The cut operation on matrix factorisations

https://doi.org/10.1016/j.jpaa.2017.08.014Get rights and content

Abstract

The bicategory LG of Landau–Ginzburg models has polynomials as objects and matrix factorisations as 1-morphisms. The composition of these 1-morphisms produces infinite rank matrix factorisations, which is a nuisance. In this paper we define a bicategory C which is equivalent to LG in which composition of 1-morphisms produces finite rank matrix factorisations equipped with the action of a Clifford algebra. This amounts to a finite model of composition in LG.

Introduction

The bicategory LGk of Landau–Ginzburg models [5] over a noetherian Q-algebra k is a pivotal bicategory defined in terms of the geometry of isolated hypersurface singularities. This bicategory has various applications [7], [6], [19] but an obstacle to further development of the theory is that, while the objects, 1-morphisms and 2-morphisms of LGk are simply described, the composition operation for 1-morphisms has an “infinitary” character which makes it hard to work with. The current paper aims to resolve this problem.

The objects of LGk are a class of polynomials W(x) called potentials, and 1-morphisms W(x)V(y) are finite-rank matrix factorisations [10] of the difference V(y)W(x) over k[x,y]. The idempotent completion of the homotopy category of such matrix factorisations is denoted LGk(W,V). For a third potential U(z), composition in LGk is a functorLGk(V,U)×LGk(W,V)LGk(W,U),(Y,X)YX defined for matrix factorisations Y of VU and X of WV byYX=(Yk[y]X,dY1+1dX). This is the analogue of convolution of Fourier–Mukai kernels in the matrix factorisation setting and suffers from the same defect, namely, the resulting matrix factorisation YX is a free module of infinite rank over k[x,z]. The problem of describing this a priori infinite object in some finite way was studied in [9] and the partial solution provided there was to explicitly describe a finite rank matrix factorisation together with an idempotent, splitting in the homotopy category of matrix factorisations to YX. However this solution is not sufficiently functorial to extent to a coherent description of the entire bicategory LGk.

The current paper refines [9] by presenting the idempotents developed there as arising from representations of Clifford algebras (see Remark 3.18). This leads us to define a new bicategory C in which the objects are again potentials and the 1-morphisms WV are finite rank matrix factorisations of VW additionally equipped with the structure of a representation of a Clifford algebra. This bicategory is equivalent to LGk, but it is simpler: the composition operation is described by a finite number of polynomial functions of the input data. More precisely, the composition of 1-morphisms in C is a functorC(V,U)×C(W,V)C(W,U),(Y,X)Y|X which we call the cut operation, and this operation is polynomial in the sense that the coefficients in the polynomials making up the matrix factorisation Y|X and its Clifford action are themselves polynomial functions of the coefficients defining X, Y.

Let us describe the cut in the most important case, where Y, X are matrix factorisations as above, not equipped with any additional structure as representations of a Clifford algebra. The Jacobi algebra JV=k[y1,,ym]/(y1V,,ymV) is a finitely generated free k-module by the assumption that V(y) is a potential, and the cut is defined to be the following finite rank matrix factorisation over k[x,z]Y|X=Yk[y]JVk[y]X equipped with a family of odd closed k[x,z]-linear operators {γi,γi}i=1m defined byγi=Ati,γi=yi(dX)12qyqyi(V)Atq where the Atq are Atiyah classes (Definition 3.8). These operators satisfy Clifford relationsγiγj+γjγi=0,γiγj+γjγi=0,γiγj+γjγi=δij up to homotopy (see Theorem 3.11). The differential on Y|X and the Clifford operators can be written explicitly as matrices over k[x,z], where the coefficients of the monomials in each row and column are polynomials in the monomial coefficients of dY, dX and V. Thus, the cut operation is a polynomial function of its inputs and, in particular, infinite rank matrix factorisations do not appear in the definition of C.

Let us sketch where the cut operation comes from, why Clifford algebras are involved, and why there is an equivalence CLGk. In Section 1.1 we describe various applications. By the results of [9] there is an isomorphism in the homotopy category of (infinite rank) matrix factorisations of UW over k[x,z] of the form where Sm=(km[1]) denotes the exterior algebra as a Z2-graded module, on the space km placed in degree one. The main theorem (Theorem 4.34) identifies the natural action of the Clifford algebra Cm=Endk(Sm) on the right hand side of (1.7) with the action of Cm via the aforementioned operators γi, γi on the left hand side.

There is a category called the Clifford thickening of the homotopy category of matrix factorisations of UW, in which objects are matrix factorisations equipped with a Cl-action for some l (i.e. l is allowed to be different for different objects). In this category Smk(YX) with its Cm-action is isomorphic to YX with no Clifford action, and so (1.7) can be read as a natural isomorphism Y|XYX. We prove this is suitably functorial in Y, X and therefore defines an equivalence CLGk.

The paper is structured as follows: in Section 2 we recall the definition of superbicategories and Clifford algebras, and introduce the Clifford thickening of a supercategory. The reason we need the formalism of supercategories is that we make extensive use of Clifford actions on objects of linear categories, and for this to make sense the categories themselves must be Z2-graded (that is, supercategories). In Section 3 we define C and in Section 4 we prove it is equivalent to LGk as a superbicategory without units.

As a special case of the deformation retract (1.7) we obtain in Section 4.5 for any potential V(y) and matrix factorisations X, X of V a deformation retract of Z2-graded complexes of k-modules where Homk[y](X,X) denotes the Hom-complex. Observe that if k is a field this is infinite-dimensional (with finite-dimensional cohomology) while (X)|X is finite-dimensional.

In the case X=X the right hand side is a DG-algebra and the deformation retract may be taken as the input to the standard algorithms for producing an A-minimal model of Endk[y](X). It turns out that this input is well-suited for actually doing computations, which we hope to return to elsewhere.

One of our motivations for introducing C is that it has an enrichment over affine schemes. This is not true of LGk, precisely because the composition is infinitary. We hope to return to this elsewhere, but for context we include a sketch assuming k is an algebraically closed field of characteristic zero. Given potentials W(x), V(y), a matrix factorisation of V(y)W(x) is after choosing a homogeneous basis just a matrixDM2r(k[x,y]) satisfying D2=(V(y)W(x))I2r where, dividing the matrix D into r×r blocks, the upper left and bottom right blocks are zero matrices. If we fix the size r of the blocks and bound the degrees of the monomials appearing in D by an integer s, then we can easily parametrise matrix factorisations by the k-points of an affine scheme.

We may add further coordinates and equations to encode closed odd operators satisfying Clifford relations (1.6) and, in this way, for a choice of parameters λ=(r,s,) define a scheme Mλ(W,V) such that as setsob(C(W,V))=λMλ(W,V)k where Zk denotes the k-points of a scheme Z. The cut operation is a polynomial function of its inputs, so there is an indexed family of morphisms of schemesMμ(V,U)×Mλ(W,V)Mf(λ,μ)(W,U) lifting the functor (1.2) on the level of objects for some function f(λ,μ) of indices. Finally, using the approach of Section 4.5, we can define for each pair of potentials W, V and tuple of parameters λ a vector bundle of Clifford representations such that for matrix factorisations X, X of VW the fiber of this bundle over the point (X,X) is the Clifford representation (X)|X of (1.8), which is isomorphic to the Hom-complex Homk[x,y](X,X) tensored with a suitable spinor representation.

The enrichment of C then consists of the family of schemes Mλ(W,V), the composition morphisms (1.11), the bundles (1.12), and various other data satisfying natural constraints which encode the structure of the bicategory.

In forthcoming work we use the above enrichment of C to define a semantics of linear logic [13] in which formulas are interpreted by tuples (X,k[x],W) where X is a scheme and Wk[x] is a potential. Proofs in the logic are interpreted by certain correspondences between these pairs. Let us briefly explain the general point which leads us to use C rather than LG, for which we will use the denotation of the Church numeral 2_ (see [21]).

Let α be a variable of the logic and suppose it has denotation α=(Spec(k),k[x],W) in the semantics. Let M=M(W,W) be the (indexed) affine scheme parametrising 1-morphisms X:WW in C, as discussed above. Then the semantics works as follows:αα=(Spec(k),k[x,x],W(x)W(x)),!(αα)=(M,k,0). A proof of the sequent !(αα)αα has for its denotation a correspondence between !(αα) and αα, which is a matrix factorisation of W(x)W(x) over the scheme M×Spec(k[x,x]). For example, the Church numeral 2_ has for its denotation the matrix factorisation whose fiber over a loop X:WW, viewed as a point of M, is the square X|X as a matrix factorisation of W(x)W(x) over Spec(k[x,x]). For these definitions to work, the coefficients of the monomials in the matrix factorisation X|X need to be given explicitly as polynomial functions of the coefficients in X. This means we have to use C rather than LGk.

Section snippets

Background

Throughout k is a noetherian Q-algebra.1 By default categories and functors are k-linear. We

The superbicategory C

In this section we define a superbicategory without units C. Later we will prove that this bicategory is equivalent to LG, although this will not be apparent at first. The objects of C are the same as LG, namely potentials (x,W).

Definition 3.1

Given potentials (x,W) and (y,V) we defineC(W,V)=(hmf(k[x,y],VW)ω), where ()ω denotes the idempotent completion (see Remark 2.5) and () is the Clifford thickening (see Section 2.5).

Thus a 1-morphism WV in C is a finite rank matrix factorisation of V(y)W(x)

The equivalence of C and LG

We prove that the cut functor defined in Section 3 models composition in LG by providing an equivalence of this superbicategory with C. This is done by presenting the cut Y|X with its Clifford action as the solution of the problem of finding a finite model for the matrix factorisation YX over k[x,z].

As was mentioned in Section 2.2, YX is a matrix factorisation of infinite rank over k[x,z]. The obvious notion of a “finite model” would be a finite rank matrix factorisation to which YX is

Acknowledgements

Thanks to Nils Carqueville for helpful comments on the draft.

References (23)

  • T. Dyckerhoff et al.

    Pushing forward matrix factorisations

    Duke Math. J.

    (2013)
  • View full text