### On BV Quantization. Part I.

#### Posted by Urs Schreiber

Of course, one of the questions I heard after my talk on the right structure 3-group of Chern-Simons theory here at the ESI workshop Poisson-Sigma Models, Lie Algebroids, Deformations and Higher Analogues in Vienna was

Why would we need to realize Chern-Simons theory as a 3-functor?

The true answer is

Because that’s how the dao works. It simply is the way it will turn out to be and we shouldn’t fight it.

But I am not supposed to argue like that. I am supposed to give answers that point out that it is *useful* to do this. That it helps to solve problems which otherwise cannot be solved.

Okay, sure. If it follows the dao, it is bound to help us solve problems. So here we go: I point out that we have a pretty good idea how to better understand 2-dimensional CFT starting from 3-functorial CFT, which crucially depends on that 3-functoriality. And in order to understand that still better, it would be helpful to understand how the Chern-Simons 3-functor comes to us from first principles.

I could go on about that point, but that’s not what my topic shall be right at the moment. Rather, there is yet another good reply:

Since (so I claim), $n$-dimensional QFT

isbest understood as an $n$-functor, it follows thatifyou already have a better than average understanding of $n$-dimensional QFT, then chances are that you are alreadysecretlyusing $n$-functors yourself. Without noticing so. And in that case, it would immensely boost our overall understanding if we’d made that explicit. Since it will add to your computational prowess the glory of conceptual understanding.

That’s actually why I am here at this workshop in the first place: a priori I am not all that interested in $L_\infty$-algebras, dG-manifolds, Courant algebroids and all the other highly involved algebraic structures that people here like to fill their blackboards with. These structures all look rather involved and somewhat messy. I wouldn’t really want to care about them – unless I knew that all this is really the shadow of very sensible structures: $L_\infty$-algebras are just Lie $n$-algebras, dG-manifolds are just a funny way to talk about higher Lie algebroids, Courant algebroids are just certain Lie 2-algebroids coming from something like sections of an Atiyah-2-bundle. And so on.

Only with this interpretation in mind does all this here make sense to me, and I’d dare to claim that, while the differential algebraic realization is very helpful for efficient computations, only with these interpretations do we know what the right things to do with these gadgets are.

Okay. Now there is one big topic here which is probably the most general, most powerful and most fascinating of all of these. And it is the one where I don’t yet quite know the “true” interpretation of, in the above sense:

Batalin-Vilkovisky formalism, also known as *BV-Quantization*.

Or, at least I didn’t. Until I heard Jae-Suk Park and Dmitry Roytenberg give talks on this.

Now I think I am beginning to see on the horizon that, possibly, BV-Quantization in fact is *secretly* precisely about what Isham called Quantization on a category, or rather on an $n$-category, and what I keep referring to as the program of studying the the charged quantum $n$-particle.

The key is once again to keep in mind the equivalence $\mathrm{qDGCA}s \stackrel{\sim}{\to} L_{\infty} \stackrel{\sim}{\to} \omega\mathrm{Lie}$ between (quasi-free) differential graded algebras, $L_{\infty}$-algebras and Lie $n$-algebras for arbitrary $n$, and to keep in mind that Lie $n$-algebras are the differential version of Lie $n$-group – or in fact that Lie $n$-algebroids are the differential version of Lie $n$-groupoids – and to use that in order to translate all that differential algebra (and BV quantization is graded differential algebra taken to the extreme) back into something of manifest intrinsic meaning.

Of course I am far from fully understanding BV-quantization, let alone that translation which i would like to perform. But that won’t stop me from talking about it.

Here I start, instead of beginning to comprehensively describe BV-formalism itself (but Jim Stasheff just writes in to tell me to read

Jim Stasheff
*The (secret?) homological algebra of the Batalin-Vilkovisky approach*

hep-th/9712157

), by pointing out that there are indications that people in BV-theory have *already*, without admitting it, realized Chern-Simons theory as a 3-functor. And it seems they have even found, implicitly, just that Lie 3-algebra which I am claiming is the right Lie 3-algebra of Chern-Simons theory.

My main source currently is the very insightful review

Dmitry Roytenberg
*AKSZ-BV formalism and Courant algebroid topological field theories*

hep-th/0608150

which is in based in particular on

M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky
*The Geometry of the Master Equation and Topological Quantum Field Theory*

hep-th/9502010 .

All this takes place in the world of “differential graded manifolds”. And we should admit that this is really nothing but defined to be the dual to the world of differential graded algebras.

(I cannot force you into thinking one way or another, but myself, I will think of dG manifolds as essentially being a way of talking about Lie $n$-algebroids, hence about Lie $n$-groupoids. At least “at the end of the day”, whenever that may be.)

**The general story of quantization – in dG language**

Now, as Dmitry Roytenberg discusses on p 5, we want to fix two duch differential graded manifolds. He calls them “source” and “target”. But I will, to emphasize the correspondence with the $n$-particle , instead call them

$\mathrm{par}$ for “parameter space”

and $\mathrm{tar}$ for “target space”.

Moreover, Dmitry Roytenberg writes $P$ for the space (the “internal hom”, actually) of maps from the first to the second. Following my $n$-particle conventions, I will instead call this
$\mathrm{conf} := \mathrm{hom}(\mathrm{par},\mathrm{tar})
\,,$
the *configuration space* of maps (“fields”!) from parameter space to target space.

We imagine $\mathrm{par}$ to be a model for the $n$-particle (a particle, a string, a membrane…), which propagates in target space $\mathrm{tar}$. From the point of view of $\mathrm{par}$ an object in $\mathrm{conf}$ is a “field configuration” on $\mathrm{par}$. From the point of view of $\mathrm{tar}$ an object of $\mathrm{conf}$ is a “position configuration” of $\mathrm{par}$ in $\mathrm{tar}$.

(By the magic of the $n$-functorial description of $n$-bundles with connection, target space may be rather $n$-orbifold-like, in which case a “field configuration” on $\mathrm{par}$ will be nothing but a choice of $n$-bundle with connection on $\mathrm{par}$. This will be important for understanding how Chern-Simons theory fits into this picture here: it is like the membrane propagating on a 3-orbifold. But I’ll get to that later.)

Whenever we have a situation as just described, we have in fact also this situation: $\array{ &&&& \mathrm{conf} \\ &&& \nearrow \\ \mathrm{tar} & \leftarrow & \mathrm{conf} \times \mathrm{par} & \\ &&& \searrow \\ &&&& \mathrm{par} } \,.$

On top of this kinematical setup, there is now a bit of *dynamics*. That’s some structure on $\mathrm{tar}$, possibly also some structure on $\mathrm{par}$ (but the inclined reader might remember that there are supposed to be arguments that when done *really* right, the latter isn’t actually extra structure).

In the context of the charged $n$-particle, that extra piece of dynamical data is an $n$-functor $\mathrm{tra} : \mathrm{tar} \to D$ encoding the “background field” (that’s a different kind of “field”, different from the objects of $\mathrm{conf}$. The space $\mathrm{conf}$ conbtains the fields of the theory, while $\mathrm{tra}$ will become a field of the second quantized theory. But that’s another story…)

Here, in the BV context, this extra dynamical structure is – not too surprisingly for the cognoscenti – a *symplectiv 2-form* (but in the dG context!)
$\Theta
\,.$

(In section 4 of his article Dmitry Roytenberg spells out how this symplectic 2-form does indeed encode background $n$-bundles with connection.)

Whichever way we think of the extra structure present on $\mathrm{tra}$, the star-shaped diagram above makes us want to pull-push that structure through to either $\mathrm{par}$ or to $\mathrm{conf}$. The story of that journey is the drama of the charged $n$-particle.

Dmitry Roytenberg discusses this on p. 9 of his article. He wants to get an *action functional* on the space of all field configurations, so he pulls back $\Theta$ to $\mathrm{conf} \times \mathrm{par}$ and then pushes forward to $\mathrm{conf}$ by integrating over $\mathrm{par}$.

There is a measure on $\mathrm{par}$ needed to do that: the push-forward is then simply the integration, with respect to that measure, over the fibers of $\mathrm{conf} \times \mathrm{par} \to \mathrm{par}$.

(And I cannot refrain from saing it once again: there are indications that when everything is done following the dao, there is actually a measure arising here canonically all by itself.)

**Focusing on a special case: Chern-Simons**

The joy of this formalism (I haven’t even mentioned most of the further ingredients that go into this. See Dmitry’s review!!) now is its generality. Choosing different dG manifolds $\mathrm{par}$ and $\mathrm{tar}$ and different dG-symplectic 2-forms $\Theta$, feeding these into the machinery and turning the crank, out drop lots of classical BV-action functionals which are held in high esteem.

Among them, the Chern-Simons functional.

And that’s an interesting example to have a closer look at.

Recall, Jim Stasheff and I are claiming that for any Chern-Simons theory in $(2n+1)$-dimensions, coming from a Lie algebra $g$ with transgressive invariant polynomial $k$ on it, there is a Lie $(2n+1)$-algebra $\mathrm{cs}_k(g)$ such that the Chern-Simons functional (at least for trivial $G$-bundles) is a $(2n+1)$-connection with values in this Lie $(2n+1)$-algebra.

Moreover, we are claiming that this Lie $(2n+1)$-algebra is actually *isomorphic* (not just equivalent, really ismorphic, that’s not unimportant here)
to the inner derivation Lie $(2n+1)$-algebra of the Baez-Crans-type Lie $2n-algebra$ which comes from the corresponding $(2n+1)$-cocycle:
$\mathrm{cs}_k(g) \simeq \mathrm{inn}( g_{\mu_k})
\,.$

To drive home this point: this means that in the context of $n$-functorial QFT, on the differential level, Chern-Simons theory should be that QFT obtained by setting $\mathrm{tar} = \mathrm{cs}_k(g) \simeq \mathrm{inn}(g_{\mu_k}) \,.$ (I am slightly oversimplifying now, to get the message across. Really we’d want to pass to a Lie $(2n+1)$-group integrating that Lie $(2n+1)$-algebra and we might want to account for the presence of nontrivial $G$-bundles and things like that. But at the moment this is just distraction and will be ignored.)

Okay, so let’s see what Dmitry Roytenberg, in his review of the work by M. Alexandrov, M. Kontsevich, A. Schwarz and O. Zaboronsky, says we should use for $\mathrm{tar}$:

look at his “example 4.1” and remember to interpret that using the prescription on the bottom of p. 10.

Then notice that the *Courant algebroid over a point* mentined in example 4.1 is precisely the Baez-Crans Lie 2-algebra
$g_{\mu = \langle \cdot, [\cdot, \cdot]\rangle}$
which we like to call the skeletal version of the String Lie 2-algebra. But think of it as the corresponding Koszul dual differential graded algebra, hence as a dG-manifold (whose even part is just a point).

Now go back to the bottom of p. 10, to see what the target space is supposed to be: it’s supposed to be essentially the *shifted cotangent bundle*
$\Pi T^* g_\mu$

of this dG-manifold.

Maybe open the AKSZ paper at p. 7 to learn more about this.

Then notice the following:

**Claim**. *As a free graded commutative algebra,
$\Pi T^* g_\mu$ is precisely that free garded commutative algebra underlying
$\mathrm{inn}(g_\mu)$*.

Proof: Just unwrap the definitions.

AKSZ are a little coy about defining the differential structure on $\Pi T^* g_\mu \,.$ They mention (p. 7 still) that there is one. And seem to assume that it is obvious. In fact, I think, too, that it is obvious. Hopefully we are agreeing with our feelings here, because I am thinking that the only obvious differential structure on $\Pi T^* g_\mu$ is precisely that on $\mathrm{inn}(g_\mu)$.

So it seems I am saying:

**Claim**. *When *translated* to the world of Lie $n$-algebras, the claim of AKSZ-Roytenberg is that the right target for Chern-Simons theory is the Lie 3-algebra
*
$\mathrm{inn}(g_\mu)
\,.$

Which coincides, up to isomorphism, with our statement about the 3-functorial realization of Chern-Simons.

**Disclaimer**

I am not claiming to fully understand all the BV details.While all what I said above looks quite right to me, I might be wrong somewhere. I will need to talk to a couple of people about this.

## Re: On BV Quantization. Part I.

I’ll need to refine my claim above, on how the AKSZ target space for Chern-Simons is $\mathrm{inn}(g_{\mu})^*$. Now that I had a more careful look at what Roytenberg writes, I realize that it is a little more indirect than that: the information encoded in $g_{\mu}$ is in this formalism in some way split into a “kinetic” and an “interaction” part. More on that later.