Nils Carqueville

Computing Khovanov–Rozansky homology and defect fusion

We compute the categorified sl(N) link invariants as defined by Khovanov and Rozansky, for various links and values of N. This is made tractable by an algorithm for reducing tensor products of matrix factorisations to finite rank, which we implement in the computer algebra package Singular.

Adjunctions and defects in Landau-Ginzburg models

Abstract We study the bicategory of Landau–Ginzburg models, which has polynomials as objects and matrix factorisations as 1-morphisms. Our main result is the existence of adjoints in this bicategory and formulas for the evaluation and coevaluation maps in terms of Atiyah classes and homological perturbation. The bicategorical perspective offers a unified approach to Landau–Ginzburg models: we show how to compute arbitrary correlators and recover the full structure of open/closed TFT, including the Kapustin–Li disc correlator and a simple proof of the Cardy condition, in terms of defect operators which in turn are directly computable from the adjunctions.

Orbifolds and Topological Defects

AbstractWe study orbifolds of two-dimensional topological field theories using defects. If the TFT arises as the twist of a superconformal field theory, we recover results on the Neveu–Schwarz and Ramond sectors of the orbifold theory, as well as bulk-boundary correlators from a novel, universal perspective. This entails a structure somewhat weaker than ordinary TFT, which however still describes a sector of the underlying conformal theory. The case of B-twisted Landau–Ginzburg models is discussed in detail, where we compute charge vectors and superpotential terms for B-type branes. Our construction also works in the absence of supersymmetry and for generalised “orbifolds” that need not arise from symmetry groups. In general, this involves a natural appearance of Hochschild (co)homology in a 2-categorical setting, in which among other things we provide simple presentations of Serre functors and a further generalisation of the Cardy condition.

Rigidity and defect actions in Landau-Ginzburg models

Studying two-dimensional field theories in the presence of defect lines naturally gives rise to monoidal categories: their objects are the different (topological) defect conditions, their morphisms are junction fields, and their tensor product describes the fusion of defects. These categories should be equipped with a duality operation corresponding to reversing the orientation of the defect line, providing a rigid and pivotal structure. We make this structure explicit in topological Landau-Ginzburg models with potential xd, where defects are described by matrix factorisations of xd − yd. The duality allows to compute an action of defects on bulk fields, which we compare to the corresponding

Orbifold equivalent potentials

{\mathcal N = 2}

Bulk deformations of open topological string theory

conformal field theories. We find that the two actions differ by phases.

Introductory lectures on topological quantum field theory

We investigate tensor products of matrix factorizations. This is most naturally done by formulating matrix factorizations in terms of bimodules instead of modules. If the underlying ring is we show that bimodule matrix factorizations form a monoidal category. This monoidal category has a physical interpretation in terms of defect lines in a two-dimensional Landau?Ginzburg model. There is a dual description via conformal field theory, which in the special case of W = xd is an minimal model and which also gives rise to a monoidal category describing defect lines. We carry out a comparison of these two categories in certain subsectors by explicitly computing 6j-symbols.

Orbifold completion of defect bicategories

Abstract To a graded finite-rank matrix factorisation of the difference of two homogeneous potentials one can assign two numerical invariants, the left and right quantum dimensions. The existence of such a matrix factorisation with non-zero quantum dimensions defines an equivalence relation between potentials, giving rise to non-obvious equivalences of categories. Restricted to ADE singularities, the resulting equivalence classes of potentials are those of type { A d − 1 } for d odd, { A d − 1 , D d / 2 + 1 } for d even but not in { 12 , 18 , 30 } , and { A 11 , D 7 , E 6 } , { A 17 , D 10 , E 7 } and { A 29 , D 16 , E 8 } . This is the result expected from two-dimensional rational conformal field theory, and it directly leads to new descriptions of and relations between the associated (derived) categories of matrix factorisations and Dynkin quiver representations.