Correction to the definition of the String crossed module, part 3: arriving at the contradiction, and the fix

Continuing on from the previous post, I need to finish off the calculation that shows the definition of the 1-form $\beta_p = -2\int_0^{2\pi} \langle \Theta,p(\theta)^{-1}\partial_\theta p(\theta)\rangle\,d\theta \in \Omega^1(P\Omega G)$

in BCSS leads to a contradiction, for $G$ a suitable compact connected Lie group, taken to be $SU(2)$ for concreteness. Recall that I had calculated the difference between a pair of connections on the $U(1)$-bundle $\widehat{\Omega SU(2)}\to \Omega SU(2)$, one arising from the construction of Murray and Stevenson, and the other being the pullback of this along the lift of $Ad_p\colon \Omega SU(2) \to \Omega SU(2)$ for $p$ a path in $SU(2)$ starting at the identity matrix. This lift is defined in BCSS using $\beta_p$. In actual fact, we did the calculation on the trivial bundle over $P\Omega SU(2)$ that is the pullback of the one I care about, and arrived at the formula $ev_{2\pi}^*(\widehat{Ad}_p^*\mu - \mu - \beta_p) = 2\int_{[0,2\pi]}ev^*(d\beta_p)$

where $ev_{2\pi}\colon P\Omega SU(2)\to \Omega SU(2)$ evaluates a path at the endpoint. I can show that for a careful choice of $p$, path $f\colon [0,2\pi]\to \Omega SU(2)$ and tangent vector at $f$ that is vertical for $ev_{2\pi}$, the right hand side fails to vanish. The left hand side is identically zero for vertical tangent vectors, and this will be the contradiction.

To start, we need to calculate $d\beta_p$ when evaluated on a pair of tangent vectors. Since $\beta_p \in \Omega^1(\Omega SU(2))$ is left invariant, this simplifies matters, so that we have $(d\beta_p)_\gamma(\gamma X,\gamma Y) \propto (\beta_p)_1([X,Y])$

where since I merely need to show this is nonzero, I ignore all prefactors in the calculation here and below. But taking the definition of $\beta_p$, this gives $(d\beta_p)_\gamma(\gamma X,\gamma Y) \propto \int_0^{2\pi} \langle [X,Y] ,p(\theta)^{-1}\partial_\theta p(\theta)\rangle\,d\theta$

But, however, I have to pull this back along the evaluation map $ev\colon P\Omega G\times [0,2\pi]\to \Omega G$, integrate this over the fibre of the projection $P\Omega G\times [0,2\pi]\to P\Omega G$, and evaluate it on a tangent vector $fZ \in T_fP\Omega SU(2)$. This gives the following:

Now at this point we need to pick some actual paths and so on. Since we are in $SU(2)$, I can write things out in the basis for the Lie algebra $\mathfrak{su}(2)$ given by the (anti-Hermitian version of the) Pauli matrices, namely

Now a vertical tangent vector $Z$ will be a loop $Z\colon [0,2\pi] \to \Omega \mathfrak{su}(2)$ based at the constant loop at $0$, so let us take $Z(t,\theta) = t(2\pi-t)\theta(2\pi-\theta)\sigma_1$

For the path $p\colon [0,2\pi]\to SU(2)$, based at the identity matrix $I$, let us take $p(\theta) = \exp(\theta\sigma_2)$.

And for the path $f\colon [0,2\pi]\to \Omega SU(2)$, let us take $f(t,\theta) = \exp(t\theta(2\pi-\theta)\sigma_3)$

From these definitions we get $p(\theta)^{-1}\partial_\theta p(\theta) = \sigma_2$ and $f(t)^{-1}\partial_tf(t) = \theta(2\pi-\theta)\sigma_3$. Finally, we can plug these in to our integral and get $\left(\int_{[0,2\pi]}ev^*(d\beta_p)\right)_f(fZ) \propto \int_0^{2\pi} \int_0^{2\pi} \theta^2(2\pi-\theta)^2 t(2\pi-t)tr\{[\sigma_3,\sigma_1]\sigma_2\}\,d\theta dt$

Using the commutation relations for the Pauli matrices $[\sigma_3,\sigma_1]=2\sigma_2$, the trace term becomes $2tr(\sigma_2\sigma_2)= -4$, and since we are merely looking to find something nonzero, we are left with the integral $\int_0^{2\pi} \int_0^{2\pi} \theta^2(2\pi-\theta)^2 t(2\pi-t)\,d\theta dt = \int_0^{2\pi} 2\pi t-t^2\,dt \cdot \int_0^{2\pi} 4\pi^2\theta^2 - 4\pi\theta^3 + \theta^4\,d\theta$

which is manifestly positive. We thus find that for our choice of $ev_{2\pi}$-vertical tangent vector $fZ$ and path $p$, $\left(\int_{[0,2\pi]}ev^*(d\beta_p)\right)_f(fZ) \neq 0$, contradicting the fact $ev_{2\pi}^*(\widehat{Ad}_p^*\mu - \mu - \beta_p)_f(fZ)=0$.

So what is the fix? Basically, the issue arose because for the given definition of $\beta_p$, BCSS get the identity $Ad_p^*R- R = -d\beta_p$. If instead we had $Ad_p^*R- R = d\beta_p$, then in the calculation in the previous post, Stokes’ theorem for integrating over the fibre would instead give us $ev_{2\pi}^*(\widehat{Ad}_p^*\mu - \mu) = ev_{2\pi}^*\beta_p$. Making the definition $\beta_p^{new} = 2\int_0^{2\pi} \langle \Theta,p(\theta)^{-1}\partial_\theta p(\theta)\rangle\,d\theta \in \Omega^1(P\Omega G)$

avoids the contradiction. In follows that in the statement of Proposition 3.1 of BCSS, the resulting crossed module of Lie algebras has the action $d\alpha(p)(\ell,c) = ([p,\ell],2k \int_0^{2\pi}\langle \ell(\theta),p'(\theta)\rangle\,d\theta)$

where I’ve used integration by parts to rearrange slightly. This means that there will have to be some sign changes elsewhere in the article. This is constrained by having to get the definition of (weak) morphism of 2-term $L_\infty$-algebras to agree with the definition of Lada and Markl, and to make sure the theorem of HDA VI relating the 2-cateories of Lie 2-algebras and 2-term $L_\infty$-algebras still holds. I am confident this can be done, since in fact only the special case relating crossed modules of Lie algebras and 2-term dg-Lie algebras needs to applied here, leaving some extra flexibility in how the definitions for Lie 2-algebras need to be arranged, if at all, to ensure the equivalence of 2-categories exists.

2 thoughts on “Correction to the definition of the String crossed module, part 3: arriving at the contradiction, and the fix”

This site uses Akismet to reduce spam. Learn how your comment data is processed.