On the Loewner energy of a welding homeomorphism

To any Jordan curve one may associate a circle homeomorphism $\varphi : \mathbb S^1 \to \mathbb S^1$ via conformal welding. Through this correspondence, the Loewner energy $I^L$, also known as the universal Liouville action, is a K\"ahler potential for the unique homogeneous K\"ahler metric on the universal Teichm\"uller space. Despite this, explicit expressions for $I^L$ in terms of $\varphi$ alone do not seem to be available in the literature. In this paper, we obtain such formulas. For this, we introduce an operator ${\bf \Lambda}_\varphi$ defined using the Fourier coefficients of the function $$(z,w) \mapsto \log \left|\frac{\varphi(z)-\varphi(w)}{z-w}\right|, \qquad (z,w) \in \mathbb{S}^1 \times \mathbb{S}^1.$$ We relate ${\bf \Lambda}_\varphi$ to the single-layer potential and composition operator, and prove an analog of the classical Grunsky inequalities for quasisymmetric $\varphi$. We show moreover that $\varphi$ is Weil--Petersson if and only if ${\bf \Lambda}_\varphi$ is Hilbert--Schmidt, and we express $I^L$ as several related Fredholm determinants as well as a regularized Fredholm determinant. We also treat Schatten classes, and we obtain formulas in terms of Dirichlet integrals involving $\log \varphi'$ and in terms of the composition operator induced by $\varphi$.