1. What is the moduli space of a QFT?

  2. What does it mean exactly that there are different inequivalent vacua?

  3. Can someone give a precise definition of the moduli space, and some easy examples?

  4. And why is it so important and studied nowadays?


I think it would be useful to migrate questions 1,3 to math.SE; as for the mathematical side of the story, I think that all gravitates around the idea of moduli space (a modulus being -as far as I know- the old name for "the set of values a certain variable takes as soon as it runs over a certain set (often a continuum)").

The idea of a "moduli problem" is central to modern Algebraic Geometry, and it is deeply intertwined with Physics: I am not an expert on the second field, but I can give a (fairly heuristic) idea of the first. You will realize in a while that the topic is really huge; all that follows is not intended to strive for rigor.

The rough idea is fairly simple, indeed. As a starting example consider the generic conic in the projective plane $\mathbb P^2(k)$; as you may know, such a geometric locus is the zero-set of a polynomial which depends on a certain set of coefficients: more precisely, you have to consider the polynomial $$ a X_0^2 + 2 b X_0 X_1 + 2 c X_0 X_2 + d X_1^2 + 2 e X_1X_2 + f X_2^2=0 $$ where $(a,b,c,d,e,f)$ can on its own right be identified to a point of the projective space $\mathbb P^5(k)$. And non-degenerate conics, i.e. those conics whose matrix has nonzero determinant, can be identified with an hypersurface in $\mathbb P^5(k)$.

This identification is the key-step in understanding the definition of moduli space: a set of geometric objects can often be regarded as a geometric object of the same kind.

Now that you have your moduli-glasses on, you will notice that lots of things are indeed moduli spaces: the set of nondegenerate conics in the projective plane happens to be a cubic hypersurface; the set of iso classes of vector bundles over a given manifold can be thought as a space on its own right; and in the same vein, for example, there are cases when the set of degree-zero complex line bundles over a smooth projective curve is not only a group (the Picard group $Pic^0(\mathcal C)$ of the curve $\cal C$) but also a complex torus (this is an old theorem due to Jacobi).

The formalism of moduli problems offers you a way to express this in full generality: to formulate a "moduli problem" you need the following gadgets.

  1. A class of geometric objects (if this is not painful, think of it as a category $\cal A$ of spaces -manifolds, bundles, Lie groups,...-)
  2. A rule which assigns to every space $S$ an $S$-parametric family of objects, i.e. a bundle of $\cal A$-objects. In other words, you want to give a function $\rho\colon s\in S\mapsto F_s\in \cal A$: in the case where $\cal A=$manifolds, any space $X$ gives rise to such a structure, taking as $\rho$ a rule which assigns to every $s\in S$ a bundle $F_s\to X$ over $X$.
  3. The assignment $\rho=\rho_S$ has to be functorial and contravariant in $S$, namely any transformation $S\to T$ must induce a morphism between the bundle $F'$ of $T$-parametric objects and the bundle $F$ of $S$-parametric object; again in the case of bundles over a manifold, this is given by pull-back operation.

To package all this stuff in a single, nifty request: we would like to have a contravariant functor ${\cal A}\to Set$ which sends an object $S$ to the set of all $S$-parametric objects $F_*=\{F_s\}_s$ of $\cal A$: understanding this functor is your moduli problem.

This is the place where a key step for the solution of your moduli problem comes into play: denote this functor as $\Phi$ and suppose you can find a space $M\in\cal A$ such that $$\Phi(S)\cong {\rm Map}(S,M);$$ this request is not so absurd, since for example you can impose $M=Ob(\cal A)$ endowed with a suitable topology, and define a map of sets $F_*\mapsto (s\mapsto F_s)$. This map is natural in $S$, so that if $M$ exists, then it is unique up to a unique isomorphism in $\cal A$. In a more categorical jargon, the functor $\Phi$ is represented by the space $M$, which is called the (fine) moduli space of your moduli problem.

The problem is that moduli problems are seldom solvable (if all of them were, Algebraic Geometry would be far more easy; it's up to you to decide if this is a good thing)! In most cases of interest no moduli space exists, in the fine sense. To workaround this problem, one needs to define a coarse notion of solution for a moduli problem, but this would turn out to be too technical, and I think I've been far too verbose for a single response.

Instead I would try to sum up: often in geometry you have to cope with families of spaces which can be turned into another space, called the moduli space of the family: each point in the moduli space corresponds to a space on its own right. The geometry of the moduli space tells you things about the geometry of the spaces/points in it. This is precisely what I read when I see

the space of vacua for the quantum field theory is a manifold (or orbifold), usually called the vacuum manifold. This manifold is often called the moduli space of vacua, or just the moduli space, for short.


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