Brane actions for coherent $\infty$-operads

TL;DR

This paper extends the brane action construction to general coherent $ $-operads, clarifies the relationship between different models for extension spaces, and applies results to $B$-framed little disks operads to produce new brane operations.

Contribution

It proves the extension of Mann-Robalo's brane action to broader coherent $ $-operads and clarifies the homotopy-theoretic nature of extension spaces, correcting previous assumptions.

Findings

Models for extension spaces are equivalent.

The space of extensions is a homotopy quotient, not a fiber.

$B$-framed little disks operads are coherent, enabling new brane operations.

Abstract

We prove that Mann-Robalo's construction of the brane action extends to general coherent $\infty$-operads, with possibly multiple colors and non-contractible spaces of unary operations. This requires to establish two results regarding spaces of extensions that were left unproven in the aforementioned construction. First, we show that Lurie's and Mann-Robalo's models for such spaces are equivalent. Second, we prove that the space of extensions in the sense of Lurie is not in general equivalent to the homotopy fiber of the associated forgetful morphism, but rather to its homotopy quotient by the $\infty$-groupoid of unary operations, correcting an oversight in existing literature. As an application, we obtain that the $\infty$-operads of $B$-framed little disks are coherent and therefore yield new operations on spaces of branes of perfect derived stacks.