Odd structures are odd

TL;DR

This paper investigates odd algebraic structures in graded vector spaces, focusing on odd modular operads relevant to string field theory, and reveals that different constructions can yield conflicting results, challenging common sign conventions.

Contribution

It introduces two canonical constructions of odd modular endomorphism operads and demonstrates that only one aligns with correct algebraic signs, questioning standard sign conventions.

Findings

Two different constructions of odd modular endomorphism operads produce conflicting results.

The choice of monoidal structure significantly affects odd algebraic structures.

The common Koszul sign convention may not always lead to correct signs in odd structures.

Abstract

By an odd structure we mean an algebraic structure in the category of graded vector spaces whose structure operations have odd degrees. Particularly important are odd modular operads which appear as Feynman transforms of modular operads and, as such, describe some structures of string field theory. We will explain how odd structures are affected by the choice of the monoidal structure of the underlying category. We will then present two `natural' and `canonical' constructions of an odd modular endomorphism operad leading to different results, only one being correct. This contradicts the generally accepted belief that the systematic use of the Koszul sign convention leads to correct signs.