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Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations

Alexander Polishchuk, Arkady Vaintrob

TL;DR

The paper develops a concrete algebraic framework for matrix factorizations of isolated hypersurface singularities by computing their Hochschild homology, a canonical bilinear form, and explicit Chern characters. It derives a Hirzebruch-Riemann-Roch-type formula for the Euler characteristics of Hom-spaces between matrix factorizations, and extends all results to G-equivariant and graded settings, connecting to Milnor rings, stabilized diagonals, and boundary-bulk maps. The work provides explicit formulas for ch(E) and τ^E, and shows how these yield nondegenerate pairings and Cardy-type equalities, enabling a purely algebraic realization of open-closed TFT structures in this noncommutative context. Overall, it gives a complete, computable picture of the Hochschild-categorical data associated with MF(R,w) and its equivariant/graded variants, with direct applications to singularity theory and mirror-symmetric frameworks.

Abstract

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.

Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations

TL;DR

The paper develops a concrete algebraic framework for matrix factorizations of isolated hypersurface singularities by computing their Hochschild homology, a canonical bilinear form, and explicit Chern characters. It derives a Hirzebruch-Riemann-Roch-type formula for the Euler characteristics of Hom-spaces between matrix factorizations, and extends all results to G-equivariant and graded settings, connecting to Milnor rings, stabilized diagonals, and boundary-bulk maps. The work provides explicit formulas for ch(E) and τ^E, and shows how these yield nondegenerate pairings and Cardy-type equalities, enabling a purely algebraic realization of open-closed TFT structures in this noncommutative context. Overall, it gives a complete, computable picture of the Hochschild-categorical data associated with MF(R,w) and its equivariant/graded variants, with direct applications to singularity theory and mirror-symmetric frameworks.

Abstract

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.

Paper Structure

This paper contains 19 sections, 36 theorems, 322 equations.

Table of Contents

  1. Hochschild homology for smooth proper dg-categories
  2. Modules and Hochschild homology for dg-categories
  3. Functoriality
  4. Generalized abstract Hirzebruch-Riemann-Roch Theorem
  5. Matrix factorizations
  6. Categories of matrix factorization s
  7. Tensor product and duality
  8. Matrix factorizations and modules over hypersurface singularities
  9. Generators and Hochschild homology for matrix factorization s
  10. $G$-equivariant Hochschild homology
  11. Chern character and boundary-bulk maps
  12. Technical lemmas
  13. Formula for the Chern character and the boundary-bulk map
  14. $G$-equivariant Chern character
  15. Hirzebruch-Riemann-Roch formula
  16. ...and 4 more sections

Key Result

Lemma 1.1.3

(i) For dg-categories ${\cal C},{\cal D}$ and objects $F\in\operatorname{Per}({\cal C}^{op}\otimes {\cal D})$ and $G\in\operatorname{Per}({\cal D}^{op}\otimes {\cal C})$ there is a canonical functorial isomorphism in $D(k)$. (ii) For $E\in\operatorname{Per}_{dg}({\cal D})$ one has a canonical isomorphism where $E^\vee(D)=\operatorname{Hom}_{{\cal D}^{op}-\operatorname{mod}}(E,h_D)$ corresponds t

Theorems & Definitions (52)

  • Remark 1.1.1
  • Remark 1.1.2
  • Lemma 1.1.3
  • Proposition 1.1.4
  • Lemma 1.2.1
  • Remark 1.2.2
  • Definition 1.2.3
  • Proposition 1.2.4
  • Lemma 1.2.5
  • Theorem 1.3.1
  • ...and 42 more