Table Of ContentMEMOIRS
of the
American Mathematical Society
Volume 248 • Number 1179 • Forthcoming
Applications of Polyfold Theory I: The
Polyfolds of Gromov-Witten Theory
H. Hofer
K. Wysocki
E. Zehnder
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 248 • Number 1179 • Forthcoming
Applications of Polyfold Theory I: The
Polyfolds of Gromov-Witten Theory
H. Hofer
K. Wysocki
E. Zehnder
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
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DOI:http://dx.doi.org/10.1090/memo/1179
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Contents
Chapter 1. Introduction and Main Results 1
1.1. The Space Z of Stable Curves 2
1.2. The Bundle W 6
1.3. Fredholm Theory 8
1.4. The GW-invariants 8
Chapter 2. Recollections and Technical Results 11
2.1. Deligne–Mumford type Spaces 11
2.2. Sc-smoothness, Sc-splicings, and Polyfolds 28
2.3. Polyfold Fredholm Sections of Strong Polyfold Bundles 39
2.4. Gluings and Anti-Gluings 43
2.5. Implanting Gluings and Anti-gluings into a Manifold 52
2.6. More Sc-smoothness Results. 55
Chapter 3. The Polyfold Structures 61
3.1. Good Uniformizing Families of Stable Curves 61
3.2. Compatibility of Good Uniformizers 73
3.3. Compactness Properties of M(G,G(cid:3)) 78
3.4. The Topology on Z 83
3.5. The Polyfold Structure on the Space Z 89
3.6. The Polyfold Structure of the Bundle W →Z 91
Chapter 4. The Nonlinear Cauchy-Riemann Operator 99
4.1. Fredholm Sections of Strong Polyfold Bundles 99
4.2. The Cauchy-Riemann Section: Results 101
4.3. Some Technical Results 104
4.4. Regularization and Sc-Smoothness of ∂ 110
J
4.5. The Filled Section, Proof of Proposition 4.8 116
4.6. Proofs of Proposition 4.23 and Proposition 4.25 123
Chapter 5. Appendices 135
5.1. Proof of Theorem 2.56 135
5.2. Proof of Lemma 3.4 146
5.3. Linearization of the CR-Operator 149
5.4. Consequences of Elliptic Regularity 150
5.5. Proof of Proposition 4.11 160
5.6. Banach Algebra Properties 164
5.7. Proof of Proposition 4.12 165
5.8. Proof of Proposition 4.16 170
5.9. Proof of Lemma 4.19 171
iii
iv CONTENTS
5.10. Orientations for Sc-Fredholm Sections 172
5.11. The Canonical Orientation in Gromov-Witten Theory 187
Bibliography 213
Index 217
Abstract
In this paper we start with the construction of the symplectic field theory
(SFT). As a general theory of symplectic invariants, SFT has been outlined in
Introduction to symplectic field theory(2000),byY.Eliashberg, A.GiventalandH.
Hofer who have predicted its formal properties. The actual construction of SFT is
a hard analytical problem which will be overcome be means of the polyfold theory
due to the present authors. The current paper addresses a significant amount of
the arising issues and the general theory will be completed in part II of this paper.
To illustrate the polyfold theory we shall use the results of the present paper to
describe an alternative construction of the Gromov-Witten invariants for general
compact symplectic manifolds.
ReceivedbytheeditorJuly21,2011and,inrevisedform,July29,2013,November30,2014,
andDecember4,2014.
ArticleelectronicallypublishedonMarch20,2017.
DOI:http://dx.doi.org/10.1090/memo/1179
2010 MathematicsSubjectClassification. 58B99,58C99,57R17.
Keywordsandphrases. sc-smoothess,polyfolds,polyfoldFredholmsections,GW-invariants.
ResearchpartiallysupportedbyNSFgrantDMS-0603957andDMS-1104470.
ResearchpartiallysupportedbyNSFgrantDMS-0906280.
(cid:2)c2017 American Mathematical Society
v
CHAPTER 1
Introduction and Main Results
In this paper we start with the application of the polyfold theory to the sym-
plectic field theory (SFT) outlined in [5]. It turns out that the polyfold struc-
tures near noded stable curves lead also naturally to a polyfold description of the
Gromov-Witten theory which is a by-product of the analytical foundation of SFT,
presented here. The polyfold constructions for SFT will be completed in [8]. The
Gromov-Witten invariants (GW-invariants) are invariants of symplectic manifolds
deduced from the structure of stable pseudoholomorphic maps from noded Rie-
mann surfaces to the symplectic manifold. The construction of GW invariants of
general symplectic manifolds goes back toFukaya-Ono in [12] and Li–Tian in [33].
Cieliebak–Mohnke studied the genus zero case in [2]. Earlier work for special sym-
plecticmanifoldsareduetoRuanin[42]and[43]. Wesuggest[38]foradiscussion
of some of the inherent difficulties in these approaches.
Our approach to the GW-invariants is quite different from the approaches in
the literature. We shall apply the general Fredholm theory developed in [22–24]
and surveyed in [18] and [17]. A comprehensive discussion of the abstract theory
will be contained in the upcoming lecture note [29]. The theory is designed for the
analytical foundations of the SFT in [8].
We recall from[24] thatapolyfoldZ is ametrizable space equippedwithwith
an equivalence class of polyfold structures. A polyfold structure [X,β] consists of
an ep-groupoid X which one could consider as a generalization of an ´etale proper
Lie groupoid whose object and morphism sets have M-polyfold structures instead
of manifold structures, and whose structure maps are sc-smooth maps. Moreover,
β :|X|→Z isahomeomorphism betweentheorbit space ofX andthetopological
space Z. The relevant concepts here are recalled in Section 2.2 below.
Our strategy to obtain the GW-invariants is as follows. We first construct the
ambient space Z of stable curves, from noded Riemann surfaces to the symplec-
tic manifold, which are not required to be pseudoholomorphic. The space Z has
a natural paracompact Hausdorff topology and we construct an equivalence class
of natural polyfold structures [X,β] on Z. The second step constructs a so called
strongbundlep:W →Z whichwillbeequippedwithansc-smoothstrongpolyfold
bundle structure. In the third step we shall show that the Cauchy-Riemann oper-
ator ∂ defines an sc-smooth section of the bundle p which is a particular case of
J
SFT.Weshallprovethat∂ isansc-smoothproperFredholmsectionofthebundle
J
p:W →Z. The solution sets of the section ∂ are the Gromov compactified mod-
J
uli spaces which, as usual, are badly behaving sets. However, the three ingredients
alreadyestablishedatthispointimmediatelyallowsonetoapplytheabstractFred-
holm perturbation theory from [23,24]. After a generic perturbation, the solution
sets of the perturbed Fredholm problem become smooth objects, namely compact,
weighted, smooth branched sub-orbifolds. They also have a natural orientation,
1
2 1. INTRODUCTION AND MAIN RESULTS
so that the branched integration theory from [25,26] allows one to integrate the
sc-differential forms over the perturbed solution sets to obtain the GW-invariants
in the form of integrals.
Ourmainconcerninthefollowingistheconstructionofthepolyfoldstructures
which allows one to deal with noded objects in a smooth way. For this purpose we
describe, in particular, the normal forms for families of noded Riemann surfaces
in the Deligne-Mumford theory used in our constructions. We also include some
related technical results needed for the SFT in [8].
1.1. The Space Z of Stable Curves
We start with the construction of the ambient space Z of stable curves. The
stablecurvesarenotrequiredtobepseudoholomorphic. Weconsider maps defined
onnodedRiemannsurfacesS havingtheirimagesintheclosedsymplecticmanifold
(Q,ω) and possessing various regularity properties.
We shall denote by
u:O(S,x)→Q
agermofamapdefinedon(apiece)ofRiemannsurfaceS aroundx∈S. Through-
out the paper we identify S1 with R/Z unless otherwise noted. Moreover, smooth
(in the classical sense) means C∞-smooth.
Definition 1.1. Let m ≥ 2 be an integer and δ ≥0. A germ of a continuous
map u : O(S,x) → Q is called of class (m,δ) near the point x, if for a smooth
chartψ :U(u(x))→R2n mapping u(x)to0andforholomorphic polar coordinates
σ :[0,∞)×S1 →S\{x} around x, the map
v(s,t)=ψ◦u◦σ(s,t)
which is defined for s large, has weak partial derivatives up to order m, which
weighted by eδs belong to the space L2([s ,∞)×S1,R2n) for s sufficiently large.
0 0
We call the germ of class m around the point z ∈S, if u is of class Hm near z.
loc
One easily verifies that if σ is a germ of biholomorphic map mapping x∈S to
y ∈ S(cid:3) then u is of class (m,ε) near x if and only if the same is true for u◦σ−1
near y. Moreover, the above definition does not depend on the choices involved,
like charts and holomorphic polar coordinates.
Definition 1.2. A noded Riemann surface with marked points is a tuple
(S,j,M,D) in which (S,j) is an oriented closed smooth surface S equipped with
a smooth almost complex structure j. The subset M of S is a finite collection of
marked points which can be ordered or un-ordered, and D is a finite collection of
un-ordered pairs {x,y} of points in S so that x (cid:8)= y and two pairs which intersect
areidentical. Theunionofallsets{x,y}belongingtoD,denotedby|D|,isdisjoint
from M. We call D the set of nodal pairs and |D| the set of nodal points.
It is a classical result, proved for example in [4], Theorem 3.2, that the pair
(S,j) determines a unique compatible Riemann surface structure in the sense of
complex manifolds.
TheRiemannsurfaceS canconsistofdifferentconnectedcomponentsC called
domaincomponents. The nodedRiemann surface (S,j,M,D)iscalledconnected
if the topological space S, obtained by identifying the points x and y in the nodal
pairs {x,y}∈D, is connected.
1.1. THE SPACE Z OF STABLE CURVES 3
So in our terminology it is possible that the noded surface (S,j,M,D) is con-
nected, but the Riemann surface S has several connected components, namely its
domain components.
The arithmetic genus g of a connected noded Riemann surface (S,j,M,D)
a
is the integer g defined by
a
(cid:2)
g =1+(cid:9)D+ [g(C)−1]
a
C
where (cid:9)D is the number of nodal pairs in D and where the sum is taken over the
finitely many domain components C of the Riemann surface S, and where g(C)
denotes the genus of C. The arithmetic genus g agrees with the genus of the
a
connected closed Riemann surface obtained by taking disks around the nodes in
everynodalpairandreplacingthetwodisksbyaconnectingtube. Inthefollowing
we refer to the elements of M ∪|D| as to the special points. The set of special
points lying on the domain component C is abbreviated by Σ :=C∩(M ∪|D|).
C
Two connected noded Riemann surfaces
(cid:3) (cid:3) (cid:3) (cid:3)
(S,j,M,D) and (S ,j ,M ,D )
are called isomorphic (or equivalent) if there exists a biholomorphic map
φ:(S,j)→(S(cid:3),j(cid:3))
(i.e., the diffeomorphism satisfies Tφ◦j = j(cid:3) ◦Tφ) mapping the marked points
onto the marked points and the nodal pairs onto the nodal pairs, hence satisfying
φ(M)=M(cid:3) and φ∗(D)=D(cid:3) where
(cid:3) (cid:4)
φ∗(D)= {φ(x),φ(y)}∈D(cid:3)|{x,y}∈D .
IfthemarkedpointsM andM(cid:3) areordereditisrequiredthatφpreservestheorder.
If the two noded Riemann surfaces are identical, the isomorphism above is called
an automorphism of the noded surface (S,j,M,D). In the following we denote by
[(S,j,M,D)]
the equivalence class of all connected noded Riemann surfaces isomorphic to the
connected noded Riemann surface (S,j,M,D).
Definition 1.3. The connected noded Riemann surface (S,j,M,D) is called
stable if its automorphism group G is finite
One knows that a connected noded Riemann surface (S,j,M,D) is stable if
and only if every domain component C of S satisfies
2·g(C)+(cid:9)Σ ≥3
C
where g(C) is the genus of C.
Next we describe the tuples α = (S,j,M,D,u) in which (S,j,M,D) is a, not
necessarilystable,nodedRiemannsurfacewithorderedmarkedpoints,andu:S →
Q a continuous map, in more detail.
Definition 1.4 (Stable maps and stable curves). The tuple
α=(S,j,M,D,u)
is called a stable map (of class (m,δ), where m ≥ 3 and δ ≥ 0), if it has the
following properties,
