pivotal symbols

The *pivotal symbols* or *$3j$ symbols* are a certain collection of signs attached to a fusion category arising from analyzing how invariant vectors in a $3$-particle state transform under rotation by a full revolution.

In a fusion category, the nature of duals for objects in the category can be encoded in principal ${\u2102}^{\times}$-bundle over the simple objects. Firstly, suppose one goes ahead and chooses an arbitrary dual object ${V}_{i}^{*}$ for every simple object ${V}_{i}$. Write

(1)$$\mathrm{Dual}({V}_{i}\u22a3{V}_{i}^{*})$$

for the set of ways that ${V}_{i}^{*}$ can be expressed as a right dual of ${V}_{i}$. Notice that $\mathrm{Dual}({V}_{i}\u22a3{V}_{i}^{*})$ is a ${\u2102}^{\times}$-torsor (you can multiply the unit by a nonzero complex number, following which you are forced to multiply the counit by the reciprocal of this number if you want to obey the snake equations).

In a fusion category, if ${V}_{i}^{*}$ is a right dual of ${V}_{i}$, then it is also a left dual of ${V}_{i}$, though not in a canonical way. This gives us the principal ${\u2102}^{\times}$-bundle $L$ over the simple objects — the fiber above ${V}_{i}$ is

(2)$${L}_{{V}_{i}}=\{\mathrm{Dual}({V}_{i}\u22a3{V}_{i}^{*})\stackrel{\sim}{\to}\mathrm{Dual}({V}_{i}^{*}\u22a3{V}_{i})\},$$

the set of isomorphisms between $\mathrm{Dual}({V}_{i}\u22a3{V}_{i}^{*})$ and $\mathrm{Dual}({V}_{i}^{*}\u22a3{V}_{i})$.

A *pivotal structure* or *even-handed structure* is a section of this bundle satisfying some naturality properties. In other words, it is a consistent way to turn right duals into left duals. The most stringent condition is that the invariant vectors in a tensor product of three particles must either be totally ‘bosonic’ or totally ‘fermionic’ when you rotate them through a full rotation.

That is, the endomorphism

(3)$${T}_{\mathrm{ijk}}:\mathrm{Hom}(1,{V}_{i}\otimes {V}_{j}\otimes {V}_{k})\to \mathrm{Hom}(1,{V}_{i}\otimes {V}_{j}\otimes {V}_{k})$$

(which can be proved to be an involution) given by sending

must be equal to $\pm \mathrm{id}$. Assuming this is satisfied (if not, there is no consistent way to turn right duals into left duals), we can call this collection of signs the **$3j$ symbols** or the **pivotal symbols** ${\u03f5}_{\mathrm{ijk}}$ of the fusion category:

(4)$${T}_{\mathrm{ijk}}={\u03f5}_{\mathrm{ijk}}\mathrm{id}.$$

That is to say, the $3j$ symbols are the signs arising from choosing a trivialization of the ‘duality bundle’ and analyzing how invariant $3$-particle tensors transform when they go through a full revolution.

It turns out that one can express the concept of a pivotal structure directly in terms of the pivotal symbols ${\u03f5}_{\mathrm{ijk}}$. Namely, a pivotal structure on the category amounts to an $\u03f5$-twisted monoidal natural tranformation? of the identity on the category. That is to say, a collection of nonzero complex numbers ${t}_{i}$ satisfying

(5)$${t}_{j}{t}_{k}={\u03f5}_{\mathrm{ijk}}{t}_{i}$$

whenever ${V}_{i}$ appears in ${V}_{j}\otimes {V}_{k}$. If all the $3j$ symbols were equal to $1$, this would just be an ordinary monoidal natural transformation of the identity. Perhaps this phenomenon can be understood in a more enlightening way using the language of planar algebras.

- P. Etingof, D. Nikshych and V. Ostrik, On fusion categories.
- T. J. Hagge and S-M Hong, Some non-braided fusion categories of rank 3.
- B. Bartlett, On unitary 2-representations of finite groups and topological quantum field theory (chapter 6).

Revised on July 1, 2009 08:31:30
by Toby Bartels
(71.104.230.172)