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In this letter, we stress that the simplest cosmological model consisting ina massless scalar field minimally coupled to homogeneous and isotropic gravityhas an in-built $\SL(2,\mathbb{R})$ symmetry. Protecting this symmetrynaturally provides an efficient way to constrain the quantization of thiscosmological system whatever the quantization scheme and allows in particularto fix the quantization ambiguities arising in the canonical quantizationprogram. Applying this method to the loop quantization of the FLRW cosmologyleads to a new loop quantum cosmology model which preserves the$\SL(2,\mathbb{R})$ symmetry of the classical system. This new polymerregularization consistent with the conformal symmetry can be derived as anon-linear canonical transformation of the classical FLRW phase space, whichmaps the classical singular dynamics into a regular effective bouncingdynamics. This improved regularization preserves the scaling properties of thevolume and Hamiltonian constraint. 3d scale transformations, generated by thedilatation operator, are realized as unitary transformations despite theminimal length scale hardcoded in the theory. Finally, we point out that theresulting cosmological dynamics exhibits an interesting duality between shortand long distances, reminiscent of the T-duality in string theory, with thenear-singularity regime dual to the semi-classical regime at large volume. Thetechnical details of the construction of this model are presented in a longercompanion paper \cite{BenAchour:2019ywl}.

Polymer Quantum Cosmology: Lifting quantization ambiguities using a $SL(2,\mathbb{R})$ conformal symmetry