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.


  • $\begingroup$ Does the "moduli space of Einstein metrics" come from a moduli problem in the sense of algebraic geometry (notions of family, pullback and isomorphism of families, which give a fibered category or a stack over a suitable base site)? $\endgroup$
    – Qfwfq
    Mar 7 '17 at 22:03
  • $\begingroup$ See Costello's perspective on supergravity. I don't know how much still holds up without the extra super. $\endgroup$
    – AHusain
    Mar 7 '17 at 22:18
  • $\begingroup$ @Qfwfq I don't know about that, but if the "philosophy" that I mention in the question is valid only for "moduli problems in the sense of algebraic geometry" then it is not true that it can be applied to every moduli or deformation problem. Most deformation or moduli problems of interest do not belong to algebraic geometry, which studies a rather particular class of these models. $\endgroup$
    – Bilateral
    Mar 8 '17 at 18:06

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.

  • $\begingroup$ Thanks a lot for the explanation, very nice. I have a question though. As you know, what you have described is the standard "three-term" deformation complex, consisting on infinitesimal symmetries, first order deformations and obstructions. However, I thought that the DGLA of a deformation problem was an "extension" of the standard "three-term" deformation complex. What is to be gained by obtaining the extended DGLA? Does it contain more info than the standard three-term complex? $\endgroup$
    – Bilateral
    Mar 11 '17 at 17:28
  • 2
    $\begingroup$ Absolutely. The point here is that when one looks at deformations modulo gauge one is actually looking at the set $\pi_0MC$ of connected components of a space $MC$: the Maurer-Cartan equation cuts a subspace $MC$ of $\mathfrak{g}^1$, and $\mathfrak{g}^0$ knows about paths in this space. But we could go a step deeper: $\mathfrak{g}^{-1}$ knows about paths between paths, i.e., homotopies between paths (these are usually called gauge of gauge transformations in the Physics literature), and so on. $\endgroup$ Mar 11 '17 at 18:27

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

  • 1
    $\begingroup$ Thanks for the reference. It is indeed very relevant to the question that I posed, although already in the first page, the following: "The equations of motion of a classical field theory are a system of elliptic differential equations..." is not correct. $\endgroup$
    – Bilateral
    Mar 8 '17 at 21:28
  • $\begingroup$ Do you want to stay in Lorentzian signature? No rotation allowed? $\endgroup$
    – AHusain
    Mar 8 '17 at 22:18
  • $\begingroup$ @Bilateral am not aware of any work in Lorentzian signature. $\endgroup$ Mar 10 '17 at 19:47
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    $\begingroup$ @JustinHilburn: Even in Riemannian signature the equations of motion of classical field theories of interest are not always elliptic. That was a minor comment anyway, I am not particularly interested in Lorentzian signature. The reference is indeed relevant to my question, as it partially address it. However, I still have the feeling that this "general philosophy" applies to a restricted set of moduli problems (perhaps to all of interest in algebraic geometry), leaving aside many many moduli problems of interest in other areas of mathematics and mathematical physics. $\endgroup$
    – Bilateral
    Mar 10 '17 at 21:49

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