The Adjoint of the Clifford Multiplication

Introduction

On a Riemannian spin manifold the Clifford multiplication is the bundle map $c : TM\otimes\mathcal{S}\to\mathcal{S}$, $(v,\sigma)\mapsto v\cdot\sigma$, and its adjoint for the metric and the fibre form is the negative of itself,

$$ c(v)^{*}=-c(v) , $$

the fibre form being the one for which the Clifford coeffients are skew. This is the manifold form of the abstract statement $L_u^*=-L_u$ for the left multiplication by a vector, and it is the reason the Cauchy–Riemann operator $D=c\circ\nabla^{\mathcal{S}}$ is formally self-adjoint and the Clifford contraction $c : T^*M\otimes\mathcal{S}\to\mathcal{S}$ has the adjoint $c^*=-\sum_i\theta^i\otimes(e_i\cdot)$ with $cc^*=n\operatorname{id}$. The article collects the pointwise adjoint, the adjoint of the contraction, and the consequences for the first-order operators of the category.

The boundaries. The Clifford multiplication and its kernel and image are The Clifford Multiplication Operator; the contraction and the projector $\pi$ are The Twistor Operator; the Cauchy–Riemann operator, its self-adjointness and the Lichnerowicz formula are The Spinor Operator; the adjoint of the spin connection and the connection Laplacian are The Codifferential on a Spinor Bundle; the abstract one-sided adjoint is The Adjoint of the Left Multiplication on a Clifford Algebra and The Twisted Adjoint on a Clifford Algebra. The body is a Riemannian spin manifold $(M,g)$ with $c(v)^2=-g(v,v)\operatorname{id}$, $c(v)^*=-c(v)$, and the fibre form $h$.

The Pointwise Adjoint

Theorem. For the fibre form $h$ and the metric $g$, the Clifford multiplication by a tangent vector is skew-adjoint,

$$ h(c(v)\sigma,\tau)=-h(\sigma,c(v)\tau) , \qquad c(v)^{*}=-c(v) , $$

and consequently $h(c(v)\sigma,c(v)\tau)=g(v,v)h(\sigma,\tau)$: the multiplication by a unit vector is an isometry of the fibre.

Proof. The Clifford relations give $c(v)^2=-g(v,v)\operatorname{id}$, and the fibre form is invariant under the local action of $\operatorname{Spin}(n)$ generated by the unit vectors; for a unit $e$, $c(e)^2=-\operatorname{id}$ makes $c(e)$ a complex structure, which is skew for a form invariant under the orthogonal group it generates, so $c(e)^*=-c(e)$; the general vector is a scalar multiple, and the isometry statement is the second display. The pointwise computation is The Clifford Multiplication Operator.

Remark (the abstract origin). The manifold computation is the special case of $L_u^*=-L_u$ for the left multiplication in the Clifford algebra with the standard form of The Twisted Adjoint on a Clifford Algebra: the fibre is a Clifford module, the multiplication is the module action, and the form is the invariant inner product. The corpus records the two statements in their own settings and identifies them here.

The Adjoint of the Contraction

Definition. The Clifford contraction is $c : T^*M\otimes\mathcal{S}\to\mathcal{S}$, $c(\xi\otimes\sigma)=\xi^{\sharp}\cdot\sigma$, whose adjoint for the tensor product of $g^{-1}$ and $h$ is

$$ c^* : \mathcal{S}\longrightarrow T^*M\otimes\mathcal{S} , \qquad c^*\sigma = -\sum_{i=1}^{n}\theta^i\otimes(e_i\cdot\sigma) . $$

Proposition. The contraction satisfies $cc^*=n\operatorname{id}_{\mathcal{S}}$ and $\ker c=\operatorname{im}c^*$; the contraction is a surjection with the symbol $c(\xi)=$ Clifford multiplication by $\xi^{\sharp}$, and the adjoint has the symbol $c^*(\xi)=-(\xi\wedge\text{-type})$ contraction, so the two symbols are adjoint at every covector.

Proof. $cc^*\sigma=-\sum_ie_i\cdot(e_i\cdot\sigma)=\sum_ig(e_i,e_i)\sigma=n\sigma$; the surjectivity of $c$ and the identity give the orthogonal decomposition; the symbol statements are the linear-algebra part of the two displays, and they are adjoint because the bundle maps are. This is The Twistor Operator, where the orthogonality is used to split the covariant derivative.

Remark (the two contractions). The map $c$ of this section (contraction of a covector with a spinor) and the skew-adjointness of $c(v)$ of the previous section are two aspects of the same bundle $\mathrm{Hom}(T^*M\otimes\mathcal{S},\mathcal{S})$: the first displays the adjoint $c^*$ and the trace identity, the second the pointwise sign. Both are used in the construction of the twistor and Penrose operators.

Consequences for the First-Order Operators

Proposition. The Cauchy–Riemann operator $D=c\circ\nabla^{\mathcal{S}}$ is formally self-adjoint for the $L^2$ form, $D^*=D$, and the twistor operator $\mathcal{T}=\pi\circ\nabla^{\mathcal{S}}$ satisfies $c\circ\mathcal{T}=0$; the adjoint of the spin connection is the codifferential $\delta$, and $\delta\circ c^*=D$ holds as the statement of $D^*=D$ after the frame-derivative terms.

Proof. The self-adjointness of $D$ is computed from the adjoint of the composition and the metric compatibility of the spin connection in The Spinor Operator and The Codifferential on a Spinor Bundle; the identity $c\circ\mathcal{T}=0$ is the definition of $\pi$ and is The Twistor Operator; the last statement is the adjoint identity of the codifferential, quoted.

Remark. The whole adjoint theory of the first-order operators of the category follows from the pointwise adjoint $c(v)^*=-c(v)$: the skew-adjointness of the multiplication makes $D$ self-adjoint, and the adjoint of the contraction makes the codifferential the natural partner of the connection. The geometric operators of the next articles are obtained by composing these adjoints with the twisted and graded operators of the algebra.

Worked Cases

A Surface

On a surface with an orthonormal frame, $c(e_1)$ and $c(e_2)$ are two anticommuting complex structures of the fibre, both skew-adjoint, and the contraction $c^*$ sends a spinor to $-\theta^1\otimes e_1\sigma-\theta^2\otimes e_2\sigma$; the identity $cc^*=2$ is the rank-two statement of the surface.

The Flat Space

On $\mathbb{R}^n$ with the standard spinor module, $c(v)^*=-c(v)$ in the standard Hermitian form, $c^*\sigma=-\sum_i\theta^i\otimes e_i\sigma$, $cc^*=n$, and $D=\sum_ic(e_i)\partial_i$ is self-adjoint on the flat spinor bundle by integration by parts.

The Exterior Bundle

For $c(v)=v^{\flat}\wedge-\iota_v$ on $\Lambda^\bullet T^*M$, the adjoint for the metric is $c(v)^*=-(v^{\flat}\wedge-\iota_v)$, since $\iota_v$ and $v^{\flat}\wedge$ are adjoint to within the identity; the identity $cc^*=n$ then holds with the same proof, and the pointwise adjoint of the exterior Clifford multiplication is the negative of the operator.

Summary

The adjoint of the Clifford multiplication on a Riemannian spin manifold is the negative of the operator, $c(v)^*=-c(v)$, and the multiplication by a unit vector is an isometry of the fibre, $h(c(v)\sigma,c(v)\tau)=g(v,v)h(\sigma,\tau)$; this is the manifold form of the abstract $L_u^*=-L_u$. The Clifford contraction $c:T^*M\otimes\mathcal{S}\to\mathcal{S}$ has adjoint $c^*\sigma=-\sum_i\theta^i\otimes(e_i\cdot\sigma)$, so $cc^*=n\operatorname{id}$ and $\ker c=\operatorname{im}c^*$, which is the decomposition used by the twistor and Penrose operators. The consequences for the first-order operators are the formal self-adjointness $D^*=D$ of the Cauchy–Riemann operator, the identity $c\circ\mathcal{T}=0$ of the twistor operator, and the fact that the adjoint of the spin connection is the codifferential, with $\delta c^*=D$. The abstract adjoints are The Adjoint of the Left Multiplication on a Clifford Algebra and The Twisted Adjoint on a Clifford Algebra; the contraction is The Twistor Operator, and the self-adjointness is The Spinor Operator and The Codifferential on a Spinor Bundle.

Summary of Notation

Symbol	Meaning
$c(v)$, $h$	Clifford multiplication and fibre form
$c(v)^*=-c(v)$	Pointwise skew-adjointness
$h(c(v)\sigma,c(v)\tau)=g(v,v)h(\sigma,\tau)$	Isometry up to the metric scale
$c:T^*M\otimes\mathcal{S}\to\mathcal{S}$	Clifford contraction
$c^*\sigma=-\sum_i\theta^i\otimes(e_i\cdot\sigma)$	Its adjoint
$cc^=n\operatorname{id}$, $\ker c=\operatorname{im}c^$	Trace identity and decomposition
$D^*=D$	Self-adjointness of the Cauchy–Riemann operator
$\delta c^*=D$	Codifferential and the operator
$L_u^*=-L_u$	The abstract one-sided statement