The Landau–Ginzburg/conformal field theory correspondence is the conjecture which states that the behavior of the conformal fixed point under the renormalization group flow of a Landau–Ginzburg model is captured by its topological B-twist. This project aims at studying the LG/CFT correspondence using the insights provided by the string-net construction of CFT correlators, with the potential to produce the first rigorous solution of non-rational, non-logarithmic CFTs in the functorial approach.
The topological defects in a CFT are described by a pivotal bicategory B, whose objects are the phases of the theory. Given a rigid separable Frobenius 2-functor F from B to the category C of chiral data (viewed as a pivotal bicategory with a single object), the image of any object of B is endowed by the functor F with the structure of a Δ-separable symmetric Frobenius algebra and the image of every 1-morphism that of a bimodule. The string-net construction then provides a consistent system of correlators in the sense that the mapping class group invariance and the sewing constraints are fulfilled. An important feature of the string-net construction is that the geometric expression of the correlators can be converted into algebraic ones in terms of compositions of internal natural transformations in the Drinfeld center Z(C).
We would like to apply this procedure to the pivotal bicategory LG of Laundau-Ginzburg models and matrix bi-factorizations which, by the conjectured LG/CFT correspondence, is preserved under the renormalization group flow and describes the topological defects of a CFT. To this end, we look for a spherical tensor category C that is naturally constructed from LG and a canonical rigid separable Frobenius functor with domain LG and codomain C. The structures provided by the rigid separable Frobenius functor would then allow us to calculate the defect field contents of the theory in terms of internal natural transformations.
|Yang Yang||Oberwolfach Research Institute for Mathematics|