Traces for factorization homology in dimension 1

TL;DR

The paper constructs a circle-invariant trace in factorization homology for dualizable objects in symmetric monoidal $$-categories, confirming a conjecture and revealing a Poincaré duality in 1D.

Contribution

It proves a conjecture of Toën--Vezzosi by constructing a circle-invariant trace and calculates the factorization homology of the walking adjunction, demonstrating a form of Poincaré duality.

Findings

Construction of a circle-invariant trace from factorization homology.

Calculation of factorization homology over the circle of the walking adjunction.

Establishment of a Poincaré duality in 1-dimensional factorization homology.

Abstract

We construct a circle-invariant trace from the factorization homology of the circle $ {\sf trace} \colon \int^\alpha_{{\mathbb S}^1} \\underline{\sf End}(V) \longrightarrow \uno $ associated to a dualizable object $V\in {\boldsymbol{\mathfrak X}}$ in a symmetric monoidal $\infty$-category. This proves a conjecture of To\"en--Vezzosi on existence of circle-invariant traces. Underlying our construction is a calculation of the factorization homology over the circle of the walking adjunction in terms of the paracyclic category of Getzler--Jones: $ \int_{{\mathbb S}^1} {\sf Adj} ~\simeq~ {\bDelta_{\circlearrowleft}}^{\triangleleft\!\triangleright} ~. $ This calculation exhibits a form of Poincar\'e duality for 1-dimensional factorization homology.