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When applying the techniques of Loop Quantum Gravity (LQG) tosymmetry-reduced gravitational systems, one first regularizes the scalarconstraint using holonomy corrections, prior to quantization. In inhomogeneoussystem, where a residual spatial diffeomorphism symmetry survives, suchmodification of the gauge generator generating time reparametrization canpotentially lead to deformations or anomalies in the modified algebra of firstclass constraints. When working with self-dual variables, it has already beenshown that, for spherically symmetric geometry coupled to a scalar field, theholonomy-modified constraints do not generate any modifications to generalcovariance, as one faces in the real variables formula- tion, and can thusaccommodate local degrees of freedom in such inhomogeneous models. In thispaper, we extend this result to Gowdy cosmologies in the self-dual Ashtekarformulation. Furthermore, we show that the introduction of a $\bar{\mu}$-schemein midisuperspace models, as is required in the `improved dynamics' of LQG, ispossible in the self-dual formalism while being out of reach in the currenteffective models suing real-valued Ashtekar-Barbero variables. Our resultsindicate the advantages of using the self-dual variables to obtain a covariantloop regularization prior to quantization in inhomogeneous symmetry reducedpolymer models, implementing additionally the crucial $\bar{\mu}$-scheme, andthus a consistent semiclassical limit.

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