Modified Trace in Tensor Categories

• Modified Trace is a generalized categorical trace that extracts nonzero invariants and modified dimensions in non-semisimple tensor categories.
• It enables recovery of classical topological invariants, such as the knot writhe, by leveraging graded ribbon structures and ambidextrous trace properties.
• Its applications span representation theory, low-dimensional topology, and quantum invariants, offering tools for analyzing complex tensor categories and modular data.

A modified trace is a generalization of the standard categorical trace designed to yield nontrivial invariants and dimension functions in settings—particularly non-semisimple or interpolating tensor categories—where the usual trace vanishes on large tensor ideals. In contemporary mathematics, modified traces have become central to the paper of representation theory, low-dimensional topology, quantum invariants, and the structural theory of monoidal categories.

1. Definition and Characterization in Ribbon and Pivotal Categories

In a ribbon category $\mathcal{C}$, the standard notion of trace is inadequate in the presence of objects with vanishing categorical dimension. The modified trace is defined on a proper tensor ideal $I \subset \mathcal{C}$ (that is, a full subcategory closed under tensor product and retracts). Given a family of linear functions

$\{ t_V : \mathrm{End}(V) \to F \}_{V \in I}$

the modified trace satisfies two compatibility conditions for any $f : V \to U$, $g : U \to V$:

• Tensorial compatibility:

$t_{U \otimes W}(f) = t_U(\mathrm{tr}^r(f))$

where $\mathrm{tr}^r$ denotes the right partial trace (see equation (2) in the data).

• Cyclicity:

$t_V(gf) = t_U(fg)$

A crucial ingredient in the construction is the notion of an ambidextrous trace on an object $J$: a linear map $t_J : \mathrm{End}(J) \to F$ such that for any $h \in \mathrm{End}(J \otimes J)$,

$t_J(\mathrm{tr}_l(h)) = t_J(\mathrm{tr}_r(h))$

where $\mathrm{tr}_l$, $\mathrm{tr}_r$ are left/right partial traces. Such a trace uniquely extends to the tensor ideal generated by $J$.

In Deligne’s interpolating category $\mathrm{Rep}(S_t)$ for the symmetric group—where the categorical trace often vanishes on negligible objects—this approach yields a unique, nontrivial, modified trace on the ideal of negligible objects for integral values $t \in \mathbb{Z}_+$ (1103.2082). For $M_n = ([n], s_n)$ with $s_n$ the normalized antisymmetrizer, the trace is characterized on the basis elements (e.g., $t_n(s_n) = t_n(s_n x_n s_n) = 1$), and its ambidextrous property is established via diagrammatic arguments.

2. Extension to Graded Categories and Recovery of the Writhe Invariant

Standard ribbon categories with symmetric braiding induce only trivial topological invariants. To circumvent this, a graded variant, $g(S_t)_q$, is introduced:

• Objects are labeled $[a,b]$ with degree $a-b$.
• Morphisms are defined only when degrees match.
• Braiding is deformed via a parameter $q$:

$c_{V,W} = q^{rs}\beta$

for homogeneous $V,W$ of degrees $r,s$.

• The twist morphism is $\theta_{[a,b]} = q^{(a-b)^2}\mathrm{Id}$.

A full and faithful “degrading” functor $F : g(S_t)_q \to S_t$ projects back to the symmetric category, enabling the pull-back of the modified trace to the graded context. This graded setting is essential for encoding nontrivial quantum invariants of framed knots.

Labeling a framed knot $K$ with $M_{a,b}$ ($a+b = n$, $a-b \neq 0$), the constructed invariant is

$K \mapsto q^{(a-b)^2 \cdot \omega(K)}$

where $\omega(K)$ denotes the knot’s writhe. This invariant, which arises through evaluation using the categorical ribbon structure and the modified trace, recovers exactly the classical writhe invariant, a fundamental topological quantity that cannot be extracted from symmetric categories (1103.2082).

3. Modified Trace and Representation Theory of Tensor Categories

The existence of a (unique) nontrivial modified trace on the ideal of negligible objects in (non-semisimple) categories such as $\mathrm{Rep}(S_t)$ is significant in representation theory. The modified trace yields nonzero invariants (called modified dimensions) even for objects (such as $M_n$) of vanishing categorical dimension. This is instrumental in:

• Understanding representation categories of quantum groups or superalgebras where standard dimensions vanish (see also Kac–Wakimoto conjecture and generalizations).
• Providing new tools for character theory and Grothendieck ring analysis.
• Formulating “Verlinde-like” formulas for decomposition of tensor products, notably for projective objects, within nonsemisimple and modular settings (Gainutdinov et al., 2017).

In pivotal and factorisable finite tensor categories, the modified trace on the ideal of projective objects:

• Is unique up to scaling under mild assumptions.
• Enables the internal characters of projectives to form modules under the projective $SL(2,\mathbb{Z})$-action and facilitates diagonalization of Grothendieck ring actions.
• Admits explicit descriptions in terms of S-matrix elements, connecting topological invariants to modular data.

4. Applications in Low-Dimensional Topology and Quantum Invariants

Modified traces are foundational in constructing quantum invariants in nonsemisimple settings, particularly those derived from nontrivial ribbon or modular categories:

• In graded versions of Deligne’s category, the recovery of the knot writhe as an invariant reveals that modified traces enable nontrivial 3-manifold and link invariants even when the categorical trace is trivial (1103.2082).
• In quantum group settings, the modified trace corresponds to a symmetrized integral on the underlying Hopf algebra; this perspective unifies algebraic and topological constructions and is essential for defining renormalized Hennings invariants and nonsemisimple TQFTs (Ha, 2018, Geer et al., 2020).
• The modified trace is also used in the construction of modular categories relevant for logarithmic conformal field theories, leading to new quantum invariants and advances in categorified representation theory.

5. Categorical and Algebraic Structures Underpinning Modified Trace

The theory of modified traces is deeply intertwined with Calabi–Yau and Frobenius structures in categories:

• In finite pivotal tensor categories, nondegenerate modified traces can be interpreted as defining twisted Calabi–Yau structures on the projective ideal (Beliakova et al., 2017, Geer et al., 2018).
• The categorical framework ties the existence and uniqueness of modified traces to the Nakayama functor and, in the unimodular and pivotal cases, to sphericity and ribbon structures (Shibata et al., 2021).
• Modified traces are classified and constructed via ambidextrous traces, partial trace properties, and cyclicity conditions, with their nondegeneracy translating to natural isomorphisms or the existence of unique (up to scale) symmetric bilinear forms on suitable ideals.

6. Summary Table: Core Concepts

Context	Modified Trace Role	Key Property
Deligne's $\mathrm{Rep}(S_t)$	Nontrivial trace on negligible ideal	Extension from ambidextrous trace; recovers writhe
Graded category $g(S_t)_q$	Ribbon structure induces knot invariants	Pull-back/faithful functor enables grading
Pivotal tensor categories	Trace on projective ideal determines modified dimension functions	Uniqueness (up to scale); relates to Calabi–Yau structure
Quantum groups, Hopf alg.	Correspondence to symmetrized integral; computes renormalized invariants	Non-semisimple TQFTs/TQFT for roots of unity cases
Kernel for topology	Building nontrivial 3-manifold/knot invariants in nonsemisimple setting	Writhe, link invariants, modular S-matrix connections

7. Broader Impact and Future Directions

The modified trace framework bridges representation theory, low-dimensional topology, and categorical algebra:

• Enables construction of invariants in situations where standard quantum methods fail due to vanishing categorical trace.
• Opens new pathways for Verlinde-type classification in nonsemisimple settings.
• Offers computational tools for quantum invariants in topological and physical contexts, as well as applications to higher representation theory and categorified quantum field theories.
• Stimulates further investigations into ambidextrous and nondegenerate traces in broader categorical and algebraic frameworks, crucial for the development of new quantum field theories and invariants, and the understanding of the representation theory of interpolating and deformation-categorified objects.

In summary, the modified trace is an indispensable categorical and algebraic tool for extracting nontrivial invariants and constructing quantum topological structures in settings where standard dimension and trace methods are trivial or degenerate. Its unifying role continues to drive advances at the intersection of modern algebra, topology, and quantum field theory.