Higher dimensional Zamolodchikov

Sjoerd Crans (20/11/96)

1. Relevant papers: three of Kapranov and Voevodsky on braided monoidal 2-categories and Zamolodchikov equations, one of mine, Central observations on \omega-teisi, in preparation. The aim of the talk was to explain the Zamolodchikov equation by using teisi, and derive higher-dimensional generalizations as a consequence.
2. Recalled Yang-Baxter operator, Yang-Baxter system, two proofs of the lemma that in a braided monoidal category the braiding is a YB system, both of them in string diagram and commutative diagram formulation.
3. Recalled "one-dimensional" Zamolodchikov operator and system, and waved my hands a bit about pulling strings to prove the lemma that in a braided monoidal category the braiding gives a 1-dim Zam system.
4. In a braided monoidal 2-category the two proofs of YB turn into two cells which are required to be equal: "S+ = S-". The data for a "two-dimensional" Zam system are 1-dim R's and 2-dim S's filling YB hexagons.
5. To formulate Zam equation, look at how R and S operate on four factors; it turns out that the equation states that the two sides of the 3-dimensional permutohedron are equal.
6. By investigating

     ---   ---   ---   ---
    /   \ /   \ /   \ /   \
   .  D  .  C  .  B  .  A  .
    \   / \   / \   / \   /
     ---   ---   ---   ---
   

   in an \omega-tas with one arrow, get a clear and complete picture: in a braided monoidal 2-category the braiding is a Zam system, by taking S either one of the fillings of the YB hexagon (which are equal anyway), there are three different proofs of the Zam equation, which in a braided monoidal Gray-category turn into cells, and it follows from the axioms for \omega-teisi that these cells are equal.

7. It would also be possible to do things slightly weaker, by having a cell between S+ and S-, and elsewhere, and then there are five different proofs of Zam which again turn into cells. Careful analysis gives a beautiful five-dimensional diagram relating these 3-dim cells.
8. I gave the decompositions of the permutohedron for the different proofs; the decompositions are the same as Kapranov and Voevodsky give, except that 3 of their 8 don't give a legitimate proof (to wit: only a+++ = +++a, ---a = a--- and ++-- = --++ are legitimate).
9. It is now straightforward to define a three-dimensional Zam system as R's, S's and 3-dim T's filling the permutohedron; the 3-dim Zam equation can be summarized by

     ---   ---   ---   ---   ---
    /   \ /   \ /   \ /   \ /   \
   .  E  .  D  .  C  .  B  .  A  .   ,
    \   / \   / \   / \   / \   /
     ---   ---   ---   ---   ---
   

   and it is trivial that the braiding in a braided 4-dim tas gives rise to a 3-dim Zam system by taking T to be any filling corresponding to a proof of the 2-dim Zam equation (which are equal anyway). This continues in higher dimensions, and also slightly weaker.

10. Can also generalize Yang-Baxter by investigating

      -----   -----   -----
     / / \ \ / / \ \ / / \ \
    . | C | . | B | . | A | .
     \ \ / / \ \ / / \ \ / /
      -----   -----   -----
    

    in a sylleptic 4-dim tas, and the syllepsis gives rise to a 4-dim YB system. This also continues in higher dimensions.

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Steve Lack

Last modified: Tue May 19 10:54:10 EST 1998