### BV-Formalism, Part IV

#### Posted by Urs Schreiber

Here are some further thoughts on BV-formalism and its interpretation in terms of groupoids. My main point here is to promote the slogan

What is called the“space of fields and ghosts”in BRST/BV formalism is nothing but the Lie algebroid of theaction groupoid of the gauge group acting on the space of fields.

This is based on

a) the claim, which I make hereby (and which should to be well known in one incarnation or other), that the Lie algebroid of the action Lie groupoid of a group $G$ acting on a space $X$ is Koszul dual to the Chevalley-Eilenberg cochain complex for Lie algebra cohomology of $g = \mathrm{Lie}(G)$ with values in the module $C^\infty(X)$.

b) the observation, nicely emphasized in

Kevin Costello
*Renormalization and the Batalin-Vikovisky Formalism*

0706.1533

(p. 16) that with $V$ the space of fields of some theory and with $G$ the gauge group acting on it, “the space of fields and ghosts” of the corresponding BRST construction is itself dual to that Lie algebra cochain complex.

This Koszul-duality between Lie $n$-algebras and differential graded algebras I have been talking about a lot here. Useful references might be Zoo of Lie $n$-algebras, Lie $n$-algebra cohomology and String- and Chern-Simons $n$-Transport.

Here we need the generalization of this to the many-object version of Lie $n$-algebras, namely Lie $n$-algebroids. One way to conceive these dually is as dg-manifolds. Another way is, I think, as differential algebras with values in nontrivial modules.

More precisely:

**Claim** Let $G$ be a Lie group acting on a manifold $X$, such that $g = \mathrm{Lie}(G)$ acts on the algebra of functions $C^\infty(X)$ by algebra derivations. Then the Lie algebroid of the action groupoid
$X // G$
is dually given by the graded commutative differential algebra generated by $C^\infty(X)$ in degree 0 and by $g^*$ in degree 1, with the differential on $g^*$ being the Chevalley-Eilenberg differential
$d \omega(x,y) = - \omega([x,y])$
and with the differential on $C^\infty(X)$ given by the action of $g$ on $C^\infty(X)$.

**Example.** Not quite an action groupoid but the example to keep in mind:

For $\Pi_1(X)$ the fundamental groupoid of $X$, the corresponding Lie algebroid is the tangent algebroid $T X$ whose Koszul dual is simply the differential algebra of differential forms on $X$. This we can think of as being the cochain complex for computing the Lie algebra cohomology of the Lie algebra $diff(X)_0 := \Gamma(T X)$ with values in the module $C^\infty(X) \,.$ Then the formula for the deRham differential $d \omega(x,y) = \omega([x,y]) + x(\omega(y)) - y(\omega(x))$ is a special case of the differential described in the claim above.

Somehow all what BV formalism is about should be something like making sense of colimits over functors over infinite dimensional action Lie groupoids.

For consider $X$ to be the space of fields (configuration space) of some physical system with gauge group $G$ acting on $X$ $X \times G \to X \,.$

We know (see Canonical Measures on Configuration spaces) that we should be looking at the *configuration category* rather, whose objects are the configurations and whose morphism are processes between configurations. Here we just consider processes which describe gauge transformations and hence address the action groupoid

$\mathrm{conf} = X // G$

as the “configuration category”.

More generally, there might not be a global gauge group acting and we’d take $\mathrm{conf}$ to be an arbitrary (Lie, maybe) groupoid.

Supposing that we have a gauge invariant “action functional” $e^S$ – some functor on $\mathrm{conf}$ which is constant on isomorphism classes, we want to understand its colimit or the like

$\int_{\mathrm{conf}} e^S$

over this configuration category.

When everything is finite we know how this works: the Leinster measure on finite categories takes control, which for the groupoid case is Baez-Dolan groupoid cardinality – and which indeed is the right measure for Dijkgraaf-Witten theory, the finite group version of Chern-Simons.

When things are not finite, like for Chern-Simons theory, something has to be done about regularizing the Leinster measure, as described in The return of the Euler characteristic of a category. When things are Lie, it seems like BV-formalism is what takes care of this.

At least you can see, with the above claim in mind, that this is how Kevin Costello describes the situation on his p. 16.

## Re: BV-Formalism, Part IV

This sounds very beautiful!

The light is starting to dawn on me.

I guess I have the following confusion about all this differential graded algebra, Koszul duality stuff : it all looks very

algebraic, but I want to dogeometry!If the “space of fields and ghosts” is just the Lie algebroid of the action of the gauge group on the fields (which I hope it is)… then howcome people do all this supersymmetry dg-algebra cochain complex homology/cohomology stuff?

Why can’t they just talk about curves and tangent vectors and derivatives and flat connections and things?

Is one point of view more powerful than the other?