Spin-Invariance of Fusion Supercategories

• The paper establishes that every fusion supercategory is automatically Spin(3)-invariant, resolving longstanding conjectures on categorical symmetries.
• The methodology involves constructing an extended 3-category (ESF) and analyzing the central homomorphism μ, which plays a key role in trivializing the spin-dependent phase.
• The results connect spin structures with state sum TQFT models, ensuring that both fusion rules and topological data are robust to alterations in framing.

A fusion supercategory is a semisimple, rigid, $\mathbb{Z}/2$-graded monoidal supercategory with finitely many simple objects and finite-dimensional Hom-superspaces, equipped with an enriched tensor product over the symmetric monoidal category of supervector spaces, sVect. Spin-invariance of fusion supercategories captures a precise insensitivity of categorical and topological data to changes in spin structure; it has deep consequences for the construction of invariants in low-dimensional topology and the classification of topological quantum field theories (TQFTs).

1. Structure and Properties of Fusion Supercategories

A fusion supercategory is defined as an algebra object in the symmetric monoidal $2$-category of finite semisimple module categories over $\mathrm{SV}$, the category of complex supervector spaces. Every object in a fusion supercategory possesses left and right duals, and the morphism spaces are $\mathbb{Z}/2$-graded. The Drinfeld center of a simple (indecomposable, simple unit) fusion supercategory is a nondegenerate braided fusion supercategory, called a braided fusion supercategory (BFSC). The set of their isomorphism classes modulo Drinfeld centers forms the super-Witt group, $SW$ (Freed et al., 9 Jan 2026).

The fusion rules in such categories are of the form $X_i \otimes X_j \cong \bigoplus_k N_{i,j}^k X_k$ with integers $N_{i,j}^k$, and a supersymmetric braiding supplies degree-dependent isomorphisms $$\beta_{X_i, X_j} : X_i \otimes X_j \to \Pi^{|i||j|}(X_j \otimes X_i)$$, enforcing the parity structure. The fusion coefficients are invariant under simultaneous parity (“spin”) flips of the simple objects: $$N_{i,j}^k = N_{i^{\mathrm{spin}}, j^{\mathrm{spin}}}^{k^{\mathrm{spin}}}$$ (Sam et al., 2020).

2. The Spin-Invariance Theorem

The central result is the Spin(3)-invariance theorem for fusion supercategories: For any fusion supercategory $F$, the action of the framing group Spin(3) on the symmetric monoidal 3-category $ESF$ (an enlargement of the 3-category SF of fusion supercategories) is scalar on objects and, for ordinary fusion supercategories, this action is trivial, i.e., any $F$ admits a unique Spin(3)-invariance datum. Formally, there is a canonical group homomorphism $\mu: \widetilde{SW} \to \mathbb{C}^\times$ from a central $\mathbb{Z}/24$-extension $\widetilde{SW}$ of $SW$, controlling invertible objects in $ESF$, with every $F$ sitting in the zero-class and thus obeying $\mu(F) = 1$ (Freed et al., 9 Jan 2026).

The corollary is that every fusion supercategory is automatically Spin(3)-invariant (i.e., the change-of-framing phase vanishes). This confirms longstanding conjectures about categorical symmetries and their invariance with respect to spin structure manipulations.

3. Mechanism and Proof Outline

The construction of $ESF$ is pivotal: $ESF$ is obtained by formally adjoining Morita inverses for each nontrivial class in the super-Witt group, resulting in

$$ESF \simeq \bigoplus_{\widetilde{w} \in \widetilde{SW}} SF_{T(\widetilde{w})}$$

where $T(\widetilde{w})$ runs over representative BFSCs, and $SF_{T(\widetilde{w})}$ denotes fusion supercategories admitting a central action of $T(\widetilde{w})$. The group $$\ker(\widetilde{SW} \rightarrow SW) \cong \mathbb{Z}/24$$ realizes the 24 invertible framed 3d-TQFTs (Freed et al., 9 Jan 2026).

Under the Cobordism Hypothesis, changes in manifold framings (through Spin(3)) act on objects of $ESF$ by multiplication with $\mu(X) \in \mathbb{C}^\times$ on 3-endomorphisms of $X$. For ordinary fusion supercategories, this weight $\mu(F)$ is exactly $1$, as the Spin(2)-action can always be lifted and trivialized, completing the proof of automatic Spin-invariance.

4. Spin Structures, State Sums, and Fermionic Signatures

The intimate connection between spin structures and fusion supercategories is concretely realized in state sum constructions for TQFTs. For any superfusion category $C$, the associated fusion category over sVect (via the $\Pi$-envelope and even morphism subcategory) provides the input for a spin-refined Turaev–Viro or Gaiotto–Kapustin state sum. The presence of a central simple object $\pi$ in the center with a braiding that imparts a $-1$ sign upon crossing (mimicking the fermion line) ensures that the resulting 3-manifold invariant $Z_C(M, \sigma)$ depends only on the spin structure $\sigma$ and is spin-invariant. The requisite fermionic $6j$-symbols encode the $\mathbb{Z}/2$-grading, and the extra phase factors from the super-pentagon translate to correct global cancellations when the triangulation is equipped with a spin structure (Usher, 2016).

5. Tangential Structures and Central Charges

Given a simple object $X \in ESF$, the "topological central charge" is defined by $$\mu(X) = \exp(2\pi i \tilde{c}(X)/6) \in \mathbb{C}^\times$$, with $\tilde{c}(X) \in \mathbb{Z}/6$ the reduced central charge. Anomalous framed TQFTs can be extended to Spin$^{p_1/4}$-structured (or finer) TQFTs by extracting suitable roots of $\mu(X)$. Consequently, the realization of a TQFT from a fusion supercategory can be thought of as a boundary field theory for a 4-dimensional invertible Spin(3)-TQFT, whose invariants precisely encode the spin dependence via powers of $\mu(X)$ taken to the $p_1/4$ (Freed et al., 9 Jan 2026).

6. SO-Invariance, Modular, and Spherical Structures

If the state space $T_X(S^1)$ admits a modular tensor structure (notably, a ribbon trivialization of the square of the braiding), the 3d theory associated to $X$ factors through oriented manifolds with $p_1$-structure, which canonically selects $\mu(X)^{1/4}$, erasing explicit Spin dependence. In the bosonic setting, the existence of a spherical structure on a fusion category $F$ is equivalent to granting $F$ an SO(3)-invariance structure with compatible invariance on the regular module. The canonical $p_1$-orientation aligns with the modularity of the center and recovers the canonical spherical structure conjectured earlier (Freed et al., 9 Jan 2026).

7. Consequences and Categorical Rigidity

Spin-invariance has immediate consequences for categorical classification and topological invariants. The structure constants of the fusion algebra are invariant under spin flips of simple objects, enforced by the supersymmetric braiding, and the fusion data are preserved under these automorphisms (Sam et al., 2020). Ocneanu rigidity extends: given a fixed $\pi$-Grothendieck ring for a superfusion category, there exist only finitely many such categories up to superequivalence, paralleling the rigidity of spin-TQFTs (Usher, 2016).

Moreover, the proof of spin-invariance resolves conjectures, notably that fusion supercategories are Spin(3)-invariant and that spherical structures correspond bijectively to SO(3)-invariance data (Freed et al., 9 Jan 2026). The categorical origin of this invariance is the parity-twisted (Clifford) symmetry, mathematically formalized by the equivalence between certain 2-categories of symmetric monoidal and supersymmetric monoidal supercategories via Clifford eversion (Sam et al., 2020).

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References:

(Freed et al., 9 Jan 2026) "Fully local Reshetikhin-Turaev theories" (Sam et al., 2020) "Supersymmetric monoidal categories" (Usher, 2016) "Fermionic 6$j$-symbols in superfusion categories"