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. Nils Carqueville and Daniel Murfetboundary-bulk map (which reduces to the Chern character (−1) ( m+1 2 ) str(∂ z 1 D . . . ∂ zm D) for Φ = 1) and the Kapustin-Li disc correlator.Another application of our construction of adjunctions in LG k is a new proof of the Cardy condition (see Section 9 for its precise statement). This generalisation of the Hirzebruch-Riemann-Roch theorem is the most "quantum" among the axioms for open/closed TFTs (as it stems from a one-loop diagram) and may accordingly be viewed as a particularly deep structure. In the case of Landau-Ginzburg models it was proved only recently in [PV12] when k is a field of characteristic zero. Our proof works for any ring k and simply follows from the fact that the 2-morphism in LG k to be read off from the diagram(which is to be identified with an annulus correlator) can be evaluated in two ways: either by first contracting the inner X-loop and then contracting the outer Y -loop, or by first fusing X with Y and then contracting the fused (X ∨ ⊗ Y )-loop. Applying special cases of our evaluation and coevaluation maps then immediately produces the Cardy condition, see Theorem 9.1.Let us conclude with future applications of our results. One of the most intriguing properties of Landau-Ginzburg models is that they are on one side of the CFT/LG correspondence. This roughly states that many aspects of a large class of conformal field theories (CFTs) can be described in terms of (non-conformal) Landau-Ginzburg models, and one may wonder which structures encountered in rational CFT can also be found in the theory of Landau-Ginzburg models.One example is the generalised orbifold procedure of [FFRS09] which constructs all rational CFTs of fixed central charge and with identical left and right chiral algebras from any given single such CFT. Carried over to Landau-Ginzburg models this leads to the following picture: under the right circumstances a Landau-Ginzburg model with potential V can be obtained from a model with potential W by identifying an object A in the monoidal category LG k (W, W ) that can be equipped with the structure of a special symmetric Frobenius algebra (see e. g. [FRS02, Section 3]). Then the category of matrix factorisations of V is equivalent to the category of A-modules. The results of the present paper facilitate the construction of suitable algebras; the details appear in [CR].The rest of the prese...

No abstract

In a recent paper, Iyama and Yoshino considered two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen–Macaulay modules in terms of linear algebra data. In this paper, we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov’s result on the graded singularity category.

We describe the pushforward of a matrix factorisation along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and use this construction to study the convolution of kernels defining integral functors between categories of matrix factorisations. We give an elementary proof of a formula for the Chern character of the convolution generalising the Hirzebruch-Riemann-Roch formula of Polishchuk and Vaintrob. IntroductionA linear factorisation of an element W in a ring R (all our rings are commutative) is a Z/2-graded R-module X together with an odd R-linear map d : X −→ X, called the differential, which squares to W · 1 X . The pair (X, d) is called a matrix factorisation if each X i is free over R, and a finite rank matrix factorisation if each X i is free of finite rank. We write HMF(R, W ) for the homotopy category of matrix factorisations and hmf(R, W ) for the full subcategory of finite rank matrix factorisations. These objects were introduced by Eisenbud [Eis80] as a way of representing a special class of modules over local rings of hypersurface singularities, and they have played an important role in singularity theory and more recently in homological mirror symmetry and string theory. Because of this latter connection W is now often referred to as the potential.Suppose that ϕ : S −→ R is a ring morphism and that W = ϕ(V ). Viewing X as an S-module via ϕ we obtain a Z/2-graded S-module ϕ * (X) with a differential squaring to V ·1 ϕ * (X) , and we call this linear factorisation of V over S the pushforward along ϕ. In this paper we study the pushforward of finite rank matrix factorisations along ring morphisms. Even when X is finite rank the pushforward is typically not a finitely generated S-module, but we can ask: when is ϕ * (X) homotopy equivalent to a finite rank matrix factorisation, and can such a finite model for the pushforward be described concretely? We refer to this as the construction of finite pushforwards.Before stating our results we want to mention some interesting examples of pushforwards. Our first motivation was to understand the composition of integral functors between categories of matrix factorisations; let us begin with a special case, which involves the Hirzebruch-Riemann-Roch theorem for matrix factorisations.Hirzebruch-Riemann-Roch. Let k be a field of characteristic zero, W ∈ R = k x 1 , . . . , x n a polynomial and X, Y two finite rank matrix factorisations of W over R. The R-module Hom R (X, Y ) has a natural Z/2-grading and a differential making it into a matrix factorisation of zero and if the zero locus of W has an isolated singularity at the origin then Hom R (X, Y ) has finite-dimensional cohomology over k. One then defines the Euler characteristicThe Hirzebruch-Riemann-Roch theorem for matrix factorisations, recently proven by Polishchuk and Vaintrob [PV10], expresses this Euler characteristic in terms of the Chern characters of X and Y using the general Riemann-Roch theorem for dg-algebras of Shklyarov [Shk07]. Our point of view is that pushing Hom...

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.2000 Mathematics Subject Classification 57M27 1 We do not use the (representation theoretically more appropriate) "wide edge" depiction of [KR08a] for the second diagram. This is done to facilitate the interpretation in terms of Landau-Ginzburg models below. Nils Carqueville