The Adjoint of the Twistor Operator

Introduction

The twistor operator $\mathcal{T}=\pi\circ\nabla^{\mathcal{S}}$ is not an operator on one bundle: it maps the spinor sections to the one-form valued spinors, $\Gamma(\mathcal{S})\to\Gamma(T^*M\otimes\mathcal{S})$, and it is the projection of the covariant derivative onto the kernel of the Clifford contraction. Its adjoint for the $L^2$ form is therefore an operator in the reverse direction, and the shape of the answer is simple: the adjoint of the projection is the projection, and the adjoint of the covariant derivative is the covariant divergence, so

$$ \mathcal{T}^{*} = \nabla^{*}\circ\pi , $$

with $\nabla^*$ the adjoint connection of The Codifferential on a Spinor Bundle. The article computes the adjoint, derives the orthogonal decomposition of the connection Laplacian that it produces,

$$ \nabla^{*}\nabla = \mathcal{T}^{*}\mathcal{T}+\tfrac1n D^{2} , $$

with $D$ the self-adjoint Cauchy–Riemann operator, and reads from it the consequences for the twistor equation $\mathcal{T}\sigma=0$: the equivalent form $\nabla_X\sigma=-\frac1nc(X)D\sigma$ of the equation and the scalar identity satisfied by the twistor spinors.

The boundaries. The operator, its projection $\pi$ and the contraction $c$ are The Twistor Operator; the Cauchy–Riemann operator, its self-adjointness $D^*=D$ and the Lichnerowicz formula are The Spinor Operator; the adjoint connection and its calculus are The Codifferential on a Spinor Bundle; the Clifford multiplication and its adjoint are The Clifford Multiplication Operator and The Adjoint of the Clifford Multiplication. The involutions, the conjugate symmetry and the Hermitian pairing of the kernel are Involutions of the Twistor Operator, the preceding entry of the group. The spectral theory of $\mathcal{T}^*\mathcal{T}$ is Part III's. The body is a Riemannian spin manifold with the fibre form $h$, $c(v)^*=-c(v)$ and the twistor operator of The Twistor Operator.

The Formal Adjoint

Definition. The formal adjoint $\mathcal{T}^*$ of the twistor operator is the differential operator $\Gamma(T^*M\otimes\mathcal{S})\to\Gamma(\mathcal{S})$ with

$$ \int_M h(\mathcal{T}\sigma,\psi)\,\mu_g = \int_M h(\sigma,\mathcal{T}^{*}\psi)\,\mu_g $$

for all compactly supported smooth sections, $h$ extended to the twisted bundle by the metric on the cotangent factor.

Theorem. $\mathcal{T}^{*}=\nabla^{*}\circ\pi$, where $\pi=\mathrm{id}-\frac1nc^*c$ is the $h$-orthogonal projection onto $\ker c$ and $\nabla^*$ is the adjoint of the covariant derivative; the principal symbol of $\mathcal{T}^*$ is the adjoint Clifford contraction, and $\mathcal{T}$ and $\mathcal{T}^*$ are elliptic of order one.

Proof. The adjoint of a composite is the composite of the adjoints in the reverse order, so $\mathcal{T}^*=(\pi\nabla)^*=\nabla^*\pi^*$, and $\pi$ is an orthogonal projection for the pointwise form, hence $\pi^*=\pi$. The symbol statements are the symbol of the contraction and of its adjoint, and the ellipticity is the invertibility of the symbol of $\mathcal{T}$ on the trace-free part, which is the content of the symbol computation of The Twistor Operator.

Proposition (the components). Writing a twisted section as $\psi=\sum_i\theta^i\otimes\psi_i$ with $\psi_i\in\Gamma(\mathcal{S})$, the projection acts by

$$ \pi\psi = \sum_i\theta^i\otimes\Bigl(\psi_i+\tfrac1n c(e_i)(c\psi)\Bigr) , \qquad c\psi=\sum_j c(e_j)\psi_j , $$

and the adjoint is $\mathcal{T}^{*}\psi=-\sum_i\nabla_i\bigl(\psi_i+\tfrac1nc(e_i)(c\psi)\bigr)$ up to the lower-order terms of $\nabla^*$; in particular $\mathcal{T}^*\psi$ depends on $\psi$ through its trace-free part and the divergence of its components.

Proof. The formula for $\pi$ is the definition $\pi\psi=\psi-\frac1nc^*c\psi$ with the adjoint contraction $c^*\sigma=-\sum_i\theta^i\otimes c(e_i)\sigma$ of The Adjoint of the Clifford Multiplication; the divergence is the leading part of $\nabla^*$, which is $-\sum_i\nabla_i$ on the components plus the curvature terms recorded in The Codifferential on a Spinor Bundle.

Remark (the pairing as the twistor form). The $L^2$ pairing of the definition is the integrated form of the pointwise Hermitian form of Hermitian Clifford Structures; the anti-linearity in the second argument and the reality of $\nabla$ make the pairing conjugate-symmetric, which is the conjugate symmetry of Involutions of the Twistor Operator read at the level of the adjoint.

The Decomposition of the Connection Laplacian

Theorem (the orthogonal decomposition). The connection Laplacian decomposes as

$$ \nabla^{*}\nabla = \mathcal{T}^{*}\mathcal{T}+\tfrac1n D^{2} , $$

where $D$ is the Cauchy–Riemann operator of The Spinor Operator; both summands are non-negative operators on the compactly supported sections, and the decomposition is orthogonal for the $L^2$ form.

Proof. Let $\pi=\mathrm{id}-\frac1nc^*c$ be the projection onto $\ker c$ and write the identity $\mathrm{id}=\pi+\frac1nc^*c$. Then

$$ \nabla^{*}\nabla=\nabla^{*}\bigl(\pi+\tfrac1nc^{*}c\bigr)\nabla = (\nabla^{*}\pi)\nabla+\tfrac1n\nabla^{*}c^{*}c\nabla = \mathcal{T}^{*}\mathcal{T}+\tfrac1n\nabla^{*}c^{*}c\nabla . $$

Now $\nabla^*c^*=(c\nabla)^*$, because the adjoint of a composite is the composite of the adjoints and $(\nabla^*)^*=\nabla$; and $(c\nabla)^*=D^*=D$ by the formal self-adjointness of the Cauchy–Riemann operator of The Spinor Operator. Hence $\nabla^*c^*c\nabla=D\circ D=D^2$, which gives the identity. The non-negativity is $\langle\nabla^{*}\nabla\sigma,\sigma\rangle=\lVert\nabla\sigma\rVert^2\ge0$ and the corresponding statements for the two summands, since each is a composite of an operator with its adjoint: $\mathcal{T}^*\mathcal{T}=(\mathcal{T})^*(\mathcal{T})$ and $D^2=D^*D$.

Corollary. The kernel of $\mathcal{T}$ is the kernel of $\mathcal{T}^*\mathcal{T}$, and it is a subspace of the kernel of $\nabla^*\nabla$; the twistor spinors are the parallel spinors of the decomposition's first summand, and the identity shows how the twistor equation sits inside the equation $\nabla^*\nabla\sigma=\frac1nD^2\sigma$.

Proof. For a compactly supported section, $\langle\mathcal{T}^*\mathcal{T}\sigma,\sigma\rangle=\lVert\mathcal{T}\sigma\rVert^2$, so $\mathcal{T}\sigma=0$ exactly when $\mathcal{T}^*\mathcal{T}\sigma=0$; the identity gives the second statement.

The Twistor Equation

Theorem. For a spinor $\sigma$ the twistor equation $\mathcal{T}\sigma=0$ is equivalent to

$$ \nabla_X\sigma = -\tfrac1n c(X)D\sigma \qquad\text{for all vector fields } X , $$

and then $\sigma$ satisfies the scalar identity

$$ \nabla^{*}\nabla\sigma = \frac{\operatorname{scal}}{4(n-1)}\,\sigma , \qquad D^{2}\sigma = \frac{n}{4(n-1)}\,\operatorname{scal}\,\sigma , $$

where $\operatorname{scal}$ is the scalar curvature.

Proof. $\mathcal{T}\sigma=0$ means that $\nabla\sigma$, a one-form valued spinor, lies in the kernel of the contraction $c$, that is in the image of $c^*$, and the component form of that membership is $\nabla_X\sigma=-\frac1nc(X)D\sigma$: indeed $c(\nabla\sigma)=0$ means $\sum_ic(e_i)\nabla_i\sigma=0$, and the trace-free condition identifies $\nabla_X\sigma+\frac1nc(X)D\sigma$ as the orthogonal projection onto the trace-free part. For the scalar identity, substitute into the decomposition of the connection Laplacian: $\nabla^*\nabla\sigma=\frac1nD^2\sigma$ because the first summand vanishes, and the Lichnerowicz formula $D^2=\nabla^*\nabla+\frac14\operatorname{scal}$ of The Spinor Operator gives $\nabla^*\nabla\sigma=\frac1n\nabla^*\nabla\sigma+\frac{\operatorname{scal}}{4n}\sigma$, which is the display.

Corollary (the domain of the solutions). A twistor spinor is determined by $\sigma(x)$ and $\nabla\sigma(x)$ at one point, so the space of twistor spinors has dimension at most $2^{\lfloor n/2\rfloor+1}$; the bound is attained on the conformally flat manifolds by The Penrose Operator, where the space is a representation of the conformal algebra.

Proof. The equation $\nabla_X\sigma=-\frac1nc(X)D\sigma$ is a first-order system whose initial data are the values of $\sigma$ and of $\nabla\sigma$ modulo the relation; the dimension of the spinor fibre gives $2^{\lfloor n/2\rfloor}$, and the one-form constrained to be the image of $c^*$ contributes one further spinor, whence $2^{\lfloor n/2\rfloor+1}$; the attainment is the quoted statement of The Penrose Operator.

Remark. The scalar identity is the reason a twistor spinor with a parallel $D\sigma$ off a zero of the scalar curvature is forced to vanish or to be special: integrating the identity against the volume form gives $\int\lvert\nabla\sigma\rvert^2=\frac1{4(n-1)}\int\operatorname{scal}\,\lvert\sigma\rvert^2$, and a definite sign of the scalar curvature then excludes or forces the solutions; the argument is the analytic counterpart of the Bochner method and its rigour is Part III's.

Worked Cases

The Flat Space

For $\mathbb{R}^n$ with the standard spin structure, $\mathcal{T}\sigma=0$ has the solutions $\sigma=\sigma_0+\sum_ix^ic(e_i)\sigma_0$ with $\sigma_0$ parallel, the dimension is $2^{\lfloor n/2\rfloor+1}$, and $\mathcal{T}^*\mathcal{T}$ is $-\Delta$ on the trace-free part; the identity $\nabla^*\nabla=\mathcal{T}^*\mathcal{T}+\frac1nD^2$ is the flat decomposition of the connection Laplacian, and the scalar identity is vacuous because $\operatorname{scal}=0$.

The Round Sphere

For $S^n$ with the round metric, $\operatorname{scal}=n(n-1)$, the twistor spinors are the sums of the two Killing families, the dimension attains the bound $2^{\lfloor n/2\rfloor+1}$, and the scalar identity reads $\nabla^*\nabla\sigma=\frac n4\sigma$; the first eigenvalue of the connection Laplacian on the twistor spinors is $\frac n4$.

A Compact Einstein Manifold

For a compact Einstein manifold of positive scalar curvature the twistor operator has finite-dimensional kernel by ellipticity, the identity gives the bound on the dimension in terms of the eigenvalue of the twistor Laplacian, and the rigidity statements — that a compact manifold with the maximal number of twistor spinors is conformally flat — are quoted from the literature.

Summary

The formal adjoint of the twistor operator is $\mathcal{T}^{*}=\nabla^{*}\circ\pi$, the composition of the covariant divergence with the orthogonal projection onto $\ker c$; with the corpus's signs $c(v)^*=-c(v)$ and $D^*=D$, it yields the orthogonal decomposition of the connection Laplacian

$$ \nabla^{*}\nabla=\mathcal{T}^{*}\mathcal{T}+\tfrac1n D^{2} , $$

proved from $\mathrm{id}=\pi+\frac1nc^*c$ and $\nabla^*c^*=D$. There follows the equivalent form of the twistor equation, $\nabla_X\sigma=-\frac1nc(X)D\sigma$, the scalar identity $\nabla^*\nabla\sigma=\frac{\operatorname{scal}}{4(n-1)}\sigma$ with $D^2\sigma=\frac{n}{4(n-1)}\operatorname{scal}\sigma$, and the bound $2^{\lfloor n/2\rfloor+1}$ on the dimension of the space of twistor spinors, attained on the conformally flat manifolds. The operator is The Twistor Operator, its involutions and the conjugate symmetry of the pairing are Involutions of the Twistor Operator, the Cauchy–Riemann operator and the Lichnerowicz formula are The Spinor Operator, the adjoint connection is The Codifferential on a Spinor Bundle, and the spectral theory of $\mathcal{T}^*\mathcal{T}$ is Part III's.

Summary of Notation

Symbol	Meaning
$\mathcal{T}=\pi\circ\nabla$, $\pi=\mathrm{id}-\frac1nc^*c$	Twistor operator
$\mathcal{T}^{}=\nabla^{}\pi$	Formal adjoint
$D=c\circ\nabla$, $D^*=D$	Self-adjoint Cauchy–Riemann operator
$\nabla^{}\nabla=\mathcal{T}^{}\mathcal{T}+\frac1nD^2$	Orthogonal decomposition of the connection Laplacian
$\nabla_X\sigma=-\frac1nc(X)D\sigma$	Equivalent form of the twistor equation
$\nabla^{*}\nabla\sigma=\frac{\operatorname{scal}}{4(n-1)}\sigma$	Scalar identity for twistor spinors
$2^{\lfloor n/2\rfloor+1}$	Bound on the dimension of the space of twistor spinors