As proposed by Quillen, Drinfeld, and Deligne and other important mathematicians, there is supposed to be a philosophy that, at least over a field of characteristic zero, assigns to every "deformation problem" a differential graded Lie algebra or $L\_{\infty}$-algebra that controls it. 


I've seen this idea realized in various situations, like for example the deformation theory of a compact complex manifold, which is "controlled" by the Kodaira-Spencer DGLA, or the deformation theory of Dirac structures in exact Courant algebroids. However, from my naive point of view as an outsider, I see this set of techniques completely disconnected from a different type of moduli problems of more "analytic" character. I refer for example to the moduli of Einstein metrics or the moduli of instantons in Donaldson's theory. It looks like the DGLA-philosophy has played virtually no role in the study of the moduli spaces of Einstein metrics and instantons. Why is this so? Does the DGLA-principle still applies to these problems but it does not add anything interesting? What is the DGLA controlling the deformation theory of these problems? It looks like the more analysis requires a moduli space problem, the less of a relevant role is played by the DGLA-philosophy, which seems to be more "algebraically inclined". I wonder because sometimes it looks like the moduli-literature is very polarized in different communities using different techniques to study moduli problems, so I would like to know to which extent the methods of one community apply to the problem considered by a different community.


Thanks.



DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons
The Quillen-Drinfeld-Deligne-etc. philosopy should not be looked at as something too mysterious.


Namely, it reduces to the fact that if the set of objects one is interesting in the infinitesimal deformations of is not too wild, then it can be described in the form $f(v)+Q(v)=0$, where $f:V\to W$ is a linear function and $Q:V \to W$ is a quadratic function. Let me make a very simple example of how this works: assume we have a degree four polynomial $p(x)$ and assume $x=a$ is a zero of $p$. We wanto to describe the other zeroes. To do so, we consider the polynomial $q(x)=p(x+a)$. this will be a fourth degree polynomial vanishing at $x=0$so it will have the form, say, $q(x)=x^4+2x^3-x^2+7x$. We are now interested in the zeroes of $x$. We can add a new variable $y$ and set $x^2=y$. Now the equation $q(x)=0$ has become the two degree two equations $y^2+2xy-x^2+7x=0$ and $x^2-y=0$. So it is of the form $f(v)+Q(v)=0$ with $v=(x,y)$, $f(x,y)=(7x,-y)$ and $Q(x,y)=(y^2+2xy-x^2,x^2)$.


Once the set we want to describe is written in the form $f(v)+Q(v)=0$ we are done. The linear map $f\colon V\to W$ can be seen as a degree 1 map from $V[-1]$ to $W[-2]$, where $V[-1]$ is the same vector space as $V$ but now with all of its elements seen as if they were in degree 1, and $W[-2]$ is the same vector space as $W$ but with all of its elements seen as if they were in degree 2. Also, the quadratic map $Q: Sym^2(V) \to W$ can be seen as a degree zero map $V[-1]\wedge V[-1] \to W[-2]$. It is then immediate to see that setting $\mathfrak{g}=V[-1]\oplus W[-2]$, the graded vector space $\mathfrak{g}$ is a differential graded Lie algebra with differential defined by $f$ and bracket defined by $2Q$ (here we use the fact that we are in characteristic zero or at least not in characteristic 2). The set of deformations we are intersted in is then described by the equation $dx+\frac{1}{2}[x,x]=0$ for degree 1 elements in $\mathfrak{g}$, i.e., by the solutions of the Maurer-Cartan equation for $\mathfrak{g}$. 


Moreover, if there is a Lie algebra $\mathfrak{h}$ of symmetries for our set, we can include $\mathfrak{h}$ in our differential graded Lie algebra by setting $\mathfrak{g}^0=\mathfrak{h}$ and extending the definition of the differential and of the bracket so to encode both the Lie algebra structure on $\mathfrak{h}$ and its Lie action on $V$. Doing this one sees that our infinitesimal deformations modulo the action of $\exp(\mathfrak{h})$ are precisely the Maurer-Cartan elements of $\mathfrak{g}$ modulo the gauge action of $\exp(\mathfrak{g}^0)$.


All this to say that describing every (regular enough) deformation problem by means of a DGLA is something that should not be seen as something particularly deep. What is particular is the fact that sometimes the DGLA governing the problem is naturally attached to the geometry of the problem, e.g., for (almost) complex structures the equation defining them is $J^2=0$, so it is quadratic on the nose, and being quadratic is not an artifact. 


For Einstein metrics or instantons I have not thought of which the DGLA description would be. It could indeed be interesting to figure out. Yet, as I tried to explain above, if such an approach is not taken in the literature, the reason it is not because it in principle cannot be taken, but if the DGLA governing these two problems is to be build artificially then DGLA point of view adds very little (if anything) with respect to a direct approach to the problem. 


