The n-Category Café

Instead, we just focus on understanding the critical points of our function $S$ on $X$. These are the zeros of the differential $$d S \,.$$ They form the space $$\{x \in X | d S(x) = 0\}$$ called the shell.

If that were it, we’d be done. The point is that we want more: what we really want are the connected components of the space of critical points. $$\pi_0(\{x \in X | d S(x) = 0\}) \,.$$ Namely if there is, for instance, a whole line of critical points, we say that all points on this line are isomorphic configurations. What we are interested in, then, are the isomorphism classes of critical points.

A function on such isomorphism classes would be a ‘gauge invariant observable’. These are one of the main objects of interest.

So we need to do two things:

a) first identify the functions on the space of critical points among all functions on $X$

b) then identify the gauge invariant functions among all functions on critical points.

We want to tackle a) by quotienting the space of all functions by those vanishing on the critical points. One assumes ‘sufficient regularity’ of $S$, such that the algebra of functions vanishing on the space of critical points is generated from all functions of the form $$d S(v)$$ for $v$ any vector field on $X$. Think of this as picking out all the different ‘components’ of the ‘covector’ $d S$.

We can say this elegantly as follows: forming the 2-term complex of vector spaces $$BV_a = (0 \to \Gamma(T X) \stackrel{d S(\cdot)}{\to} C^\infty(X) \to 0 )$$ which we think of as being concentrated in degree -1 and 0, with the differential being of degree +1, we find that its cohomology in degree 0 is precisely the space of functions on the critical points:

$$H^0(BV_a) = C^\infty(X)/\mathrm{im}(d S(\cdot)) \,.$$

We know that we can think of this little complex as a Baez-Crans type 2-vector space (a $\mathrm{disc}(\mathbb{R})$ module category) – and we should in fact do that, I belive. Because then passing to the cohomology is not a trick pulled out of thin air – but something that just is: 2-vector spaces are equivalent to their cohomology. Meaning that any 2-vector space of the form

$$0 \to V_{-1} \stackrel{f}{\to} V_0 \to 0$$ is equivalent to $$0 \to \mathrm{ker}(f) \stackrel{0}{\to} \mathrm{coker}(f) \to 0 \,.$$ And for vector spaces we simply have $$\mathrm{coker}(f) = V_0/\mathrm{im}(f) \,.$$

Of course it doesn’t matter whether you think of this happeneing in the 2-category of 2-vector spaces or, more traditionally, in the 2-category $2\mathrm{Term}$, of 2-term chain complexes (which is equivalent to $2\mathrm{Vect}$, even as a symmetric monoidal 2-category) – but you need to take care of the 2-morphisms (otherwise you’ll end up talking about “quasi-isomorphisms”, and who wants a quasi 1-thing if we can have an honest 2-thing?).

So we would like to say: whether or not we pass to the cohomology explicitly, i.e. whether or not we pass to the skeleton of our 2-vector space, it is in any case equivalent to the (1-)vector space of functions on the critical points.

That would be what we want. But we are not quite there yet.

Trouble is: we would want the cohomology in degree -1 to vanish, for our intended statement to be true. If that vanished, our 2-vector space would indeed be equivalent to the 1-vector space of functions on the critical points.

But $H^{-1}(BV_a)$ does in general, not vanish. It is equal to $$H^{-1}(BV_a) = \mathrm{ker}(d S(\cdot)) \,.$$ This is equal to the space of all those vector fields on $X$, which have the property that flowing along them leaves our function $S$ invariant.

This is the space of Noether relations. These vector fields detect to which degree the different ‘equations of motion’ given by the components of the 1-form $d S$ are not independent.

Notice the subtle but crucial difference between Noether relations and symmetries, conceptually:

Noether relations are those vector fields along which our functions $S$ is invariant.

Symmetries are those vector fields along which the equations of motion are invariant, on shell.

In our case these two concepts actually coincide. But for more general setups they need not. See page number 8 (below equation (10)) of the article by Kazinski, Lyakhovich & Sharapov for a very clear-sighted discussion.

(They are interested also in the case where we have a system defined by its equations of motion for which no Lagrangian description exists. Like self-dual $(2k)$-form theories. By considering this generalization, they are lifting accidental degeneracies of concepts, which gives them a clearer view.)

In any case, it follows that, in general, our 2-vector space of functions in not yet equivalent to the 1-vector space of functions on the critical points of $S$.

Instead, we have an extension $$(\mathrm{ker}(d S(\cdot)) \to 0) \;\;\;\; \stackrel{t}{\hookrightarrow} \;\;\;\; ( \Gamma(T X) \stackrel{d S(\cdot)}{\to} C^\infty(X)) \;\;\;\;\to\;\;\;\; (0 \to C^\infty(X)/\mathrm{im}(d S(\cdot)))$$ of the space we are interested in by the 2-vector space which has a trivial space of objects and the Noether identities of our function $S$ as its space of morphisms.

We would like to quotient the 2-vector space that we have by that of Noether identites. So we should just take the cokernel of the inclusion $t$. But let’s be sophisticated about: let’s take this cokernel not in the world of 2-vector spaces, but in the world of 3-vector spaces. In other words, let us take the weak cokernel (aka homotopy quotient) of $t$.

The result is, simple enough, the 3-vector space which is given by the 3-term chain complex $$BV_b := \mathrm{wcoker}(t) = ( 0 \to \mathrm{ker}(d S(\cdot)) \hookrightarrow \Gamma(T X) \stackrel{d S(\cdot)}{\to} C^\infty(X) \to 0 ) \,.$$

By its very construction, this 3-vector space now is equivalent to the 1-vector space of functions on the critical points. The additional part in degree -2 kills the cohomology in degree -1.

Of course the construction the way I presented here is highly revisionistic (even when compared to Jim Stasheff’s already revisionistic desciption). People have plenty of other names for this beast.

Mathematicians know this construction as the Koszul-Tate resolution, while physicists call it the antifield-antighost construction. Jim Stasheff emphasized this point in The (secret?) homological algebra of the BV approach.

Here is the table that assigns the names to the symbols

$$\array{ \mathrm{ker}(d S(\cdot)) &\hookrightarrow& \Gamma(T X) &\stackrel{d S(\cdot)}{\to}& C^\infty(X) \\ \\ Tate && Koszul \\ \\ antighosts && antifields && fields \\ \\ deg = -2 && deg = -1 && deg = 0 } \,.$$

Whichever way you like to look at it, we obtain a 3-vector space which is a puffed-up version of the 1-vector space of functions on critical points that we are looking for.

And I should maybe admit that this space of critical points is called the shell in physics (since in the case that $S$ is the action functional of a relativistic particle, this space is a standard hyperbola in Minkowski space, of all vectors whose Minkowski norm is a given value called the mass. This hyperbola is traditionally called the mass shell, for not so un-obvious reasons.)

So we obtain a 3-vector space which is a puffed-up version of the 1-vector space of functions on the shell.

All right. That solves part a). Next there is part b): we want to restrict to those functions that are gauge invariant, i.e those that are defined on shell and are invariant under the flow along the vector fields sitting in $\mathrm{ker}(d S(\cdot))$.

We know from BV, Part IV that the Lie algebroid which is obtained by differentiation from the action Lie groupoid which is associated to the action of the group of symmetries of $S$ (the subgroup of the group of diffeomorphisms of $X$ leaving $S$ invariant) is, dually, the Chevalley-Eilenberg algebra of the corresponding Lie algebra with values in the module of functions on $X$.

Somehow we want to be acting with that action Lie algebroid on our 3-vector space of on-shell functions and then pass to the subspace of functions acted on trivially.

I still don’t quite know the nice abstract $n$-categorical way to say this. It seems there should be one, related to a general concept of actions of Lie $n$-groups, but right now I feel I am still missing something.

So next I’ll need to fall back to what everybody is used to do in this BV business anyway: we simply go ahead proposing constructions and check in the end that these constructions are justified by the results they produce for us.

After that disclaimer, let’s agree that it is kind of obvious that if we want to incorporate the action of the symmetry algebra into our context, we simply add the vector space underlying the corresponding Chevalley-Eilenberg action Lie algebroid in degree 1, to form the 4-term chain complex:

$$BV = ( 0 \to \mathrm{ker}(d S(\cdot)) \hookrightarrow \Gamma(T X) \stackrel{d S(\cdot)}{\to} C^\infty(X) \stackrel{\rho}{\to} C^\infty(X) \otimes g^* \to 0 ) \,. )$$

Here $g$ is the Lie algebra of the group of symmetries, such that the space of vector fields in the image of $$\rho : g \to \Gamma(T X)$$ completed as a $C^\infty(X)$-module yields the space of all vector fields fixing the function $S$. We may think of $\rho$ as an element in $$\rho \in \Gamma(T X) \otimes g^* \,.$$ Then the map denoted $$C^\infty(X) \stackrel{\rho}{\to} C^\infty(X) \otimes g^*$$ consists of first tensoring with $\rho$ and then postcomposing with the obvious action $$\Gamma(T X ) \otimes C^\infty(X) \to C^\infty(X) \,.$$

You might wonder if I am now thinking about this 4-term complex as a 4-vector space. But I do not. Rather, I think we are seeing some notion of action Lie 3-algebroid which is slightly more general than usually considered:

Recall that ordinary Lie algebroids are the same as Lie-Rinehart pairs (these are pairs consisting of a Lie algebra and an associative algebra, both modules over each other in a consistent way that models the archetypical obvious Lie-Rinehart pair $(C^\infty(C),\Gamma(T X))$). If you try to categorify that, you clearly expect to find pairs consisting of Lie $n$-algebras acting – not on 1-algebras but on $n$-algebras.

It seems that we are seeing here something of this sort. But, as I said, I am not entirely sure yet what the best abstract way yo say this is. (On the other hand, you can see that the kind of Lie $n$-algebroid that I am talking about here, which is more like categorified Lie-Rinehart pairs than like that underlying the usual notion of NPQ-manifolds, is related to the construction of Lie algebras of inner automorphism groups of Lie $n$-groupoids that I talked about in The $G$ and the $B$ (and yesterday with Prof. Nigel Hitchin). Maybe more on that later, when the time is ripe.)

In any case, we end up looking at this 4-term complex

$$\array{ \mathrm{ker}(d S(\cdot)) &\hookrightarrow& \Gamma(T X) &\stackrel{d S(\cdot)}{\to}& C^\infty(X) &\stackrel{\rho}{\to}& C^\infty(X) \otimes g^* \\ \\ Tate && Koszul && && Chevalley-Eilenberg \\ \\ antighosts && antifields && fields && ghosts \\ \\ deg -2 && deg -1 && deg 0 && deg 1 } \,.$$

Its cohomology in degree 0 is now that of on-shell gauge-invariant functions.