Logarithmic extensions of minimal models: Characters and modular transformations

We study logarithmic conformal field models that extend the (p, q) Virasoro minimal models. For coprime positive integers p and q, the model is defined as the kernel of the two minimal-model screening operators. We identify the field content, construct the W-algebra W_{p, q} that is the model symmetry (the maximal local algebra in the kernel), describe its irreducible modules, and find their characters. We then derive the SL (2, Z)-representation on the space of torus amplitudes and study its properties. From the action of the screenings, we also identify the quantum group that is Kazhdan-Lusztig-dual to the logarithmic model.