
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…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Black Holes and Theoretical Physics
