Pseudotrace Construction in Algebra & VOAs

• Pseudotrace construction is a trace-like method that transforms symmetric linear functionals on algebras into cyclic functionals on endomorphism rings using projective modules.
• It employs the Hattori–Stallings trace and coordinate systems to achieve Morita invariance, extending from finite-dimensional algebras to almost unital and vertex-operator-algebra contexts.
• This framework unifies diverse formulations—including conformal block sewing and algebraic invariants—demonstrating broad applicability across algebra, VOA theory, and even contrasting areas like cosmology.

Pseudotrace construction is a trace-like procedure that starts from a symmetric linear functional on an algebra and produces, via a projective module or projective generator, a symmetric linear functional on an endomorphism algebra or on a related algebra of operators. In the finite-dimensional setting, it is the composition of the Hattori–Stallings universal trace with a symmetric linear functional; in the almost unital and finite-dimensional setting, it is formulated using local units and coordinate systems; and in the vertex-operator-algebra setting, it is identified with vacuum torus conformal blocks through an end/coend object and sewing-factorization. The recent literature shows that these formulations are not separate constructions but compatible realizations of the same basic mechanism (Arike, 2010, Gui et al., 1 Aug 2025, Gui et al., 6 Aug 2025).

1. Classical algebraic definition

In the finite-dimensional associative-algebra setting, let $A$ be a finite-dimensional associative algebra over an algebraically closed field of characteristic $0$, and let $\phi \in \mathrm{SLF}(A)$ be a symmetric linear function, meaning

$\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$

If $P_A$ is a finitely generated projective right $A$-module, then $P$ admits an $A$-coordinate system $\{u_i,\alpha_i\}$ with $u_i\in P$ and $0$0 such that

$0$1

The Hattori–Stallings trace is then

$0$2

Composing with $0$3 gives the induced symmetric linear function

$0$4

and $0$5 for all $0$6 (Arike, 2010).

This formulation makes precise the sense in which a pseudotrace is a trace only after passage through a symmetric linear functional on the ground algebra. The universal part is $0$7, which lands in $0$8; the scalar-valued part is the evaluation against $0$9. The construction is independent of the chosen coordinate system because differences between coordinate-system expressions lie in $\phi \in \mathrm{SLF}(A)$0, which is annihilated by every symmetric linear functional (Arike, 2010).

In the literature summarized here, this is the basic algebraic template from which later generalizations proceed. A central point is that pseudotraces are not arbitrary linear forms on endomorphism rings: they are induced functorially from cyclic classes in the source algebra and therefore inherit cyclicity, additivity on direct sums, and compatibility with Morita-type passage to basic algebras (Arike, 2010).

2. Coordinate systems, projectivity, and Miyamoto’s formulation

The construction becomes especially explicit when the algebra is symmetric. In the setting of a basic indecomposable symmetric algebra $\phi \in \mathrm{SLF}(A)$1 with primitive idempotents $\phi \in \mathrm{SLF}(A)$2 and a symmetric linear function $\phi \in \mathrm{SLF}(A)$3 inducing a nondegenerate associative symmetric bilinear form, Arike shows that Miyamoto’s pseudotrace map is exactly the induced symmetric linear function $\phi \in \mathrm{SLF}(A)$4 on $\phi \in \mathrm{SLF}(A)$5 for finitely generated projective modules $\phi \in \mathrm{SLF}(A)$6 (Arike, 2010).

The key structural notion in that setting is that of an interlocked module. A finitely generated right $\phi \in \mathrm{SLF}(A)$7-module $\phi \in \mathrm{SLF}(A)$8 is interlocked with $\phi \in \mathrm{SLF}(A)$9 if, for each $\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$0,

$\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$1

where $\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$2 is the canonical basis constructed in the paper and $\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$3 are the socle basis elements dual to the primitive idempotents. The main result is that $\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$4 is interlocked with $\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$5 if and only if $\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$6 is projective, and the multiplicity of the indecomposable projective $\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$7 in $\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$8 is $\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.$9 (Arike, 2010).

For projective $P_A$0, one chooses basis elements $P_A$1 corresponding to the summand generators and defines maps $P_A$2. This yields a $P_A$3-coordinate system, and Miyamoto’s pseudotrace is

$P_A$4

Arike proves

$P_A$5

thereby identifying the pseudotrace map with the classical coordinate-system construction (Arike, 2010).

This equivalence dispels a common misconception that Miyamoto’s pseudotraces are intrinsically tied to a special basis-level combinatorial formula. The basis formula is present, but the paper shows that it is a special case of the Hattori–Stallings/symmetric-linear-function mechanism. A plausible implication is that many formal properties of pseudotraces are better understood at the level of $P_A$6 and Morita invariance than at the level of basis expansions alone.

3. Generalization to almost unital and finite-dimensional algebras

The 2025 generalization by Gui–Zhang extends pseudotraces from unital finite-dimensional algebras to almost unital and finite-dimensional (AUF) algebras, which may be non-unital or infinite-dimensional as vector spaces but possess sufficiently many idempotents (Gui et al., 1 Aug 2025). An associative algebra $P_A$7 is almost unital if every element has a local idempotent unit and finite sets of idempotents admit a common dominating idempotent. It is AUF if there exists a family of mutually orthogonal idempotents $P_A$8 such that

$P_A$9

and

$A$0

The natural finiteness category is $A$1, the category of coherent left $A$2-modules: finitely generated modules that are quotients of finite direct sums of modules of the form $A$3. If $A$4 is strongly AUF and $A$5 is a projective generator, then

$A$6

is the endomorphism algebra controlling the right action on $A$7 (Gui et al., 1 Aug 2025).

The left-coordinate-system version of pseudotrace is formulated for an $A$8–$A$9 bimodule $P$0 that is projective as a right $P$1-module. A left coordinate system is a family

$P$2

satisfying local finiteness and

$P$3

Then the left $P$4-trace is

$P$5

and for $P$6 the pseudotrace is

$P$7

It is independent of the chosen coordinate system, and $P$8 (Gui et al., 1 Aug 2025).

The theory also includes a right-coordinate-system construction yielding the map in the opposite direction,

$P$9

and under the hypotheses that $A$0 is strongly AUF and $A$1 is a projective generator, these two constructions are inverse linear isomorphisms

$A$2

The non-degeneracies on the two sides are equivalent as well (Gui et al., 1 Aug 2025).

This result is a Morita-type invariance statement for spaces of symmetric linear functionals, but formulated in the non-unital AUF context rather than the ordinary finite-dimensional unital context. Because the algebra may be infinite-dimensional while still decomposing into finite-dimensional blocks, the construction isolates the genuinely finite part needed for trace theory: local units, coherent modules, and blockwise finite support.

4. Ends, coends, and the universal algebra in VOA theory

In the vertex-operator-algebra setting, the construction is recast geometrically and categorically. Let

$A$3

be an $A$4-graded $A$5-cofinite VOA, not necessarily rational or self-dual. Then $A$6, the linear category of grading-restricted generalized $A$7-modules, is finite abelian. For a grading-restricted generalized $A$8-module $A$9, the contragredient module is

$\{u_i,\alpha_i\}$0

and the paper studies the end

$\{u_i,\alpha_i\}$1

The main identification is

$\{u_i,\alpha_i\}$2

where $\{u_i,\alpha_i\}$3 is the default fusion product of $\{u_i,\alpha_i\}$4 along the standard two-pointed sphere $\{u_i,\alpha_i\}$5, and dually

$\{u_i,\alpha_i\}$6

as a coend (Gui et al., 6 Aug 2025).

For each module $\{u_i,\alpha_i\}$7 there is a canonical $\{u_i,\alpha_i\}$8-morphism

$\{u_i,\alpha_i\}$9

and the family $u_i\in P$0 is dinatural and universal. This realizes $u_i\in P$1 as the universal pairing object for modules and their contragredients (Gui et al., 6 Aug 2025).

The paper then equips $u_i\in P$2, and therefore $u_i\in P$3, with a natural associative $u_i\in P$4-algebra structure compatible with the $u_i\in P$5-module structure. Using the canonical conformal blocks

$u_i\in P$6

the multiplication is

$u_i\in P$7

Associativity follows from a special case of the sewing-factorization theorem, which equates different parenthesizations of conformal-block compositions (Gui et al., 6 Aug 2025).

There is also a canonical involution

$u_i\in P$8

satisfying

$u_i\in P$9

and $0$00 is an anti-automorphism of the algebra: $0$01 With the idempotents

$0$02

the algebra decomposes into finite-dimensional corners: $0$03 and every $0$04 is a finite sum

$0$05

Thus $0$06 is an almost-unital finite-dimensional algebra in the sense of Gui–Zhang (Gui et al., 6 Aug 2025).

5. Symmetric linear functionals, pseudo-$0$07-traces, and torus blocks

The bridge from the universal algebra $0$08 to conformal blocks is given by symmetric linear functionals. For a $0$09-module $0$10,

$0$11

The paper proves that

$0$12

that is, symmetric linear functionals on $0$13 are precisely conformal blocks on the standard sphere $0$14. Specializing to $0$15 gives

$0$16

and sewing with the canonical insertion

$0$17

yields the isomorphism

$0$18

where the target is the space of vacuum torus conformal blocks (Gui et al., 6 Aug 2025).

Now let $0$19 be a projective generator in $0$20 and set

$0$21

Because $0$22 is a projective right $0$23-module, one may choose a left coordinate system

$0$24

with

$0$25

and the required finiteness properties. For $0$26, the pseudotrace on $0$27 is

$0$28

This is independent of the choice of left coordinate system, and the main theorem of Gui–Zhang gives a linear isomorphism

$0$29

preserving non-degeneracy (Gui et al., 1 Aug 2025, Gui et al., 6 Aug 2025).

Composing with the sewing-factorization isomorphism gives the pseudo-$0$30-trace construction

$0$31

and explicitly

$0$32

Equivalently,

$0$33

where $0$34 and the sum converges absolutely by $0$35-cofiniteness and grading restrictions (Gui et al., 6 Aug 2025).

The paper uses the normalization without an explicit $0$36 shift. It states that incorporating $0$37 is a standard physics normalization, whereas the geometric sewing choice here implements $0$38 via the outgoing local coordinate $0$39 (Gui et al., 6 Aug 2025).

6. The main isomorphism, conjectures, and scope of the term

The central VOA theorem proves the Gainutdinov–Runkel conjecture: for any projective generator $0$40 in $0$41, the pseudo-$0$42-trace construction yields a linear isomorphism

$0$43

The proof combines three ingredients: the AUF pseudotrace isomorphism $0$44, the identification $0$45, and the sewing-factorization isomorphism from sphere blocks to torus blocks (Gui et al., 6 Aug 2025).

A corollary confirms the Arike–Nagatomo conjecture. If $0$46 is a unital finite-dimensional $0$47-algebra such that $0$48 as linear categories, then

$0$49

The proof uses the end/coend description together with the identification of the Deligne tensor product with finite-dimensional bimodules and the realization

$0$50

cited in the paper (Gui et al., 6 Aug 2025).

The same framework yields geometric interpretations of other algebraic structures. The Zhu algebra $0$51 and the higher Zhu algebras $0$52 can be realized as quotients of $0$53, with their multiplications encoded by the $0$54-module structure of $0$55 and truncations by the idempotents $0$56 (Gui et al., 6 Aug 2025). In strongly-finite logarithmic CFT, when $0$57 is self-dual, $0$58 is finite-dimensional, and $0$59 is rigid and factorizable, a distinguished modified trace exists, unique up to scale and non-degenerate; it yields an isomorphism

$0$60

which the paper states is consistent with unimodularity (Gui et al., 6 Aug 2025).

The literature also shows that the term “pseudotrace” is not entirely uniform across fields. In cosmology, a 2020 paper defines a “pseudotrace” density by

$0$61

with phase difference

$0$62

to describe the hydrodynamics of first-order phase transitions (Giese et al., 2020). That usage is formally unrelated to the algebraic pseudotrace construction discussed above. This suggests that, in current arXiv usage, “pseudotrace” may denote either a trace-like algebraic functional built from symmetric linear forms and projective modules, or a specific thermodynamic linear combination of energy density and pressure, and the distinction is terminologically important.

Across the algebraic and VOA settings, however, the construction has a consistent conceptual core: a universal cyclic functional is transported through a projective generator and then identified either with a symmetric linear functional space or, after sewing, with a space of conformal blocks. In the formulation of the 2025 VOA paper, this is summarized by the chain

$0$63

which geometrically explains why pseudo-$0$64-traces give all vacuum torus conformal blocks for $0$65-cofinite VOAs without assuming rationality or self-duality (Gui et al., 6 Aug 2025).