arXiv subjects

Nils Carqueville

Publications and source records attributed to Nils Carqueville.

At least 37 records · Page 2Linked to original sources

arXiv2017

Introductory lectures on topological quantum field theory

These notes offer a lightening introduction to topological quantum field theory in its functorial axiomatisation, assuming no or little prior exposure. We lay some emphasis on the connection between the path integral motivation and the definition in terms symmetric monoidal categories, and we highlight the algebraic formulation emerging from a formal generators-and-relations description. This allows one to understand (oriented, closed) 1- and 2-dimensional TQFTs in terms of a finite amount of algebraic data, while already the 3-dimensional case needs an infinite amount of data. We evade these complications by instead discussing some aspects of 3-dimensional extended TQFTs, and their relation to braided monoidal categories.

math.QA↗

arXiv2016

Lecture notes on 2-dimensional defect TQFT

These notes offer an introduction to the functorial and algebraic description of 2-dimensional topological quantum field theories `with defects', assuming only superficial familiarity with closed TQFTs in terms of commutative Frobenius algebras. The generalisation of this relation is a construction of pivotal 2-categories from defect TQFTs. We review this construction in detail, flanked by a range of examples. Furthermore we explain how open/closed TQFTs are equivalent to Calabi-Yau categories and the Cardy condition, and how to extract such data from pivotal 2-categories.

math.QA↗

arXiv2015

Adjunctions and defects in Landau-Ginzburg models

We study the bicategory of Landau-Ginzburg models, which has potentials as objects and matrix factorisations as 1-morphisms. Our main result is the existence of adjoints in this bicategory and a description of 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 disk correlator and a simple proof of the Cardy condition, in terms of defect operators which in turn are directly computable from the adjunctions.

math.AG↗

arXiv2014

Discrete torsion defects

Orbifolding two-dimensional quantum field theories by a symmetry group can involve a choice of discrete torsion. We apply the general formalism of `orbifolding defects' to study and elucidate discrete torsion for topological field theories. In the case of Landau-Ginzburg models only the bulk sector had been studied previously, and we re-derive all known results. We also introduce the notion of `projective matrix factorisations', show how they naturally describe boundary and defect sectors, and we further illustrate the efficiency of the defect-based approach by explicitly computing RR charges. Roughly half of our results are not restricted to Landau-Ginzburg models but hold more generally, for any topological field theory. In particular we prove that for a pivotal bicategory, any two objects of its orbifold completion that have the same base are orbifold equivalent. Equivalently, from any orbifold theory (including those based on nonabelian groups) the original unorbifolded theory can be obtained by orbifolding via the `quantum symmetry defect'.

hep-th↗

arXiv2014

Orbifold completion of defect bicategories

Orbifolds of two-dimensional quantum field theories have a natural formulation in terms of defects or domain walls. This perspective allows for a rich generalisation of the orbifolding procedure, which we study in detail for the case of topological field theories. Namely, a TFT with defects gives rise to a pivotal bicategory of "worldsheet phases" and defects between them. We develop a general framework which takes such a bicategory B as input and returns its "orbifold completion" B_orb. The completion satisfies the natural properties B \subset B_orb and (B_orb)_orb = B_orb, and it gives rise to various new equivalences and nondegeneracy results. When applied to TFTs, the objects in B_orb correspond to generalised orbifolds of the theories in B. In the example of Landau-Ginzburg models we recover and unify conventional equivariant matrix factorisations, prove when and how (generalised) orbifolds again produce open/closed TFTs, and give nontrivial examples of new orbifold equivalences.

math.QA↗

arXiv2014

Orbifolds and topological defects

We 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.

hep-th↗

arXiv2014

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.

math.QA↗

arXiv2013

Orbifold equivalent potentials

To a graded finite-rank matrix factorisation of the difference of two homogeneous potentials one can assign two numbers, the left and right quantum dimension. 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.

math.QA↗

arXiv2013

A quick guide to defect orbifolds

We provide a lightning review of the construction of (generalised) orbifolds [arXiv:0909.5013, arXiv:1210.6363] of two-dimensional quantum field theories in terms of topological defects, along the lines of [arXiv:1307.3141]. This universal perspective has many applications, some of which we sketch in the examples of 2d Yang-Mills theory, Landau-Ginzburg models, and rational CFT.

hep-th↗

arXiv2013

A toolkit for defect computations in Landau-Ginzburg models

We review the results of arXiv:1208.1481 on orientation reversal and duality for defects in topological Landau-Ginzburg models, with the intention of providing an easily accessible toolkit for computations. As an example we include a proof of the main result on adjunctions in a special case, using Pauli matrices. We also explain how to compute arbitrary correlators of defect-decorated planar worldsheets, and briefly discuss the relation to generalised orbifolds.

hep-th↗

arXiv2012

Remarks on quiver gauge theories from open topological string theory

We study effective quiver gauge theories arising from a stack of D3-branes on certain Calabi-Yau singularities. Our point of view is a first principle approach via open topological string theory. This means that we construct the natural A-infinity-structure of open string amplitudes in the associated D-brane category. Then we show that it precisely reproduces the results of the method of brane tilings, without having to resort to any effective field theory computations. In particular, we prove a general and simple formula for effective superpotentials.

hep-th↗

arXiv2012

Bulk deformations of open topological string theory

We present a general method to construct bulk-deformed open topological string theories from Landau-Ginzburg models. To this end we obtain a weak version of deformation quantisation, and we show how this together with the technique of homological perturbation allows to explicitly compute all bulk-deformed open topological string amplitudes at tree-level before tadpole-cancellation. Our approach is based on a coherent treatment of the problem in terms of the fundamental A-infinity- and L-infinity-structures involved.

hep-th↗

arXiv2012

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 x^d, where defects are described by matrix factorisations of x^d-y^d. The duality allows to compute an action of defects on bulk fields, which we compare to the corresponding N=2 conformal field theories. We find that the two actions differ by phases.

hep-th↗

arXiv2011

Algorithmic deformation of matrix factorisations

Branes and defects in topological Landau-Ginzburg models are described by matrix factorisations. We revisit the problem of deforming them and discuss various deformation methods as well as their relations. We have implemented these algorithms and apply them to several examples. Apart from explicit results in concrete cases, this leads to a novel way to generate new matrix factorisations via nilpotent substitutions, and to criteria whether boundary obstructions can be lifted by bulk deformations.

hep-th↗

arXiv2011

An invitation to algebraic topological string theory

The purpose of this note is to provide a short invitation to the universal algebraic approach to topological string theory. In the first section we make an attempt to explain the origin of this approach and how it fits into the bigger picture of full string theory, while in the second half of this note we will introduce the relevant notions in more detail and discuss some of our main results on bulk-deformed open topological string amplitudes.

hep-th↗

arXiv2010

On the monoidal structure of matrix bi-factorisations

We investigate tensor products of matrix factorisations. This is most naturally done by formulating matrix factorisations in terms of bimodules instead of modules. If the underlying ring is C[x_1,...,x_N] we show that bimodule matrix factorisations 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=x^d is an N=2 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.

math-ph↗

arXiv2009

Matrix factorisations and open topological string theory

Amplitudes in open topological string theory may be described completely by certain A-infinity-categories. We detail a general construction of all cyclic minimal models for a given A-infinity-algebra and apply this result to the case of N=2 supersymmetric Landau-Ginzburg models. This allows to solve the tree-level theory in the sense that all amplitudes and hence the effective superpotential can be computed algorithmically. Furthermore, the construction provides a novel derivation of the topological metric of such models.

hep-th↗

arXiv2008

Triangle-generation in topological D-brane categories

Tachyon condensation in topological Landau-Ginzburg models can generally be studied using methods of commutative algebra and properties of triangulated categories. The efficiency of this approach is demonstrated by explicitly proving that every D-brane system in all minimal models of type ADE can be generated from only one or two fundamental branes.

hep-th↗