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The fermionic zero modes to the "standard" $SU(2)$ 't Hooft instantons are known as follows (see page 42 in this article):

For a 't Hooft instanton (self-dual field strength tensor) in the regular gauge, centered at the origin, the massless Dirac equation admits a unique right-handed normalizable fundamental-representation Weyl spinor solution (up to an overall constant):

$$\psi_{\alpha,a}(x)=\begin{pmatrix}0 \\ \lambda_{\alpha,a}(x) \end{pmatrix}\,\,\, \textrm{(Right-handed)}$$

$$\lambda_{\alpha,a}(x)=\frac{\epsilon_{\alpha a}}{(x^2+\rho^2)^{3/2}}$$

where $\alpha=1,2$ is the spinor index, $a=1,2$ is the color/gauge index, and $\epsilon_{\alpha a}$ is the standard Levi-Civita tensor which surprisingly mixes spinor and color indices. By definition, this solution satisfies the equation:

$$(\sigma^{\mu})^{\alpha \alpha'}(\partial_\mu \lambda_{\alpha', a}-iA^{m}_{\mu}(T_m)_{ab}\lambda_{\alpha', b})=0$$

where $A^m_{\mu}$ is the 't Hooft instanton gauge field, $T_m$ are the 3 $SU(2)$ infinitesimal generators (here in the fundamental representation, so they're just Pauli matrices), and $\sigma^\mu=(\sigma^i,i)$.

In the singular gauge, the corresponding solution is:

$$\lambda_{\alpha,a}(x)=\frac{1}{x (x^2+\rho^2)^{3/2}}x_{\mu}(\sigma^{\dagger \mu})_{\alpha,a}$$

[My Question] What about non-zero modes? In other words, what is the spectrum of the Dirac operator in the background of an $SU(2)$ self-dual instanton? ($\gamma^{\mu}D_{\mu}\psi=\lambda \psi$)

Do the corresponding fermionic non-zero modes have analytic solutions? What about orthogonality relations between these eigenfunctions?

I'm having a hard time finding answers to this online...

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I am not aware of any attempts to write down the non-zero mode solutions. However, we are (typically) not really interested in individual non-zero mode solutions, but in the non-zero mode propagator $$ S(x,y)=\sum_{\lambda\neq 0} \frac{\psi_\lambda(x)\psi_\lambda^\dagger(y)}{\lambda} $$ and this object has been determined, see here.

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