I came across the following statement in a paper:
On $S^3$, the eigenvalues of the vector Laplacian on divergenceless vector fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell \in \mathbb{ Z}$.
Is it possible to prove the spectrum and degeneracy using the representation theory of $SO(4)$? Perhaps there is a general result for the n-sphere.
The paper then proceeds to make the non-sense statement (RHS is divergent):
$$ \det \big(-\Delta + a\big) = \prod_{\ell=1}^\infty \big((\ell + 1)^2 + a \big)^{2\ell(\ell+2)} $$
How do we make sense of the determinant of the Laplacian on the space of divergenceless vector fields?
The spectral determinant of $-\Delta+a$ is defined as $e^{-\zeta_{-\Delta+a}'(0)}$ where
$$\zeta_{-\Delta+a}(s) = \sum_{n = 0}^{\infty} (\lambda_n+a)^{-s} $$ is the zeta function defined in a region of the complex plane, it admits analytic continuation to negative values of s and this is the way to write the "nonsense" infinite product. To compute the RHS you need to find the way to define a function which gives the analytic continuation of the series (usually is similar as in the Riemann zeta function, otherwise is too difficult), and then find the derivative at $s = 0$.
For rigorous detais on convergence I recommend Collected Papers volume V - J. Jorgenson, S. Lang.
For the physics behind these definitions (most of them related with quantum field theory and string theory) see Quantum Fields and Strings: A Course for Mathematicians: Volume 2: (American Mathematics Society non-series title) for example.