Correction to the definition of the String crossed module, part 2: triangulation

After the previous post was shared on Twitter, and I tagged Urs Schreiber, one of the authors of the paper in question (‘BCSS‘), he rightly pointed out that I had merely found a discrepancy in the coherence equations between BCSS and HDA VI. The true source of equation (5) of BCSS (which I claimed last time was in error) should be, he says, the theory of weak maps of $L_\infty$ algebras, going back to Lada and Markl’s Strongly homotopy lie algebras (or free arXiv version). I totally agree with this point, but I felt daunted by having to unwind the various definitions that unravel to equation (5). So I wanted to explore the issue a bit more. I know that the sign in front of the 1-form $\beta_p = -2\int_0^{2\pi} \langle \Theta,p(\theta)^{-1}\partial_\theta p(\theta)\rangle\,d\theta$

in the proof of Proposition 3.1 cannot be correct, since it leads to a contradiction in a geometric calculation independent of all the $L_\infty$-algebra material. I should note that this geometric calculation is nothing exotic; a correspondent let me know I muddied the water in the previous post by discussing it in the context of a larger project with more novel constructions. As a result, I will give the calculation later below the fold.

What I did not realise when I wrote the first post is that the sign error in $\beta_p$ is linked not only the an overall sign in equation (5) of BCCS, but also to a plus sign implicit in the definition of the comparison functor from 2-term $L_\infty$-algebras to (semistrict) Lie 2-algebras, which are possibly weak Lie algebra objects in the 2-category of groupoids internal to $\mathbf{Vect}$. This functor is constructed in HDA VI, in Theorem 4.3.6 (=Theorem 36 in the arXiv version), and here is the relevant version Part of the proof of Theorem 4.3.6 of HDA VI, constructing a functor from 2-term -algebras to Lie 2-algebras

This was invisible to me, until I checked a different article generalising HDA VI, namely Roytenberg’s On weak Lie 2-algebras (or free arXiv version), where a generalisation of the comparison functor to weaker objects is given. Here is included a minus sign on what turns out to be the troublesome piece of data arising from the crossed module of Lie algebras, arising from the sign on $\beta_p$. Discussion in Roytenberg about the relation between weak 2-term -algebra maps and morphisms between weak Lie 2-algebras

So now in fact one can see the sign on $\beta_p$ as being a combination of two signs: one from the comparison functor, and one from the coherence equation (5), just that in BCSS, the first of these is a $+$, and the second is a $-$ .

Now the interesting thing is that Roytenberg uses a version of the comparison functor where there is explicitly a minus sign built in, where HDA VI has a plus, but also uses a version of the coherence equation that reduces to (5) in BCSS, once the additional assumptions of BCSS are applied. So we find ourselves in the position of having three papers, none of which agree about two different signs:

To get a consistent version of Theorem 5.1 together with Proposition 3.1, given a choice of convention, the sign out the front of $\beta_p$ must be the product of the signs in the given row. BCSS get a minus sign, whereas if one used either of the other two conventions, it would be a plus sign. As we will see below, the minus sign in front of $\beta_p$ leads to a contradiction. Urs Schreiber’s point is that the sign in the first column should be dictated by the definition of morphism of $L_\infty$-algebra, so this is going to force the sign in the comparison functor. Before doing this, however, I want to lay out my own computation, for why I think the sign on $\beta_p$ is incorrect. This will just be the first part, as it takes some background in geometry to set up. I will conclude with the actual contradiction in the next post, before trying to tackle the $L_\infty$-algebra material.

The construction in BCSS takes a generic simply-connected, simple compact Lie group $G$, but we can just take $G=SU(2)$. Now $PSU(2)$ is the infinite-dimensional Lie group of smooth functions $\gamma\colon [0,2\pi]\to SU(2)$ such that $\gamma(0)=I$. Then there is a (closed, Lie) subgroup $\Omega SU(2)$ satisfy additionally $late \gamma(2\pi)=I$, making $\Omega SU(2)$ the kernel of the smooth homomorphism $ev_{2\pi}\colon PSU(2) \to SU(2)$. In both of these the group structure is pointwise, and we can identify tangent vectors with left-invariant vector fields, and hence with elements of the respective Lie algebras, which are $P \mathfrak{su}(2)$ and $\Omega \mathfrak{su}(2)$, of paths and loops based at the origin. We also need to use the group $P\Omega SU(2)$, which can be considered as functions $f\colon [0,2\pi]\times [0,2\pi]\to SU(2)$ such that $f(0,\theta) = I$, and $f(t,0)=f(t,2\pi)=I$. There is a principal $U(1)$-bundle $\widehat{\Omega SU(2)}\to \Omega SU(2)$ that is also a central extension of Lie groups, with a connection $\mu$ defined in various places, for instance by Murray and Stevenson.

Now the geometry in question is that $\Omega SU(2)$ is a normal subgroup in $PSU(2)$, and one can lift the conjugation action to the central extension $\widehat{\Omega SU(2)}$, as shown in Proposition 3.1 of BCSS. This is where the 1-form $\beta_p$ comes in. The definition of both the central extension and the connection on it go via working up on $P\Omega SU(2)$ and then by showing everything descends along $ev_{2\pi}$. Since $P\Omega SU(2)$ is contractible, the principal bundle has a section, and the central extension even becomes a semidirect product of Lie groups, namely $P\Omega SU(2)\ltimes U(1)$. The connection $\mu$ up on $P\Omega SU(2)\ltimes U(1)$ I will denote as $\widetilde{\mu}$, as is defined to be $\widetilde{\mu}:=\theta^{-1}d\theta + \int_{[0,2\pi]} ev^*R$

where: $ev\colon P\Omega SU(2) \times [0,2\pi]\to \Omega SU(2)$ is the evaluation map, $\int_{[0,2\pi]}$ is integration over the fibre, and $R$ is a left-invariant 2-form given by $R(\gamma X,\gamma Y) = \int_0^{2\pi} \langle X,\partial_\theta Y\rangle d\theta$.

Recall that $X,Y\colon [0,2\pi]\to \mathfrak{su}(2)$ are loops based at the identity. It should be noted that the definition of integration over the fibre of Murray–Stevenson uses contraction of a lifted vector in the first slot of the 2-form, not the last slot, as is sometimes the case.

The lift of the function $Ad\colon PSU(2)\times \Omega SU(2)\to \Omega SU(2)$ to $\widehat{Ad}\colon PSU(2)\times \widehat{\Omega SU(2)} \to \widehat{\Omega SU(2)}$ is defined first as a map $\widetilde{Ad}\colon PSU(2)\times P\Omega SU(2)\ltimes U(1) \to P\Omega SU(2)\ltimes U(1)$, and then shown to descend. This map is given by $\widetilde{Ad}(p;f,z) = (Ad_p f, z \exp(i (\int_{[0,2\pi]} \beta_p)(f)))$

although in BCSS it is written in more elementary terms. At this point it is not important what $\beta_p$ is, all we need to know is that BCSS establishes that $Ad_p^*R - R = -d\beta_p$ on $\Omega SU(2)$ (I have checked this myself, it is a direct computation, as stated in BCSS). I shall write $\widehat{Ad}_p$ for the restriction of $\widehat{Ad}$ to $\{p\}\times \Omega SU(2)$ and similarly for $\widetilde{Ad}_p$.

Where the contradiction arises is when one tries to establish the analogue of this last identity, linking $R$ and $\beta_p$, at the level of connections. It is true on general principles that $\widehat{Ad}_p^*\mu - \mu$ descends to a 1-form on $\Omega SU(2)$, but the question is, what is this 1-form? We can more easily answer the question for the lifted connection $\widetilde{\mu}$, since we have explicit formulas for it and for $\widetilde{Ad}_p$. The answer should be a 1-form on $P\Omega SU(2)$ that descends along $ev_{2\pi}$.

We can calculate as follows:

where in the second-to-last line I have used Stokes theorem for integration over the fibre, and the fact $ev_0^*$ is the zero map on forms (be aware though that with the definition of $\mu$ from Murray–Stevenson, this means that the sign is different to the version of Stokes at, for example, this math.SE answer). Rearranging this we get $ev_{2\pi}^*(\widehat{Ad}_p^*\mu - \mu - \beta_p) = 2\int_{[0,2\pi]}ev^*(d\beta_p)$

The left side vanishes when evaluated on tangent vectors vertical for $ev_{2pi}$, so to arrive at a contradiction I can evaluate the right side at a specific point on specific vertical vectors and get something nonzero. Despite the form of the identity, it isn’t immediately obvious this will happen, so I will plug in some actual paths and matrices next time to check.

2 thoughts on “Correction to the definition of the String crossed module, part 2: triangulation”

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