Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Mukai pairing, I: a categorical approach

Andrei Caldararu, Simon Willerton

TL;DR

This work develops a fully categorical treatment of the Mukai pairing on Hochschild homology for smooth spaces via the 2-category ${\mathscr{V}\textsf{ar}}$, where spaces are objects and integral kernels are 1-morphisms. It establishes functorial maps on Hochschild homology induced by kernels, proves that adjoint kernels yield pairing-adjoint maps, and defines a Chern character from $\mathrm{K}_0(X)$ to $\mathrm{HH}_0(X)$ that satisfies a semi-Hirzebruch-Riemann-Roch relation. The paper then connects these algebraic structures to open-closed 2d TQFTs, showing how the Cardy Condition emerges naturally in Calabi–Yau settings and proposing a general Baggy Cardy Condition for non-Calabi–Yau spaces. Overall, it provides a broad, robust framework linking Mukai-type pairings, Chern characters, and topological quantum field theory in a versatile categorified setting.

Abstract

We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and integral kernels as its 1-morphisms.

The Mukai pairing, I: a categorical approach

TL;DR

This work develops a fully categorical treatment of the Mukai pairing on Hochschild homology for smooth spaces via the 2-category , where spaces are objects and integral kernels are 1-morphisms. It establishes functorial maps on Hochschild homology induced by kernels, proves that adjoint kernels yield pairing-adjoint maps, and defines a Chern character from to that satisfies a semi-Hirzebruch-Riemann-Roch relation. The paper then connects these algebraic structures to open-closed 2d TQFTs, showing how the Cardy Condition emerges naturally in Calabi–Yau settings and proposing a general Baggy Cardy Condition for non-Calabi–Yau spaces. Overall, it provides a broad, robust framework linking Mukai-type pairings, Chern characters, and topological quantum field theory in a versatile categorified setting.

Abstract

We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and integral kernels as its 1-morphisms.

Paper Structure

This paper contains 29 sections, 27 theorems, 145 equations.

Table of Contents

  1. The 2-category of kernels
  2. A reminder on 2-categories.
  3. The 2-category ${\mathscr{V}\textsf{ar}}$.
  4. Serre functors
  5. The Serre functor on $\mathbf{D}(X)$.
  6. Serre kernels and the Serre functor for $\mathbf{D}(X\times Y)$.
  7. Adjoint kernels
  8. Adjunctions in 2-categories.
  9. Left and right adjoints of kernels.
  10. Partial traces.
  11. Adjunction as a 2-functor.
  12. Induced maps on homology
  13. Hochschild cohomology.
  14. Hochschild homology.
  15. Push-forward and pull-back.
  16. ...and 14 more sections

Key Result

Proposition 1

For spaces $X$ and $Y$ the Serre functor $\mathsf{S}_{X\times Y}\colon \mathbf{D}(X\times Y)\to \mathbf{D}(X\times Y)$ can be taken to be $\Sigma_Y\circ{-}\circ\Sigma_X$.

Theorems & Definitions (56)

  • Definition
  • Proposition 1
  • proof
  • Definition
  • Proposition 2
  • proof
  • Lemma 3
  • Definition
  • Theorem 4
  • Proposition 5
  • ...and 46 more