### r-flatness = 2-associativity

#### Posted by Urs Schreiber

I am further discussing loop space-, surface- and and 2-group-holonomy with those interested. While trying to convince the 2-group-theorists that 2-associativity admits more solutions than currently considered in the literature, Orlando Alvarez keeps coming up with ever more such solutions to loop space r-flatness(which ensures that loop space connections assign well-defined surface holonomy)! But what is the systematic way to find these solutions? Well, the answer should be given by 2-Lie algebra theory…

It seems to me that hence some effort demonstrating the *equivalence* of loop space r-flatness with 2-group 2-associativity is in order.

And Ithink that I have now figured out how to do the higher order expansion of (differential) 2-group 2-associativity, demonstrating that it is indeed (essentially) the same as r-flatness in loop space.

One key observation is that for the purpose of defining surface holonomies the 2-associativity law (‘exchange law’)

(where a simple dot indicates the horizontal and an open dot the vertical product)

needs only be required to hold in the limit where the surface elements ${f}_{2}$ and ${f}_{2}^{{\textstyle \prime}}$ become infinitesimal, since that’s sufficient to consistently integrate their contributions up to a macroscopic surface holonomy.

(Note that this makes good sense also in the light of the fact that the right factor in the horizontal product is parallel transported to the left factor along a *single* edge, something which manifestly should not in general be done to an extended object.)

For a surface element ${f}_{1}$ and another one ${f}_{2}^{{\textstyle \prime}}$ to the right of it the 2-associativity condition says that the commutator

must vanish (when all elements take values in the same group and $t$ is trivial).

Because ${f}_{2}^{{\textstyle \prime}}$ should be infinitesimal we write (following Girelli & Pfeiffer)

and let ${f}_{1}$ be the horizontal product of many small surface elements as in this figure

What looks like TESLA superconducting cavities is a ‘horizontal’ chain of small surface elements (drawn, however, diagonally) making up the total surface strip ${f}_{1}$. ${x}^{\mu}$ and ${x}^{\nu}$ are the coordinates parallel to the sides of the little squares and $\sigma =\frac{1}{2}({x}^{\mu}+{x}^{\nu})$ runs ‘horizontally’ along the diagonal.

The whole trick is now to get an expression for ${f}_{1}$ in the limit where it becomes the horizontal product of all these squares

and where the left factor ${g}_{2}^{-1}{f}_{1}{g}_{1}$ becomes

This problem however becomes very simple once it is realized that the terms obtained by truncating ‘$\cdots $’ are recursively related as

where $X$ is the same expression with at most single primes, and so on.

By performing the lattice computations similar to those done by Girelli&Pfeiffer, but taking one more order in $\u03f5$ into account one finds that

and

Plugging this into the above recursion formula one finds that ${g}_{2}^{-1}{f}_{1}{g}_{1}$ for ${f}_{1}$ a long thin ‘horizontal’ strip becomes

where $W$ is the holonomy of $A$ along the strip.

Inserting this result into the 2-associativity condition finally yields

This, I claim, is the ‘exchange law’ in 2-group theory in terms of the target space fields $A$ and $B$ which ensures that 2-group holonomy of continuous surfaces is well-defined.

It is *essentially* the same as the r-flatness condition found by Orland Alvarez. The latter involves another integral with $\sigma $ and ${\sigma}^{{\textstyle \prime}}$ exchanged.

*But* for all solutions to r-flatness found so far the integrand actually vanishes for all ${\sigma}^{{\textstyle \prime}}$ seperately. All these solutions hence also solve the above 2-associativity condition.

The interesting question arises whether the above is actually equivalent to the vanishing of the integrand all by itself.

## Re: r-flatness = 2-associativity

This is what you wrote in the pdf file of your notes, right? I am reading that at the moment, and will get back to you with some comments.