Reconstruction of modular tensor categories from strongly rational vertex operator algebras (Möller–Rayhaun, Conjecture 3.11)

The reconstruction conjecture of Möller and Rayhaun (Equivalence Relations on Vertex Operator Algebras, I: Genus, arXiv:2408.07117, Conjecture 3.11) asks whether every modular tensor category $\mathcal{C}$ arises, up to ribbon equivalence, as the representation category $\mathrm{Rep}(V)$ of some strongly rational vertex operator algebra $V$; equivalently, whether for every modular tensor category $\mathcal{C}$ there is a sufficiently large rational central charge $c$ with $c \equiv c(\mathcal{C})\ (\mathrm{mod}\ 4)$ for which the bulk genus $(\mathcal{C}, c)$ is nonempty. One direction is a theorem of Huang (a strongly rational VOA has a modular tensor category of modules); the converse is known only for special classes such as quantum-group categories, pointed categories, and (conditionally) twisted group doubles, and is open in general.

Background

A vertex operator algebra (VOA) is the algebraic axiomatization of the chiral part of a two-dimensional conformal field theory: a graded vector space with vertex operators satisfying locality, Jacobi-identity, and creation-axiom rules. A VOA is called strongly rational if it is simple, of CFT type (grading starts at $0$ with one-dimensional degree-$0$ piece), self-contragredient (canonically isomorphic to its graded contragredient dual), $C_2$-cofinite (a finiteness condition making the category of modules tractable), and rational (it has finitely many irreducible modules and every module is a direct sum of irreducibles). A celebrated theorem of Yi-Zhi Huang (built on Huang–Lepowsky–Zhang tensor product theory) states that for a strongly rational VOA $V$ the category $\mathrm{Rep}(V)$ of modules carries the structure of a modular tensor category: a semisimple ribbon (braided with compatible duality twists) category with finitely many simple objects and nondegenerate braiding, whose Grothendieck ring and modular $S$- and $T$-matrices encode the fusion and braiding of the theory. The central charge $c(V) \in \mathbb{Q}$ of $V$ satisfies the congruence $c(V) \equiv c(\mathcal{C})\ (\mathrm{mod}\ 4)$, where $c(\mathcal{C}) \in \mathbb{Q}/8\mathbb{Z}$ is the categorical central charge determined by the ribbon twist; for pseudo-unitary categories (those in which the categorical dimensions of simple objects are positive and hence coincide with the Frobenius–Perron dimensions of the fusion ring) this strengthens to a congruence mod 8.

In 2003 Gerald Höhn introduced the bulk genus of a strongly rational VOA, directly generalizing the classical genus of an even lattice: two strongly rational VOAs lie in the same bulk genus if their representation categories are equivalent as ribbon categories (and their central charges agree, in the appropriate sense). This groups rational chiral conformal field theories by the $2+1$-dimensional topological quantum field theory (Reshetikhin–Turaev type) they induce on the boundary. Sven Möller and Brandon C. Rayhaun (2024) systematized this picture and proved, among other results, that the finer hyperbolic genus (testing isomorphy after tensoring with the lattice VOA of the hyperbolic plane, in the sense of Moriwaki) refines the bulk genus. A bulk genus is thus a symbol $(\mathcal{C}, c)$, and the natural existence question — the analogue for lattices of Nikulin's theorem that the obvious necessary invariants are sufficient in large rank — is which symbols are actually realized.

The realization direction is known only for special classes: quantum-group (Chern–Simons) categories $(X_n, k)$ are realized by simple affine VOAs; pointed modular tensor categories (all simple objects invertible) are realized by lattice VOAs; Deligne products of realized categories are realized by tensor products; twisted Drinfeld doubles of finite groups (the modular categories $D^\omega(G)$ built from a finite group $G$ and a 3-cocycle twist $\omega$) are realized conditionally on the regularity of fixed-point subalgebras (proved for solvable automorphism groups by Carnahan–Miyamoto) by work of Evans–Gannon; and, when a quantum-group category $(X_n, k)$ admits no nontrivial condensable algebras (étale algebra objects inside the category, the categorical counterpart of conformal extensions), its ribbon-reversed category $\overline{(X_n, k)}$ can be reached through cosets (commutant subalgebras of embedded subalgebras) inside holomorphic VOAs (strongly rational VOAs with only one irreducible module, namely themselves) via the gluing principle of Creutzig–Kanade–McRae. Conversely, Rayhaun showed that the specific symbol $(\overline{(A_1, 5)_{1/2}}, 8/7)$ is not realized, where $(A_1,5)_{1/2}$ denotes the subcategory of $(A_1, 5)$ generated by the objects labeled by integer $SU(2)$ spins and the overline denotes ribbon reversal: there is no positive strongly rational VOA (positivity meaning that every irreducible module other than the vacuum has strictly positive minimal conformal weight) of central charge $8/7$ with representation category $\overline{(A_1,5)_{1/2}}$ (and essential positivity of $\overline{(A_1,5)_{1/2}}$ excludes non-positive realizations), although the same category is realized at every central charge $8/7 + 8n$ with $n \geq 1$. So the naive fixed-central-charge reading is false, and the correct question allows arbitrarily large central charge. The reconstruction conjecture (Conjecture 3.11 of Möller–Rayhaun) asserts that this flexibility is enough for every modular tensor category. A potentially easier variant (their Conjecture 3.13) asks only that the ribbon-reversed category $\overline{\mathcal{C}}$ (the same monoidal category with braiding and twist inverted) of a realized category $\mathcal{C}$ again be realized by a positive strongly rational VOA — the existence of a 'seed' algebra for the gluing principle — and is likewise open.

Problem Statement

Conjecture (Reconstruction, Möller–Rayhaun, Conjecture 3.11): For every modular tensor category $\mathcal{C}$, there exists a strongly rational vertex operator algebra $V$ such that $\mathrm{Rep}(V)$ is equivalent to $\mathcal{C}$ as a ribbon category. Equivalently: for every modular tensor category $\mathcal{C}$ there is a sufficiently large rational central charge $c$ with $c \equiv c(\mathcal{C})\ (\mathrm{mod}\ 4)$ such that the bulk genus $(\mathcal{C}, c)$ is nonempty, i.e. contains at least one isomorphism class of strongly rational VOAs. Establish the conjecture, or exhibit a modular tensor category $\mathcal{C}$ and prove that no strongly rational VOA has representation category ribbon-equivalent to $\mathcal{C}$ at any admissible central charge.

References

1. Sven Möller and Brandon C. Rayhaun, Equivalence Relations on Vertex Operator Algebras, I: Genus, arXiv:2408.07117 (2024). https://arxiv.org/abs/2408.07117
2. Gerald Höhn, Genera of Vertex Operator Algebras and Three Dimensional Topological Quantum Field Theories, Fields Inst. Commun. 39, AMS (2003), arXiv:math/0209333. https://arxiv.org/abs/math/0209333
3. Yuto Moriwaki, Genus of vertex algebras and mass formula, Math. Z. 299 (2021), 1473–1505, arXiv:2004.01441. https://arxiv.org/abs/2004.01441, doi:10.1007/s00209-021-02702-0
4. Sven Möller and Brandon C. Rayhaun, Equivalence Relations on Vertex Operator Algebras, II: Witt Equivalence and Orbifolds, arXiv:2410.18166 (2024). https://arxiv.org/abs/2410.18166
5. Brandon C. Rayhaun, Bosonic Rational Conformal Field Theories in Small Genera, Chiral Fermionization, and Symmetry/Subalgebra Duality, J. Math. Phys. 65(5):052301 (2024), arXiv:2303.16921. https://arxiv.org/abs/2303.16921, doi:10.1063/5.0167192
6. David E. Evans and Terry Gannon, Reconstruction and Local Extensions for Twisted Group Doubles, and Permutation Orbifolds, Trans. Amer. Math. Soc. 375 (2022), 2789–2826, arXiv:1804.11145. https://arxiv.org/abs/1804.11145, doi:10.1090/tran/8575
7. Yi-Zhi Huang, Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math. 10 (2008), suppl. 1, 871–911, arXiv:math/0502533. https://arxiv.org/abs/math/0502533, doi:10.1142/S0219199708003083
8. Thomas Creutzig, Shashank Kanade and Robert McRae, Gluing vertex algebras, Adv. Math. 396 (2022), Paper No. 108174, arXiv:1906.00119. https://arxiv.org/abs/1906.00119, doi:10.1016/j.aim.2021.108174
9. Scott Carnahan and Masahiko Miyamoto, Regularity of fixed-point vertex operator subalgebras, arXiv:1603.05645 (2016). https://arxiv.org/abs/1603.05645