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.

With Domenico's clear explanation, I can actually write down more or less explicitly the DGLA describing the deformations of Einstein metrics.

First, some notation. Let $\bar{g}_{ab}$ denote a given (background) Einstein metric, with corresponding Levi-Civita connection $\bar{\nabla}_a$, Riemann tensor $\bar{R}_{abc}{}^d$, and Ricci tensor $\bar{R}_{ac} = \bar{R}_{abc}{}^b$. It is given that $\bar{\nabla}_a \bar{g}_{bc} = 0$ and $\bar{R}_{ac} = k \bar{g}_{ac}$ for a fixed constant $k$.

Let $\nabla_a$ denote an arbitrary symmetric (torsion free) affine connection, which differs from the background Levi-Civita connection as $\nabla_a v^b = \bar{\nabla}_a v^b + C^b_{ac} v^c$. Let me call $C^b_{ac} = C^b_{(ac)}$ the corresponding Christoffel tensor (or connection coefficients). The equation for an Einstein metric $g_{ab} = \bar{g}_{ab} + h_{ab}$ (with the same constant $k$) can be written in the form \begin{align*} \nabla_a g_{bc} &= \bar{\nabla}_a h_{bc} - C^d_{ab} \bar{g}_{dc} - C^d_{ac} \bar{g}_{bd} - C^d_{ab} h_{dc} - C^d_{ac} h_{bd} , \\ R_{ac}[C] - k h_{ac} &= -kh_{ac} - \bar{\nabla}_a C^b_{bc} + \bar{\nabla}_b C^b_{ac} + C^b_{ac} C^d_{db} - C^b_{ad} C^d_{cb} , \end{align*} where $R_{ac}[C] = R_{abc}{}^b[C]$ is the usual Ricci contraction of the curvature tensor $R_{abc}{}^d[C]$ of $\nabla_a$. The first of the above equations is the $g$-compatibility condition for $\nabla_a$. Solving it for the Christoffel tensors identifies $\nabla_a$ with the Levi-Civita connection of $g_{ab}$ and plugging that solution into the second equation gives the equation $R_{abc}[g] - k g_{ac} = 0$ for an Einstein metric, due to the identity $R_{ab}[g] = \bar{R}_{ab} + R_{ab}[C]$.

Now the equations are precisely in the form that Domenico described, $f[h,C] + Q[h,C;h,C] = 0$, with $f$ linear and $Q$ quadratic \begin{align*} \begin{pmatrix} f_{abc}[h,C] \\ f_{ac}[h,C] \end{pmatrix} &= \begin{pmatrix} \bar{\nabla}_a h_{bc} - C^d_{ab} \bar{g}_{dc} - C^d_{ac} \bar{g}_{bd} \\ -kh_{ac} - \bar{\nabla}_a C^b_{bc} + \bar{\nabla}_b C^b_{ac} \end{pmatrix} \\ \begin{pmatrix} Q_{abc}[h,C;h',C'] \\ Q_{ac}[h,C;h',C'] \end{pmatrix} &= \frac{1}{2} \begin{pmatrix} - C^d_{ab} h'_{dc} - C^d_{ac} h'_{bd} - C'^d_{ab} h_{dc} - C'^d_{ac} h_{bd} \\ + C^b_{ac} C'^d_{db} - C^b_{ad} C'^d_{cb} + C'^b_{ac} C^d_{db} - C'^b_{ad} C^d_{cb} \end{pmatrix} \end{align*} It remains to describe how infinitesimal symmetries (diffeomorphisms) act on the $h_{ab}$ and $C^b_{ac}$ tensor fields, as well as on $f$ and $Q$. They are generated by vector fields $u^a$. They act on tensors via the usual Lie derivative, which we find convenient to express via $\bar{\nabla}_a$, so that $\mathcal{L}_u X^a = u^b \bar{\nabla}_b X^a - X^b \bar{\nabla}_b u^a$ and $\mathcal{L}_u Y_a = u^b \bar{\nabla}_b Y_a + Y_b \bar{\nabla}_a u^b$, and they act on each other via the usual Lie bracket $[u,u'] = \mathcal{L}_u u' = -\mathcal{L}_{u'} u$. But special attention must be paid to the identities \begin{align*} \mathcal{L}_u \bar{\nabla}_b X^a - \bar{\nabla}_b \mathcal{L}_u X^a & = u^c \bar{\nabla}_c \bar{\nabla}_b X^a - (\bar{\nabla}_b X^c) \bar{\nabla}_c u^a + (\bar{\nabla}_c X^a) \bar{\nabla}_b u^c \\ & \quad {} -\bar{\nabla}_b (u^c \bar{\nabla}_c X^a - X^c \bar{\nabla}_c u^a) \\ &= u^c \bar{R}_{bcd}{}^{a} X^d + X^c \bar{\nabla}_b \bar{\nabla}_c u^a = (\bar{\nabla}_{(b} \bar{\nabla}_{c)} u^a - u^d \bar{R}_{d(bc)}{}^a) X^c , \\ % &= u^c \bar{R}_{bcd}{}^{a} X^d - X^c \bar{R}_{bcd}{}^a u^d + X^c \bar{\nabla}_c \bar{\nabla}_b u^a \\ % &= X^c \bar{R}_{cdb}{}^a u^d + X^c \bar{\nabla}_c \bar{\nabla}_b u^a , \\ \mathcal{L}_u \bar{\nabla}_b Y_a - \bar{\nabla}_b \mathcal{L}_u Y_a & = u^c \bar{\nabla}_c \bar{\nabla}_b Y_a + (\bar{\nabla}_b Y_c) \bar{\nabla}_a u^c + (\bar{\nabla}_c Y_a) \bar{\nabla}_b u^c \\ & \quad {} -\bar{\nabla}_b (u^c \bar{\nabla}_c Y_a + Y_c \bar{\nabla}_a u^c) \\ &= -u^c \bar{R}_{bca}{}^{d} Y_d - Y_c \bar{\nabla}_b \bar{\nabla}_a u^c = (\bar{\nabla}_{(b} \bar{\nabla}_{a)} u^c - u^d \bar{R}_{d(ba)}{}^c) Y_c , \end{align*} which identify the action of $[\mathcal{L}_u, \bar{\nabla}_b]$ as a derivation on the algebra of tensors. This means that infinitesimal diffeomorphisms generate the infinitesimal transformation $(h,C) \mapsto (h,C) + \epsilon K[u;h,C] + O(\epsilon^2)$, where \begin{equation*} \begin{pmatrix} K_{ab}[u;h,C] \\ K_{ab}^c[u;h,C] \end{pmatrix} = \begin{pmatrix} \bar{g}_{ac} \bar{\nabla}_b u^c + \bar{g}_{cb} \bar{\nabla}_a u^c \\ \bar{\nabla}_{(a} \bar{\nabla}_{b)} u^c - u^d \bar{R}_{d(ab)}{}^c \end{pmatrix} . \end{equation*}

Finally, we can put these formulas together in the definition of a DLGA $(L,d,[-,-])$. $L$ itself will break down into a sum of sections of certain tensor bundles. For simplicity, I will write $T$ to denote the space of sections of the bundle of vectors, $S^2T^*$ for symmetric covariant 2-tensors, etc. The breakdown by degree is \begin{gather*} \begin{array}{c|ccccc} & 0 && 1 && 2 \\ \hline L & T &\to& S^2 T^*\oplus S^2T^*\otimes T &\to& S^2T^* \otimes T^* \oplus S^2 T^* \\ d & & K & & f & \end{array} , \\ \begin{array}{c|ccc} [-,-] & 0 & 1 & 2 \\ \hline 0 & [u,u'] & K[u;h',C'] & \mathcal{L}_u \\ 1 & -K[u';h,C] & 2Q[h,C;h',C'] & 0 \\ 2 & -\mathcal{L}_{u'} & 0 & 0 \end{array} \end{gather*} I believe that this DGLA could be extended by one more degree to take the Bianchi identities into account. But I will stop here.

There are of course other ways to present the same DGLA and one can find explicit attempts in the literature of writing it down. Here's one that uses a somewhat different presentation:

Michael Reiterer, Eugene Trubowitz The graded Lie algebra of general relativity arXiv:1412.5561

I haven't thought about Einstein metrics but, as AHusain mentioned, Kevin Costello has written down many examples. The keyword is elliptic moduli problem. Look at https://arxiv.org/abs/1111.4234