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Pushing forward matrix factorisations

Tobias Dyckerhoff, Daniel Murfet

TL;DR

This work develops a residue-centric framework for pushing forward matrix factorisations along ring morphisms and for convolving kernels in the homotopy category of matrix factorisations. Central to the approach is an explicit idempotent on $X/\bold{t}X$ constructed from the relative Atiyah class, yielding finite pushforwards and enabling concrete computation of convolutions. The authors derive elementary, residue-based formulas for Chern characters, prove a Cardy-type condition, and provide explicit convolution formulas that extend to power-series rings; Knörrer periodicity is treated with a transparent residue-based inverse. Collectively, these results supply practical tools for defect fusion in Landau–Ginzburg models and for related categorified invariants, with direct computational potential and connections to Hirzebruch–Riemann–Roch-type theorems in matrix factorisation theory.

Abstract

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.

Pushing forward matrix factorisations

TL;DR

This work develops a residue-centric framework for pushing forward matrix factorisations along ring morphisms and for convolving kernels in the homotopy category of matrix factorisations. Central to the approach is an explicit idempotent on constructed from the relative Atiyah class, yielding finite pushforwards and enabling concrete computation of convolutions. The authors derive elementary, residue-based formulas for Chern characters, prove a Cardy-type condition, and provide explicit convolution formulas that extend to power-series rings; Knörrer periodicity is treated with a transparent residue-based inverse. Collectively, these results supply practical tools for defect fusion in Landau–Ginzburg models and for related categorified invariants, with direct computational potential and connections to Hirzebruch–Riemann–Roch-type theorems in matrix factorisation theory.

Abstract

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.

Paper Structure

This paper contains 22 sections, 38 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Fundamentals
  3. Outline
  4. Koszul complexes on null-homotopic maps
  5. Perturbation
  6. From homotopies to morphisms
  7. The idempotent
  8. Residues and Traces
  9. Contracting the Koszul complex
  10. The Atiyah class
  11. Residues and the Atiyah class
  12. A formula for the idempotent
  13. The Chern character
  14. The Cardy condition
  15. The general pushforward
  16. ...and 7 more sections

Key Result

Theorem 1

The $n$-fold suspension of $\varphi_*(X)$ is a direct summand of $X/\bold{t} X$ in the homotopy category of linear factorisations of $W$ over $S$, and the associated idempotent is $e$. That is, there are morphisms in the homotopy category \xymatrix@+3pc{ X/\bold{t} X \ar@<-0.8ex>[r]_(0.5){\psi} & \v

Theorems & Definitions (104)

  • Theorem
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: (Folding)
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 4.1
  • ...and 94 more