Conformal Invariance in Higher Spin Spaces

• Conformally invariant states are structures that remain unchanged under local angle-preserving transformations, crucial for understanding higher spin function spaces.
• They are constructed via fundamental solutions and convolution operators that ensure equivariance under the conformal group while distinguishing bosonic and fermionic cases.
• These states are pivotal in quantum field theory, Clifford analysis, and representation theory by providing explicit solutions to complex higher-spin differential equations.

A conformally invariant state is a mathematical or physical structure—such as a field configuration, a differential operator, a metric, or a class of functions—that remains unchanged (up to prescribed weights or scaling factors) under conformal (i.e., local angle-preserving) transformations of the underlying space. In the context of differential geometry and representation theory, such states are realized as the solutions or kernels of conformally invariant differential operators, and, in higher spin analysis, as functions valued in irreducible representations of the Spin group. The classification and explicit construction of such states require a rigorous understanding of the interplay between the conformal group’s action and the representation spaces, especially in dimensions larger than two and for non-scalar (higher spin) fields.

1. Foundations: Conformal Invariance and Higher Spin Spaces

A differential operator $D$ acting on a space of functions (or sections) is said to be conformally invariant—equivariant with respect to the conformal group—if there exists a collection of conformal weights $J_t(\varphi, x)$ (typically expressed as powers of $\|cx+d\|$ for Möbius transformations $\varphi$) such that for all $\varphi \in \mathrm{Conf}(\mathbb{R}^m)$ and all admissible $f$,

$$J_t^{-1}(\varphi, y) D_{t, y, v} f(y, v) = D_{t, x, u} [J_t(\varphi, x) f(\varphi(x), \cdot)],$$

where $x$ denotes the original variable, $y = \varphi(x)$ its image, and $u, v$ the auxiliary variables tracking spinorial or harmonic polynomial degrees of freedom. In higher spin spaces—function spaces valued in irreducible representations of $\operatorname{Spin}(m)$ such as homogeneous harmonic polynomials ($\mathcal{H}_k$; integer spin) and monogenic polynomials ($\mathcal{M}_k$; half-integer spin)—the construction and classification of such operators are highly nontrivial, relying on deep results from Clifford analysis and representation theory.

2. Explicit Construction via Fundamental Solutions and Convolution Operators

The methodology for constructing arbitrary order conformally invariant operators proceeds by specifying a fundamental solution (kernel) for each operator and then inverting the convolution to obtain the differential operator itself. The fundamental solution of order $t$ takes the general form

$E_t(x, u, v) = c \|x\|^{p(t) - m} Z_k(xux, v),$

where $Z_k$ is a reproducing kernel for the relevant representation ($\mathcal{H}_k$ or $\mathcal{M}_k$), and $p(t)$ is an explicit function of order $t$ (positive for even/bosonic orders, negative for odd/fermionic orders). The associated convolution-type (Knapp–Stein) operator acts as

$$\Phi(f)(y, v) = \int_{\mathbb{R}^m} \int_{\mathbb{S}^{m-1}} E_t(x-y, u, v) f(x, u) dS(u)\,dx,$$

and in the distributional sense,

$D \cdot E_t = \mathrm{Id},$

so the operator $D$ can be realized as the convolution inverse.

The Knapp–Stein intertwining theory asserts that if the fundamental solution kernel is conformally invariant, so is the associated operator, and vice versa. Additionally, the convolution operators themselves (and their inverses, when defined as pseudo-differential operators) are conformally invariant, broadening the class of conformally invariant states one can construct.

3. Classification of Bosonic and Fermionic Conformally Invariant Operators

The operators naturally bifurcate into two families based on parity:

Order	Domain Space	Target Space	Kernel Structure
Even $2j$	$C(\mathbb{R}^m, \mathcal{H}_k)$	$C(\mathbb{R}^m, \mathcal{H}_k)$	$E_{2j}(x,u,v) = c\|x\|^{2j-m} Z_k(xux, v)$
Odd $2j-1$	$C(\mathbb{R}^m, \mathcal{M}_k)$	$C(\mathbb{R}^m, \mathcal{M}_k)$	$E_{2j-1}(x,u,v) = c\|x\|^{-2j+2} Z_k(xux, v)$

• Bosonic operators (even order): These act on integer-spin ($\mathcal{H}_k$) spaces and generalize the even-order Laplacian; higher spin Laplace operators are constructed using twistor and dual-twistor (Stein–Weiss) operators.
• Fermionic operators (odd order): These act on half-integer-spin ($\mathcal{M}_k$) spaces and generalize the Dirac and Rarita–Schwinger operators.

A typical higher-order bosonic operator $D_{2j}$ may be expressed as a product

$$D_{2j} = D_2 \prod_{s=2}^j \left( D_2 - \frac{4s(2s-2)}{(m+2k-2)(m+2k-4)} \Delta \right),$$

where $D_2$ is a second-order operator constructed from Stein–Weiss gradients and their duals.

4. Intertwining Operators and Equivariance

The intertwining property ensures that under a conformal change of variables via a Möbius transformation $\varphi(x)$, the operator $D_t$ transforms according to its conformal weight: $$J_t^{-1}(\varphi, y) D_{t, y, v} f(y, v) = D_{t, x, u}\left[ J_t(\varphi, x) f(\varphi(x), \cdot) \right],$$ with $J_t(\varphi, x) = \|cx + d\|^{\alpha}$, where $\alpha$ is determined by $t$, the order of the operator, and the dimension $m$. This property is essential for the classification and guarantees that the space of solutions—the conformally invariant states—is preserved under the conformal group action.

Underlying this is the representation-theoretic fact that the spaces $\mathcal{H}_k$ and $\mathcal{M}_k$ are irreducible under $\mathrm{Spin}(m)$; thus, all equivariant operators between such representation spaces are uniquely determined up to normalization once their intertwining property is specified.

5. Applications: Conformally Invariant States in Higher Spin Theories

Explicitly constructed conformally invariant operators in higher spin spaces play a crucial role in the analysis of higher-spin field equations, Clifford analysis, boundary value problems, and the representation theory of the conformal group. The conformally invariant states are realized as the kernels of these operators: functions or sections annihilated by $D_t$, preserved under the conformal group’s action.

Some key applications and implications are:

• Quantum field theory: These operators model free field equations (Dirac, Rarita–Schwinger, higher-spin Laplacians) in backgrounds with conformal symmetry, relevant to higher-spin gauge theories and AdS/CFT correspondence.
• Clifford analysis and boundary problems: The explicit knowledge of fundamental solutions and intertwining properties permits the solution of Cauchy problems, spectral analysis, and examination of boundary regularity for conformally invariant PDEs in the higher spin setting.
• Representation theory: The operators realize explicit realization of intertwining maps between principal series representations of the conformal group and classify admissible conformally invariant function spaces (the “states”) in the context of the local and global theory (e.g., on spheres or the Minkowski space and their compactifications).

6. Advanced Aspects: Stein–Weiss Gradients and Motivation from Representation Theory

The construction leans heavily on the generalization of the gradient (the Stein–Weiss gradient), which projects the ordinary Euclidean gradient onto subspaces determined by irreducible representations. For integer-spin, this yields the higher-spin Laplace and twistor operators; for half-integer spin, one obtains generalizations of the Dirac and Rarita–Schwinger operators.

Twistor and dual-twistor operators ($T_k$, $T_k^*$) are central to the structure of these differential operators and their fundamental solutions. Their algebraic relations, projection properties, and explicit expressions fix both the nature of the operators and of their conformally invariant states.

7. Broader Context and Significance

By completing the classification of arbitrary order conformally invariant operators in higher spin spaces, extending results previously confined to locally conformally flat manifolds and lower spin, the framework establishes a robust foundation for the paper of higher-spin field equations, conformally invariant functional analysis, and the explicit representation theory of the conformal group.

The explicit form of the intertwining operators, the structure of fundamental solutions (including their parity and conformal weights), and the convolution inverse construction combine to provide a unique and authoritative picture of conformally invariant states in the landscape of higher spin mathematics and physics.

References: (Ding et al., 2016)