration

The [[octonionic Hopf fibration]] is [[equivariant function|equivariant]] with respect to the [[Spin(9)]]-[[action]], the one on $S^8 = S(\mathbb{R}^9)$ induced from the canonical action of $Spin(9)$ on $\mathbb{R}^9$, and on $S^{15} = S(\mathbb{R}^{16})$ induced from the canonical inclusion $Spin(9) \hookrightarrow Spin(16)$.

This equivariance is made fully manifest by realizing the octonionic Hopf fibration as a map of [[coset spaces]] as follows ([Ornea-Parton-Piccinni-Vuletescu 12, p. 7](#OrneaPartonPiccinniVuletescu12)):

$$
  \array{
    S^7
      &\overset{fib(h_{\mathbb{O}})}{\longrightarrow}&
    S^{15}
      &\overset{h_{\mathbb{O}}}{\longrightarrow}&
    S^8
    \\
     = && = && =
    \\
    \frac{Spin(8)}{Spin(7)}
    &\longrightarrow&
    \frac{Spin(9)}{Spin(7)}
    &\longrightarrow&
    \frac{Spin(9)}{Spin(8)}
  }
$$

### Relation to standard model gauge group

The exact [[gauge group]] of the [[standard model of particle physics]] (see [there](standard+model+of+particle+physics#GaugeGroup)) is [[isomorphism|isomorphic]] to the [[subgroup]] of the [[Jordan algebra]] [[automorphism group]] of the [[octonions|octonionic]] [[Albert algebra]] that "[[stabilizer subgroup|stabilizes]] a 4d sub-[[Minkowski spacetime]]" (see [there](Albert+algebra#StabilizerOf4dMinkowskiInsideOctonionicAlbertAlgebra) for details).

   More concretely, it is identified with the [[subgroup]] of [[Spin(9)]] which respects a splitting $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$ ([Krasnov 19](#Krasnov19), see also at [SO(10)-GUT](GUT#TheGUTKnownAsSO10))

## Related concepts

[[!include low dimensional rotation groups -- table]]

\linebreak

## References

* Thomas Friedrich, _Weak Spin(9)-Structures on 16-dimensional Riemannian Manifolds_, ([arXiv:math/9912112](https://arxiv.org/abs/math/9912112))

* Maurizio Parton, Paolo Piccinni, _Spin(9) and almost complex structures on 16-dimensional manifolds_, ([arXiv:1105.5318](https://arxiv.org/abs/1105.5318))

* {#OrneaPartonPiccinniVuletescu12} Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, _Spin(9) geometry of the octonionic Hopf fibration_, ([arXiv:1208.0899](http://arxiv.org/abs/1208.0899), [doi:10.1007/s00031-013-9233-x]( https://doi.org/10.1007/s00031-013-9233-x))

* Maurizio Parton, Paolo Piccinni, _The Role of Spin(9) in Octonionic Geometry_, ([arXiv:1810.06288](https://arxiv.org/abs/1810.06288))

* {#Krasnov19} [[Kirill Krasnov]], _$SO(9)$ characterisation of the Standard Model gauge group_ ([arXiv:1912.11282](https://arxiv.org/abs/1912.11282))

[[!redirects Spin9]]

[[!redirects SO(9)]]
[[!redirects SO9]]


Spin(9)/Spin(7) is the 15-sphere

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## Statement

The [[coset space]] of [[Spin(9)]] by [[Spin(7)]] is [[homeomorphism|homeomorphic]] to the [[n-sphere|15-sphere]]:

$$
  Spin(9)/Spin(7)
  \;\simeq\;
  S^15
  \,.
$$

This coset realization features in the [[octonionic Hopf fibration]], making manifest its $Spin(9)$-[[equivariance]] ([Ornea-Parton-Piccinni-Vuletescu 12, p. 7](#OrneaPartonPiccinniVuletescu12)).

## Related concepts

[[!include coset space structure on n-spheres -- table]]

## References

* {#GrayGreen70} [[Alfred Gray]], Paul S. Green, p. 2 of _Sphere transitive structures and the triality automorphism_, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 ([euclid:1102976640](https://projecteuclid.org/euclid.pjm/1102976640))

* {#OrneaPartonPiccinniVuletescu12} Liviu Ornea, Maurizio Parton, [[Paolo Piccinni]], Victor Vuletescu, _Spin(9) geometry of the octonionic Hopf fibration_, Transformation Groups (2013) 18: 845 ([arXiv:1208.0899](http://arxiv.org/abs/1208.0899), [doi:10.1007/s00031-013-9233-x]( https://doi.org/10.1007/s00031-013-9233-x))


Spin(9,1)

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## Idea

$Spin(9,1)$ denotes the [[spin group]]-cover of the [[Lorentz groups]] in [[Minkowski spacetime|Minkowski]] [[signature of a quadratic form|signature]] $(9,1)$.

## Properties

### Relation to the octonions

[[!include exceptional spinors and division algebras -- table]]

## Related entries

* [[Spin(10,2)]]

* [[Spin(11,3)]]

[[!include low dimensional rotation groups -- table]]


spin-isospin transmutation

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[spin]]

* [[isospin]]

## References

The original articles:

* [[Roman Jackiw]], C. Rebbi, _Spin from Isospin in a Gauge Theory_, Phys. Rev. Lett. 36, 1116, 1976 ([doi:10.1103/PhysRevLett.36.1116](https://doi.org/10.1103/PhysRevLett.36.1116))

* P. Hasenfratz, [[Gerard 't Hooft]], _Fermion-Boson Puzzle in a Gauge Theory_, Phys. Rev. Lett. 36, 1119 (1976) ([doi:10.1103/PhysRevLett.36.1119](https://doi.org/10.1103/PhysRevLett.36.1119))

Review:

* Maciej A Nowak, [[Mannque Rho]], [[Ismail Zahed]], Chapter 7 of: _[[Chiral Nuclear Dynamics]]_, World Scientific 1996 ([doi](https://doi.org/10.1142/1681))

In the context of the [[Skyrmion model]] with [[heavy mesons]]:

* [[Mannque Rho]], _Massive-quark baryons as Skyrmions_,  Modern Physics Letters A, Vol. 06, No. 23 (1991) ([doi:10.1142/S0217732391002268](https://doi.org/10.1142/S0217732391002268))

[[!redirects spin-isospin transmutations]]


spin-orbit coupling

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## Idea

An [[interaction]] between a [[particle]]'s [[angular momentum]] and intrinsic [[spin]].

## Related concepts

* [[spin Hall effect]]

* [[quantum spin Hall effect]]

## References

See also:

* Wikipedia, _[Spin-orbit interaction](https://en.wikipedia.org/wiki/Spin%E2%80%93orbit_interaction)_

[[!redirects spin-orbit couplings]]

[[!redirects spin-orbit interaction]]
[[!redirects spin-orbit interactions]]


spin-statistics theorem

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#Contents#
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{:toc}

## Idea ##
In [[quantum field theory]] on [[Minkowski space]] all fields must transform according to a definite (finite dimensional) [[representation]] of the universal cover of the [[Poincare group]], which determines the [[spin]] of each field. From the [[representation theory]] of the [[Poincare group]] it is known that the spin $s$ is a number $s = \frac{n}{2}$ with $n \in \mathbb{N}$.

On the other hand, if we take fields to be pointwise localized in the sense of the [[Wightman axioms]], then the locality axiom (also known as _Einstein microcausality_ ) says that spacelike separated field operators either commute or anticommute: Two [[Fermionic field]]s anticommute, two [[Bosonic field]]s commute, a Fermionic and a Bosonic field commute.

The **spin-statistics theorem** states that fields with integer spin $s$ (n  is even) are Bosonic fields, fields with half-integer spin (n is uneven) are [[Fermionic field]]s. A better name for the theorem would therefore be spin- _commutation_ theorem, the name spin- _statistics_ theorem stems from the fact that [[Boson]]s (the particles associated to [[Bosonic field]]s) are social, multiple particles can exist in the same quantum state, while [[Fermion]]s are not social: The [[Pauli exclusion principle]] says maximally one [[Fermion]] can exist in a given quantum state. This leads to different [[partition functions]] in [[statistical mechanics]] of systems consisting of [[Boson]]s only and of [[Fermion]]s only, hence the name.

The statement and proof of the theorem depend on the framework for [[quantum field theory]] that is used, therefore there are, strictly speaking, several versions of the spin-statistics theorem, but the _physical interpretation_ is always the same.

## Statement

### In Algebraic quantum field theory

In the [[Haag-Kastler approach]] ("[[algebraic quantum field theory]]") the [[Bisognano-Wichmann theorem]] states a relation of the representation of the Poincare group on certain local algebras of [[local nets of algebras of observables]] and their [[modular groups]]. If this relation holds for a given net, then this net is said to fulfill the Bisognano-Wichmann property.

([GuidoLongo 94](#GuidoLongo94), [Verch 01](#Verch01))

...

## Related concepts

* [[Atiyah-Sutcliffe conjecture]]

* [[PCT theorem]]

## References

The original formulation of the spin-statistics theorem:

* {#Fierz39} [[Markus Fierz]],  _Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin_. Helvetica Physica Acta. 12 (1): 3–37. (1939) [doi:10.5169/seals-110930](https://dx.doi.org/10.5169%2Fseals-110930)

* {#Pauli40} [[Wolfgang Pauli]], _The connection between spin and statistics_, Phys. Rev. 58, 716–722 (1940)

Textbook accounts:

* [[Raymond F. Streater]], [[Arthur S. Wightman]], *PCT, Spin and Statistics, and All That*, Princeton University Press (1989, 2000) [[ISBN:9780691070629](https://press.princeton.edu/books/paperback/9780691070629/pct-spin-and-statistics-and-all-that), [jstor:j.ctt1cx3vcq](https://www.jstor.org/stable/j.ctt1cx3vcq)]

* {#Strocchi13} [[Franco Strocchi]], §4.2 in: _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

The proof, which goes back originally to Fermi. There is also a more intuitive approach based on topology. One can see hints of it in Feynman's lecture here:

* [[Richard Feynman]] and [[Steven Weinberg]], _Elementary Particles and the Laws of Physics_ , the 1986 Dirac Memorial Lectures, Cambridge U. Press, Cambridge, 1987.

Exposition:

* [[John Baez]], _Spin, Statistics, CPT and All That Jazz_ ([web](http://math.ucr.edu/home/baez/spin_stat.html))

A statement and proof of both a spin-statistics and a [[PCT theorem]] in the [[axiom|axiomatic]] of [[algebraic quantum field theory]] is in

* {#GuidoLongo94} [[Daniele Guido]], [[Roberto Longo]], _An Algebraic Spin and Statistics Theorem_ ([arXiv:funct-an/9406005] (http://arxiv.org/abs/funct-an/9406005))

Geometric proofs of spin-statistics via [[configuration spaces of points]] (related to the [[Atiyah-Sutcliffe conjecture]]):

* {#BerryRobbins97} [[Michael V. Berry]], [[Jonathan M. Robbins]], *Indistinguishability for quantum particles: spin, statistics and the geometric phase*, Proceedings of the Royal Society A **453** 1963 (1997) 1771-1790 ([doi:10.1098/rspa.1997.0096](https://doi.org/10.1098/rspa.1997.0096))

* A.F. Reyes-Lega, C. Benavides, *Remarks on the Configuration Space Approach to Spin-Statistics*,  Found Phys (2010) 40: 1004-1029 ([arXiv:0911.0579](https://arxiv.org/abs/0911.0579))

Generalization for [[AQFT on curved spacetime]] is in

* {#Verch01} [[Rainer Verch]], _A spin-statistics theorem for quantum fields on curved spacetime manifolds in a generally covariant framework_, Communications in Mathematical Physics October 2001, Volume 223, Issue 2, pp 261–288 ([arXiv:math-ph/0102035](http://arxiv.org/abs/math-ph/0102035), [doi:10.1007/s002200100526](http://dx.doi.org/10.1007/s002200100526))

Discussion relating the spin-statistics theorem to [[extended topological field theory]], [[categorification]] (via [[2-rings]]) and [[Deligne's theorem on tensor categories]] is in

* {#JohnsonFreyd15} [[Theo Johnson-Freyd]], _Spin, statistics, orientations, unitarity_, Algebraic and Geometric Topology [[arXiv:1507.06297](https://arxiv.org/abs/1507.06297)]

and via [[hermitian functorial field theory]]:

* [[Lukas Müller]], [[Luuk Stehouwer]], _Reflection Structures and Spin Statistics in Low Dimensions_ [[arXiv:2301.06664](https://arxiv.org/abs/2301.06664)]


spin^c

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# Contents
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## Definition

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

By definition of the [[spin group]] $Spin(n)$ there is a canonical inclusion

$$
  \mathbb{Z}/2\mathbb{Z}\hookrightarrow Spin(n)
$$

of the [[group of order 2]]. For $Spin(n)\hookrightarrow GL_1(Cl(\mathbb{R}^n))$ canonically realized by even [[Clifford algebra]] elements of unit norm, this is given by the inclusion of $\{+1,-1\}$.

=--

> We frequently write $\mathbb{Z}_2$ as shorthand for $\mathbb{Z}/2\mathbb{Z}$.

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

For $n \in \mathbb{N}$, the [[Lie group]] $Spin^c(n)$ is the [[quotient group]]

$$
  \begin{aligned}
    Spin^c(n) & \coloneqq Spin(n) \times_{\mathbb{Z}_2} U(1)
    \\
    & = (Spin(n) \times U(1))/{\mathbb{Z}_2}
    \,,
  \end{aligned}
$$

of the [[product]] of the [[spin group]] with the [[circle group]] by the common [[subgroup|sub]]-[[group of order 2]] $\mathbb{Z}_2 \hookrightarrow Spin$ and $\mathbb{Z}_2 \hookrightarrow U(1)$ (i.e.: the [[central product group]]).

=--

Usually the only the case $n \geq 3$ is considered.

Some authors (e.g. [Gompf 97, p. 2](#Gompf97)) denote this as

$$
  \begin{aligned}
    Spin^c(n)
     & \coloneqq
    Spin(n)\cdot Spin(2)
     \\
    & \simeq Spin(n) \cdot U(1)
  \end{aligned}
$$

following the notation [[Sp(n).Sp(1)]] (see [there](SpnSp1#SpinnSpin2IsSpinc)).

## Examples

* For $n=3$, the [[exceptional isomorphism]] between [[Spin(3)]] and [[SU(2)]] extends to an isomorphism between $Spin^c(3)$ and the [[unitary group]] $U(2)$:
  $$
    Spin^c(3)
    \;\simeq\;
    U(2)
  $$
over the exceptional isomorphism [[SO(3)|$SO(3) \simeq PU(2)$]], as both of these [[quotient groups]] are quotients by the respective [[centers]], both identifiable with the [[circle group]] $U(1)$) ([Ozbagci–Stipsicz 2004](#OzbagciStipsicz) Section 6.2). This isomorphism follows from considering the surjective homomorphism [[SU(2)|$SU(2) \times U(1) \to U(2)$]] given by $(A,z) \mapsto z A$, and noticing its [[kernel]] is precisely $\{\pm(I,1)\}$, hence $U(2)$ satisfies the [[universal property]] defining $Spin^c(3)$ as a quotient.

* For $n=4$, we have
$$
    Spin^c(4)
      = \big(SU(2)\times SU(2)\times U(1)\big) / \{\pm(I,I,1)\}
      \simeq U(2)\times_{U(1)} U(2).
$$
This latter group is the [[fibre product]] of groups over $U(1)$, namely $\{(A,B)\in U(2)\times U(2)\mid \det(A) = \det(B)\}$ ([Ozbagci–Stipsicz 2004](#OzbagciStipsicz) Section 6.3). The construction can be seen by considering the surjective homomorphism
$$
    SU(2) \times SU(2) \times U(1)
     \to SU(2) \times SU(2) \times U(1) \times U(2)
      \to U(2) \times U(2)
$$
defined by $(A,B,z) \mapsto (z A,z B)$, which has image $U(2)\times_{U(1)} U(2)$. Similarly to the case of $Spin^c(3)$, the kernel consists of triples $(A,B,z)$ such that $z A = I = z B$, hence that $A = B = (1/z) I$. Since $\det(A) = 1/z^2 = 1$, we must have $z=\pm 1$, and hence $A=B=\pm I$ with the same sign as $z$. Thus the kernel is precisely $\{\pm(I,I,1)\}$, and so again by the universal property we get the isomorphism as stated.

  The [[Spin(4)|$Spin(4)$]] subgroup can be seen as the subgroup of pairs of unitary matrices with both of them having determinant 1.

* Using the exceptional isomorphism [[Spin(6)|$Spin(6) \simeq SU(4)$]], and the multiplication map $SU(4)\times U(1) \to U(4)$ analogous to the above, it can be seen that the group $Spin^c(6)$ is the connected [[double cover]] of $U(4)$ corresponding to the (unique) index-2 subgroup $2\mathbb{Z} \hookrightarrow \mathbb{Z} \simeq \pi_1(U(4))$. This is because the multiplication map is surjective and has kernel canonically isomorphic to $\mu_4$, the fourth [[roots of unity]], via the map $\mu_4\to SU(4)\times U(1)$ sending $\zeta\mapsto (\zeta^{-1}I,\zeta)$. Hence there are a pair of 2:1 surjective homomorphisms
$$
SU(4) \times U(1) \to Spin^c(6) \to U(4),
$$
and hence the result.

The last two examples can also be considered as [[exceptional isomorphisms]], even if not to one of the more [[classical Lie groups]].

## Properties

### Group extension

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

We have a [[short exact sequence]]

$$
 U(1) \to Spin^c \to SO
 \,,
$$

where $U(1) \to Spin^c$ is the canonical inclusion into the defining product $U(1) \to Spin \times U(1) \to Spin \times_{\mathbb{Z}_2} U(1)$.

=--

### General

### As the homotopy fiber of the smooth $\mathbf{W}_3$
 {#AsHomotopyFiberOfSmoothW3}

We discuss in the following that

1. the universal third [[integral Stiefel-Whitney class]] $W_3$ has an essentially unique lift from [[∞Grpd]] $\simeq$ [[Top]] to [[Smooth∞Grpd]];

1. the smooth [[delooping]] $\mathbf{B}Spin^c \in Smooth\infty Grpd$ is the [[homotopy fiber]] of $\mathbf{W}_3$, hence is the [[circle n-bundle|circle 2-bundle]] over $\mathbf{B} SO$ classified by $\mathbf{W}_3$.

+-- {: .num_prop #SpinCAsHomotopyPullbackOfW2AndC1}
###### Proposition

We have a [[homotopy pullback]] diagram

$$
  \array{
     \mathbf{B} Spin^c
     &\stackrel{\mathbf{B}det}{\longrightarrow}&
     \mathbf{B}U(1)
     \\
     \big\downarrow
     &&
     \big\downarrow{}^{\mathrlap{\mathbf{c}_1 mod 2}}
     \\
     \mathbf{B} SO
     &\stackrel{\mathbf{w}_2}{\longrightarrow}&
     \mathbf{B}^2 \mathbb{Z}_2
  }
$$

in [[Smooth∞Grpd]], where

* the right morphism is the universal first [[Chern class]] modulo 2;

* the bottom morphism is the universal second [[Stiefel-Whitney class]].

=--

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

We present the sitation as usual in the projective [[model structure on simplicial presheaves]] over [[CartSp]] by [[∞-anafunctors]].

The first [[Chern class]] is given by the [[∞-anafunctor]]

$$
  \array{
     \mathbf{B}(\mathbb{Z} \to \mathbb{R})
     &\stackrel{\mathbf{c}_1}{\to}&
     \mathbf{B}(\mathbb{Z} \to 1)
     =
     \mathbf{B}^2 \mathbb{Z}
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     \mathbf{B} U(1)
  }
  \,,
$$

where $(G_1 \to G_0)$ denotes a presentation of a [[strict 2-group]] by a [[crossed module]].

The second [[Stiefel-Whitney class]] is given by

$$
  \array{
    \mathbf{B}(\mathbb{Z}_2 \to Spin)
    &\stackrel{\mathbf{w}_2}{\to}&
    \mathbf{B}(\mathbb{Z}_2 \to 1)
    =
    \mathbf{B}^2 \mathbb{Z}_2
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B} SO
  }
  \,.
$$

Notice that the top horizontal morphism here is a [[fibration]].

Therefore the [[homotopy pullback]] in question is (as discussed there) given by the ordinary [[pullback]] $Q$ in

$$
  \array{
    Q
    &\to&
    \mathbf{B}(\mathbb{Z} \to \mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}(\mathbb{Z}_2 \to Spin)
    &\to&
    \mathbf{B}^2 \mathbb{Z}_2
  }
  \,.
$$

This pullback is $\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})$, where

$$
  \partial\colon n \mapsto ( n \,mod\, 2 , n)
  \,.
$$

This is equivalent to

$$
  \begin{aligned}
   (\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})
   & \simeq
  (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times (\mathbb{R}/2\mathbb{Z}))
   \\
  & \simeq
  (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1))
  \end{aligned}
  \,,
$$

(notice the non-standard identification $U(1) \simeq \mathbb{R}/(2\mathbb{Z})$ here, which is important below in prop. \ref{UniversalDeterminantLineBundleMap} for the identification of $det$)
where now $\partial'$ is the [[diagonal]] embedding of the [[subgroup]]

$$
  \partial'\colon \sigma \mapsto (\sigma, \sigma)
  \,.
$$

This in turn is equivalent to

$$
  \mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1))
  \,,
$$

which is def. \ref{DirectDefinitionOfSpinC}.

=--

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

Compare this with the similar but different [[homotopy pullback]] that defines the [[spin group]]

$$
  \array{
    \mathbf{B}Spin &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2
  }
$$

=--

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

Under the identificaton $Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times} U(1)$ the "universal [[determinant line bundle]] map"

$$
  det \colon Spin^c \to U(1)
$$

is given in components by

$$
  (g,c) \mapsto 2 c
$$

(where on the right we write the group structure additively).

=--

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

By the proof of prop. \ref{SpinCAsHomotopyPullbackOfW2AndC1} the $U(1)$-factor in $Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times}U(1)$ arises from the identification $U(1) \simeq \mathbb{R}/2\mathbb{Z}$. But under the horizontal map as it appears in the homotopy pullback in that proof this corresponds to multiplication by 2.

=--

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

The third _[[integral Stiefel-Whitney class]]_

$$
  W_3
   \coloneqq
  \beta_2 \circ w_2
   \colon
  B SO \stackrel{w_2}{\to} B^2 \mathbb{Z}_2
  \stackrel{\beta_2}{\to}
  B^3 \mathbb{Z}
$$

has an essentially unique lift through [[geometric realization]] ${\vert-\vert}\colon $ [[Smooth∞Grpd]] $\stackrel{\Pi}{\to}$ [[∞Grpd]] $\stackrel{\simeq}{\to}$ [[Top]]

given by

$$
  \mathbf{W}_3 = \mathbf{\beta}_2 \circ \mathbf{w}_2
  \colon
  \mathbf{B} SO(n)
   \stackrel{w_2}{\to}
  \mathbf{B}^2 \mathbb{Z}_2
    \stackrel{\mathbf{\beta}_2}{\to}
  \mathbf{B}^2 U(1)
  \,,
$$

where $\mathbf{\beta}_2$ is simply given by the canonical subgroup embedding.

=--

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

Once we establish that this is a lift at all, the essential uniqueness follows from the respective theorem at [[smooth ∞-groupoid -- structures]].

The ordinary [[Bockstein homomorphism]] $\beta_2$ is presented by the [[∞-anafunctor]]

$$
  \array{
    \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})
    &\to&
    \mathbf{B}^2 (\mathbb{Z} \to 1)
    =
    \mathbf{B}^3 \mathbb{Z}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}^2 \mathbb{Z}_2
  }
  \,.
$$

Accordingly we need to lift the canonical presentation of $\mathbf{\beta}_2$ to a comparable $\infty$-anafunctor. This is accomplished by

$$
  \array{
    \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})
    &\stackrel{\hat \mathbf{\beta}_2}{\to}&
    \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R})
    \\
    \downarrow^{\mathrlap{\simeq}}
     &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}^2 \mathbb{Z}_2
     &\stackrel{\mathbf{\beta}_2}{\to}&
    \mathbf{B}^2 U(1)
  }
  \,.
$$

Here the top horizontal morphism is induced from the morphism of [[crossed module]]s that is given by the commuting diagram

$$
  \array{
    \mathbb{Z} &\stackrel{id}{\to}& \mathbb{Z}
    \\
    \downarrow^{\mathrlap{\cdot 2}} && \downarrow^{\mathrlap{\cdot 2}}
    \\
    \mathbb{Z}
    &\stackrel{}{\hookrightarrow}&
    \mathbb{R}
  }
  \,.
$$

Since $\mathbb{R}$ is contractible, we have indeed under [[geometric realization]] an equivalence

$$
  \array{
    \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert
    &\stackrel{\vert \hat {\mathbf{\beta}}_2\vert}{\to}&
    \vert
      \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R})
    \vert
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert
    &\to&
    \vert\mathbf{B}^2(\mathbb{Z} \to 1)\vert
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \vert B^2 \mathbb{Z}_2\vert
     & \stackrel{\beta_2}{\to}&
    \vert B^3 \mathbb{Z}\vert
  }
  \,.
$$

=--

+-- {: .num_prop #HomotopyFiberOfSmoothBeta2}
###### Proposition

The sequence

$$
  \mathbf{B} U(1)
    \stackrel{\mathbf{c}_1 mod 2}{\to}
  \mathbf{B}^2 \mathbb{Z}_2
    \stackrel{\mathbf{\beta}_2}{\to}
  \mathbf{B}^2 U(1)
  \,,
$$

where $\mathbf{\beta}_2$ is the smoothly refined [[Bockstein homomorphism]] from prop. \ref{SmoothRefinementOfBockstein},
is a [[fiber sequence]].

=--

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

The homotopy fiber of $\mathbf{B} \mathbb{Z}_2 \to \mathbf{B}U(1)$ is $U(1)/\mathbb{Z}_2 \simeq U(1)$. Thinking of this is $(\mathbb{Z} \stackrel{\cdot 1/2}{\to} \mathbb{R})$ one sees that the inclusion of this fiber is indeed $\mathbf{c}_1 mod 2$.

=--

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

The [[delooping]] $\mathbf{B}Spin^c$ of the Lie group $Spin^c$ in [[Smooth∞Grpd]] is the [[homotopy fiber]] of the universal third smooth [[integral Stiefel-Whitney class]] from \ref{SmoothRefinementOfBockstein}.

$$
  \mathbf{B}Spin^c
   \to
  \mathbf{B} SO
   \stackrel{\mathbf{W}_3}{\to}
  \mathbf{B}^2 U(1)
  \,,
$$

=--

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

Then consider the [[pasting diagram]] of [[homotopy pullbacks]]

$$
  \array{
     \mathbf{B}Spin^c
       &\to&
     \mathbf{B} U(1)
     &\to&
     {*}
     \\
     \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} && \downarrow
     \\
     \mathbf{B} SO
       &\stackrel{\mathbf{w}_2}{\to}&
     \mathbf{B}^2 \mathbb{Z}_2
       &\stackrel{\mathbf{\beta}_2}{\to}&
     \mathbf{B}^2 U(1)
  }
  \,.
$$

The right square is a homotopy pullback by prop. \ref{HomotopyFiberOfSmoothBeta2}.
The left square is a homotopy pullback by
prop. \ref{SpinCAsHomotopyPullbackOfW2AndC1}.
The bottom composite is the smooth $\mathbf{W}_3$ by prop \ref{SmoothRefinementOfBockstein}.

This implies by claim by the [[pasting law]].
=--

### Relation to metaplectic group $Mp^c$

There is a direct analogy between [[Spin]], [[Spin^c]] and the [[metaplectic groups]] [[Mp]] and [[Mp^c]] (see there for more).

## Related concepts

* [[spin group]], **$spin^c$-group**, [[spin^h group]]

* [[spin structure]]

* [[MSpin^c]]

* [[spin^c structure]], [[twisted spin^c structure]]

* [[spin^h structure]]

* [[string 2-group]], [[string structure]]

* [[string^c 2-group]]

## References

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], Appendix D in: _[[Spin geometry]]_, Princeton University Press (1989)

For more see the references at _[[spin^c structure]]_.

The exceptional isomorphisms in low dimension are described in

* {#OzbagciStipsicz} Ozbagci, B., Stipsicz, A.I. (2004). _$Spin^c$ Structures on 3- and 4-Manifolds._ In: Surgery on Contact 3-Manifolds and Stein Surfaces. Bolyai Society Mathematical Studies, vol 13. Springer, Berlin, Heidelberg. doi:[10.1007/978-3-662-10167-4_6](https://doi.org/10.1007/978-3-662-10167-4_6).

[[!redirects Spin^c]]

[[!redirects spin^c group]]
[[!redirects spin^c groups]]
[[!redirects spin^c-group]]
[[!redirects spin^c-groups]]

[[!redirects Spin^c group]]
[[!redirects Spin^c groups]]
[[!redirects Spin^c-group]]
[[!redirects Spin^c-groups]]


Spin^c Dirac operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A type of [[Dirac operator]] defined on [[manifolds]] equipped with [[Spin^c structure]].

## Properties

### Relation to the Dolbeault operator

Every [[almost complex manifold]] carries a canonical [[spin^c structure]] (as discussed there). If $X$ is a [[complex manifold]], then under the identification

$$
  S(X) \simeq \wedge^{0,\bullet} T^\ast X
$$

of the [[spinor bundle]] with that of [[holomorphic differential forms]], the corresponding $Spin^c$-Dirac operator $D$ is identified with the [[Dolbeault-Dirac operator]]

$$
  D^+  \simeq \overline\partial + \overline \partial^\ast
  \colon
  \Gamma(\wedge^{0,even} T^\ast X)
  \to
  \Gamma(\wedge^{0,odd} T^\ast X)
  \,.
$$

### Genus

The [[A-hat genus]] for the $Spin^c$-operator is the [[Todd genus]] (e.g. [Kitada 75](#Kitada75)).

## Applications

* [[Freed-Witten anomaly]]

* [[geometric quantization by push-forward]]

## Related concepts

* [[spin^c structure]], [[twisted spin^c structure]]

* [[K-orientation]]

* [[spin^c quantization]]

* [[Dolbeault-Dirac operator]], [[Dirac operator]]

## References

Original articles include

* {#Kitada75} Yasuhiko Kitada, _Semi-free circle actions on $Spin^c$-manifolds_, Publ. RIMS, Kyoto Univ. 10(1975), 601-617 ([[Kitada75.pdf:file]])

A quick statement of the definition is in

* Ulrich Krämer, _$Spin^{\mathbb{C}}$-Dirac structures and Dirac operators_ (2009) ([pdf](https://web.archive.org/web/20150909233527/http://www.maths.gla.ac.uk/~ukraehmer/Tehran0409c.pdf))

Detailed accounts include

* J. J. Duistermaat, _The Spin-c Dirac Operator_, in _The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator_
Modern Birkhäuser Classics, 2011, 41-51, DOI: 10.1007/978-0-8176-8247-7_5

* [[Eckhard Meinrenken]], _Symplectic Surgery and the Spin-C Dirac operator_ ([arXiv:dg-ga/9504002](http://arxiv.org/abs/dg-ga/9504002))

* Charles Epstein, _Subelliptic $Spin_{\mathbb{C}}$-Dirac operators, I_ ([pdf](http://www.math.upenn.edu/~cle/papers/spinc1_rev2.pdf))

[[!redirects Spin^c Dirac operators]]

[[!redirects spin^c Dirac operator]]
[[!redirects spin^c Dirac operators]]

[[!redirects Spin-c Dirac operator]]
[[!redirects Spin-c Dirac operator]]

[[!redirects spin-c Dirac operator]]
[[!redirects spin-c Dirac operator]]


spin^c structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Cohomology
+-- {: .hide}
[[!include cohomology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

### Topological

For $n \in \mathbb{N}$ the [[Lie group]] [[spin^c|spinc]] is a [[central extension]]

$$
  U(1) \to Spin^c(n) \to SO(n)
$$

of the [[special orthogonal group]] by the [[circle group]]. This comes with a long [[fiber sequence]]

$$
  \cdots \to B U(1) \to B Spin^c(n)  \to B SO(n) \stackrel{W_3}{\to} B^2 U(1)
  \,,
$$

where $W_3$ is the _third [[integral Stiefel-Whitney class]]_ .

An [[oriented]] manifold $X$ **has $Spin^c$-structure** if the [[characteristic class]] $[W_3(X)] \in H^3(X, \mathbb{Z})$

$$
  W_3(X) \coloneqq W_3(T X)
  \;\colon\;
  X \stackrel{T X}{\to} B SO(n) \stackrel{W_3}{\to} B^2 U(1) \simeq K(\mathbb{Z},3)
$$

is trivial. This is the [[Dixmier-Douady class]] of the [[circle n-bundle|circle 2-bundle]]/[[bundle gerbe]] that [[obstruction|obstructs]] the existence of a $Spin^c$-[[principal bundle]] [[lift of structure group|lifting]] the given [[tangent bundle]].

A manifold $X$ is **equipped with $Spin^c$-structure** $\eta$ if it is equipped with a choice of trivializaton

$$
  \eta : 1 \stackrel{\simeq}{\to} W_3(T X)
  \,.
$$

The [[homotopy type]]/[[∞-groupoid]] of $Spin^c$-structures on $X$ is the [[homotopy fiber]] $W_3 Struc(T X)$ in the [[pasting diagram]] of [[homotopy pullback]]s

$$
  \array{
    W_3 Struc(T X) &\to& W_3 Struc(X) &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    * &\stackrel{T X}{\to}& Top(X, B SO(n))
    &\stackrel{W_3}{\to}&
    Top(X,B^2 U(1))
  }
  \,.
$$

If the class does not vanish and if hence there is no $Spin^c$-structure, it still makes sense to discuss the structure that remains as _[[twisted spin^c structure|twisted spinc structure]]_ .

### Smooth

Since $U(1) \to Spin^c \to SO$ is a sequence of [[Lie group]]s, the above may be lifted from the [[(∞,1)-topos]] $L_{whe}$ [[Top]] $\simeq$ [[∞Grpd]] of [[discrete ∞-groupoids]] to that of [[smooth ∞-groupoids]], [[Smooth∞Grpd]].

More in detail, by the discussion at [[Lie group cohomology]] (and [[smooth ∞-groupoid -- structures]]) the characteristic map $W_3 : B SO \to B^2 U(1)$ in $\infty Grpd$ has, up to equivalence, a unique lift

$$
  \mathbf{W}_3 : \mathbf{B} SO \to \mathbf{B}^2 U(1)
$$

to [[Smooth∞Grpd]], where on the right we have the [[delooping]] of the smooth [[circle n-group|circle 2-group]].

Accordingly, the [[2-groupoid]] of _smooth $spin^c$-structures_ $\mathbf{W}_3 Struc(X)$ is the joint [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{W}_3 Struc(T X) &\to& \mathbf{W}_3 Struc(X) &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    * &\stackrel{T X}{\to}& Smooth \infty Grpd(X, \mathbf{B} SO(n))
    &\stackrel{\mathbf{W}_3}{\to}&
    Smooth \infty Grpd(X,\mathbf{B}^2 U(1))
  }
  \,.
$$

### Higher $spin^c$-structures
 {#Higher}

In parallel to the existence of [higher spin structures](spin+structure#Higher) there are higher analogs of $Spin^c$-structures, related to [[quantum anomaly]] cancellation of theories of higher dimensional [[branes]].

* [[string^c structure|stringc structure]]

## Properties

### Of $Spin^c$
 {#PropertiesOfSpinC}

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

The group $Spin^c$ is the [[fiber product]]

$$
  \begin{aligned}
    Spin^c & := Spin \times_{\mathbb{Z}_2} U(1)
    \\
    & = (Spin \times U(1))/{\mathbb{Z}_2}
    \,,
  \end{aligned}
$$

where in the second line the [[action]] is the diagonal action induced from the two canonical embeddings of [[subgroups]] $\mathbb{Z}_2 \hookrightarrow \mathbb{Z}$
and $\mathbb{Z}_2 \hookrightarrow U(1)$.

=--

+-- {: .num_prop #SpinCAsHomotopyFiberProductW2C1}
###### Proposition

We have a [[homotopy pullback]] diagram

$$
  \array{
     \mathbf{B} Spin^c &\to& \mathbf{B}U(1)
     \\
     \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}}
     \\
     \mathbf{B} SO &\stackrel{w_2}{\to}&
     \mathbf{B}^2 \mathbb{Z}_2
  }
  \,.
$$

=--

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

We present this as usual by [[simplicial presheaves]] and [[∞-anafunctors]].

The first [[Chern class]] is given by the [[∞-anafunctor]]

$$
  \array{
     \mathbf{B}(\mathbb{Z} \to \mathbb{R})
     &\stackrel{c_1}{\to}&
     \mathbf{B}(\mathbb{Z} \to 1)
     =
     \mathbf{B}^2 \mathbb{Z}
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     \mathbf{B} U(1)
  }
  \,.
$$

The second [[Stiefel-Whitney class]] is given by

$$
  \array{
    \mathbf{B}(\mathbb{Z}_2 \to Spin)
    &\stackrel{w_2}{\to}&
    \mathbf{B}(\mathbb{Z}_2 \to 1)
    =
    \mathbf{B}^2 \mathbb{Z}_2
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B} SO
  }
  \,.
$$

Notice that the top horizontal morphism here is a [[fibration]].

Therefore the [[homotopy pullback]] in question is given by the ordinary pullback

$$
  \array{
    Q
    &\to&
    \mathbf{B}(\mathbb{Z} \to \mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}(\mathbb{Z}_2 \to Spin)
    &\to&
    \mathbf{B}^2 \mathbb{Z}_2
  }
  \,.
$$

This pullback is $\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})$, where

$$
  \partial : n \mapsto ( n mod 2 , n)
  \,.
$$

This is equivalent to

$$
  \mathbf{B}(\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1))
$$

where now

$$
  \partial' : \sigma \mapsto (\sigma, \sigma)
  \,.
$$

This in turn is equivalent to

$$
  \mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1))
  \,,
$$

which is the original definition.

=--

This factors the above characterization of $\mathbf{B}Spin^c$ as the homotopy fiber of $\mathbf{W}_3$:

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

We have a [[pasting diagram]] of [[homotopy pullbacks]] of [[smooth infinity-groupoids]] of the form

$$
  \array{
    \mathbf{B} Spin^c &\to& \mathbf{B}U(1) &\to& \ast
    \\
    \downarrow && \downarrow^{\mathrlap{c_1 \, mod\, 2}}
    && \downarrow
    \\
   \mathbf{B}SO
    &\stackrel{\mathbf{w}_2}{\to}&
   \mathbf{B}^2 \mathbb{Z}_2
   &\stackrel{\mathbf{\beta}_2}{\to}&
   \mathbf{B}^2 U(1)
  }
  \,.
$$

=--

This is discussed at _[Spin^c -- Properties -- As the homotopy fiber of smooth w3](spin%5Ec#AsHomotopyFiberOfSmoothW3)_.

### As $KU$-orientation

For $X$ an [[orientation|oriented]] [[manifold]], the map $X \to \ast$ is [[orientation in generalized cohomology|generalized oriented]] in [[periodic complex K-theory]] precisely if $X$ has a $Spin^c$-structure.

See at _[[K-orientation]]_ for more.

### Relation to metaplectic structures
 {#RelationToMetaplecticStructures}

Let $(X,\omega)$ be a [[compact topological space|compact]] [[symplectic manifold]] equipped with a [[Kähler polarization]] $\mathcal{P}$ hence a [[Kähler manifold]] structure $J$. A [[metaplectic structure]] of this data is a choice of square root $\sqrt{\Omega^{0,n}}$ of the [[canonical line bundle]]. This is equivalently a [[spin structure]] on $X$ (see the discussion at _[[Theta characteristic]]_).

Now given a [[prequantum line bundle]] $L_\omega$, in this case the [Dolbault quantization](geometric+quantization#IndexOfDolbeaultDiracOperator) of $L_\omega$ coincides with the [[spin^c quantization|spinc quantization]] of the [[spin^c structure|spinc structure]] induced by $J$ and $L_\omega \otimes \sqrt{\Omega^{0,n}}$.

This appears as ([Paradan 09, prop. 2.2](#Paradan09)).

## Examples

### From almost complex structures
 {#FromAlmostComplexStructures}

An [[almost complex structure]] canonically induces a $Spin^c$-structure:

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

For all $n \in \mathbb{N}$ we have a homotopy-[[commuting diagram]]

$$
  \array{
    \mathbf{B}U(n) &\to& \mathbf{B}U(1)
    \\
    \downarrow
     &\swArrow&
    \downarrow^{\mathrlap{\mathbf{c_1} mod 2}}
    \\
    \mathbf{B}SO(2n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2
  }
  \,,
$$

where the vertical morphism is the canonical morphism induced from the identification of real vector spaces $\mathbb{C} \to \mathbb{R}^2$, and where the top morphism is the canonical projection $\mathbf{B}U(n) \to \mathbf{B}U(1)$ (induced from $U(n)$ being the [[semidirect product group]] $U(n) \simeq SU(n) \rtimes U(1)$).

=--

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

By the general relation between $c_1$ of an [[almost complex structure]] and $w_2$ of the underlying orthogonal structure, discussed at _[Stiefel-Whitney class -- Relation to Chern classes](#Stiefel-Whitney+class#RelationToChernClasses)_.

=--

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

By prop. \ref{SpinCAsHomotopyFiberProductW2C1} and the [[universal property]] of the [[homotopy pullback]] this induces a canonical morphism

$$
  k \colon \mathbf{B}U(n) \to \mathbf{B}Spin^c
  \,.
$$

=--

and this is the universal morphism from almost complex structures:

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

For $c \colon X \to \mathbf{B}U(n)$ modulating an [[almost complex structure]]/[[complex vector bundle]] over $X$, the composite

$$
  k c \colon X \stackrel{c}{\to} \mathbf{B}U(n) \stackrel{k}{\to} \mathbf{B}Spin^c
$$

is the corresponding $Spin^c$-structure.

=--

## Related concepts

* [[spin^c|spinc]], [[MSpin^c|MSpinc]]

* [[twisted differential c-structures|(twisted, differential) c-structures]]

  * [[orientation]]

  * [[spin structure]], [[twisted spin structure]], [[differential spin structure]]

    **$spin^c$ structure**, [[twisted spin^c structure|twisted spinc structure]],

    [[K-orientation]]

    [[Spin^c Dirac operator|Spinc Dirac operator]]

  * [[Spin^h structure]]

  * [[string structure]], [[twisted differential string structure]],

  * [[string^c 2-group|stringc 2-group]], [[string^c structure|stringc structure]]

  * [[fivebrane structure]], [[twisted differential fivebrane structure]]

* [[spin^c quantization|spinc quantization]]

## References

### General

A canonical textbook reference is

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], Appendix D in: _[[Spin geometry]]_, Princeton University Press (1989)

Other accounts include

* Blake Mellor, _$Spin^c$-manifolds_ ([[BlakeSpinC.pdf:file]])

* _Stable complex and $Spin^c$-structures_ ([[StableComplexSpinC.pdf:file]])

* [[Peter Teichner]], Elmar Vogt, _All 4-manifolds have $Spin^c$-structures_ ([pdf](http://people.mpim-bonn.mpg.de/teichner/Papers/spin.pdf))

See also

* {#Gompf97} [[Robert Gompf]], _$Spin^c$ structures and homotopy equivalences_, Geom. Topol. 1 (1997) 41-50 ([arXiv:math/9705218](https://arxiv.org/abs/math/9705218))

### As $KU$-orientation/anomaly cancellation in type II string theory

That the $U(1)$-[[gauge field]] on a [[D-brane]] in [[type II string theory]] in the absense of a [[B-field]] is rather to be regarded as part of a $spin^c$-structure was maybe first observed in

* [[Edward Witten]], _Baryons and Branes In Anti de Sitter Space_, JHEP 9807:006 (1998) ([arXiv:hep-th/9805112](http://arxiv.org/abs/hep-th/9912086)).

The [[twisted spin^c structure|twisted spinc structure]] (see there for more details) on the [[worldvolume]] of [[D-branes]] in the presence of a nontrivial [[B-field]] was discussed in

* [[Daniel Freed]], [[Edward Witten]], _Anomalies in String Theory with D-Branes_ ([arXiv:hep-th/9907189](http://arxiv.org/abs/hep-th/9907189))

See at _[[Freed-Witten-Kapustin anomaly cancellation]]_.

A more recent review is provided in

* Kim Laine, _Geometric and topological aspects of Type IIB D-branes_ ([arXiv:0912.0460](http://arxiv.org/abs/0912.0460))

See also

* [[Hisham Sati]], _Geometry of $Spin$ and $Spin^c$ structures in the M-theory partition function_ ([arXiv:1005.1700](http://arxiv.org/abs/1005.1700))

The relation to [[metaplectic corrections]] is discussed in

* {#Paradan09} [[Paul-Emile Paradan]], _Spin-quantization commutes with reduction_ ([arXiv:0911.1067](http://arxiv.org/abs/0911.1067))

See also

* O. Hijazi, S. Montiel, F. Urbano, _$Spin^c$-geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds_ ([pdf](http://hal.archives-ouvertes.fr/docs/00/09/81/71/PDF/lagrangian0412.pdf))

[[!redirects spin^c structure]]
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spin^h structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Cohomology
+-- {: .hide}
[[!include cohomology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

With [[Sp(1)]] denoting the [[quaternion unitary group]],
we define the [[Spin^h group|Spin$^h$ group]]
$$Spin^h(n) = (Spin(n) \times Sp(1))/(\mathbf{Z}/2),$$
in complete analogy to the [[Spin^c group|Spin$^c$ group]]
$$Spin^c(n) = (Spin(n) \times U(1))/(\mathbf{Z}/2),$$
and the [[Spin group]]
$$Spin(n) = (Spin(n) \times O(1))/(\mathbf{Z}/2).$$

We have a canonical [[double covering]], which is a [[homomorphism]] of [[Lie groups]]:
$$Spin^h(n) \to SO(n)\times SO(3).$$
It induces canonical homomorphisms of Lie groups
$$Spin^h(n) \to SO(n)$$
and
$$Spin^h(n) \to SO(3).$$

A __spin$^h$-structure__ on a [[principal bundle]] $P\to B SO(n)$
is a [[lift]] through the canonical map $B Spin^h(n) \to B SO(n)$.

Thus, in concrete terms, a spin$^h$-structure on $P$ is a principal $SO(3)$-bundle $E$ together with a principal $Spin^h(n)$-bundle $Q$ and a double covering
map $Q\to P\times E$ equivariant with respect to the homomorphism $Spin^h(n) \to SO(n)\times SO(3)$.

The canonical inclusions
$$Spin(n)\to Spin^c(n)\to Spin^h(n)$$
allow promotions of [[spin-structures]] to [[spin^c-structures]] to [[spin^h-structures]]. The converse is not true: just as $\mathbb{CP}^2$ is a spin$^c$ manifold with no spin structure, the _Wu manifold_ $SU(3)/SO(3)$ is a spin$^h$ manifold with no spin$^c$ structure ([MathOverflow discussion](https://mathoverflow.net/questions/304471/spin-h-structures)).

## Obstructions to existence

The homotopy fiber of $B Spin^h(n) \to B SO(n)$ is not an [[Eilenberg-MacLane space]],
so we cannot expect a single cohomological class to control the existence of spin$^h$-structures.

The first obstruction is the vanishing of the fifth [[integral Stiefel-Whitney class]].

## In physics

[Freed-Hopkins](#FH16) use spin$^h$ [[invertible field theory|invertible field theories]] to model and classify [[SPT order|SPT phases]] in Altland-Zirnbauer class C.

[Wang-Wen-Witten](#WWW19) study an anomaly in 4d $SU(2)$ gauge theory that can appear when the theory is placed in spin$^h$ manifolds.

## References

The original definition is due to

* [[Christian Bär]], _Elliptic symbols_.  Mathematische Nachrichten, 201(1), 7–35.

A survey is given in

* Michael Albanese, Aleksandar Milivojevic, _Spin^h and further generalisations of spin_.  [arXiv:2008.04934](https://arxiv.org/abs/2008.04934)

Applications in physics:

* {#FH16} [[Dan Freed]] and [[Mike Hopkins]], _Reflection positivity and invertible topological phases_ ([arXiv:1604.06527](https://arxiv.org/abs/1604.06527)).

* {#WWW19} [[Juven Wang]], [[Xiao-Gang Wen]], [[Edward Witten]], _A New SU(2) Anomaly_, Journal of Mathematical Physics **60**  (2019) 052301 [[arXiv:1810.00844](https://arxiv.org/abs/1810.00844), [doi:10.1063/1.5082852](https://doi.org/10.1063/1.5082852)]

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spine

+-- {: .rightHandSide}
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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

If $S$ is an [[(∞,n)-category]] (including the case of [[∞-groupoids]], [[(∞,1)-categories]], ...) or a model thereof (including [[Kan complexes]] or [[quasi-categories]], ...) then
the _spine_ of $S$ is the maximal list of _composable_ edges ([[morphisms]]) of $S$.

Similarly, if $S$ is an [[(∞,1)-operad]] or a model thereof (given by a [[dendroidal set]] or [[Segal operad]], etc. ), the spine $Sp(T) \hookrightarrow T \to S$ of a [[tree]] $T$ in $S$ is a collection of composable operations in  the $(\infty,1)$-operad.

## Definition

### In simplicial sets

Let $S : \Delta^{op} \to Set$ be a [[simplicial set]].

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

The **spine** of an $n$-[[simplex]] $\sigma \in S_n$, also called its **backbone**, is the [[union]] of the edges (1-cells) $0 \to 1$, $1 \to 2$, $\cdots$ , $(n-1) \to n$ between the successive vertices of $\sigma$.

=--

When $n \gt 2$, an $(n,i)$-[[horn]] in $S$ has the same edges as any of its fillers, so we may speak of the spine of a horn as well.

### In dendroidal sets

The above notion generalizes to [[dendroidal set]]s

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

The **spine** of a [[tree]] $T$ is the [[union]] over its [[corollas]]

$$
  Sp(T) = \bigcup_{C_{k} \to T} C_k
  \,.
$$

=--

In ([Cisinski-Moerdijk](#CisinskiMoerdijk)) this is called the _Segal core_ of $T$.

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

For a linear tree this reproduces the above definition of spines of simplices.

=--

## Properties

### Characterizations of compositions

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

A [[simplicial set]] $X$ is the [[nerve]] of a [[category]] precisely if for all $n \in \mathbb{N}$ all the morphisms induced from the spine inclusion

$$
  X^{Sp[n] \hookrightarrow \Delta[n]} : X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1
$$

are [[bijections]].

More generally, a [[dendroidal set]] $X$ is the [[dendroidal nerve]] of a [[symmetric operad]] over [[Set]] (a [[symmetric multicategory]]), precisely if for all [[trees]] $T$ the morphisms induced from the spine inclusion
$X^{Sp[T] \hookrightarrow \Delta[n]}$ are bijections.

=--

For simplicial sets, this is a classical statement (Grothendieck / Segal). Its homotopical weakening leads to the notion of [[Segal category]] and [[complete Segal space]]. For dendroidal sets this is ([Cisinski-Moerdijk, cor. 2.7](#Cisinski-Moerdijk)).

### Closure and lifting properties

In the following, _[[tree]]_ means "finite non-planar rooted tree" as used in the definition of _[[dendroidal set]]_ .

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

For any [[tree]] $T$, the spine inclusion $Sp[T] \hookrightarrow \Omega[T]$ is an [[inner anodyne morphism]].

=--

This is ([Cisinski-Moerdijk, prop. 2.4](#CisinskiMoerdijk)).

## Related concepts

* [[horn]], [[boundary of a simplex]]

## References

Dendroidal spines are discussed in section 1 of

* [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal Segal spaces and infinity-operads_ ([arXiv:1010.4956](http://arxiv.org/abs/1010.4956))
 {#CisinskiMoerdijk}

[[!redirects spines]]

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spinning particle

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#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _spinning relativistic particle_ is a variant of the plain [[relativistic particle]] which has an "internal degree of freedom" called [[spin]]: it is a _[[spinor]]_ , a [[fermion]]. Examples that appear in the [[standard model of particle physics]] are [[electron]]s, and [[quarks]].

As a 1-dimensional [[sigma-model]], the _spinning_ relativistic particle is like the [[relativistic particle]] but with [[fermion field]]s on the [[worldline]]. This worldline action always happens to have worldline [[supersymmetry]], entirely independent of whether there is any supersymmetry on [[target space|target]] [[spacetime]].

## Properties

### Worldline supersymmetry
 {#WorldlineSupersymmetry}

We discuss how spinning particles automatically have [[supersymmetry]] in their [[worldline formalism]]. For more see the [references below](#WorldlineSupersymmtryReferences) for more and see also at _[string theory FAQ -- Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)_.

In the [[Polyakov action]]-formulation the [[action functional]] of the [[relativistic particle]] [[sigma model]] on the 1-dimensional [[worldline]] is actually 1-dimensional [[gravity]] coupled to "worldline matter fields", where the latter are the embedding fields $\phi : \mathbb{R} \to X$ into the [[target space]].

It turns out that the generalization of this 1-dimensional gravity action to [[supergravity]] yields the action functional that describes ordinary Dirac [[spinors]] -- spinning particles like [[electron]]s -- propagating on [[target space]] $X$. See the [references on worldline supersymmetry](#WorldlineSupersymmtryReferences) below.

The worldline supersymmetry of fermions comes down to the fact that their [[Hamiltonian]](-constraint) operator $H$ has a square root: the
[[Dirac operator]] $D$. Its defining equations

$$
  [D,D] = 2 D^2 = H\,,\;\;\; [H,H] = 0\;,\;\; [D,H] = 0
$$

characterize the $d = 1$ [[translation super-Lie algebra]].

For appreciating this fact it is important to keep the ingredients of [[sigma-model]] theory sorted out correctly: a supersymmetric theory on the worldline describes a spinning particle on some [[target space|spacetime]] coupled to some [[background gauge field]]s. That background geometry need not have a "global supersymmetry" (a covariant constant spinor), hence under [[second quantization]] the [[perturbation theory]] _on target space_ induced by the worldline theory need not have any global supersymmetries (in particular no superpartners to the effective particle excitations). What will happen, though, is that the full target space theory induced under [[second quantization]] will be a [[supergravity]] theory on target space. Some of its solutions may have covariantly constant spinors (and hence global supersymmetry), but generically they will not, just like the generic solution to ordinary [[Einstein equations]] does not have a [[Killing vector]].

### Paritition function

[[!include genera and partition functions - table]]

## History

That some [[particle]]s have a property called [[spinor|spin]] was found in 1922 in the [[Stern-Gerlach experiment]].

## Related concepts

* [[fermion]], [[spinor]]

* [[sigma model]], [[brane]]

  * [[relativistic particle]], **spinning particle**, [[superparticle]]

    * [[A-hat genus]], [[(1,1)-dimensional Euclidean field theories and K-theory]]

  * [[string]], [[spinning string]], [[superstring]]

    [string theory FAQ -- Does string theory predict supersymmetry?](string%20theory%20FAQ#DoesSTPredictSupersymmetry)

  * [[membrane]]

## References

### General

Lecture notes in view of application to [[perturbative quantum field theory]] via [[worldline formalism]]:

* Olindo Corradini, [[Christian Schubert]], James P. Edwards, Naser Ahmadiniaz, *Spinning Particles in Quantum Mechanics and Quantum Field Theory* ([arXiv:1512.08694](https://arxiv.org/abs/1512.08694))

Discussion that cancellation of the [[quantum anomaly]] of the spinning particle precisely requires [[Spin-structure]] on its [[target spacetime]]:

* {#Witten85} [[Edward Witten]], _Global anomalies in string theory_, in: W. Bardeen and A. White (eds.). _Symposium on Anomalies, Geometry, Topology_, pp. 61–99. World Scientific, 1985  ([[WittenGlobalAnomaliesInStringTheory.pdf:file]], [spire:214913](https://inspirehep.net/literature/214913))

Discussion of worldline dynamics of spinning particles in [[background fields]]:

* [[Jan-Willem van Holten]], *Relativistic Dynamics of Spin in Strong External Fields* [[arXiv:hep-th/9303124](http://arxiv.org/abs/hep-th/9303124)]

* A. Pomeranskii, R A Sen'kov, I.B. Khriplovich, _Spinning relativistic particles in external fields_ Acta Physica Polonica B Proceedings Supplement Vol. 1 (2008) ([pdf](http://iopscience.iop.org/1063-7869/43/10/R03))

* Krzysztof Andrzejewski, Cezary Gonera, Joanna Goner, Piotr Kosinski, Pawel Maslanka, _Spinning particles, coadjoint orbits and Hamiltonian formalism_ ([arXiv:2008.09478](https://arxiv.org/abs/2008.09478))

Discussion via [[coadjoint orbits]]:

* Thomas Basile, Euihun Joung, TaeHwan Oh, *Manifestly Covariant Worldline Actions from Coadjoint Orbits. Part I: Generalities and Vectorial Descriptions* [[arXiv:2307.13644](https://arxiv.org/abs/2307.13644)]

### Worldline supersymmetry
 {#WorldlineSupersymmtryReferences}

Among the early references that describe the observation that the supersymmetric extension of the worldline theory of the [[relativistic particle]] describes ordinary Dirac [[fermion]]s are

* F.A. Berezin and M.S. Marinov, Ann. Phys. (NY) 104 (1977) 336

* [[Lars Brink]], [[Paolo Di Vecchia]], [[Paul Howe]], *A Lagrangian formulation of the classical and quantum dynamics of spinning particles*, Nucl. Phys. B **118** (1977) 76 [doi:10.1016/0550-3213(77)90364-9]

* R. Casalbuoni, Phys. Lett. B62 (1976), 49

* A. Barducci, R. Casalbuoni and L. Lusanna, Nuov. Cim. 35A (1976), 377
Nucl. Phys. B124 (1977), 93; id. 521

A survey of constructions of [[worldline]] supersymmetric action functional for spinning particles in various [[background gauge field|background fields]] is given in

* J. van Holten, _$D = 1$ Supergravity and Spinning Particles_ ([pdf](http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/28/002/28002527.pdf))

An argument that for arbitrary backgrounds the spinning particle's worldline action is supersymmetric is given in

* Darius Gagné, _Worldline Supersymmetry and Dimensional Reduction_ ([arXiv:hep-th/9604149](http://arxiv.org/abs/hep-th/9604149))

Textbook surveys of worline supersymmetry include

the beginning of section 14.1.1 in

* Tomás Ortín, _Gravity and strings_ Cambridge University Press (2004)

Derivation of the supersymmetric [[worldline]] action of the spinning particle in the _[[worldline formalism]]_ of [[QFT]] [[scattering amplitudes]] is around (3.6) of

* [[Matthew Strassler]], _Field Theory Without Feynman Diagrams: One-Loop Effective Actions_, Nucl. Phys. B385:145-184,1992 ([arXiv:hep-ph/9205205](http://arxiv.org/abs/hep-ph/9205205))

See also:

* Rachel H. Rietdijk, _Spinning particles in general relativity_ Theoretical and Mathematical Physics, Vol. 98, No. 3 (1994)

* [[Warren Siegel]], p. 194 of: _Fields_ ([arXiv:hep-th/9912205](http://arxiv.org/abs/hep-th/9912205))

  > There, first exercise IIIB1.3 gives the action expanded out in all components, and then the following exercise IIIB1.4 gives the reformulation in [[superfield]] formalism which makes manifest that the action is invariant under worldline [[supersymmetry]].

* Roberto Casalbuoni, Joaquim Gomis, Kiyoshi Kamimura, Giorgio Longhi, _Space-time Vector Supersymmetry and Massive Spinning Particle_ ([web](http://www.mendeley.com/research/spacetime-vector-supersymmetry-and-massive-spinning-particle/))

* E. Boffo, *Particles in the superworldline and BRST* [[arXiv:2207.02041](https://arxiv.org/abs/2207.02041)]

[[!redirects spinning particles]]

[[!redirects worldline supersymmetry]]


spinning string

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _spinning string_ is a variant of the [[string]] in direct analogy (but one [[worldvolume]] [[dimension]] higher) to how the [[spinning particle]] is a variant of the [[particle]].

(Which means that "spinning" here is _not_ in the sense of "rotating", but in the sense of the kind of "intrinsic" spin as exhibited by [[fermion]] [[particle]]s. A spinning string can rotate _in addition_ to its intrinsic spin.)

And just as as the [[spinning particle]] automatically exhibits local [[worldline supersymmetry]], the spinning string automatically exhibits local [[worldsheet]] [[supersymmetry]]. For that reason it is today almost exclusively known as the _[[superstring]]_ .

In fact, the very notion of [[supersymmetry]] was realized (in parallel to its independent discovery in Russia) when people noticed that the spinning string [[sigma-model]] -- also called the _Neveu-Schwarz-Ramond model_ (**NSR model**) -- automatically exhibits this extra graded symmetry.

When supplemented with an operation called the _[[GSO projection]]_, the [[second quantization]] of the spinning string -- its effective [[target space|target]] [[spacetime]] theory -- also becomes a locally supersymmetric theory: [[spacetime]] [[supergravity]].

There are other formulations of the [[superstring]] which are thought to be equivalent in some sense. Notably the [[Green-Schwarz superstring]] is a sigma-model which is not supersymmetric on the [[worldsheet]] but instead takes values in a genuine [[spacetime]] [[supermanifold]].

## Related concepts

* [[picture number]], [[picture changing operator]]

* [[sigma model]], [[brane]]

  * [[relativistic particle]], [[spinning particle]], [[superparticle]]

  * [[string]], **spinning string** , [[superstring]]

  * [[Goddard-Thorn no-ghost theorem]], [[GSO projection]]

    [string theory FAQ -- Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)

  * [[membrane]]

* [[string theory]]

[[!include worldvolume-target supersymmetry of brane sigma-models]]

## References
 {#References}

### General

The original articles that formulated the NSR model:

* [[André Neveu]], [[John Schwarz]], _Factorizable dual model of pions_, Nucl. Phys. B31, 86 (1971) (doi:10.1016/0550-3213(71)90448-2)

* [[Pierre Ramond]], _Dual theory for Free Fermions_, Phys Rev. D3  2415 (1971) ([doi:10.1103/PhysRevD.3.2415](https://doi.org/10.1103/PhysRevD.3.2415))

  > (introducing the [[Dirac-Ramond operator]])

* {#FriedanMartinecShenker86} [[Daniel Friedan]], [[Emil Martinec]], [[Stephen Shenker]], _Conformal invariance, supersymmetry and string theory_, Nucl. Phys., B271 (1986) 93 (doi:10.1016/S0550-3213(86)80006-2)

Original articles that realize the local [[worldsheet]] [[supersymmetry]] of the spinning string (via the supersymmetric [[Polyakov action]]):

* [[Stanley Deser]], [[Bruno Zumino]], *A complete action for the spinning string*, Physics Letters B **65** 4 (1976) 369-373 [doi:10.1016/0370-2693(76)90245-8[pdf](http://cdsweb.cern.ch/record/201648/files/CM-P00061857.pdf?version=1)]

* [[Lars Brink]], [[Paolo Di Vecchia]], [[Paul Howe]], _A locally supersymmetric and reparametrization invariant action for the spinning string_ , Physics Letters B **65** Issue 5, 20 (1976) 471-474 [doi:10.1016/0370-2693(76)90445-7]

Definition of the NSR string in [[RR-field]] [[background field|backgrounds]] is subtle (in contrast to the classical [[Green-Schwarz superstring]]), discussion is in

* {#BerensteinLeigh99} [[David Berenstein]], Robert Leigh, _Superstring Perturbation Theory and Ramond-Ramond Backgrounds_, Phys. Rev. D 60, 106002 (1999) ([arXiv:hep-th/9904104](https://arxiv.org/abs/hep-th/9904104))

See also

* Nathan Benjamin, Ying-Hsuan Lin, _Lessons from the Ramond sector_ ([arXiv:2005.02394](https://arxiv.org/abs/2005.02394))

Relation to the [[Berkovits superstring]]:

* [[Nathan Berkovits]], *Manifest Spacetime Supersymmetry and the Superstring* ([arXiv:2106.04448](https://arxiv.org/abs/2106.04448))

### GSO projection and spacetime supersymmetry

The [[GSO projection]] and the conjecture that this makes also the target space theory of the spinning string [[supergravity|locally supersymmetric]] is due to

* F. Gliozzi, [[Joël Scherk]], D. I. Olive, _Supergravity and its spinor dual model,_  Phys. Letters B65, 282 (1976)

* F. Gliozzi, [[Joël Scherk]], D. I. Olive, _Supersymmetry, supergravity theories and the dual spinor model_ , Nucl. Phys, B122, 253 (1977)

The demonstration of this fact is due to

* [[Michael Green]], [[John Schwarz]],

  _Supersymmetrical dual string theory_ , Nucl. Phys. B181 (1981) 502;

  _Supersymmetrical string theories_ , Phys. Lett. 109B (1982) 444.

Superconformal invariance of the spinning string was discussed in

* [[Paul Howe]], _Super Weyl transformations in two dimensions_ J. Phys. 12 (1979) 393.

A decent account is in

* {#Majer} Imre Majer, _Superstrings_ ([pdf](http://edu.itp.phys.ethz.ch/fs13/cft/SuperS2_Majer.pdf), [[MajerSuperstrings.pdf:file]])

A review of the history of these developments is in

* [[John Schwarz]], _String theory origins of supersymmetry_, Nucl. Phys. Proc. Suppl. 101 (2001) 54-61 ([arXiv:hep-th/0011078](http://arxiv.org/abs/hep-th/0011078))

An exposition of some technical aspects is in the wiki

* [The Tangent Bundle](http://www.physics.thetangentbundle.net/wiki/Main_Page), _[RNS superstring](http://www.physics.thetangentbundle.net/wiki/String_theory/RNS_superstring)_

Reviews and survays are also in any text on [[string theory]]. See the references there for more detail.

### BRST description

Discussion of the [[BRST complex]] of the NSR string:

* [[José Figueroa-O'Farrill]], [[Takashi Kimura]], _The BRST cohomology of the NSR string: vanishing and ``no-ghost'' theorems_, Comm. Math. Phys. **124** 1 (1989) 105-132. [[euclid:cmp/1104179078](http://projecteuclid.org/euclid.cmp/1104179078)]

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spinor bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _spinor bundle_ on a [[smooth manifold]] with [[spin structure]] is a $\rho$-[[associated bundle]] associated to the [[spin group]]-[[principal bundle]] lifting the [[tangent bundle]], for $\rho : \mathbf{B} Spin \to $ [[Vect]] a [[spin representation]].

A [[section]] of a spinor bundle is called a **spinor** (a [[fermion]] [[field (physics)|field]])

## Related concepts

* [[Clifford bundle]]

* [[stringor bundle]]

* [[fiber bundles in physics]]

* [[rotor]]

* [[spinor]], [[spin representation]]
* [[pure spinor]]
* [[Dirac field]]

A [[Dirac operator]] acts on [[sections]] of a spinor bundle.

In [[physics]], sections of spinor bundles model matter [[relativistic particle|particle]]s: [[fermion]]. See [[spinors in Yang-Mills theory]].

* [[spin]]

* [[fermion]]

  * [[electron]], [[quark]]

  * [[gravitino]], [[gluon]]

* [[Velo-Zwanziger problem]]

* [[symplectic spinor]]

* [[twistor]]

* [[stringor bundle]]

[[!include standard model of fundamental physics - table]]

## References

The term "spinor" is due to [[Paul Ehrenfest]], see the historical references at *[[spin]]*.

* {#Cartan66} [[Élie Cartan]], _Theory of Spinors_, Dover,  first edition 1966

* [[Roger Penrose]], [[Wolfgang Rindler]], _Spinors and space time_, in 2 vols. Cambridge Univ. Press 1984/1988.

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], chapter II, section 3 _[[Spin geometry]]_, Princeton University Press (1989)

* [[Daniel Freed]], _[[Five lectures on supersymmetry]]_

* [pdf](http://verbit.ru/MATH/TALKS/Israel/kahler-3.pdf)

Spinors in [[classical field theory]] ([[fermions]]):

* {#DeligneFreed99} [[Pierre Deligne]], [[Daniel Freed]], §3.4 of _Classical field theory_ (1999) ([pdf](https://publications.ias.edu/sites/default/files/79_ClassicalFieldTheory.pdf))

  this is a chapter in

  [[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], J. Morgan, D.R. Morrison, [[Edward Witten|E. Witten]] (eds.)  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

* [[Radovan Dermisek]] _qft I-8_ ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-I-8-4p.pdf), [[DermisekSpinorField.pdf:file]])

Discussion relating manifolds with spinor bundles to [[supergeometry]] includes

* [[Dmitri Alekseevsky]], [[Vicente Cortés]], [[Chandrashekar Devchand]], [[Uwe Semmelmann]], _Killing spinors are Killing vector fields in Riemannian Supergeometry_ ([arXiv:dg-ga/9704002](http://arxiv.org/abs/dg-ga/9704002))

## History

* B. L.Van derWaerden, _Exclusion principle and spin_, in _Theoretical Physics in the Twentieth Century: A Memorial Volume to Wolfgang Pauli_, ed. M. Fierz and V. F. Weisskopf, New York: Interscience, 1960

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spinor field

+-- {: .rightHandSide}
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### Context
#### Spin geometry
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#### Fields and quanta
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[[!include fields and quanta - table]]
=--
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=--

## Idea

[[spinor]] [[field (physics)|field]]

## Examples

* [[Dirac field]]

## References

* [[Radovan Dermisek]] _qft I-8_ ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-I-8-4p.pdf), [[DermisekSpinorField.pdf:file]])

* L. Bonora, R. Soldati, S. Zalel, _Dirac, Majorana, Weyl in 4d_ ([arXiv:2006.04546](https://arxiv.org/abs/2006.04546))

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