Zamolodchikov systems

Sjoerd Crans, International Category Theory Conference, Como, 17 July 2000

Why are solutions to Zamolodchikov equations interesting? Because they give examples of braided monoidal 2-categories.

For a (weak or strict) Zamolodchikov system (Z, R, S) on a monoidal 2-category C, let br(C, Z, R, S) be the smallest submonoidal 2-category of st(C) containing the objects (A) for A

Z, the arrows R_A,B for A, B

Z, and the 2-arrows S_A,B,C for A, B, C

Z.
Define for objects A = (A₁, ..., A_k) and B = (B₁, ..., B_k) of br(C, Z, R, S) an arrow R_A,B in br(C, Z, R, S) by

-                                    -
| (A₁, ..., A_k-1, R_{A_k,B₁}, B₂, ..., B_l), |
|                ...,                |
|(A₁, ..., A_k-1, B₁, ..., B_l-1, R_{A_k,B_l}),|
|                 .                  |
|                 .                  | .
|                 .                  |
| (R_A₁,B₁, B₂, ..., B_l, A₂, ..., A_k),  |
|                ...,                |
| (B₁, ..., B_l-1, R_{A₁,B_l}, A₂, ..., A_k) |
-                                    -

Theorem The arrows R_A,B form part of the data for a (somewhat weak or strict) braiding on br(C, Z, R, S).

Because a (somewhat weak or strict) braiding on a (strict) monoidal 2-category gives a (weak or strict) Zamolodchikov system on C consisting of all objects of C, the arrows R_A,B (, the 2-arrows R^~_(A|B,C) and R^~_(A,B|C)) and the 2-arrows S_A,B,C = R_{A,R_B,C} = R_{R_A,B,C} [hmpz], the theorem completes the lowest-dimensional part of the

Conjecture There is a triadjunction

           <--
br: ZA2-Cat     BM2-Cat :za
           -->

which restricts to a triequivalence between (strict) monoidal 2-categories with a (weak or strict) Zamolodchikov system on all non-unit objects and (somewhat weak or strict) braided monoidal 2-categories in which the arrows and 2-arrows are generated by the R_A,B's and R_{A,R_B,C}'s.

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