Functorial, operadic and modular operadic combinatorics of circuit algebras

TL;DR

This work unifies diverse quantum-algebraic and representation-theoretic structures under the umbrella of circuit algebras. It establishes three equivalent viewpoints—algebras over a wiring-diagram operad, symmetric lax monoidal functors from Brauer diagram categories, and modular operads with an extra product—and uses this dictionary to prove a circuit-algebra characterisation of algebras for the orthogonal and symplectic groups, as well as a parallel description for the infinite-dimensional limit groups. The companion paper RayCA2 further develops a monad, graphical calculus, and nerve theorem, enriching the combinatorial and categorical toolkit for circuit algebras. By linking Brauer, oriented Brauer, and wiring-diagram formalisms, the results illuminate Schur–Weyl dualities, invariant theory, and modular operad structures within a single, broadly applicable framework, with potential applications from quantum topology to networked systems.

Abstract

Circuit algebras are a symmetric analogue of Jones's planar algebras introduced to study finite-type invariants of virtual knotted objects. Circuit algebra structures appear, in different forms, across mathematics. This paper provides a dictionary for translating between their diverse incarnations and describing their wider context. A formal definition of a broad class of circuit algebras is established and three equivalent descriptions of circuit algebras are provided: in terms of operads of wiring diagrams, modular operads and categories of Brauer diagrams. As an application, circuit algebra characterisations of algebras over the orthogonal and symplectic groups are given.

Figures (8)

• Figure 1: String diagram representation of the triangle identities.
• Figure 2: (a) ${\lceil{f}\rceil}\colon y^* \otimes x \to I$; (b)${\lfloor{f}\rfloor}\colon I \to y \otimes x^*$; (c) $f^*\colon y^*\to x^*$ .
• Figure 3: (a) Composition of pairings on $X \amalg Y$ and $Y \amalg Z$; (b) the resulting pairing on $X \amalg Z$, together with the single closed component formed in the composition.
• Figure 4: Composing coloured pairings.
• Figure 5: (a) Composing oriented Brauer diagrams. (b) Up to a shuffle permutation, this is equivalent to a composition of walled Brauer diagrams, where horizontal arrows go from left to right.

Theorems & Definitions (90)

• Theorem 1.1: \ref{['thm. lax functor ca']} & \ref{['prop CA MO']}
• Theorem 1.2: \ref{['thm. CA inv']}
• Remark 2.2
• Example 2.3
• Definition 2.4
• Example 2.5
• Example 2.6
• Example 2.7
• Example 2.8
• Definition 2.9