Nonlinear Traces

TL;DR

This paper develops a unifying geometric and algebraic framework for fixed point and character formulas using traces in higher algebra, extending classical identities to nonlinear and categorical contexts.

Contribution

It introduces a nonlinear geometric approach to traces and fixed point formulas, unifying classical and modern higher categorical theories within a derived algebraic geometry setting.

Findings

Universal nonlinear versions of Grothendieck-Riemann-Roch theorems

Derived loop space interpretations of fixed points and traces

Extension of classical character formulas to higher categories

Abstract

We combine the theory of traces in homotopical algebra with sheaf theory in derived algebraic geometry to deduce general fixed point and character formulas. The formalism of dimension (or Hochschild homology) of a dualizable object in the context of higher algebra provides a unifying framework for classical notions such as Euler characteristics, Chern characters, and characters of group representations. Moreover, the simple functoriality properties of dimensions clarify celebrated identities and extend them to new contexts. We observe that it is advantageous to calculate dimensions, traces and their functoriality directly in the nonlinear geometric setting of correspondence categories, where they are directly identified with (derived versions of) loop spaces, fixed point loci and loop maps, respectively. This results in universal nonlinear versions of Grothendieck-Riemann-Roch theorems, Atiyah-Bott-Lefschetz trace formulas, and Frobenius-Weyl character formulas. On the one hand, we can then linearize by applying sheaf theories, such as the theories of coherent sheaves and D-modules, developed by Gaitsgory and Rozenblyum, as functors out of correspondence categories (in the spirit of topological field theory). This recovers the familiar classical identities, in families and without any smoothness or transversality assumptions. On the other hand, the formalism also applies to higher categorical settings not captured within a linear framework, such as characters of group actions on categories.