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 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

in the proof of Proposition 3.1 cannot be correct, since it leads to a contradiction in a geometric calculation independent of all the -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 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 -algebras to (semistrict) Lie 2-algebras, which are possibly weak Lie algebra objects in the 2-category of groupoids internal to . This functor is constructed in HDA VI, in Theorem 4.3.6 (=Theorem 36 in the arXiv version), and here is the relevant version

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 .

So now in fact one can see the sign on 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:

coherence equation | comparison functor | |

HDA VI | ||

BCSS | ||

Roytenberg |

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 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 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 -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 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 -algebra material.

#### The contradiction, part 1

The construction in BCSS takes a generic simply-connected, simple compact Lie group , but we can just take . Now is the infinite-dimensional Lie group of smooth functions such that . Then there is a (closed, Lie) subgroup satisfy additionally $late \gamma(2\pi)=I$, making the kernel of the smooth homomorphism . 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 and , of paths and loops based at the origin. We also need to use the group , which can be considered as functions such that , and . There is a principal -bundle that is also a central extension of Lie groups, with a connection defined in various places, for instance by Murray and Stevenson.

Now the geometry in question is that is a normal subgroup in , and one can lift the conjugation action to the central extension , as shown in Proposition 3.1 of BCSS. This is where the 1-form comes in. The definition of both the central extension and the connection on it go via working up on and then by showing everything descends along . Since is contractible, the principal bundle has a section, and the central extension even becomes a semidirect product of Lie groups, namely . The connection up on I will denote as , as is defined to be

where: is the evaluation map, is integration over the fibre, and is a left-invariant 2-form given by

.

Recall that 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 to is defined first as a map , and then shown to descend. This map is given by

although in BCSS it is written in more elementary terms. At this point it is not important what is, all we need to know is that BCSS establishes that on (I have checked this myself, it is a direct computation, as stated in BCSS). I shall write for the restriction of to and similarly for .

Where the contradiction arises is when one tries to establish the analogue of this last identity, linking and , at the level of connections. It is true on general principles that descends to a 1-form on , but the question is, what is this 1-form? We can more easily answer the question for the lifted connection , since we have explicit formulas for it and for . The answer should be a 1-form on that descends along .

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 is the zero map on forms (be aware though that with the definition of 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

The left side vanishes when evaluated on tangent vectors vertical for , 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.

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