[Paper Review] Symplectic level-rank duality via tensor categories

Victor Ostrik, Eric C. Rowell+1 more|arXiv (Cornell University)|Feb 18, 2020

Algebraic structures and combinatorial models23 references4 citations

TL;DR

This paper establishes symplectic level-rank duality for braided fusion categories arising from quantum groups of type C at roots of unity, proving a braid-reversing equivalence between the category ${\mathcal{C}}(\mathfrak{sp}_{2n})_{k}$ and the reversed category ${\mathcal{C}}(\mathfrak{sp}_{2k})_{n}^{-}$. The result is proven via two methods: conformal embeddings and classification of braided fusion categories, extending to non-unitary cases at odd roots of unity.

ABSTRACT

We give two proofs of a level-rank duality for braided fusion categories obtained from quantum groups of type $C$ at roots of unity. The first proof uses conformal embeddings, while the second uses a classification of braided fusion categories associated with quantum groups of type $C$ at roots of unity. In addition we give a similar result for non-unitary braided fusion categories quantum groups of types $B$ and $C$ at odd roots of unity.

Motivation & Objective

• To establish level-rank duality for braided fusion categories associated with quantum groups of type C at roots of unity.
• To provide two distinct proofs of the duality: one using conformal embeddings and another using classification of braided fusion categories.
• To extend the duality to non-unitary braided fusion categories for types B and C at odd roots of unity.
• To clarify the relationship between finite and affine Lie algebra representations through braided monoidal equivalences.
• To generalize the type A level-rank duality to the symplectic case using categorical and Lie-theoretic tools.

Proposed method

• Utilizes conformal embeddings of affine Lie algebras: $(\widehat{\mathfrak{sp}}_{2n})_{k} \oplus (\widehat{\mathfrak{sp}}_{2k})_{n} \subset (\widehat{\mathfrak{so}}_{4nk})_{1}$ to construct the duality.
• Applies the construction of the reversed braided fusion category ${\mathcal{C}}^{-}$ by modifying the braiding on odd-graded components of a $\mathbb{Z}/2$-graded category.
• Employs the diagonal construction of $\mathcal{C} \boxtimes s\mathrm{Vec}$ to realize the $\mathbb{Z}/2$-graded braided structure.
• Uses combinatorial data from partitions in $n \times k$ rectangles to parametrize irreducible representations and weights.
• Applies the Kac-Peterson formula in the symplectic setting to compute characters and verify duality.
• Leverages reconstruction techniques for braided fusion categories with $C$-type fusion rules to prove the duality in the non-unitary case.

Experimental results

Research questions

• RQ1 Is there a braid-reversing tensor equivalence between the braided fusion categories ${\mathcal{C}}(\mathfrak{sp}_{2n})_{k}$ and ${\mathcal{C}}(\mathfrak{sp}_{2k})_{n}^{-}$?
• RQ2 Can level-rank duality for type C quantum groups at roots of unity be proven using conformal embeddings?
• RQ3 How does the duality extend to non-unitary categories when the level is an odd integer?
• RQ4 What is the role of the $\mathbb{Z}/2$-grading and the $\mathcal{C}^{-}$ construction in realizing the dual category?
• RQ5 Can the duality be established via categorical classification of braided fusion categories with $C$-type fusion rules?

Key findings

• There exists a braid-reversing tensor equivalence between ${\mathcal{C}}(\mathfrak{sp}_{2n})_{k}$ and ${\mathcal{C}}(\mathfrak{sp}_{2k})_{n}^{-}$, establishing symplectic level-rank duality.
• The duality is proven via conformal embeddings into $\widehat{\mathfrak{so}}_{4nk}$ at level 1, providing a direct geometric construction.
• A second proof uses the classification of braided fusion categories with $C$-type fusion rules, relying on advanced categorical machinery.
• The duality extends to non-unitary categories for odd levels $\ell$, where ${\mathcal{C}}(\mathfrak{sp}_{2n},\ell)$ with $\ell = 2k + 2n + 1$ plays the dual role.
• The weight computation $s\rho - \rho_{\mathfrak{t}}$ for $s \in S_{n+k}$ matches the combinatorics of partitions and their complements $\lambda^{c}$, confirming the duality at the representation-theoretic level.
• The set $W_{\mathfrak{t}}^{1}$ is shown to be isomorphic to $I_{n,k}$ via a bijection with black-and-white dot diagrams, linking group theory to partition combinatorics.