Zamolodchikov Tetrahedron Equation

The Zamolodchikov tetrahedron equation, illustrated above by J. Scott Carter and Masahico Saito, is a fundamental law governing surfaces embedded in 4-dimensional space. It also arises purely algebraically in the theory of braided monoidal 2-categories.

Given an object $x$ in a monoidal category, we say a morphism

$$B \colon x \otimes x \to x \otimes x$$

obeys the Yang–Baxter equation if

$$(B \otimes 1)(1 \otimes B)(B \otimes 1) = (1 \otimes B)(B \otimes 1)(1 \otimes B)$$

We can understand this using the technique of string diagrams if we draw $B$ as a ‘braiding’, the process of switching two copies of the object $x$:

The Yang–Baxter equation then says this:

In other words, we can slide a crossing of two strands under a third strand. In topology this is called the third Reidemeister move, one of three basic ways of changing a picture of a knot without changing the topology of the knot.

Given an object $x$ in a monoidal 2-category and a morphism

$$B \colon x \otimes x \to x \otimes x,$$

we can demand that the Yang–Baxter equation hold up to a 2-morphism. This means that there is a 2-morphism

$$Y \colon (B \otimes 1)(1 \otimes B)(B \otimes 1) \Rightarrow (1 \otimes B)(B \otimes 1)(1 \otimes B)$$

called the Yang–Baxterator. We think of this as the process of sliding a crossing of strands under a third strand:

Using topology, we can see that it is natural for the Yang–Baxterator to satisfy an equation of its own, a higher-dimensional analogue of the Yang–Baxter equation. This is called the Zamolodchikov tetrahedron equation:

$$\begin{array}{c} [Y \circ (1 \otimes 1 \otimes B)(1 \otimes B \otimes 1)(B \otimes 1 \otimes 1)] [(1 \otimes B \otimes 1)(B \otimes 1 \otimes 1) \circ Y \circ (B \otimes 1 \otimes 1)] \\ [(1 \otimes B \otimes 1)(1 \otimes 1 \otimes B) \circ Y \circ (1 \otimes 1 \otimes B)] [Y \circ (B \otimes 1 \otimes 1)(1 \otimes B \otimes 1)(1 \otimes 1 \otimes B)] \\ = \\ [(B \otimes 1 \otimes 1)(1 \otimes B \otimes 1)(1 \otimes 1 \otimes B) \circ Y ] [(B \otimes 1 \otimes 1) \circ Y \circ (B \otimes 1 \otimes 1)(1 \otimes B \otimes 1)] \\ [(1 \otimes 1 \otimes B) \circ Y \circ (1 \otimes 1 \otimes B)(1 \otimes B \otimes 1)] [(1 \otimes 1 \otimes B)(1 \otimes B \otimes 1)(B \otimes 1 \otimes 1) \circ Y]. \end{array}$$

To see the significance of this complex but beautifully symmetrical equation, one should think of $Y$ as the surface in 4-dimensional space traced out by the process of performing the third Reidemeister move. Then the Zamolodchikov tetrahedron equation says the surface traced out by first performing the third Reidemeister move on a threefold crossing and then sliding the result under a fourth strand:

can be deformed to the surface traced out by first sliding the threefold crossing under the fourth strand and then performing the third Reidemeister move:

So, the Zamolodchikov tetrahedron equation says this:

Here is another picture of it created by Carter and Saito:

The numbers indicate which three strands are involved in each appearance of the Yang–Baxterator.

To learn more about the role of the Zamolodchikov tetrahedron equation in topology, see these papers:

• J. Scott Carter, Seiichi Kamada and Masahico Saito, Surfaces in 4-Space, Springer, Berlin, 2004.

• J. Scott Carter and Masahico Saito, Knotted Surfaces and Their Diagrams, AMS, Providence, Rhode Island, 1998.

Just as the Yang–Baxter equation is a consequence of the definition of braided monoidal category, the Zamolodchikov tetrahedron equation automatically follows from the definition of ‘braided monoidal 2-category’. For details and connections to other algebraic structures, see:

• John Baez and Martin Neuchl, Higher-dimensional algebra I: braided monoidal 2-categories, Adv. Math. 121 (1996), 196–244.

• Sjoerd Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Adv. Math. 136 (1998), 183–223.

• John Baez and Laurel Langford, Higher-dimensional algebra IV: 2-tangles, Adv. Math. 180 (2003), 705–764.

• John Baez and Alissa Crans, Higher-dimensional algebra VI: Lie 2-algebras, Theory and Applications of Categories, 12 (2004), 492–528.

The uncaptioned pictures above come from the paper by Baez and Crans; most of them were created by Aaron Lauda.

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