### Hodge and Gelfand Theory in Clifford Analysis and Tomography

Clayton Shonkwiler

http://shonkwiler.org

August, 2022

/qipa22

This talk!

Colin Roberts

Most of what I will talk about today is work of Colin Roberts.

### Motivations

Calderón Problem.

Given a Riemannian manifold $$(M,g)$$ with boundary, determine $$g$$ from the Dirichlet-to-Neumann operator $$\Lambda$$.

Belishev’s 2D Solution.

On a surface, $$\Lambda$$ determines the algebra $$\mathcal{A}(M)$$ of holomorphic functions on $$M$$, which determines $$(M,g)$$ up to conformal equivalence.

### Clifford Algebras

If $$V$$ is a vector space with symmetric bilinear form $$g$$, the tensor algebra is

and the associated Clifford algebra is

\mathcal{T}(V) := \bigoplus_{j=0}^\infty V^{\otimes j} = \mathbb{R} \oplus V \oplus (V \otimes V) \oplus \dots
C\ell(V,g) := \mathcal{T}(V)/\langle v \otimes v - g(v,v)\rangle

$$\dim(C\ell(V,g)) = 2^{\dim(V)}$$

### Examples

$$\bullet$$ $$V = \mathbb{R}^2$$, $$g(u,v) = u \cdot v$$.

(a e_1 + b e_2) (c e_1 + d e_2) = (ac + bd) + ad e_1 e_2 + bc e_2 e_1 \\ = (ac + bd) + (ad - bc)e_1e_2

$$\bullet$$ $$g = 0$$.

C\ell(V,0) = \bigwedge(V)

$$\bullet$$ If $$g$$ is non-degenerate, $$C\ell(V,g)$$ is called a geometric algebra. Use the notation $$\mathcal{G}_{p,q}:=C\ell(V,g)$$, when $$g$$ is the standard non-degenerate form of type $$(p,q)$$.

If $$A \in \mathcal{G}_{p,q}$$, let $$\langle A\rangle_r \in \mathcal{G}_{p,q}^r$$ be the grade-$$r$$ part of $$A$$.

On homogeneous elements $$A \in \mathcal{G}_{p,q}^r$$ and $$B \in \mathcal{G}_{p,q}^s$$,

A B = \langle A B\rangle_{|r-s|} + \langle AB\rangle _{|r-s|+2} + \dots + \langle AB\rangle_{r+s}
A \wedge B := \langle AB \rangle_{r+s}
A \,\lrcorner\, B := \langle AB \rangle_{s-r}

The subset $$\mathcal{G}_{p,q}^+$$ of even-graded elements forms a subalgebra, called the spinor algebra.

### Examples

• $$\mathcal{G}_{0,2}^+ \simeq \mathbb{C}$$: generated by $$1$$ and $$e_1e_2$$, with $$(e_1e_2)^2 = -1$$.
• $$\mathcal{G}_{0,3}^+ \simeq \mathbb{H}$$: generated by $$1,e_1e_2, e_2e_3, e_3e_1$$.
• $$\mathcal{G}_{1,3}^2 \simeq \mathfrak{spin}(1,3)$$, the Lie algebra of the Lorentz group.

### Multivector Fields

$$(M,g)$$ smooth, compact, connected, oriented.

Definition.

The geometric algebra bundle over $$M$$ is

\mathcal{G}M:= \bigsqcup_{p \in M} C\ell(T_pM,g_p).

The multivector fields $$\mathfrak{X}(M)$$ are smooth sections of $$\mathcal{G}M$$.

### Hodge–Dirac Operator

$$g$$ induces the Levi–Civita connection $$\nabla$$ and covariant derivative $$\nabla_v$$ on $$M$$, which extend to multivectors.

Definition.

The Hodge–Dirac operator is defined in local coordinates by

\nabla = \sum_{i=1}^n e^i \nabla_{e_i}.

For $$A,B \in \mathfrak{X}(M)$$, $$\nabla(AB) = \dot{\nabla}\dot{A}B + \dot{\nabla}A\dot{B}$$.

$$\nabla^2$$ is the Laplace–Beltrami operator.

### Examples

• For $$A \in \mathfrak{X}^+(\mathbb{R}^2)$$, $$\nabla A = 0$$ iff $$A$$ is a holomorphic function.
• For $$v \in \mathfrak{X}^1(\R^3)$$,
\nabla v = \underbrace{\nabla \,\lrcorner\, v}_{\operatorname{div}} + \underbrace{\nabla \wedge v}_{\operatorname{curl}}

### Monogenic Fields

Definition.

The space of monogenic fields $$\mathcal{M}(M) := \ker \nabla$$.

Monogenic fields

• can be uniquely continued
• have a Cauchy integral
• have harmonic components

### Clifford–Hodge Decomposition

Theorem [Roberts]

The space of multivector fields on $$M$$ has the orthogonal decomposition

\mathfrak{X}(M) = \mathcal{M}(M) \oplus \nabla \mathfrak{X}(M),

where the Dirac fields are defined as

\nabla\mathfrak{X}(M):=\{\nabla A : A \in \mathfrak{X}(M), \left.A\right|_{\partial M} = 0\}.

### Connections to EIT

$$M$$ an Ohmic region in $$\mathbb{R}^3$$, relate conductivity to $$g$$ in the usual way.

Ohm’s Law: $$-\nabla \wedge u = J$$

Conservation: $$\nabla \,\lrcorner\, J=0$$

Define the electric DN operator $$\Lambda_E: t \mathfrak{X}^0(M) \to t\mathfrak{X}^0(M)$$ by

\Lambda_E\phi = \nu \lrcorner \nabla \wedge u = \frac{\partial u}{\partial \nu}

where

\begin{cases} \nabla^2 u = 0 & \text{ in } M \\ u = \phi & \text{ on } \partial M.\end{cases}

### Magnetic Analog

Define the electric DN operator $$\Lambda_E: t \mathfrak{X}^0(M) \to t\mathfrak{X}^0(M)$$ by

\Lambda_E\phi = \nu \lrcorner \nabla \wedge u = \frac{\partial u}{\partial \nu}

where

\begin{cases} \nabla^2 u = 0 & \text{ in } M \\ u = \phi & \text{ on } \partial M.\end{cases}

Define the magnetic DN operator $$\Lambda_B: n \mathfrak{X}^2(M) \to n\mathfrak{X}^2(M)$$ by

\Lambda_B(\nu \wedge J) = \nu \wedge \nabla \lrcorner\, B

where

\begin{cases} \nabla^2 B = 0 & \text{ in } M \\ B = \nu \wedge J & \text{ on } \partial M.\end{cases}

$$A_+ := u+B \in \mathcal{M}(M)$$ is a monogenic spinor.

### Geometric Generalization

Generalized electric DN operator $$\Lambda_E: t \mathfrak{X}(M) \to t\mathfrak{X}(M)$$

\Lambda_E\phi = \nu \lrcorner \nabla \wedge A

Generalized magnetic DN operator $$\Lambda_B: n \mathfrak{X}(M) \to n\mathfrak{X}(M)$$

\Lambda_B\phi = \nu \wedge \nabla \lrcorner\, A

where

\begin{cases} \nabla^2 A = 0 & \text{ in } M \\ A = \phi & \text{ on } \partial M. \end{cases}

$$\Lambda_E \times \Lambda_B$$ is equivalent to the complete DN operator

[Sharafutdinov–Shonkwiler, 2013]

### Spinor DN Operator

Define the spinor DN operator $$\mathcal{J}: \operatorname{tr}\mathfrak{X}^\pm(M) \to \operatorname{tr}\mathfrak{X}^\pm(M)$$ by

\mathcal{J}\phi = \nu \nabla A.

Scalar part: $$\langle \mathcal{J} \rangle = \Lambda_E + \Lambda_B$$.

Theorem [Roberts]

$$\ker \mathcal{J} = \operatorname{tr} \mathcal{M}^\pm(M).$$

### Connections to Boundary Control

Building on ideas of Belishev and Vakulenko, define a spinor spectrum $$\mathfrak{M}(M)$$ consisting of certain grade-preserving $$\mathcal{G}_{0,n}^+$$-linear maps $$\mathcal{M}^+(M) \to \mathcal{G}_{0,n}^+$$.

E.g., point evaluation $$\delta[A_+] = A_+(x_\delta)$$.

Theorem [Roberts]

With the weak-$$*$$ topology on $$\mathfrak{M}(M)$$, the map

\gamma: \mathfrak{M}(M) \to M \qquad \delta \mapsto x_\delta

is a homeomorphism.

The Gelfand transform $$\widehat{A_+}(\delta) = \delta[A_+]$$ is an isometric isomorphism, so $$\mathcal{M}^+(M) \simeq \widehat{\mathcal{M}^+(M)}$$.

### Questions

1. Does the DN map determine $$\operatorname{tr}\mathcal{M}^+(M)$$?
2. Relationship between $$\vee \mathcal{M}^+(M)$$ and $$\vee \operatorname{tr}\mathcal{M}^+(M)$$?
3. Does $$\mathcal{M}^+(M)$$ determine the metric on $$M$$?
4. Hilbert transform?

# Thank you!

### References

Colin Roberts, Hodge and Gelfand Theory in Clifford Analysis and Tomography, Ph.D. thesis, Colorado State University, 2022, https://hdl.handle.net/10217/235741

Colin Roberts, A Gelfand transform for spinor fields on embedded Riemannian manifolds, preprint, 2022, arXiv:2203.00118

#### Hodge and Gelfand Theory in Clifford Analysis and Tomography

By Clayton Shonkwiler

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