Mode Transition Algebra

• Mode Transition Algebra is a framework that encodes structured state changes by linking universal enveloping algebras with Zhu algebras in vertex operator contexts.
• It organizes state transitions through a matrix-like, bigraded associative algebra that facilitates recursive decompositions and Morita-theoretic comparisons.
• It extends to automata, logic, and topology, providing unified algebraic models for behavioral equivalence, reachability, and the geometric smoothing of coinvariants.

Mode transition algebra denotes a family of algebraic constructions used to encode how structured states change across discrete “modes,” grades, or transition steps. In the vertex-algebra literature, it is a bigraded associative algebra built from a universal enveloping algebra and the level-zero Zhu algebra, with diagonal components controlling fixed-degree pieces of modules and linking directly to higher-level Zhu algebras, generalized Verma modules, and geometric smoothing of coinvariants (Damiolini et al., 2023, Barron et al., 22 Jan 2026, Damiolini et al., 2024). In adjacent literatures on automata, logic, topology, and systems with explicit operational modes, related transition-algebraic formalisms organize behavioral equivalence, feedback, reachability, or topological/modal structure through monoids, congruences, frames, and simplicial data types (Cruchten, 2024, Go et al., 2024, Collinson, 15 Apr 2026, Beggs et al., 2021).

1. Vertex-algebraic definitions and basic structure

In the vertex-algebraic setting, mode transition algebras are introduced for a vertex operator algebra of CFT-type or, more generally, a $\mathbb{Z}$-graded vertex algebra. One formulation defines

$$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$

with bigrading

$$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$

and $\mathcal{M}_0\cong A_0(V)$ (Damiolini et al., 2023). A parallel construction for $\mathbb{Z}$-graded vertex algebras defines

$A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$

with

$$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$

where $A_0=Zhu_0(V)$ (Barron et al., 22 Jan 2026). A third formulation writes

$$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$

and interprets $A_{i,-j}$ as encoding operators from degree $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$0 to degree $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$1 in admissible $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$2-graded modules (Damiolini et al., 2024).

These constructions are associative and explicitly matrix-like. In the $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$3 formulation, the multiplication satisfies

$$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$4

so the algebra behaves like an infinite block matrix algebra indexed by module degree (Damiolini et al., 2024). In the $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$5-graded formulation, the product is defined by

$$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$6

where the $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$7-operation collapses internal degrees to the degree-zero algebra (Barron et al., 22 Jan 2026). The geometric motivation given in the VOA setting is that these algebras refine the role of Zhu’s algebra by recording transitions between homogeneous pieces rather than only degree-zero information (Damiolini et al., 2023).

2. Relation to Zhu algebras, higher levels, and Morita theory

A central structural feature is the link between the $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$8-th mode transition algebra and the $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$9-th higher Zhu algebra. For VOAs of CFT-type, there is an exact sequence

$$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$0

and if $$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$1 admits an identity then the sequence splits, yielding

$$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$2

when identities exist in all lower degrees (Damiolini et al., 2023). In the $$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$3-graded framework this appears as

$$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$4

together with the splitting theorem

$$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$5

when unities exist for all $$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$6 (Barron et al., 22 Jan 2026).

These decompositions depend on strong unitality. In one formulation, a family of strong unities $$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$7 satisfies

$$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$8

for all $$\mathcal{M}(V)=\bigoplus_{d_1\geq 0,\ d_2\leq 0}\mathcal{M}_{d_1,d_2}, \qquad \mathcal{M}_d:=\mathcal{M}_{d,-d},$$9 and $\mathcal{M}_0\cong A_0(V)$0 (Barron et al., 22 Jan 2026). In another, a strong identity $\mathcal{M}_0\cong A_0(V)$1 is required to act as a two-sided identity on the off-diagonal bimodules $\mathcal{M}_0\cong A_0(V)$2 and $\mathcal{M}_0\cong A_0(V)$3 (Damiolini et al., 2024). These hypotheses are what make the higher-level decomposition precise rather than merely suggestive.

Mode transition algebras also organize induction and Morita-theoretic comparison. The degree-$\mathcal{M}_0\cong A_0(V)$4 part of a generalized Verma module is an $\mathcal{M}_0\cong A_0(V)$5-module, and the action factors through the higher Zhu algebra (Damiolini et al., 2023). In the Morita-equivalence framework, the zig-zag algebra

$\mathcal{M}_0\cong A_0(V)$6

interpolates between the degree-$\mathcal{M}_0\cong A_0(V)$7 transition algebra and an ideal in Zhu’s algebra. If $\mathcal{M}_0\cong A_0(V)$8 is strongly unital and the image of $\mathcal{M}_0\cong A_0(V)$9 in $\mathbb{Z}$0 is a unital ideal, then the module categories over $\mathbb{Z}$1 and over this ideal of $\mathbb{Z}$2 are equivalent (Damiolini et al., 2024). This is the mechanism by which Zhu’s algebra, originally associated to degree zero, is shown to contain information about higher homogeneous components.

3. Explicit computations: Heisenberg and Weyl vertex algebras

The most explicit examples presently described in the supplied literature are the rank-one Heisenberg VOA and the Weyl vertex algebra at central charge $\mathbb{Z}$3.

Vertex algebra	Mode transition algebra	Consequence for higher Zhu algebras
Heisenberg VOA $\mathbb{Z}$4	$\mathbb{Z}$5	$\mathbb{Z}$6 splits recursively, proving Conjecture 8.1 in [AB23a]
Weyl vertex algebra $\mathbb{Z}$7 at $\mathbb{Z}$8	$\mathbb{Z}$9	$A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$0

For the Heisenberg VOA, every $A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$1 admits a strong identity, and this leads to the recursive description

$A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$2

Iterating gives

$A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$3

which establishes the Addabbo--Barron conjecture cited in the source summary (Damiolini et al., 2023). A related presentation in the Morita-equivalence paper states that for Heisenberg VOA of rank $A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$4,

$A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$5

showing the same matrix-algebra pattern in a broader class of higher Zhu algebras (Damiolini et al., 2024).

For the Weyl vertex algebra at central charge $A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$6, the mode transition algebras are matrix algebras over the Weyl algebra

$A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$7

with basis indexed by pairs of multipartitions and multiplication described as matrix-like (Barron et al., 22 Jan 2026). Each $A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$8 has a unit $A=U/N^1U\otimes_{U_0}A_0\otimes_{U_0}U/N^1U,$9, so the higher Zhu algebras split completely: $$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$0 The same paper further shows that every weak module of the Weyl vertex algebra at central charge $$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$1 induced from Zhu algebras is already induced from the level-zero Zhu algebra, so the higher-level induction data collapses to level zero in this example (Barron et al., 22 Jan 2026).

4. Modules, interlocking, and geometric smoothing

Mode transition algebras control both module theory and the geometry of coinvariants. On the module-theoretic side, the Weyl-vertex-algebra study proves that all indecomposable reducible weight modules induced from a Zhu algebra are not weakly interlocked, where weak interlocking is defined by

$$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$2

The paper also proves that being weakly interlocked is preserved under the action of an invertible Li’s $$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$3-operator, and therefore all indecomposable reducible weight modules obtained by spectral flow of Zhu-induced modules are likewise not weakly interlocked (Barron et al., 22 Jan 2026). The modular implication recorded there is that pseudo-traces require interlocking, so in the bosonic ghost theory modularity is realized for doubly graded traces rather than pseudo-traces.

On the geometric side, mode transition algebras were introduced as objects that “reflect both algebraic properties of $$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$4 and geometric constructions on moduli of curves” (Damiolini et al., 2023). Given sheaves of coinvariants on pointed coordinatized curves built from $$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$5-modules, the main smoothing theorem states:

The algebras $$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$6 admit strong identity elements for all $$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$7 if and only if $$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$8 satisfies smoothing.

This equivalence translates the deformation behavior of coinvariants across nodal degenerations into a purely algebraic condition on the diagonal pieces of the mode transition algebra (Damiolini et al., 2023). When $$A_{d_1,d_2}:=(U/N^1U)_{d_1}\otimes_{U_0}A_0\otimes_{U_0}(U/N^1U)_{d_2}, \qquad A_d=A_{d,-d},$$9 is $A_0=Zhu_0(V)$0-cofinite and satisfies smoothing, the sheaves of coinvariants form a vector bundle on the moduli stack $A_0=Zhu_0(V)$1; the same source emphasizes that this yields local freeness, not only coherence.

The Heisenberg VOA gives a notable edge case. Although it is described as neither rational nor $A_0=Zhu_0(V)$2-cofinite in the source summary, smoothing still holds because all $A_0=Zhu_0(V)$3 admit strong identities, and the resulting sheaves of coinvariants give globally generated vector bundles on the moduli stack of stable pointed genus-zero curves (Damiolini et al., 2023). This suggests that the smoothing criterion isolates an algebraic mechanism that can operate beyond the conventional semisimple regime.

5. Categorical transition constructions in automata and systems

A different line of work studies transition constructions categorically for automata. For a deterministic automaton with state set $A_0=Zhu_0(V)$4 and transition function $A_0=Zhu_0(V)$5, the transition behavior is encoded by

$A_0=Zhu_0(V)$6

and the transition monoid is the image of $A_0=Zhu_0(V)$7 (Cruchten, 2024). The paper formalizes this via a transition functor

$A_0=Zhu_0(V)$8

and a machine functor

$A_0=Zhu_0(V)$9

with adjunction $$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$0 (Cruchten, 2024). Composing this adjunction with reachability and powerset dualities yields an idempotent comonad

$$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$1

and a monad

$$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$2

corresponding respectively to the largest set of equations and the smallest set of coequations satisfied by an automaton.

The same paper extends the construction to lasso automata and $$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$3-automata. Lasso automata use pairs $$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$4 representing ultimately periodic words $$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$5, with congruences on both sorts and an adjunction again of the form $$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$6 (Cruchten, 2024). For $$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$7-automata, the induced algebraic structure is a Wilke algebra congruence. The formal pattern is therefore consistent across finite words, lassos, and infinite words: transition behavior is summarized by an algebraic quotient, and the passage between automata and quotients is adjoint.

Related categorical work on open transition systems shows that $$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$8, so the single-vertex span-of-graphs model of open transition systems is the free feedback category over spans of sets (Lavore et al., 2020). That result is framed as a formal demonstration of the relationship between feedback and state and is generalized further to structured feedback categories, including span automata with initial and final states. While this work does not use the phrase “mode transition algebra” in the VOA sense, it belongs to the same broad algebraic program of extracting canonical transition structure from compositional systems.

The literature on systems with explicit operational modes uses yet another algebraic-geometric model. A system is classified into finitely many modes indexed by a simplicial complex $$A(V)=\bigoplus_{i,j\in\mathbb{N}}A_{i,-j}=L(U)\otimes_{U_0}R(U),$$9, with calibration map

$A_{i,-j}$0

where the $A_{i,-j}$1 form a partition of unity over the state space (Beggs et al., 2021). Transition data are organized by partial maps

$A_{i,-j}$2

and the source summary states that these local data types and morphisms form a presheaf over the simplicial complex. This is a model of systems with modes and mode transitions rather than a single associative algebra, but it shows how “mode transition” became a mathematically structured notion across several research programs.

6. Logical and topological transition algebras

In the logical literature, transition algebra is a many-sorted first-order formalism enhanced with dynamic-logic-style actions. A signature is

$A_{i,-j}$3

and actions are generated by

$A_{i,-j}$4

with sentences of the form

$A_{i,-j}$5

together with the usual logical connectives and quantification (Go et al., 2024). Composite actions are interpreted relationally by composition, union, and reflexive-transitive closure. The source summary emphasizes that this framework can finitely axiomatize both finiteness and reachability of models, which are not ordinarily possible in many-sorted first-order logic, but that compactness fails. Completeness for countable signatures is recovered only when proof systems are allowed to use countably infinite premises (Go et al., 2024).

A subsequent development replaces the non-compact treatment of $A_{i,-j}$6 by induction rules in a sequent calculus. There, actions are expanded to include additional constructors such as $A_{i,-j}$7, $A_{i,-j}$8, $A_{i,-j}$9, and $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$00, and the proof system is made compact by restricting the star rules to induction (Hashimoto, 5 May 2026). The new system is complete for a Kleene algebraic semantics rather than the ordinary relational semantics: $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$01 The same paper gives a model-theoretic proof of Craig interpolation for countable signatures under pushout conditions on signatures (Hashimoto, 5 May 2026).

Topological transition structures provide another algebraic interpretation. A transition structure is a set $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$02 with a binary relation $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$03, giving rise to operators

$$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$04

where $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$05 (Collinson, 15 Apr 2026). Generalizing from powersets to frames, the paper defines a bed as a frame equipped with a companion pair $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$06 satisfying

$$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$07

A topologically valued transition structure, or plot, is a triple $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$08 with a surjective valuation $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$09, and the lifted operators on $$\mathcal{M}(V)=(\mathcal{V}/N^L\mathcal{V})\otimes_{A_0(V)}(\mathcal{V}/N^R\mathcal{V}),$$10 make the topology into a bed (Collinson, 15 Apr 2026). The paper constructs a contravariant idempotent adjunction between the category of lentile plots and the category of gardens, thereby presenting a modal-topological version of transition algebra.

Taken together, these logical and topological works show that “transition algebra” can denote either a proof-theoretic calculus for rewriting-style relations or an algebra of modal operators on frames and topological spaces. A plausible implication is that the common thread is not a single universal definition but the repeated use of algebraic structure to control transition, reachability, and passage between local and global behavior.