Hirzebruch-Riemann-Roch theorem for DG algebras

TL;DR

This paper extends the Hirzebruch-Riemann-Roch theorem to proper DG algebras by introducing a pairing on Hochschild homology, providing explicit formulas for Chern characters, and exploring examples including quivers and orbifold singularities.

Contribution

It introduces a new pairing on Hochschild homology for proper DG algebras and derives a Riemann-Roch formula relating Euler characteristics to this pairing, with explicit Chern character formulas.

Findings

The pairing on Hochschild homology is non-degenerate for smooth DG algebras.

Explicit Riemann-Roch formulas are provided for quivers with relations and orbifold singularities.

The pairing coincides with topological field theory pairing in the Calabi-Yau case, verified for Frobenius algebras.

Abstract

For an arbitrary proper DG algebra A (i.e. DG algebra with finite dimensional total cohomology) we introduce a pairing on the Hochschild homology of A and present an explicit formula for a Chern-type character of an arbitrary perfect A-module (the Chern characters take values in the Hochschild homology of A). The Hirzebruch-Riemann-Roch formula in this context expresses the Euler characteristic of the Hom-complex between two perfect A-modules in terms of the pairing of their Chern characters. We mention two examples of proper DG algebras and the HRR formulas for them. The first example is Ringel's formula for quivers with relations. The second example is related to orbifold singularities of the form V/G where V is a complex vector space and G is a finite subgroup of SL(V). Furthermore, we prove that the above pairing on the Hochschild homology is non-degenerate when the DG algebra is smooth. We also formulate the conjecture that for a Calabi-Yau DG algebra A the pairing coincides with the one coming from the Topological Field Theory associated with A and verify it in the case of Frobenius algebras.