The n-Category Café

Table of Contents

Part 1. Basic classical mechanics, in which the concepts are briefly recalled, essentially symplectic geometry, which are to be generalized in the following.

Part 2. Something is missing, in which the failure to capture gauge symmetries is highlighted – and related to the need for isomorphisms.

Part 3. The plan from here on, in which the following line of attack is outlined.

Part 4. BV formalism: mechanics on graded vector spaces, in which finally some elements of the BV formalism are mentioned.

Part 5 Some literature, in which the author fails to summarize the relevant literature.

Part 6 — but why?, in which the relation between Lie $n$-algebroids and differential graded algebras is highlighted.

Part 7 Running out of steam, in which the author notices that he only made it through what was intended as a mere introductory remark and postpones further discussion to an unspecified point in the future.

Basic classical mechanics

In ordinary classical mechanics we are studying spaces $P$, called phase spaces. These are manifolds that come equipped with a 2-form $\omega \in \Omega^2*(P)$ that is closed and everywhere non-degenerate. Hence $P$ is a symplectic space.

Since $\omega$ is assumed to be non-degenerate, it induces an antisymmetric product on the space of functions $$\{\cdot , \cdot\} : C^\infty(P) \otimes C^\infty(P) \to C^\infty(P) \,,$$ sometimes called the space of classical observables (but that term is more strictly reserved for a subset of all such functions which are invariant under certain group actions; we’ll come to that) the Poisson bracket defined by $$\{F,G\} := \omega^{-1}(d F, d G) \,.$$ The fact that $\omega$ is closed makes this a Lie bracket and hence makes $C^{\inft}(P)$ Poisson algebra.

Locally these spaces look like cotangent bundles $T^* X$. The points of $X$ are the configurations of the classical system in question. When we are dealing with a field theory, then the points of $X$ are the field configurations.

The points in the fibers of $T^* X$ are called the canonical momenta associated to these configurations. In the context of the BV formalism, people like to call these not momenta, but antifields.

When one assumes that $X$ may be covered by a single coordinate chart, this chart is conventionally called $$q : X \to \mathbb{R}^n$$ and the induced chart on $T^* X$ is then by usual convention called $$(q,p) : T^* X \to \mathbb{R}^n \,.$$

In these coordinates the symplectic form $\omega$ reads, locally $$\omega_{X} = d( \sum_i p^i d q_i)$$ and hence the Poisson bracket reads locally $$\{F , G\}|_X = \sum_i \left( \frac{\partial F}{\partial q^i} \frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i} \frac{\partial G}{\partial q^i} \right) \,.$$

The main reason to emphasize these local formulas here is that in the BV formalism, to be described in a moment, the existing literature tends to make heavily recourse to local coordinate formulas. And not entirely with lack of good reason, actually.

Something is missing

The inclined $n$-Café reader may notice the following:

the above language for describing the physical world

a) is very explicit concerning the concept of a physical configuration, which is an object (I mean: element) in configuration space $X$;

b) is already slightly more coy about the notion of a physical process – but of course this is at least partially encoded in the cotangent space $T^* X$

c) says nothing about isomorphisms between physical configurations.

To the $n$-categorical eye this may seem like an oddity. (You know, Space and State, Spacetime and Process). And indeed, while they didn’t think of it from this point of view, physicist went through decade-long intensive trouble with figuring out how to best add the information of isomorphisms between physical configurations to the above classical description of classical mechanics.

The length of these struggles is reflected to some degree in the plethora of terms, often weird-sounding, that have been introduced and considered over the years: when you hear somebody ponder things like the master action for the ghosts of ghosts and their antifields you know he or she is on some major quest. And indeed – that of trying to come to grips with the notion of isomorphism of physical configurations, otherwise known as the issue of gauge theory.

The current endpoint of these developments is known as

Batalin-Vilkovisky (BV) formalism.

The plan from here on

I shall now give a quick impression of what BV-formalism is and give some pointers to useful literature. Then I make a remark on why and how this should secretly be nothing but the Lie version of physics on Lie $n$-groupoids. After that ideosyncratic detour I come back to the BV-formalism and talk about more of its details.

BV-formalism: mechanics on graded spaces

The quick answer to what BV formalism is about is this:

(Classical) BV theory is the internalization of the formulation of classical mechanics in terms of symplectic spaces and Poisson algebras from the world of smooth manifolds into that of graded and differential graded (“dG”) manifolds.

Sometimes, in the BV-literature, people talk about “supermanifolds” instead of graded manifolds. I shall try not to do that, since I think it is slightly misleading, or at least not helpful. (This becomes particularly manifest when one starts discussing supersymmetric physics using the BV formalism.)

For my purposes here (and indeed for the bigger part of existing literature), a graded manifold is nothing but the dual to a $\mathbb{Z}$-graded-commutative algebra, whose degree-0 part is an ordinary algebra of (smooth) functions on some ordinary space.

The crucial example to keep in mind is the “graded space” denoted $$\Pi E$$ for $E \to X$ any vector bundle over $X$. This is really notation for the space which we imagine has the “algebra of smooth functions” which is the exterior algebra $$\Wedge^\bullet s\Gamma(E^*)$$ of sections of the dual of this vector bundle, taken to be of degree 1.

The most important special case of this is $$\Pi T X \,,$$ the “odd tangent bundle” of an ordinary space $X$. This simply corresponds to the graded commutative algebra $$\Omega^\bullet(X)$$ of differential forms on $X$.

But we just as well have the odd tangent bundle $$\Pi T X$$ of an arbitrary graded space $X$: this is given by the algebra which is freely generated as a graded-commutative algebra from the algebra “of functions” on $X$ itself, together with the duals of the graded derivations of this algebra, each of which is regarded as as a generator of degree one higher than the derivation it corresponds to.

For instance for $$X = \mathbb{R}^{0|1}$$ the “odd line” (namely the (rather boring) exterior algebra over just $\mathbb{R}$), we have that its shifted tangent space $$\Pi T X$$ is the free graded commutative algebra $$\wedge^\bullet (s \mathbb{R}^* \oplus s s \mathbb{R}^*)$$ which is generated (over our ground field, usually $\mathbb{R}$ in the context of classical mechanics) from two generators, one of them of degree 1, the other of degree 2.

Given such a graded space, there is an obvious way how to define an analogue of a symplectic form $\omega$ on it.

The graded Poisson bracket coming from that graded symplectic form is now conventionally written $$(\cdot, \cdot) \,.$$ And – just to confuse everybody – it is called the antibracket. Don’t let that confuse you. This is our Poisson bracket! (Just as the word “antifield” shouldn’t trick you in forgetting that these are nothing but canonical momenta).

These “graded symplectic spaces” are, essentially, what classical BV formalism is concerned with.

Up to two further details:

first, instead of just blindly internalizing classical symplectic theory into the graded world, BV-formalism takes care of the following point:

once we are in the graded world in the first place, we should admit that the canonical momenta which we had encountered before really ought to be of one degree higher than the canonical coordinates that they come from.

In other words, if $X$ is the graded configuration space, then we now take the graded phase space to be $$\Pi T^* X$$ instead of just $T^* X$.

Remark.This simple observation is of major importance. Below I will point out that $n$th degree in the world of graded manifolds correspoonds to $n$-morphisms in the world of Lie $n$-groupoids. Canonical momenta encode physical processes. Hence morphisms. Hence they need to be in degree one higher than the object whose process they describe.

This then leads people to invent all the following funny terminology:

- a field is an element in $X$

- an antifield is a canonical momentum associated to that, i.e. an element in the fibers of $\Pi T^*X$

- a ghost field is a degree-1 field

- a ghost-of-ghost field is a degree-2 field.

and so on.

Hence it remains to identify the physical meaning of these ghosts and higher ghosts. Being of higher degree, they should correspond, by the above remark, to gauge transformations and higher gauge transformations, respectively. And indeed, that’s how it is: in the physics literature these ghosts are the “gauge transformation parameters”.

Finally, the second further ingredient to the mere idea of internalizing classical mechanics in graded manifolds is to give a new characterization of the classical action functional:

in BV formalism, the ordinary classical action functional (I haven’t even talked about these yet) is essentially something like a functional on phase space (but there is a subtlety here – more on that elsewhere) extended to the graded phase space, where it is called $$S$$ and required to satisfy the condition that it has vanishing graded Poisson bracket (called “antibracket”, remember) $$(S,S) = 0$$ with itself.

This conditiion is called the classical master equation in this context. I should talk about that in more detail elsewhere.

Some literature

I certainly cannot give a canonical reading list, and would in fact be grateful for any further pointers from those more expert than me. Also, I will restrict to material which is freely available online.

In fact, for the moment I just give an extremely unbalanced and very short list of references. Maybe I’ll extend that later on..

Anyway, among all the sophisticated material which one finds, this paper here stood out as one which describes in great and careful detail the basic ideas and in particular their concrete application to the description of the two most important gauge theoretic systems which one encounters: the free relativistic partice (as the archetype of a gravity-like gauge theory) and Yang-Mills theory (as the gauge theory archetype per se):

Joaquim Gomis, Jordi París, Stuart Samuel
Antibracket, Antifields and Gauge-Theory Quantization
hep-th/9412228

The BV literature splits into two parts: in one part of the literature, people start with a gauge theoretic physical system, and then go through the process of finding the right “ghosts” such as to construct a suitable graded extension of the ordinary classical configuration and phase space.

In the other part of the literature, people go the other way round: accepting the premise that physics should be done on graded spaces, these authors start with their favorite examples of a graded space, and then turn the BV crank to study the physics described by them.

To this second part belong the two articles (AKSZ and Roytenberg) which I already mentioned last time.

Another text that I found helpful is section 2 and 3 of

Noriaki Ikeda
Deformation of Batalin-Vilkovisky Structures math/0604157

Generally very inspiring are the articles by Jim Stasheff on this topic. Jim Stasheff very much cares about reflecting on all these algebraic structures which are being found here. For instance, the end of his hep-th/9712157 is, so far, the only place I found where what looks to me like the crucial question concerning the BV master equation is mentioned:

[…] but why?

But here I am not talking about the master equation much yet, I have barely mentioned it. But I do shall now ask “…but why?” with respect to the appearance of graded manifolds themselves.

…but why?

Certainly the most profound message of

J. Baez, A. Crans, Lie 2-Algebras is the following slogan:

Lie $n$-groups differentially have Lie $n$-algebras. Lie $n$-algebras are the same as $L_\inft$-algebras.

Beware that this is indeed just that, a slogan. A couple of qualifications are needed to make this a precise statement. But for the moment I just want the slogan.

Notice that there is the following helpful way to understand the technology at work behind this statement:

By forming all the tangent spaces to the space of $k$-morphisms starting at some identity $(k-1)$-morphisms in some Lie $n$-groupoid, we obtain a bunch of vector spaces which encode what the Lie $n$-groupoid looks like in linear approximation. These arrange into one big graded vector space $$V \,,$$ the $\mathbb{N}$-grading being simply the order $k$ of these $k$-morphisms.

By some kind of magic (but, surely, it must be the general geometry dao at work here, somehow, I am still not completely sure if anyone really understands this, truly), it turns out that all the structure which we have on the morphisms of the Lie $n$-groupoid, all the compositions, all the source, target and identity-assigning morphisms, are encoded, equivalently, in a – take a deep breath – degree -1 graded codifferential $$D : S^c s V \to S^c s V$$ from the cofree graded-co-commutative coalgebra on $s V$ (that is, $V$ with all degrees shifted by one) which squares to 0 $$D^2 = 0 \,.$$ That’s the important fact here. We should pay great attention to this fact: this is the bridge from the world of $n$-groupoids to that of differential algebra. Such cofree graded co-comuutative coalgebras with nilpotent degree -1 differentials are called $L_\infty$-algebras (or strongly homotopy Lie algebras).

So, suppose for a moment that you’d follow the claim that $n$-particles (points, strings, membranes, etc.), i.e. certain kinds of $n$-dimensional quantum field theories, classically come from setups where:

a field configuration (in precisely the sense I have been talking about above) is an $n$-functor from an $n$-categorical “parameter space” to an $n$-categorical “target space”.

Just trick yourself for a moment into believing this. Then, what follows? It follows that if all these $n$-categories are actually smooth $n$-groupoids, then also the “space of field configurations”, namely the $n$-functor $n$-category $$\mathrm{conf} := \mathrm{Hom}_{n\mathrm{LieGrpd}}(\mathrm{par},\mathrm{tar})$$ is a Lie $n$-groupoid.

Since it is a Lie $n$-groupoid, we would be tempted to simplify our lives a little by studying not this full thing, but just the morphism of the underlying Lie $n$-algebroids which it induces. This way we expect to find our configuration space to be turned into the corresponding Lie $n$-algebroid.

I haven’t really talked about Lie $n$-algebroids here, just about Lie $n$-algebras. That’s because I don’t want to get into a long technical discussion which clarifies the heuristic picture that I am drawing only slightly but would distract us a whole lot. So let’s just think Lie $n$-algebras for the moment and keep our fingers crossed that whatever we understand this way still applies to the Lie algebroid case.

Okay, now here is the important observation: assuming that our configuration space is a Lie $n$-algebroid, hence an $L_\infty$-algebra (“$L_\infty$ algebroid”, really, but never mind for the moment), the space of functions (physical observables!) on this is a graded-commutative differential algebra.

Conversely, every graded-commutative differential algebra is the “algebra of functions” on a Lie $n$-algebra.

This is simple vector space duality in the $L_\inft$-picture:

from the graded commutative co-free coalgebra $S^c s V$ we obtain, dually, the free graded commutative exterior algebra $$\wedge^\bullet (s V^*) \,.$$ Moreover, from the nilpotent graded codifferential $D$ we obtain the nilpotent graded differential $$d : \wedge^\bullet (s V^*) \to \wedge^\bullet (s V^*)$$ which simply acts dually as $$d \omega := \omega(D(\cdot)) \,.$$

For a concise but comprehensive review of the details on all this (well known to the experts), including references to all the relevant literature, see section 3 of our preliminary article with the working title Structure of Lie $n$-algebra.

It should be quite clear now what I am getting at: the important point is that whenever we see a quasi-free (meaning the underlyin graded commutative algebra is free) differential graded-algebra, with generators concentrated in degrees $\leq n$, we are really seeing the “algebra of functions” (in a very concrete sense! – much more concrete than in the case of “graded manifolds”, which really don’t have a independent definition apart from that in terms of the “algebra of functions” on them) on a Lie $n$-algebra.

And it is pretty clear how more general graded-commutative differential algebras correspond to Lie $n$-algebroids. (But the details of that require more discussion.) For instance it is very well known that the graded-commutative differential algebra $$(\Omega^\bullet(X), d) \,,$$ namely the deRham complex of differential forms on some manifold $X$, is the “algebra of functions” on the Lie 1-algebroid of the pair groupoid of $X$ (the codiscrete groupoid over $X$). And this pattern continues.

Finally, to finish the argument which I am sketching here, recall that degree $k$-generators of these graded algebras correspond to (tangents to) $k$-morphisms of a Lie $n$-groupoid.

With this picture in mind, the entire BV formalism suddenly appears in a very clear light:

if we have a “configuration space” of some physical system where various configurations may be related by various ismomorphisms, then we would, in the $n$-categorical picture, consider not the space of configurations, but the category of configurations: its objects are the former configurations, its morphisms are these isomorphisms.

So this “configuration category” $$\mathrm{conf}$$ would be a Lie groupoid. Hence the “algebra of functions on configuration space” would be – a graded-commutative algebra generated from

a) the ordinary algebra of functions on ordinary configuration space

together with

b) the odd generators which correspond to the (duals of the tangents to) the morphisms in $\mathrm{conf}$: this are the ghosts

I should add that what I am saying here, while I really think it deserves to be emphasized, isn’t supposed to be anything but a tautology once one thinks about it. In fact, one would expect to see this statement mentioned here and there in the literature. But if that is the case, I msut have had bad luck. At the moment I am aware of one place in the literature where the statement is explicitly mentioned, and that is

Dmitry Roytenberg
On the structure of graded symplectic supermanifolds and Courant algebroids
math/0203110

where it says somewhere in the middle of p. 6:

[…] it is natural to call NQ-manifolds of degree n n-algebroids. […]

Here an NQ-manifold is what I was referring to as a differential-graded manifold (the many-object version of the Koszul dual to a Lie n-algebra), and “$n$-algebroids” is really to be read as the sloppy version of “Lie $n$-algebroids”.

Running out of steam

“Hm”, I hear you saying, “while, in this picture, all the information about the space of $(0 \leq k \leq n)$-morphism of the configuration space $n$-groupoid is in the graded-commutative algebra, all the information about source, target and composition is in the differential. But in your review of BV formalism above we had just graded algebras, not graded differential algebras.”

Right. And – lo and behold – it turns out that graded algebras alone are not the end of the story: there is crucially, in BV formalism, an odd-graded nilpotent differential around, famously related to what is known as the BRST operator. And that’s our differential.

But I am running out of steam. The BRST operator in BV formalism will have to wait until next time.