Logarithmic extensions of minimal models: characters and modular transformations

TL;DR

This paper explores logarithmic conformal field models extending minimal models, focusing on their algebraic structure, modules, characters, and modular properties, revealing new insights into their symmetries and dualities.

Contribution

It constructs the W(p,q) algebra for these models, identifies their modules and characters, and analyzes their modular transformations and quantum group dualities.

Findings

Constructed the W(p,q) algebra as the kernel of screening operators.

Derived the characters of irreducible modules.

Analyzed the SL(2,Z) modular transformations of torus amplitudes.

Abstract

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.