In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable map was proposed by Maxim Kontsevich around 1992 and published in .
Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself.
Fix a closed symplectic manifold
X
\omega
g
n
A
X
((C,j),f,(x1,\ldots,xn))
where
(C,j)
g
n
x1,\ldots,xn
f:C\toX
is a function satisfying, for some choice of
\omega
J
\nu
\bar\partialj,f:=
1 | |
2 |
(df+J\circdf\circj)=\nu.
Typically one admits only those
g
n
2-2g-n
C
C
The operator
\bar\partialj,
\omega
J
\nu
(j,J,\nu)
g
n
A
J,\nu | |
M | |
g,n |
(X,A)
of dimension given by the Atiyah-Singer index theorem,
d:=\dimRMg,(X,A)=2
X(A) | |
c | |
1 |
+(\dimRX-6)(1-g)+2n.
J,\nu | |
M | |
g,n |
(X,A)
f
One can capture the energy by rescaling the map around the concentration point. The effect is to attach a sphere, called a bubble, to the original domain at the concentration point and to extend the map across the sphere. The rescaled map may still have energy concentrating at one or more points, so one must rescale iteratively, eventually attaching an entire bubble tree onto the original domain, with the map well-behaved on each smooth component of the new domain.
Define a stable map to be a pseudoholomorphic map from a Riemann surface with at worst nodal singularities, such that there are only finitely many automorphisms of the map.
Concretely, this means the following. A smooth component of a nodal Riemann surface is said to be stable if there are at most finitely many automorphisms preserving its marked and nodal points. Then a stable map is a pseudoholomorphic map with at least one stable domain component, such that for each of the other domain components
It is significant that the domain of a stable map need not be a stable curve. However, one can contract its unstable components (iteratively) to produce a stable curve, called the stabilization
st(C)
C
The set of all stable maps from Riemann surfaces of genus
g
n
J,\nu | |
\overline{M} | |
g,n |
(X,A).
The topology is defined by declaring that a sequence of stable maps converges if and only if
\overline{M}g,
The moduli space of stable maps is compact; that is, any sequence of stable maps converges to a stable map. To show this, one iteratively rescales the sequence of maps. At each iteration there is a new limit domain, possibly singular, with less energy concentration than in the previous iteration. At this step the symplectic form
\omega
B
\omega(B)
\omega(B)\leq
1 | |
2 |
\int|df|2,
with equality if and only if the map is pseudoholomorphic. This bounds the energy captured in each iteration of the rescaling and thus implies that only finitely many rescalings are needed to capture all of the energy. In the end, the limit map on the new limit domain is stable.
The compactified space is again a smooth, oriented orbifold. Maps with nontrivial automorphisms correspond to points with isotropy in the orbifold.
To construct Gromov–Witten invariants, one pushes the moduli space of stable maps forward under the evaluation map
J,\nu | |
M | |
g,n |
(X,A)\to\overline{M}g, x Xn,
((C,j),f,(x1,\ldots,xn))\mapsto(st(C,j),f(x1),\ldots,f(xn))
to obtain, under suitable conditions, a rational homology class
X,A | |
GW | |
g,n |
\inHd(\overline{M}g, x Xn,Q).
Rational coefficients are necessary because the moduli space is an orbifold. The homology class defined by the evaluation map is independent of the choice of generic
\omega
J
\nu
X
g
n
A
\omega
The "suitable conditions" are rather subtle, primarily because multiply covered maps (maps that factor through a branched covering of the domain) can form moduli spaces of larger dimension than expected.
The simplest way to handle this is to assume that the target manifold
X
Defining Gromov–Witten invariants without assuming some kind of semipositivity requires a difficult, technical construction known as the virtual moduli cycle.