geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
spin geometry, string geometry, fivebrane geometry …
A representation of the spin group.
A quadratic vector space $(V, \langle -,-\rangle)$ is a vector space $V$ over finite dimension over a field $k$ of characteristic 0, and equipped with a symmetric bilinear form $\langle -,-\rangle \colon V \otimes V \to k$.
Conventions as in (Varadarajan 04, section 5.3).
We write $q\colon v \mapsto \langle v ,v \rangle$ for the corresponding quadratic form.
The Clifford algebra $CL(V,q)$ of a quadratic vector space, def. 1, is the associative algebra over $k$ which is the quotient
of the tensor algebra of $V$ by the ideal generated by the elements $v \otimes v - q(v)$.
Since the tensor algebra $T(V)$ is naturally $\mathbb{Z}$-graded, the Clifford algebra $Cl(V,q)$ is naturally $\mathbb{Z}/2\mathbb{Z}$-graded.
Let $(\mathbb{R}^n, q = {\vert \vert})$ be the $n$-dimensional Cartesian space with its canonical scalar product. Write $Cl^\mathbb{C}(\mathbb{R}^n)$ for the complexification of its Clifford algebra.
There exists a unique complex representation
of the algebra $Cl^\mathbb{C}(\mathbb{R}^n)$ of smallest dimension
The Spin group $Spin(V,q)$ of a quadratic vector space, def. 1, is the subgroup of the group of units in the Clifford algebra $Cl(V,q)$
on those elements which are even number multiples $v_1 \cdots v_{2k}$ of elements $v_i \in V$ with $q(V) = 1$.
Specifically, “the” Spin group is
A spin representation is a linear representation of the spin group, def. 3.
Complex representations of the spin group follow a mod-2 Bott periodicity.
In even $d = 2n$ there are two inequivalent complex-linear irreducible representations of $Spin(d-1,1)$, each of complex dimension $2^{d/2-1}$, called the two chiral representations, or the two Weyl spinor representations.
For instance for $d = 10$ one often writes these as $\mathbf{16}$ and $\mathbf{16}'$.
The direct sum of the two chiral representation is called the Dirac spinor representation, for instance $\mathbf{16} + \mathbf{16}'$.
In odd $d = 2n+1$ there is a single complex irreducible representation of complex dimension $2^{(d-1)/2}$. For instance for $d = 11$ one often writes this as $\mathbf{32}$. This is called the Dirac spinor representation in this odd dimension.
For $d = 2n$, if $\{\Gamma^1, \cdots, \Gamma^n\}$ denote the generators of the Clifford algebra $Cl_{d-1,1}$ then there is the chirality operator
on the Dirac representation, whose eigenspaces induce its decomposition into the two chiral summands.
The unique irreducible Dirac representation in the odd dimension $d+1$ is, as a complex vector space, the sum of the two chiral representations in dimension $d$, with the Clifford algebra represented by $\Gamma^1$ through $\Gamma^d$ acting diagonally on the two chiral representations, and the chirality operator $\Gamma^{d+1}$ in dimension $d$ acting on their sum, now being the representation of the $(d+1)$st Clifford algebra generator.
One may ask in which dimensions $d$ the above complex representations admit a real structure
Real spinor representations are also called Majorana representations, and an element of a real/Majorana spin representation is also called a Majorana spinor. On a Majorana representation $S$ there is a non-vanishing symmetric and $Spin(d-1,1)$-invariant bilinear form $S \otimes S \longrightarrow \mathbb{R}^d$, projectively unique if $S$ is irreducible. This serves as the odd-odd Lie bracket in the super Lie algebra called the super Poincaré Lie algebra extension of the ordinary Poincaré Lie algebra induced by $S$. This is “supersymmetry” in physics.
The above irreducible complex representations admit a real structure for $d = 1,2,3 \, mod \, 8$. Therefore in dimension $d = 2 \, mod \, 8$ there exist Majorana-Weyl spinor representations.
The above irreducible complex representations admit a quaternionic structure for $d = 5,6,7 \, mod \, 8$.
Let $V$ be a quadratic vector space, def. 1, over the real numbers with bilinear form of Lorentzian signature, hence $V = \mathbb{R}^{d-1,1}$ is Minkowski spacetime of some dimension $d$.
The following table lists the irreducible real representations of $Spin(V)$ (Freed 99, page 48).
$d$ | $Spin(d-1,1)$ | minimal real spin representation $S$ | $dim_{\mathbb{R}} S\;\;$ | $V$ in terms of $S^\ast$ | supergravity |
---|---|---|---|---|---|
1 | $\mathbb{Z}_2$ | $S$ real | 1 | $V \simeq (S^\ast)^{\otimes}^2$ | |
2 | $\mathbb{R}^{\gt 0} \times \mathbb{Z}_2$ | $S^+, S^-$ real | 1 | $V \simeq ({S^+}^\ast)^{\otimes^2} \oplus ({S^-}^\ast)^{\otimes 2}$ | |
3 | $SL(2,\mathbb{R})$ | $S$ real | 2 | $V \simeq Sym^2 S^\ast$ | |
4 | $SL(2,\mathbb{C})$ | $S_{\mathbb{C}} \simeq S' \oplus S''$ | 4 | $V_{\mathbb{C}} \simeq {S'}^\ast \oplus {S''}^\ast$ | d=4 N=1 supergravity |
5 | $Sp(1,1)$ | $S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$ | 8 | $\wedge^2 S_0^\ast \simeq \mathbb{C} \oplus V_{\mathbb{C}}$ | |
6 | $SL(2,\mathbb{H})$ | $S^\pm_{\mathbb{C}} \simeq S_0^\pm \otimes_{\mathbb{C}} W$ | 8 | $V_{\mathbb{C}} \simeq \wedge^2 {S_0^+}^\ast \simeq (\wedge^2 {S_0^-}^\ast)^\ast$ | |
7 | $S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$ | 16 | $\wedge^2 S_0^\ast \simeq V_{\mathbb{C}} \oplus \wedge^2 V_{\mathbb{C}}$ | ||
8 | $S_{\mathbb{C}} \simeq S^\prime \oplus S^{\prime\prime}$ | 16 | ${S'}^\ast {S''}^\ast \simeq V_{\mathbb{C}} \oplus \wedge^3 V_{\mathbb{C}}$ | ||
9 | $S$ real | 16 | $Sym^2 S^\ast \simeq \mathbb{R} \oplus V \wedge^4 V$ | ||
10 | $S^+ , S^-$ real | 16 | $Sym^2(S^\pm)^\ast \simeq V \oplus \wedge_\pm^5 V$ | type II supergravity | |
11 | $S$ real | 32 | $Sym^2 S^\ast \simeq V \oplus \wedge^2 V \oplus \wedge^5 V$ | 11-dimensional supergravity |
Here $W$ is the 2-dimensional complex vector space on which the quaternions naturally act.
The last column implies that in each dimension there exists a linear map
which is
symmetric;
$Spin(V)$-equivariant.
This allows to form the super Poincaré Lie algebra in each of these cases. See there and see Spinor bilinear forms below for more.
Let $(V, \langle -,-\rangle)$ be a quadratic vector space, def. 1. For $p \in \mathbb{R}$ write $\wedge^p V$ for its $p$th skew-symmetrized tensor power, regarded naturally as a representation of the spin group $Spin(V)$.
For $S_1, S_2 \in Rep(Spin(V))$ two irreducible representations of $Spin(V)$, we discuss here homomorphisms of representations (hence $k$-linear maps respecting the $Spin(V)$-action) of the form
These appear notably in the following applications:
for $p = 0$ symmetric bilinears $(-,-) \;\colon\; S \otimes S \longrightarrow k$ define a metric on the space of spinors, also known as a charge conjugation matrix. This appears for instance in the Lagrangian for a spinor field $\psi$, which is of the form $\psi \mapsto (\psi, D \psi)$, for $D$ a Dirac operator;
for $p = 1$ symmetric bilinear $Spin(V)$-homomorphisms $\Gamma \;\colon\; S \otimes S \longrightarrow V$ constitutes the odd-odd Lie bracket in a super Poincaré Lie algebra extension of the a Poincaré Lie algebra by $S$.
for $p=2$ these homomorphisms appeare as the odd-odd bracket in a superconformal super Lie algebra;
for $p \geq 2$ higher spin bilinears $S \otimes S \longrightarrow \wedge^p V$ appear in further polyvector extensions.
We discuss spinor bilinear pairings to scalars.
Let $V$ be a quadratic vector space, def. 1 over the complex numbers of dimension $d$. Then there exists in dimensions $d \neq 2,6 \; mod \, 8$, up to rescaling, a unique $Spin(V)$-invariant bilinear form
on a complex irreducible representation $S$ of $Spin(V)$, or in dimension 2 and 6 a bilinear pairing
which is non-degenerate and whose symmetry is given by the following table:
$d \, mod\, 8$ | C |
---|---|
0 | symmetric |
1 | symmetric |
2 | $S^\pm$ dual to each other |
3 | skew-symmetric |
4 | skew-symmetric |
5 | skew-symmetric |
6 | $S^\pm$ dual to each other |
7 | symmetric |
This appears for instance as (Varadarajan 04, theorem 6.5.7).
The matrix representation of the bilinear form in prop. 2 is known in the physics literature as the charge conjugation matrix. In matrix calculus the symmetry property means that the transpose matrix $C^T$ satisfies
with $\epsilon \in \{-1,1\}$ given in dimension $d$ by the following table
$d \, mod \, 8$ | $C$ |
---|---|
0 | -1 |
1 | -1 |
2 | either |
3 | +1 |
4 | +1 |
5 | +1 |
6 | either |
7 | -1 |
For instance (van Proeyen 99, table 1).
Let $V$ be a quadratic vector space, def. 1 over the real numbers of dimension $d$ with Loentzian signature. Then there exists, up to rescaling, a unique $Spin(V)$-invariant bilinear form
on a real irreducible representation $S$ of $Spin(V)$, and its symmetry is given by the following table
$d \, mod \, 8$ | $C$ |
---|---|
0 | symmetric |
1 | symmetric |
2 | $S^{\pm}$ dual to each other |
3 | skew symmetric |
4 | skew symmetric |
5 | symmetric |
6 | $S^{\pm}$ dual to each other |
7 | symmetric |
This appears for instance as (Freed 99, around (3.4), Varadarajan 04, theorem 6.5.10).
We discuss spinor bilinear pairings to vectors.
Let $V$ be a quadratic vector space, def. 1 over the complex numbers of dimension $d$.
Then there exists unique $Spin(V)$-representation morphisms
for odd $d$ and $S$ the unique irreducible representation, and
for even $d$ and $S^\pm$ the two inequivalent irreducible representations.
This is (Varadarajan 04, theorem 6.6.3).
Let $V$ be a quadratic vector space, def. 1 over the real numbers of dimension $d$.
Then there exists unique $Spin(V)$-representation morphisms
$d \,mod \, 8$ | |
---|---|
0 | $S^\pm \otimes S^\mp \to V$ |
1 | $S \otimes S \to V$ |
2 | $S^\pm \otimes S^\pm \to V$ |
3 | $S \otimes S \to V$ |
4 | $S^\pm \otimes S^\mp \to V$ |
5 | $S \otimes S \to V$ |
6 | $S^\pm \otimes S^\pm \to V$ |
7 | $S \otimes S \to V$ |
This is (Varadarajan 04, theorem 6.5.10).
For more see (Varadarajan 04, section 6.7).
In terms of a matrix representation with respect to a chosen basis as in remark 2 the pairing of prop. 5 is given by the matrices $\Gamma^a = \{(\Gamma^a)^\alpha{}_\beta\}$ that represent the Clifford algebra by raising and lowering indices with the charge conjugation matrix of remark 2 (e.g Freed 99 (3.5)).
In such a notation if $\phi = (\phi^\alpha)$ denotes the component-vector of a spinor, then the result of “lowering its index” is given by acting with the metric in form of the charge conjugation matrix. The result is traditionally denoted
hence
This yields the component formula for the pairings to scalars and to vectors which is traditional in the physics literature as follows:
and
(Recall that all this is here for Majorana spinors, as in the previous prop. 5.)
This yields the component expressions for the bilinear pairings as familiar from the physics supersymmetry literature, for instance (Polchinski 01, (B.2.1), (B.5.1))
A spinor bilinear pairing to a vector $\Gamma \;\colon\; S \otimes S \to V$ as above serves as the odd-odd bracket in a super Poincaré Lie algebra extension of $V$. Since this is also called a “supersymmetry” super Lie algebra, with the spinors being the supersymmetry generators, the decomposition of $S$ into minimal/irreducible representations is also called the number of supersymmetries. This is traditionally denoted by a capital $N$ and in even dimensions and over the complex numbers it is traditional to write
to indicate that there are $N_+$ copies of the irreducible $Spin(V)$-representation of one chirality, and $N_-$ of those of the other chirality (i.e. left and right handed Weyl spinors).
This counting however is more subtle over the real numbers (Majorana spinors) and the notation in this case (which happens to be the more important case) is not entirely consistent through the literature.
There is no issue in those dimensions in which the complex Weyl representation already admits a real structure itself, hence when there are Majorana-Weyl spinors. In this case one just counts them with $N_+$ and $N_-$ as in the case over the complex numbers.
However, in some dimensions it is only the direct sum of two Weyl spinor representations which carries a real structure. For instance for $d = 4$ and $d = 8$ in Lorentzian signature (see the above table) it is the complex representations $\mathbf{2} \oplus \mathbf{2}'$ and $\mathbf{16} \oplus \mathbf{16}'$, respectively, which carry a real structure. Hence the real representation underlying this parameterizes $N = 1$ supersymmetry in terms Majorana spinors, even though its complexification would be $N = (1,1)$. See for instance (Freed 99, p. 53).
Similarly in dimensions 5,6 and 7 mod 8, the minimal real representation is obatained from tensoring the complex spinors with the complex 2-dimensional canonical quaternionic representation $W$ (as in the above table). These are also called symplectic Majorana representations. For instance in in 6d one typically speaks of the 6d (2,0)-superconformal QFT to refer to that with a single “symplectic Majorana-Weyl” supersymmetry (e.g. Figueroa-OFarrill, p. 9), which might therefore be counted as $(1,0)$ real supersymmetric, but which involves two complex irreps and is hence often denoted counted as $(2,0)$.
The real spinor representations in dimensions $3 \leq d \leq 11$ happen to have a particularly simple expression in terms of Hermitian forms over the four real normed division algebras, the real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, the quaternions $\mathbb{H}$ and the octonions $\mathbb{O}$. For more on this see also at supersymmetry and division algebras. We follow (Baez-Huerta 09, Baez-Huerta 10).
Some notation:
For $\mathbb{K}$ one of the four real normed division algebras, write $(-)^\ast \colon \mathbb{K} \stackrel{\simeq}{\to} \mathbb{K}^{op}$ for the conjugation anti-automorphism.
the real part of an element $a \in \mathbb{K}$ is $Re(a) \coloneqq \tfrac{1}{2} (a + a^\ast)$;
say that an $n \times n$ matrix with coefficients in $\mathbb{K}$, $A\in Mat_{n\times n}(\magthbb{K})$ is a hermitian matrix if the transpose matrix equals the conjugated matrix: $A^t = A^\ast$. Hence with $(-)^\dagger \coloneqq ((-)^t)^\ast$ this is $A = A^\dagger$, as usual;
write $\tilde A \coloneqq A - (tr A) \mathbb{1}_{n\times n}$ for the matrix minus its trace times the identity matrix.
Write $V \coloneqq Mat_{2\times 2}(\mathbb{K})^{hermitian}$ for the $\mathbb{K}$-vector space of 2x2 hermitian matrices. Write $S_+, S_- \coloneqq \mathbb{K}^2$. Define a real linear map
by
The real vector space $V$ in def. 4 equipped with the inner product $\eta(-,-)$ given by the determinant (which on hermitian $\mathbb{K}$-matrices is necessarily real):
$\mathbb{R}^{2,1}$ for $\mathbb{K} = \mathbb{R}$;
$\mathbb{R}^{3,1}$ for $\mathbb{K} = \mathbb{C}$;
$\mathbb{R}^{5,1}$ for $\mathbb{K} = \mathbb{H}$;
$\mathbb{R}^{9,1}$ for $\mathbb{K} = \mathbb{O}$.
The map $\Gamma$ in def. 4 gives a representation of the Clifford algebra of
$\mathbb{R}^{2,1}$ for $\mathbb{K} = \mathbb{R}$;
$\mathbb{R}^{3,1}$ for $\mathbb{K} = \mathbb{C}$;
$\mathbb{R}^{5,1}$ for $\mathbb{K} = \mathbb{H}$;
$\mathbb{R}^{9,1}$ for $\mathbb{K} = \mathbb{O}$.
Under restriction along $Spin(n+1,1) \hookrightarrow Cl(n+1,1)$ this is isomorphic to
for $\mathbb{K} = \mathbb{R}$ the Majorana representation of $Spin(2,1)$ on $S_+ \simeq S_-$;
for $\mathbb{K} = \mathbb{C}$ the Majorana representation of $Spin(3,1)$ on $S_+ \simeq S_-$;
for $\mathbb{K} = \mathbb{H}$ the Weyl representation of $Spin(5,1)$ on $S_+$ and on $S_-$;
for $\mathbb{K} = \mathbb{O}$ the Majorana-Weyl representation of $Spin(9,1)$ on $S_+$ and on $S_-$.
(e.g. Baez-Huerta 09, p. 6)
Under the identification of prop. 6 the bilinear pairings
and
of remark 1 are given, respectively, by forming the real part of the canonical $\mathbb{K}$-inner product
and by forming the product of a column vector with a row vector to produce a matrix, possibly up to trace reversal:
and
For $A \in V$ the $A$-component of this map is
(e.g. Baez-Huerta 09, prop. 8, prop. 9).
Write $V \coloneqq Mat_{2\times 2}(\mathbb{K})^{hermitian} \oplus \mathbb{R}$.
Write $S \coloneqq \mathbb{K}^4$. Define a real linear map
given by left matrix multiplication
The real vector space $V$ in def. 5 equipped with the inner product $\eta(-,-)$ given by
$\mathbb{R}^{3,1}$ for $\mathbb{K} = \mathbb{R}$;
$\mathbb{R}^{4,1}$ for $\mathbb{K} = \mathbb{C}$;
$\mathbb{R}^{6,1}$ for $\mathbb{K} = \mathbb{H}$;
$\mathbb{R}^{10,1}$ for $\mathbb{K} = \mathbb{O}$.
The map $\Gamma$ in def. 5 gives a representation of the Clifford algebra of
$\mathbb{R}^{3,1}$ for $\mathbb{K} = \mathbb{R}$;
$\mathbb{R}^{4,1}$ for $\mathbb{K} = \mathbb{C}$;
$\mathbb{R}^{6,1}$ for $\mathbb{K} = \mathbb{H}$;
$\mathbb{R}^{10,1}$ for $\mathbb{K} = \mathbb{O}$.
Under restriction along $Spin(n+2,1) \hookrightarrow Cl(n+2,1)$ this is isomorphic to
for $\mathbb{K} = \mathbb{R}$ the Majorana representation of $Spin(3,1)$ on $S$;
for $\mathbb{K} = \mathbb{C}$ the Dirac representation of $Spin(4,1)$ on $S$;
for $\mathbb{K} = \mathbb{H}$ the Dirac representation of $Spin(6,1)$ on $S$;
for $\mathbb{K} = \mathbb{O}$ the Majorana representation of $Spin(10,1)$ on $S$.
(Baez-Huerta 10, p. 10, prop. 8, prop. 9)
Write
Under the identification of prop. 8 of the bilinear pairings
and
of remark 1, the first is given by
and the second is characterized by
(Baez-Huerta 10, prop. 10, prop. 11).
Accounts in the mathematical literature include
H. Blaine Lawson, Marie-Louise Michelsohn, Chapter I.5 of Spin geometry, Princeton University Press (1989)
Anna Engels, Spin representations (pdf)
Specifically for Lorentzian signature and with an eye towards supersymmetry in QFT, see
Daniel Freed, Lecture 3 of Five lectures on supersymmetry 1999
Veeravalli Varadarajan, section 7 of Supersymmetry for mathematicians: An introduction, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)
For the component notation traditionally used in physics see for instance
Antoine Van Proeyen, Tools for supersymmetry, Lectures in the spring school in Calimanesti, Romania, April 1998 (arXiv:hep-th/9910030)
Joseph Polchinski, part II, appendix B of String theory, Cambridge Monographs on Mathematical Physics (2001)
Friedemann Brandt, section 2 of Supersymmetry algebra cohomology I: Definition and general structure J. Math. Phys.51:122302, 2010, (arXiv:0911.2118)
For good math/physics discussion with special emphasis on the symplectic Majorana spinors and their role in the 6d (2,0)-superconformal QFT see
A clean summary of the relation of the real representation to Hermitian forms over the real normed division algebras is in
John Baez, John Huerta, Division algebras and supersymmetry I, in R. Doran, G. Friedman and Jonathan Rosenberg (eds.), Superstrings, Geometry, Topology, and $C*$-algebras, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80 (arXiv:0909.0551)
John Baez, John Huerta, Division algebras and supersymmetry II, Adv. Math. Theor. Phys. 15 (2011), 1373-1410 (arXiv:1003.34360)