This is a sub-entry of Gromov-Witten invariants. See there for further background and context.
This entry is supposed to provide an exposition of some basic ideas underlying Gromov-Witten theory.
raw material: notes taken verbatim in some seminar – needs to be polished
question: what is a moduli problem?
some kind of object;
a notion of “familiy”/“deformation” of these objects
some equivalence relation $\sim$ (possibly trivial) on the set of such families $S(B)$ over $B$
from this we get a functor (a presheaf)
terminology an object $M$ is called a fine moduli space if the corresponding represented functor
is isomorphic to $F$, i.e. if it represents $F$.
examples in the homotopy category of topological spaces we have
so $\mathbb{C}P^\infty$ is a classifying space for complex line bundle. Similarly for higher rank vector bundles and Grassmannians.
The analogue in the algebraic category is
And also similarly for higher rank vector bundles and Grassmannians.
Despite these examples, in a lot of cases the functors are not representable. We’ll see some of these examples below.
Why are fine moduli spaces desireable?
They allow us to study a single family which tells us universal things about all families. Even if you do not care about families or deformations, moduli spaces can help, because perhaps they can tell you something about trivial families, i.e. the objects that you are studying to begin with.
example studying the cohomology rings of $Gr_n(\mathbb{R}^\infty)$ or $Gr_n(\mathbb{C}^\infty)$, which are the classifying space for higher rank real and complex vector bundles gives universal relations (or, rather, the lack thereof!) among Chern classes, etc.
Let’s look at elliptic curves (we’ll work over $\mathbb{C}$).
the functor of families here is
Fact: there is no fine moduli space of elliptic curves representing this functor
why not?
there is something called the j-invariant which classifies elliptic curves up to isomorphism
let
be an [elliptic curve]] given by parameters $a,b$. Then its j-invariant is the number
so we might guess that the “$j$-line” $\mathbb{A}^1$ is a fine moduli space for elliptic curves,
i.e. that there is a “universal family” of elliptic curves $C \to \mathbb{A}^1$ such that the fiber over $j \in \mathbb{A}^1$ is the elliptic cuvre with that $j$-invariant.
so that for any family $E \to B$ we’d have a pullback
where $B \to \mathbb{A}^1$ sends a point in $B$ to the j-invariant of the elliptic curve in the fiber over $B$.
Now consider the family
now send $\mathbb{A}^1_\lambda \to \mathbb{A}^1_{\lambda}$
by dually sending $\lambda \mapsto 1-\lambda$
check that $j(\chi_\lambda) = j(\chi_{1-\lambda})$
this induces in the fibers a map
$\phi : \chi \to \chi$
this maps further to $(x,-y,\lambda)$. But this would have to be the identity for $\mathbb{A}^1$ to be a fine moduli space, which it is not. So $\mathbb{A}^1$ at least is not a fine moduli space, even though it might look like one.
so that gives some computational insight that something goes wrong
This argument does not yet prove that there exists no moduli space of elliptic curves. It merely proves that the “j-line” $\mathbb{A}^1$ can not be the moduli space of elliptic curves. However, the basic argument can be adapted, if one so desires, to in fact prove that there is no moduli space of elliptic curves.
In fact, below we will see (in an exercise), that the $j$-line is a coarse moduli space, as explained below. Moreover, if a coarse moduli space exists, then it is unique up to canonical isomorphism. Since fine moduli spaces, if they exist, are also coarse moduli spaces, and the $j$-line is a coarse but not a fine moduli space by the above argument, it follows that no fine moduli space exists.
Abstract argument Yet another, more abstract way, to see that no fine moduli space can exist it to realize that since elliptic curves have nontivial automorphisms, it is possible to construct families of elliptic curves that are locally trivial families (of the form $U \times E \to U$ for a fixed elliptic curve $U$) but which are glued together from these local pieces using nontrivial automorphisms such that the resulting family $V \to X$ is not globally trivial, i.e. not globally of the form $E \to X \to X$.
With a bit of care this alone can be used to show that a fine moduli space cannot exist. This is often summarized by a slogan of the form
Slogan : nontrivial automorphisms of objects prevent the family-assignment of these objects to be representable by a fine moduli space.
However, one has to be careful with interpreting this slogan correctly. Taken naively, the slogan alone would also seem to imply that, since vector spaces have nontrivial automorphisms, no classifying space for “families of vector spaces”, i.e. for vector bundles does exist, while of course this does exist (recall also the example further above). But if one interprets the slogan carefully, it does yield a true statement. For more on that see the discusson at moduli space.
How to “fix” these problems.
add extra structure to the objects under consideration (e.g. add marked points) to make the automorphism groups trivial.
instead of looking for representing topological spaces, look for representing groupoids / stacks.
use coarse moduli spaces $M \in Sch/\mathbb{C}$ with $\Psi_M : F \to h_M$ such that
a) $F(Spec(\mathbb{C})) \to h_M(Spec \mathbb{C}) = hom(Spec \mathbb{C}, M)$ is a bijection
b) given $M'$ and $\Psi_{M'} : F \to h_{M'}$ then there exists unique $M \to M'$ such that $\array{ F && \stackrel{\Psi_{M'}}{\to}&& h_{M'} \\ & {}_{\Psi_M}\searrow && \nearrow \\ && h_M}$
So a coarse moduli space is one that at least has the right underlying set of points as the right moduli stack has: as long as we don’t look at families but just at single things, it does give the right classification.
exercise show that the $j$-line $\mathbb{A}^1$ is, while not a fine moduli space, a coarse moduli space.
exercise Show that if a coarse moduli space exists, then it is unique up to canonical isomorphism.
fact there exists a coarse moduli space $M_{g,n}$ of Riemann surfaces of genus $g$ with $n$ marked points and a fine moduli stack? $M_{g,n}$ such that for all $g,n$ we have
$M_{g,n}$ is a smooth Deligne-Mumford stack - aka orbifold
except for $(g,n) =$ $(0,0), (0,1), (0,2), (1,0)$ (these are the cases where the automorphism group is infinite, so in these cases we don’t get a Deligne-Mumford stack)
“Issues”:
$M_{g,n}$ is not proper, meaning: not compact, so we don’t have, for example, Poincare duality on $M_{g,n}$ and no integration theory. Proper, or even better projective, schemes or stacks are just a lot easier to deal with.
Sometimes one wants to study singular curves or families with degeneracies.
Both “issues” can be “resolved” via Deligne-Mumford compactification.
which parameterizes at-most-nodal curves, that are connected, of arithmetic genus $g$, with $n$ smooth marked points, and the group of automorphisms is finite.
$\bar M_{g,n}$ is a smooth proper Deligne-Mumford stack.
smooth here means smoothness as for orbifolds.
Deligne and Mumford were able to prove many theorems about the ordinary moduli space of curves by studying instead the compactification. For example they were able to prove that $M_{g,n}$ is irreducible.
Gromov was looking for invariants of symplectic manifolds: his idea was to use $J$-holomorphic curves in compact symplectic manifolds to get symplectic invariants
Edward Witten and other physicists studied worldsheets of strings in some spacetime manifold (e.g a Calabi-Yau 3-fold)
we want to consider now genus $g$ Riemann surfaces with $n$ marked points mapping into some space $X$
For a fixed
is the space that parameterizes maps
where $\Sigma$ is smooth, and has $n$ marked points, and such that for $[\Sigma]$ the fundamental homology class? of $\Sigma$ we have
this is a smooth Deligne-Mumford stack. (Again we must exclude the cases of small $(g,n)$. Excluding these cases, the automorphisms of the surfaces and the automorphisms of the maps are automatically finite.)
similarly, write $\bar M_{g,n}(X,\beta)$ for the same setup but with $\Sigma$ from $\bar M_{g,n}$ as above in part 1 (the DM compactified moduli stack). EXCEPT here we do not require that $\Sigma$ (with its $n$ marked points) has finite automorphism group; we require instead that the MAP has finite automorphism group, which means…….(fill in)
This is a proper (compact) Deligne-Mumford stack - but not smooth (not even in the sense of smooth stacks!). This is very important. This is what makes the theory difficult/nontrivial/interesting.
we have
what do string theorists want to do?
we have evaluation maps
where $i$ labels a marked point, and we want morphisms
There should be a virtual fundamental class $[\bar M_{g,n}(x,\beta)]^{virtual}$ that makes the maps above into the correlation functions of a quantum field theory (the integral would be the path integral of the worldsheet sigma-model) with state space $H^*(X)$. We’ll explain some of what this means below.
This virtual fundamental class in algebraic geometry was constructed by Behrend-Fantechi; in symplectic geometry it was done by Li-Tian.
Why do we want to use a “virtual” fundamental class? Because \bar M_{g,n}(x,\beta) is not smooth, the actual fundamental class may not behave very well. NEEDS ELUCIDATION – it would be nice if someone could give a more complete explanation of why we need a virtual fundamental class
the mathematical structure of GW-theory was elucidated Ruan and then by Maxim Kontsevich and Manin in 1994.
where the map $\phi_*$ is some kind of “pushforward” or “integration along the fibers”; this map uses the virtual fundamental class
there is also a map
with $g = g_1 + g_2$ and $n = n_1 + n_2$ obtained by gluing the last marked point of the first curve to the first marked point of the second curve.
so consider now the combination of these two maps
where the bottom right iso uses the Künneth formula
here $\Delta_a$ is a homogeneous basis of $H^\bullet(X)$
where
so this diagram above says that this satisfies the sewing laws that defines a quantum field theory.
There are various other axioms that Gromov-Witten theory must satisfy, but the sewing law above is the most important. The hard part of all of this is constructing the virtual fundamental class, and then proving that this class indeed makes the sewing law and the other axioms (which I have omitted) satisfied.
The above discussion does not yet reveal much of the rich structure of Gromov-Witten invariants. But GW invariants indeed have a very rich and beautiful mathematical structure. Indeed, if we look at just the easiest part of the theory, namely g=0, we are lead to quantum cohomology and Frobenius manifolds.
There is also the mirror symmetry conjecture, which roughly posits that the GW invariants can be found via calculations that a priori seem completely unrelated.
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