Covariance in self dual inhomogeneous models of effective quantum geometry: Spherical symmetry and Gowdy systems

When applying the techniques of Loop Quantum Gravity (LQG) to symmetry-reduced gravitational systems, one first regularizes the scalar constraint using holonomy corrections, prior to quantization. In inhomogeneous system, where a residual spatial diffeomorphism symmetry survives, such modification of the gauge generator generating time reparametrization can potentially lead to deformations or anomalies in the modified algebra of first class constraints. When working with self-dual variables, it has already been shown that, for spherically symmetric geometry coupled to a scalar field, the holonomy-modified constraints do not generate any modifications to general covariance, as one faces in the real variables formula- tion, and can thus accommodate local degrees of freedom in such inhomogeneous models. In this paper, we extend this result to Gowdy cosmologies in the self-dual Ashtekar formulation. Furthermore, we show that the introduction of a $\bar{\mu}$-scheme in midisuperspace models, as is required in the `improved dynamics' of LQG, is possible in the self-dual formalism while being out of reach in the current effective models suing real-valued Ashtekar-Barbero variables. Our results indicate the advantages of using the self-dual variables to obtain a covariant loop regularization prior to quantization in inhomogeneous symmetry reduced polymer models, implementing additionally the crucial $\bar{\mu}$-scheme, and thus a consistent semiclassical limit.