A wall-crossing formula and the invariance of GLSM correlation functions
Gang Tian, Guangbo Xu
TL;DR
This paper proves a wall-crossing formula essential for demonstrating that GLSM correlation functions remain invariant under different perturbations, advancing understanding in mathematical physics.
Contribution
It introduces a wall-crossing formula that ensures the invariance of GLSM correlation functions across perturbation choices.
Findings
Established a wall-crossing formula for GLSM
Proved invariance of correlation functions
Supports robustness of GLSM in mathematical physics
Abstract
In this paper we prove a wall-crossing formula, a crucial ingredient needed to prove that the correlation function of gauged linear sigma model is independent of the choice of perturbations.
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Taxonomy
TopicsBlack Holes and Theoretical Physics · Advanced Algebra and Geometry · Quantum Chromodynamics and Particle Interactions
A Wall-Crossing Formula and the Invariance of GLSM Correlation Functions
Gang Tian
Beijing International Center for Mathematical Research
Peking University
Beijing, China
and
Department of Mathematics
Princeton University
Fine Hall, Washington Road
Princeton, NJ 08544 USA
and
Guangbo Xu
Department of Mathematics
Princeton University
Fine Hall, Washington Road
Princeton, NJ 08544 USA
Abstract.
In this paper we prove a wall-crossing formula, a crucial ingredient needed to prove that the correlation function of gauged linear -model is independent of the choice of perturbations.
Contents
- 1 Introduction
- 2 Virtual Orbifold Atlases
- 3 The Wall-Crossing Formula
- 4 Constructing Local Charts
- 5 Technical Details of Gluing
- 6 Constructing an Atlas
1. Introduction
The gauged linear -model (GLSM) is a two-dimensional supersymmetric quantum field theory introduced by Witten [34]. It plays a fundamental role in the physics “proof” of mirror symmetry [15], and its idea has been adopted in many mathematical studies of mirror symmetry. It provides an important framework in studying the relation between Gromov–Witten theory and the Landau–Ginzburg model, better known as the Landau–Ginzburg/Calabi–Yau correspondence.
The significance of GLSM calls for a rigorous mathematical construction. Such a mathematical framework is well-understood when the superpotential is zero. On the algebraic side, there have been the construction of stable quotient invariants [18] and quasimap invariants [5]. On the symplectic side, there have been various studies on the theory of symplectic vortex equations aiming at defining Gromov–Witten type invariants (called the gauged GW or Hamiltonian GW invariants), see for example [22, 4, 23, 3, 25, 24]. It has also been applied to Floer theory, such as [12, 35].
The case with a nontrivial superpotential is far more difficult. A simple case is when the superpotential is a nondegenerate quasihomogeneous polynomial on and the gauge group is a finite abelian group of symmetries of . Such a theory (which we call the orbifold Landau–Ginzburg A-model) was proposed by Witten in [33], and constructed rigorously by Fan–Jarvis–Ruan via [7, 8, 6]. A new feature in the presence of a superpotential, which seems to indicate the insufficiency of algebraic methods, is the separation of narrow and broad states. While the narrow states correspond to algebraic cohomological classes, the broad states are transcendental and often of odd degrees. Algebraic method can only treat the narrow case, for example, in Chang–Li–Li’s construction of Witten’s top Chern class [1], while for the broad case one needs to perturb the differential equation using non-algebraic objects (see [6] for details).
For general GLSM target spaces, while there have been a few works on the algebraic geometry construction (for example [9] [2]) focusing on the narrow case, our project starting at [29] aims at giving a symplectic geometry construction covering both narrow and broad states. In [29, 30, 31] and this paper, we consider a special case, where the gauge group is and the superpotential is of Lagrange multiplier type. Over a fixed smooth domain curve, in [29] we studied the basic analytic properties of the moduli space of perturbed gauged Witten equation, including the compactness. In [31] we constructed a virtual fundamental cycle on the moduli space and defined the GLSM correlation function which was forecasted in [30]. (See [32] and [28] for the case of geometric phases of more general GLSM target spaces.)
1.1. Main result
This paper proves a technical result needed for [31] (this result is also claimed in [30]). Let us briefly recall the setup and the state the main result. The detailed setting will be recalled in Section 3. Let be a noncompact Kähler manifold and be a holomorphic function. Assume has only one critical point and there is a -action on making homogeneous of degree . Denote with coordinates and consider the function . On the other hand, a smooth -spin curve is a compact Riemann surface with orbifold singularities , together with an orbifold line bundle and an isomorphism
[TABLE]
The existence of the above isomorphism implies that the local group of the orbifold structure at is with divides ; moreover, viewing as a subgroup of , the monodromy of the orbifold line bundle at is an element . The orbifold point (or the monodromy ) is called narrow if the fixed point set is a single point which is the critical point of ; otherwise we say it is broad.
The gauged Witten equation over is a first-order partial differential equation of two variables and . It is elliptic modulo gauge transformation. The variable is a connection on certain -bundle over , and the variable is locally a map from to . Near a broad orbifold point with cylindrical coordinates , for fixed , the equation which satisfies is asymptotic to the following Floer-type equation
[TABLE]
It follows that converges to a critical point of . Since the critical points are degenerate, the natural linearization of the gauged Witten equation is not Fredholm.
For a broad , consider perturbations of of the form
[TABLE]
where and is a -invariant function of lower degree. If is generic, then the perturbed function has nondegenerate critical points. Upon choosing for each broad orbifold point, one can perturb the gauged Witten equation over (equipped with a “rigidification” of the -spin structure) so that the linearization becomes Fredholm.
Furthermore, one can use a list of critical points of the perturbed functions to label a moduli space which consists of solutions with being asymptotic to the critical point at . When the perturbations are generic (called strongly regular perturbations), such moduli spaces are compact. One can then construct a virtual fundamental cycle of . However it turns out that the virtual cycles depend on the choice of perturbations of . To prove that the resulting correlation functions are independent of perturbations, one needs to compare the virtual cycles for two choices. This is referred to as a “wall-crossing formula” which is very similar to the wall-crossing formula in [6] for orbifold Landau–Ginzburg theory. Our main theorem is as follows.
Theorem 1.1**.**
Let be a rigidified -spin curve with broad markings with monodromies . Let be strongly regular perturbations of associated to the markings and let , be two strongly regular perturbations of associated to . Let be critical points of the perturbed superpotential restricted to the fixed point sets of respectively.
Suppose and are connected via a generic path such that there is only one for which fails to be strongly regular. Assume for , there are only two critical points and of restricted to the fixed point set of such that
[TABLE]
assume in addition
[TABLE]
Let be the virtual cardinality of the moduli space of the gauged Witten equation defined in [31]. Then we have
[TABLE]
Here is the sign of (1.1), and is the algebraic count of the number of BPS soliton solutions connecting and (where the number equals to a topological intersection number).
For the precise meanings of relevant notions and concepts in the above statement, see [31] and Section 3 of this paper. We actually rephrase the theorem in Section 3 in a much more simplified situation without losing generality. For example, it suffices to consider the case that has no narrow markings and only one broad marking, because extra markings at which the perturbations do not vary only brings in notational complexities. The simplified version of the above theorem is restated as Theorem 3.3.
Both the definition of the correlation function and the proof of the invariance rely on constructing certain virtual fundamental chains or cycles. The virtual cycle theory provides a general method of defining Euler numbers for infinite dimensional bundles where a natural section is not transverse. Our framework of virtual cycle theory is based on the original approach of Li–Tian [17] developed for defining symplectic Gromov–Witten theory for general compact targets. In the appendix of [31], we provided a detailed discussion of this approach. We remark that our approach (as well as that of [17]) is topological, hence we do not need smooth structures as in Kuranishi theory (see [13]) or polyfold theory (see [14]).
This paper is organized as follows. In Section 2 we briefly recall the abstract setting of virtual orbifold atlases of [31]. In Section 3 we briefly recall the setting of [31] and restate the wall-crossing formula we want to prove. Certain simplifications will also be adopted. In Section 4 we construct local charts for the relevant moduli space. In Section 5 we provide the details of the gluing analysis. In Section 6 we patch the local charts together to construct a virtual orbifold atlas on the moduli space.
Acknowledgments. The authors would like to thank David Morrison, Edward Witten, Kentaro Hori, and Mauricio Romo for their patience in communicating the physics about GLSM. The second named author would like to thank Huai-Liang Chang for helpful discussions.
2. Virtual Orbifold Atlases
In this section we recall the basic notions in the abstract level needed for constructing virtual fundamental chains. We refer the reader to [31] or [28] for details.
2.1. Topological orbifolds and continuous orbibundles
The basic definitions of topological orbifolds and orbibundles have been recalled in [31]. Here we first remark on a few specifics about topological manifolds. A comprehensive reference is [16].
Let be a topological manifold. A subset of is a topological submanifold if the subset with the subspace topology is a topological manifold. Let be another topological manifold. A continuous map is called an embedding if it is a homeomorphism onto a topological submanifold of . In [29] and this paper, all embeddings are assumed to be locally flat. Namely, for any , there exists a local coordinate chart of around such that . The local flatness guarantees the existence of tubular neighborhoods.
Given two continuous vector bundles and a bundle embedding is a pair where is a locally flat embedding, is a bundle embedding covering . It is easy to see that bundle embeddings can compose.
In the topological category there is still a notion of transversality. Let be an open subset and be a continuous map. We say that is transverse to if is a locally flat submanifold of and for each , each locally flat chart such that and , for all close to , the restriction of to is a local homeomorphism near the origin. The transversality notion for such maps can be generalized to the notion of a continuous map between topological manifolds being transverse to a locally flat submanifold of the target, and the notion of a continuous section of a continuous vector bundle being transverse to the zero section. Such generalization relies on the notion of microbundles invented by Milnor [21].
An important theorem used in our construction is topological transversality theorem, proved by Kirby–Siebenmann [16] and Quinn [26, 27]. The precise statement (see the main theorem of [27]) says that two properly embedded topological manifolds in an ambient manifold can be made transverse by an arbitrary small isotopy of . This implies the existence of transverse perturbations in the following situation: for a continuous vector bundle over a topological manifold , any continuous section can be made transverse to the zero section by an arbitrarily small perturbation.
All relevant notions about topological manifolds can be extended to topological orbifolds. In this paper we only need to consider effective orbifolds. Typical examples are global quotients where is a topological manifolds, is a finite group acting continuously and effectively on . The broadest category we consider is that of orbifolds with boundary.
2.2. Charts and transitions
In the following discussion of the topological virtual cycle theory, we always assume that is a compact Hausdorff topological space.
Definition 2.1**.**
A virtual orbifold chart (chart for short) of is a tuple where
- (1)
is an orbifold (with or without boundary). 2. (2)
is an orbifold vector bundle. 3. (3)
is a continuous section. 4. (4)
is an open subset. 5. (5)
is a homeomorphism.
is called the footprint of the chart . is called the virtual dimension of . If is an open subset, then we can restrict to in the obvious way, denoted by and called a subchart or a shrinking of . If is a precompact subset of , denoted as , then we say that is a precompact shrinking of .
Definition 2.2**.**
Let , be two charts of . An embedding of into consists of a bundle embedding of orbifold vector bundles such that: 1) the diagrams
[TABLE]
commute; 2) satisfy the “tangent bundle condition,” namely, there exist a neighborhood of and a subbundle extending such that is transverse to and .
It is also not hard to show that embeddings of charts can compose.
Definition 2.3**.**
Let , be two charts. A coordinate change from to is a triple where is an open set and is an embedding from to . They must satisfy the following conditions.
- (1)
. 2. (2)
If converges to , converges to , then and .
For , let be a shrinking of . Then we can restrict the coordinate change to
[TABLE]
Denote the induced coordinate change by .
2.3. Atlases
Unlike atlases of manifolds or orbifolds, coordinate changes in a virtual atlas are not bi-directional. Naturally charts in a virtual atlas are indexed by a partially ordered set.
Definition 2.4**.**
A virtual orbifold atlas of virtual dimension on is a collection
[TABLE]
where
- (1)
is a finite, partially ordered set. 2. (2)
For each , is a virtual orbifold chart of virtual dimension on . 3. (3)
For , is a coordinate change from to .
They are subject to the following conditions.
- •
(Covering Condition) is covered by all the footprints .
- •
(Cocycle Condition) For ,
[TABLE]
- •
(Overlapping Condition) For , we have
[TABLE]
All virtual orbifold atlases considered in this paper have definite virtual dimensions, although we do not always explicitly mention it.
Remark 2.5*.*
The above setting is slightly more general than what we need in our application in this paper and the companion [31]. In this paper we will see the following situation in the concrete situations.
- (1)
The index set consists of certain nonempty subsets of a finite set , which has a natural partial order given by inclusions. 2. (2)
For each , is a finite group and . is a global quotient. Moreover, are vector spaces acted by and the orbifold bundle is the quotient where . 3. (3)
For , where is a -invariant open subset and the coordinate change is induced from the following diagram
[TABLE]
Here is a covering map with group of deck transformations identical to ; then acts on and is a -equivariant embedding of manifolds, which induces an orbifold embedding and an orbibundle embedding . Moreover, there is a natural subbundle which can be used to verify the tangent bundle condition.
2.4. The virtual fundamental chain and cycle
In order to construct the virtual chain or the virtual cycle, we need some more technical preparations.
Definition 2.6**.**
Let be a virtual orbifold atlas on . A shrinking of is another virtual orbifold atlas indexed by elements of the same set such that
- (1)
For each , is a shrinking of . 2. (2)
For each , is the induced shrinking of .
Given a virtual orbifold atlas
[TABLE]
we define a relation on the disjoint union as follows. if one of the following holds.
- (1)
and ; 2. (2)
, and ; 3. (3)
, and .
If is a shrinking of , then it is easy to see that the relation on defined as above is induced from the relation for via restriction. If is an equivalence relation, we can form the quotient space
[TABLE]
with the quotient topology. is then a compact subset of .
Lemma 2.7**.**
For each virtual orbifold atlas , there exists a shrinking of such that is an equivalence relation. Moreover, the shrinking can be made such that is a Hausdorff topological space and for each , the natural map is a homeomorphism onto its image.
Definition 2.8**.**
A virtual orbifold atlas satisfying the conditions required for in Lemma 2.7 is called a good atlas or a good coordinate system.
This is the same notion of good coordinate systems in the Kuranishi approach. On the other hand, there are also notions of orientations and orientability in the topological category, and one can define orientations and orientability of virtual orbifold atlases. We do not recall them. With an oriented good atlas one can construct perturbations inductively on all charts and define the virtual cycle. More precisely, one chooses (multivalued) sections
[TABLE]
satisfying certain compatibility condition with respect to coordinate changes, such that is transverse to the zero section.
When the virtual dimension of the atlas is zero, the union of zeroes of modulo the equivalence relation are discrete. Indeed one can prove that for suitable sufficiently small perturbations , the zero locus is compact, hence finite. The details are in [31, Appendix]. Then the counting of zeroes of the perturbed sections with signs, one can define the virtual cardinality of , which is a rational number . It can be viewed as a generalization of the orbifold Euler characteristic.
2.5. Boundary
We consider the induced atlas on the “virtual” boundary.
Let be equipped with a good virtual orbifold atlas {\mathfrak{A}}=\big{(}\{C_{I}\ |\ I\in\mathbbmsl{I}\},\{T_{JI}\ |\ I\preccurlyeq J\in\mathbbmsl{I}\}\big{)}. Denote
[TABLE]
It is easy to see that is closed in and hence compact. Take
[TABLE]
This is a virtual orbifold chart of (which may be empty). Denote , with the induced partial order . Moreover, the coordinate changes can be restricted to the boundary. For each pair , the object
[TABLE]
is a coordinate change from to . Hence we obtain a virtual atlas
[TABLE]
If is oriented, then it is routine to check that has an induced orientation.
A natural and useful result about virtual cardinality is the following.
Proposition 2.9**.**
If is an oriented virtual orbifold atlas of dimension one with boundary on and let be equipped with the induced atlas , then .
3. The Wall-Crossing Formula
In this section we briefly recall the setting of [31] and the wall-crossing formula needed for the proof of the invariance of the GLSM correlation function. All details can be found in [29] [31].
Let be a Landau–Ginzburg space. That is, is a noncompact Kähler manifold and is a holomorphic function with a single critical point. Assume that there is a holomorphic -action on , often called the R-symmetry or R-charge, such that is homogeneous of degree ( 111We assumed in [29, 30, 31]. However our whole construction, most essentially the compactness theorem proved in [29], can be extended to the case.). Assume that the R-symmetry has a moment map , where we regard as the Lie algebra of . We also assumed certain conditions on the Landau–Ginzburg space in [29] which was used to prove the compactness of the moduli spaces. However we do not need to recall the explicit statements here.
The GLSM space is a triple . Here is the product whose coordinates are denoted by , is the “Lagrange multiplier” , and is a group acting on , defined as follows. acts on the coordinate via the R-symmetry, and acts on by . Then is -invariant and homogeneous with respect to . The -action has a moment map
[TABLE]
Here is a constant which we fix from now on. We also fix a metric on the Lie algebra .
We do not need to recall the general definition of -spin curves (which can be found in [8]) but only the special case of one broad marking, and we state the definition without using orbifold language. Let be a compact Riemann surface and be a point. Denote . If we take a holomorphic coordinate centered at , then the log-canonical bundle has a local holomorphic frame . An -spin structure consists of a holomorphic line bundle , an integer , and a line bundle isomorphism . Using the local coordinate , one can find a local holomorphic frame of near such that
[TABLE]
The integer is a crucial parameter. Denote , and regard it as an element of . is called narrow if the fixed point set contains only the critical point of ; otherwise it is called broad. The wall-crossing formula is only about the broad case so we assume from now on that is broad, namely, is a positive dimensional complex submanifold of .
The triple is called an -spin curve. Notice that satisfying (3.1) is well-defined up to a -action. A rigidification of the -spin curve is a choice of satisfying (3.1). From now on we fix the data , , , , a local coordinate and a rigidification satisfying (3.1). We call these data a rigidified -spin curve, denoted by . We also choose a cylindrical end around , and the coordinate identifies with a half cylinder . Denote .
We need to perturb the superpotential in order to do Fredholm theory.
Hypothesis 3.1**.**
There exists a nonzero, finite dimensional complex vector space parametrizing certain -invariant holomorphic functions satisfying [29, Hypothesis 2.8], particularly the following conditions.
- (1)
For each , there is a complex analytic proper subset such that for each , the restriction of the function
[TABLE]
to is a Morse function with finitely many critical points. 2. (2)
The set defined by the coincidence of the imaginary parts of two critical values of is a locally finite union of real analytic hypersurfaces.
Denote . An element is called a perturbation. We also identify the pair with the induced function defined by (3.2). is called regular (resp. strongly regular) if , i.e., the perturbed superpotential is Morse (resp. , i.e., the perturbed superpotential is Morse and all critical values have distinct imaginary parts).
3.1. Perturbed gauged Witten equation
We choose a smooth volume form and a cylindrical volume form such that over the cylindrical end , . determines a cylindrical type metric on , for which we can define weighted Sobolev spaces for , and .
Choose a Hermitian metric on such that for the section in (3.1), for certain and . Choose another Hermitian line bundle . Let , be the unit circle bundles over .
The variables of the perturbed gauged Witten equation are triples . Here are certain unitary connections on the principal -bundle
[TABLE]
Here should induce a holomorphic structure on isomorphic to the original one. is a section of the associated bundle
[TABLE]
is a trivialization of the fibre of at . Notice that acts on by . We extend the -torsor of trivializations of the fibre of to a -torsor of local trivializations of , which is still denoted by . The two perspectives will be applied interchangeably. Moreover, the rigidification and the choice of local coordinate induce a local trivialization of . We denote the product with , which is a local trivialization of , still by . Any section can be written with respect to the induced trivialization of as
[TABLE]
Notation 3.2*.*
We will frequently use to denote the map obtained by applying the trivialization . When is understood from the context, to simplify notations, we also denote and these two notations are used interchangeably.
Let be the projection. For any one can lift to a section as follows. Choose local holomorphic coordinate on . The -spin structure provides a local holomorphic section of with . Choose an arbitrary locally nonvanishing smooth section of . Then locally a point of can be represented by where . Then define
[TABLE]
Using the symmetries of it is easy to verify that the above expression is independent of the choices of the local coordinate and the local sections , hence defines a section of . Moreover, given a perturbation , one can lift defined by (3.2) to a section , parametrized smoothly by the connection and the local trivialization of over the cylindrical end . Using the vertical metric on , one can define the gradient 222The explicit expression of and its gradient can be found in [29]; in this paper we only need to know certain properties of it, which will be reviewed in Section 5.. Then any section pulls back to a section .
The -perturbed gauged Witten equation for variables reads
[TABLE]
We explain the terms appearing above. The complex structure on induces a complex structure on ; the connection also gives a covariant derivative ; is the part of . On the other hand, is obtained by taking the Hodge star (with respect to the volume form ) of the curvature , and is the dual of the moment map with respect to the metric on . There is a gauge symmetry on (3.3).
3.2. The virtual cardinalities and the wall-crossing formula
When is a regular perturbation, is a Morse function having finitely many critical points . We showed in [29] that for a finite energy solution to (3.3), up to gauge transformation, there exists such that
[TABLE]
Therefore one uses to label the moduli spaces. Let be the moduli space of gauge equivalence of solutions that satisfy the above asymptotic constrain. When is strongly regular, is a compact Hausdorff space (see [29, Theorem 6.5]). Moreover, for a strongly regular perturbation , in [31] we constructed an oriented virtual orbifold atlas on . It has an associated virtual cardinality (which is nonzero only when the virtual dimension, which equals to the Fredholm index computed in [29], is zero). In the current case, is an integer since there is no nontrivial automorphism group.
In order to define a correlation function, it is important to understand the dependence of the virtual cardinality on the choice of strongly regular perturbation. We need to compare two different strongly regular perturbations and . In [31] we have already reduced the comparison to the case that and for and . Since is of codimension two while is of codimension one, we only need to consider the “wall-crossing” situation. Namely, we connect and in by a path such that only intersects once at with at the smooth part of , and the intersection is transverse. Since is always regular, we obtain families of critical points . Moreover, by the definition of , there exist exactly a pair , of paths of critical points such that
[TABLE]
[TABLE]
The sign of the path , denoted by , which is the same as the sign in Theorem 1.1, is defined to be positive (resp. negative) if (3.4) is positive (resp. negative).
On the other hand, since the imaginary part of and coincide, there may exist solitons starting from to (when their imaginary parts differ, there is no soliton). Recall that a soliton connecting and is a Floer trajectory for the Hamiltonian that connects and . Namely a soliton is a solution to the equation
[TABLE]
Since the Hamiltonian is independent of the variable , there could be Floer trajectories that are -independent, namely, negative gradient flow trajectories of ; they are called BPS solitons. Let be the moduli space of BPS solitons connecting and , or equivalently, the moduli space of negative gradient flow lines of the real part of that connect and . After choosing certain orientations on this moduli space, one can define the counting , which is indeed equal to a topological intersection number.
With all above understood, one can state the wall-crossing formula, which is reformulation of Theorem 1.1.
Theorem 3.3** (Wall-crossing formula).**
[31, Theorem 4.6]** The virtual cardinalities and satisfy
[TABLE]
In other words, there is a nontrivial wall-crossing term when becomes regular but not strongly regular. Therefore, to obtain a nontrivial number, one needs to pair the virtual cardinalities with quantities which have opposite wall-crossing effect. The idea is the same as how Fan–Jarvis–Ruan define the correlation functions for Landau–Ginzburg theory.
Now we restate the definition of the correlation function as in [30]. For a broad , define the state space in the -sector to be
[TABLE]
Here , for any . This vector space is independent of the choice of . When is a homogeneous polynomial on and , this vector space is isomorphic to the primitive cohomology of the projective hypersurface defined by . Then the correlation function in the -sector is a linear function defined by
[TABLE]
Here is the topological intersection number in between the cycle and the unstable submanifold of . We have shown that satisfies a Picard–Lefschetz type formula when crossing the wall (see [31, Proposition 4.7]), which is exactly opposite to how changes. It then follows (see the end of [30]) that is independent of the choice of a strongly regular perturbation .
The main objective of the rest of this paper is to prove Theorem 3.3. First, we consider a universal moduli space over the path of perturbations . For , let (resp. ) be the moduli space of solutions to the -perturbed gauged Witten equation which converge to (resp. ) as . Define
[TABLE]
Since is not strongly regular and , there may exist solitons connecting and . By the compactness result of [29], can be compactified by adding soliton solutions at the slice. We define
[TABLE]
Here is the moduli space of soliton solutions whose domain is a nodal curve with principal component and a rational component. is a compact and Hausdorff space. The subset of soliton solutions has two parts, the moduli space of BPS soliton solutions and the moduli space of non-BPS soliton solutions . We prove the following theorem in Section 4—6.
Theorem 3.4**.**
There exists an oriented virtual orbifold atlas with boundary on , whose oriented virtual boundary is the disjoint union of three pieces
[TABLE]
Here the orientations on , and are determined by orientations on the unstable manifolds of and chosen continuously depending on .
Lastly, observe that the boundary is not a product of two moduli spaces, because the soliton equation on the rational component depends on a parameter (depending smoothly on the gauge field on the principal component, see [29, Definition 2.15] or Fact 5.1.). One can construct a “homotopy” from to a product of moduli spaces. Let . Let be the functional . Let be the moduli space of BPS soliton solutions to the gauged Witten equation over where the parameter for the soliton components is . Then consider the universal moduli space
[TABLE]
Since it is not involved with gluing, it is easy to prove the following statement.
Proposition 3.5**.**
On there exists an oriented virtual atlas with boundary such that its oriented virtual boundary is
[TABLE]
Notice that is the product . One can choose a perturbation on which is of product type on the boundary. It follows that
[TABLE]
Combining Theorem 3.4, Proposition 3.5 and Proposition 2.9, the wall-crossing formula Theorem 3.3 is proved.
4. Constructing Local Charts
Section 4, 5, and 6 are devoted to the proof of Theorem 3.4. In this section we construct virtual orbifold charts around all points in while many details of the gluing construction will be provided in Section 5. This section is organized as follows. In Subsection 4.1, we consider those points represented by smooth solutions; this part also serves for fixing notations. In Subsection 4.2, we state the main proposition (Proposition 4.3) about the local charts of non-BPS soliton solutions. The proof of Proposition 4.3 is provided in Subsections 4.3, 4.4 and 4.5. In Subsection 4.6 we finish the separate discussion of local charts around BPS soliton solutions.
4.1. Local charts for non-soliton solutions
We briefly recall the discussion of linear Fredholm theory for perturbed gauged Witten equation here. More details can be found in [31]. Choose and fix a sufficiently small . We describe a Banach manifold as follows.
Let be the space of connections such that with respect to the local trivializations over the cylindrical end (we fixed an -torsor of trivializations), and such that are of class . Given a connection , let be the number described in Item (2) of Fact 5.1 (see also Item (5) of Fact 5.1). consists of triples where , is a framing of at , and is a section of of regularity , such that with respect to the trivialization , for and , one can write
[TABLE]
Remark 4.1*.*
The reader should keep in mind that the Sobolev spaces modeling the sections have the usual un-weighted norms , while the Sobolev spaces modeling the gauge fields have the weighted norms .
There is a Banach space bundle whose fibre over is . The perturbed gauged Witten equation defines a section
[TABLE]
The abelian condition of the group allows us to introduce a global gauge fixing condition. Choose to be a finite dimensional subspace generated by smooth sections with compact supports, such that the Laplacian is bijective. Denote . Define an augmented Banach space bundle over whose fibre over is
[TABLE]
Choose a smooth reference connection so that for all , . Define the augmented equation (with respect to ) to be
[TABLE]
Here is modulo . The augmented equation gives a section . Then the moduli space is homeomorphic to , i.e., each gauge equivalence class of solutions contains exactly one element that satisfies the gauge fixing condition.
It is convenient for us to introduce the notations
[TABLE]
Now vary . Let consists of quadruples where and . This is again a Banach manifold (with boundary). One also has a Banach space bundle whose fibre over is . The augmented gauged Witten equation defines a smooth Fredholm section
[TABLE]
The zero locus of is the moduli space (not the compactified moduli). For an arbitrary 555Notice that in this case the object is the same as its equivalence class, so is a genuine point in the moduli space., we can construct a virtual orbifold chart in the same way as we did in [31] where is not varying. Notice that is a manifold.
4.2. Local charts for non-BPS soliton solutions
Under the current restricted situation, we agree with the following conventions. Whenever we talk about solitons (or objects in the same Banach manifold), we mean an object defined over the infinite cylinder which converges to at and to at , for some ; such an object is denoted by . Whenever we talk about soliton solutions (or objects in the same Banach manifold), we mean an object defined on a nodal domain with two components, whose restriction to the principal component converges to as , whose restriction to the rational component is a soliton connecting and for some ; such an object will have a label to indicate that it has two components.
4.2.1. General settings and notations
Given , it determines a vector field on with finitely many zeroes (see Fact 5.1 for related notations). For any pair of critical points, say , denote by the Banach manifold of -maps from to that approach / at / in the -sense666Here is the frame belonging to .. When and are understood, we simply denote this Banach manifold by 777Although we skipped the dependence on in this notation but the reader should keep in mind that it depends on a point .. There is a Banach space bundle whose fibres over is . The soliton equation reads
[TABLE]
In Section 5 for technical reasons we will “augment” objects in by including gauge fields on the soliton component, and the notation is reserved for the Banach manifolds of augmented objects. Here we set
[TABLE]
The product is a Banach space bundle over . The augmented section defined by
[TABLE]
On there is a free action by the infinite cylinder which is defined by reparametrize via translation. It also acts on the bundle (hence on ) equivariantly. By the symmetry of the operator we see that the section defined by (4.3) is an equivariant section. Therefore the moduli space of soliton solutions is homeomorphic to .
4.2.2. The stabilization and the universal curve
In order to deal with the unstable rational component, we need to add an extra marked point to stabilize, and first construct local charts for the moduli space of solutions with a marked point in the domain.
Let be the compactified configuration space of points in . There is a distinguished point in corresponding to the degeneration at the original marking , which is a nodal curve with principal component and a rational component with three markings , , and the extra one. Denote and the map that forgets the second marking provides a universal family. Locally around the distinguished point, we denote the universal family by where is a small disk centered at the origin. To be consistent with other notations we will regard the same as the symbol . Denote by the fibre and the central fibre is a nodal curve with two components, a principal component isomorphic to , and a rational component attached to . We identify the rational component with the infinite cylinder and the marking with . On the other hand, for each nonzero , the regular fibre is canonically identified with with a marking . As a convention, we will identify with its logarithm as , and using the cylindrical coordinate on , has coordinate . Lastly, there is a natural -action on the family by rotating the first marked point using the cylindrical coordinate, which projects to the standard -rotation on .
Now we introduce a family of Banach manifolds for all . If , then is a copy of as described in Subsection 4.1; if , then is the Banach space defined by (4.2). The disjoint union is also a Banach manifold for which we view parametrizes the extra marking .
It is a standard knowledge that for any compact subset that is away from the singularity of the central fibre , there exists sufficiently small such that for all , there is a canonical embedding .
Now let be a point that is represented by a non-BPS soliton solution. Let be the automorphism group of , which is independent of the choice of representatives. Choose for each such a representative . By a well-known result in Floer theory (see for example [11, Theorem 4.3]), we can choose where is an embedding near . Fix such a and denote the promoted object
[TABLE]
By reparametrizing the solution, one may assume that . Therefore, the domain of is identified with the central fibre .
4.2.3. Normalization conditions
We need to fix a few other choices. To simplify the notations, we assume that the injectivity radius of is and the exponential map at every point of is a diffeomorphism. Denote ; choose a hyperplane which is transverse to at . Then is transverse to . Moreover, choose a complex linear function with kernel , which is pushed forward to a smooth function that vanishes on .
For the representative , we fix the function .
For a section of , we denote by the corresponding map under the local trivialization of induced by . When is clear from the context, we also denote it by .
Definition 4.2**.**
For , is called normal to if
[TABLE]
On the other hand, is normal to if
[TABLE]
4.2.4. The obstruction spaces
We also need to choose obstruction spaces. For the principal component of , which is , the way of choosing an obstruction space is the same as in [31]. Here we recall it. Denote
[TABLE]
Here and . We write the linearization of as with respect to the above decomposition. Then we can choose finite dimensional subspaces , satisfying the following condition.
- (1)
is generated by smooth sections with compact supports. 2. (2)
is transverse to the linearization of in the directions of , and is transverse to the linearization of the operator in the directions of .
Therefore is transverse to the linearization in the directions of and . That means transversality is achieved without deforming and . Define . There is no need to discuss group actions since has trivial stabilizer.
For the soliton components . By the Fredholm property of 888This is because solitons are Floer trajectories., one can choose a finite dimensional subspace such that is transverse to . One can also choose to be -invariant in the following sense. Recall that acts on the domain of by rotating the cylindrical coordinate. Then for any and , . So we require that is an invariant subspace of under this -action. Indeed this can be done because is a finite group.
Now define . has a -action. We call the obstruction space.
Now we want to extend to the universal curve in a -equivariant way. Suppose sections in are all supported in for a compact subset , and sections in are all supported in for a compact subset . Let . For sufficiently small, one can embed into the fibre . Then gives an embedding
[TABLE]
Take . For any object , since its domain is the central fibre , and pulls back to a section
[TABLE]
For any object , since its domain is the regular fibre , one can also pull back to a section
[TABLE]
These sections are equivariant in the following sense. For , the action on is . So we have
[TABLE]
The action on is . So
[TABLE]
4.2.5. The thickened moduli space
The -thickened moduli space is defined as
[TABLE]
has a naturally defined Hausdorff topology. Moreover, acts continuously on . Notice that we have chosen a point . We first construct a local chart of this thickened moduli space near this point.
Indeed, introduce
[TABLE]
This is a vector space acted by , equipped with an invariant metric. There is a subspace consisting of pairs in which satisfies the infinitesimal normalization condition, namely
[TABLE]
Now we state the main technical technical result of this section. Its proof is given in Subsection 4.3, Subsection 4.4, and Subsection 4.5.
Proposition 4.3**.**
There exist and a -equivariant continuous map satisfying the following conditions.
- (1)
* and is a homeomorphism onto its image.* 2. (2)
Denote , then the map defined by
[TABLE]
is transversal to (in the topological sense) and .
Now we see how we can use it to construct a local chart of .
Corollary 4.4**.**
For , define
[TABLE]
let be the section induced from the map ; define by with image . Then for small enough, the tuple
[TABLE]
is a local chart of around .
Proof.
Proposition 4.3 implies that is equivariant, hence and are well-defined. It remains to show that is a homeomorphism onto an open neighborhood of . For (local) surjectivity, if sufficiently close to , then there exists such that
[TABLE]
Then can be identified with a fibre of . Moreover, is in the image of the map constructed by Proposition 4.3. Then . By Item (2) of Proposition 4.3, . Hence the local surjectivity follows. The local injectivity follows from Item (1) of Proposition 4.3. Therefore for small enough, is a homeomorphism onto an open neighborhood of . ∎
4.3. Proof of Proposition 4.3: part I
To prepare for the gluing process, we first construct the map on the lower stratum. For , take . Consider the family of elements
[TABLE]
where is the exponential map in . If we choose a right inverse of
[TABLE]
then by the standard implicit function theorem, for small enough, the family (4.9) can be corrected by adding a unique element 999Notice that we can choose the right inverse such that . such that for
[TABLE]
one has
[TABLE]
The existence of the right inverse is due to the surjectivity . In order to guarantee Item (2) of Proposition 4.3, we need to impose the following condition on : for any , if in the infinitesimal deformation of the soliton is , then we require that
[TABLE]
We call this condition the infinitesimal normalization condition. Since the exact solution is obtained from the approximate solution by adding an element in the image of the right inverse, the infinitesimal normalization condition implies
[TABLE]
4.4. Gluing
4.4.1. Pregluing
Let be a (small) gluing parameter. We first introduce a notation which will be used frequently. Let be a map/section defined on a cylinder (where could be infinity). We define , called the -twist of , to be the map/section defined on which has the expression
[TABLE]
Now given an arbitrary smooth , we would like to construct for each gluing parameter a preglued object . Recall that has a cylindrical coordinate so that is identified with . Let be the closed subset corresponding to . We cut into the following pieces
[TABLE]
By the asymptotic behavior of the section , there is a loop such that is a constant in , and such that converges to exponentially as . Then for sufficiently large, there is such that over , . For the same reason, there exists such that over , . Then the -twist is still asymptotic to , and we have
[TABLE]
Choose a cut-off function such that for , for . Let , 121212Roughly, starts to decrease at and gets to zero at ; starts to increase at and gets to one at .. Define
[TABLE]
On the other hand, we use the same connection and the same frame in the approximate solution, namely and . The preglued object in the space is defined as
[TABLE]
Apply the pregluing construction for the family constructed in Subsection 4.3, which gives a family of approximate solutions
[TABLE]
Among this family there is a central one corresponding to , denoted by .
Recall that lies in the set for some . It is easy to see that if is small enough and , we can write
[TABLE]
4.4.2. Estimates
The domain of the approximate soliton is identified with the punctured surface . We can regard the approximate solution as an element of the Banach manifold . The norm of its tangent space and the norm on the Banach vector bundles are not modified though we have a varying gluing parameter.
We look for a family of solutions to the equation which are close to the family . We first state a few technical results.
Lemma 4.5**.**
There exist such that for ,
[TABLE]
Proof.
See Subsection 5.1. ∎
Lemma 4.6**.**
There exist , and a family of bounded linear operators
[TABLE]
satisfying the following conditions.
- (1)
For each , is a right inverse of and . 2. (2)
For each and , if in the deformation of the section is , then
[TABLE]
Proof.
See Subsection 5.5. ∎
4.4.3. Exact solutions
Recall the following version of the implicit function theorem.
Proposition 4.7** (Implicit Function Theorem).**
[19, Proposition A.3.4]** Let , , , , , , be as follows. are Banach spaces, is an open neighborhood of the origin and is a continuously differentiable map such that is surjective and is a bounded right inverse to . Moreover,
[TABLE]
[TABLE]
Then, if satisfies
[TABLE]
there exists a unique such that
[TABLE]
Moreover,
[TABLE]
Let be the one of Lemma 4.6 and fix . Define . We identify points in near with tangent vectors in , using the exponential map of . This gives a small neighborhood of the origin. For each , let be the corresponding point in the Banach manifold. On the other hand, define . Parallel transport between nearby objects induces a continuous trivialization . Then consider the map given by . Let be the operator given in Lemma 4.6, which is a right inverse to . Take where is the one in Lemma 4.6.
To apply Proposition 4.7 there are two conditions yet to check. The following two lemmata can be proved via straightforward calculation.
Lemma 4.8**.**
There exist , such that
[TABLE]
Lemma 4.9**.**
For any , there exists such that for and , lies in the -neighborhood of .
Then the tuple satisfies the hypothesis of Proposition 4.7. Moreover, by Lemma 4.5 and Lemma 4.9, there exists such that for , is identified with a point satisfying (4.18). Then by the implicit function theorem, there exists a unique satisfying (4.19), i.e.,
[TABLE]
So we have actually proved
Proposition 4.10**.**
There exist such that for every and , there exists a unique , such that if we denote
[TABLE]
then and .
4.5. Proof of Proposition 4.3: part II
Proposition 4.10 allows us to define the map
[TABLE]
Now we prove that this map satisfies the conditions of Proposition 4.3.
4.5.1. Normalization
By (4.11), each member of the family for satisfies the normalization condition. So by the pregluing construction, the approximate solutions for all satisfy the normalization condition. Since the exact solutions are corrected from approximate solutions by adding elements in the image of the right inverse, by Lemma 4.6, the normalization condition persists. This proves that . On the other hand, the function is transversal to [math] and vanishes at . It follows that for sufficiently small, is also transversal to zero and . This proves Item (2) of Proposition 4.3.
4.5.2. Injectivity
We prove that is injective. Injectivity on the lower stratum follows from the usual implicit function theorem. Indeed, the difference between an approximate solution and the corresponding exact solution is in the image of the right inverse of the linearization, while by construction, different approximate solutions differ by elements in the kernel of the linearization. Hence for two different elements of , which corresponds to different approximate solutions, their corresponding exact solutions are also different.
Now we explain why injectivity holds in the higher stratum. Suppose it is not the case, then there exist , , such that
[TABLE]
Since the underlying marked curves of and are isomorphic, the gluing parameters and are identical. Then follows from the implicit function theorem.
4.5.3. Surjectivity
Now we prove the surjectivity of . Define by
[TABLE]
By shrinking a little, we may assume that is defined over the closure of .
Lemma 4.11**.**
There exist and such that for each , is contained in .
Proof.
Suppose the lemma is not true. Then there is a sequence and a sequence of points which is not in the image of , and . Then consider the segment for . By using Lemma 4.5 and Lemma 4.8, one can show that as , converges to zero uniformly in . Hence we can apply the implicit function theorem: for each , there exists a unique such that
[TABLE]
Moreover, by (4.19),
[TABLE]
Denote , whose norm converges uniformly to zero as . Note that for each , lies in the image of . Suppose is the largest number in such that . Then . Take such that . So is bounded from below141414Recall that is obtained by pregluing the singular object . Since is on the boundary of , either is bounded from below or is bounded from below. In the former case the pregluing construction implies our claim. In the latter case, a certain kind of elliptic estimate shows that the distance between the singular object and is bounded from below. Then the pregluing construction also implies our claim.. However, by Lemma 4.5 and (4.20),
[TABLE]
Then
[TABLE]
This contradicts the convergence . The lemma is proved. ∎
Suppose the surjectivity doesn’t hold, then there exists a sequence of points converging to and . It suffices to consider the case that all are in the top stratum, i.e., they are genuine objects
[TABLE]
converging in weak topology to . Then for large enough, we can write . Here is at least a continuous object whose norm converges to zero.
To use Lemma 4.11, one needs to prove that and . Indeed, the domains of both and are identified with . By the definition of convergence which requires no energy is lost, for any , there exist and such that the energy of over the neck region is smaller than for all . Here . Then over this neck region, both and are close to the critical point . One can estimate the norms of over the neck region as in the Hamiltonian Floer theory, and the norm can be arbitrarily small if is small and is big. On the other hand, over the complement of the neck region, the smallness of together with the elliptic regularity shows that the norm of can also be arbitrarily small if is big.
Then for large , , where is given by Lemma 4.11. Hence for some . This contradicts our assumption. Hence we proved the surjectivity.
A well-known theorem in topology says that a continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism. Since the domain of the map is locally compact while the target of is Hausdorff, it follows that is a homeomorphism onto its image. This finishes the proof of Proposition 4.3.
4.6. Boundary charts for BPS soliton solutions
Suppose is a BPS soliton solution representing a point . In this subsection we will construct a local chart with boundary of the compactified moduli space around .
By definition, if we set , then the image of is contained in and is a nonconstant solution to the ODE
[TABLE]
We also view as a map from which is independent of the second variable . Set be a reference point and set . Then there is a real hyperplane which is transverse to . Choose a function which is locally a linear function near vanishing on .
Lemma 4.12**.**
If the perturbation is small enough, then there exists a finite-dimensional subspace such that
[TABLE]
Proof.
is essentially
[TABLE]
The last term is the Hessian of the perturbed superpotential applied to . It is a standard fact that when the Hessian is sufficiently small (in sense), obstructions only come from sections of Fourier mode zero, i.e., one can construct from sections which are invariant in the circle direction. It is indeed the case when the perturbation is sufficiently small, since the energy is proportional to the difference of critical values of the two critical points, and hence the solution should be sufficiently close to where the Hessian of the unperturbed superpotential vanishes. ∎
Choose as in Lemma 4.12. As before, for the principal component one can find an obstruction space . Then we define
[TABLE]
which has a trivial -action. We can also define a thickened moduli space as (4.5), while we have constructed an element from by adding the marking . Notice that since is -invariant, there is an -action in a neighborhood of that has a free -action.
To construct a local chart of the thickened moduli around , introduce the following vector space as before (see (4.6))
[TABLE]
Moreover, to induce a local chart of , introduce a real hypersurface (compare to (4.7))
[TABLE]
and have trivial -actions. For , let (resp. ) be the radius ball centered at the origin. Note that there is a homeomorphism
[TABLE]
This is the local model of the boundary charts. More precisely, we have the following proposition analogous to Proposition 4.3.
Proposition 4.13**.**
There exist and an -equivariant continuous map satisfying the following conditions.
- (1)
* and is a homeomorphism onto its image.* 2. (2)
Denote , then the map defined by
[TABLE]
is transversal to (in the topological sense) and .
Proof.
By the pregluing construction and applying the implicit function theorem, one can obtain and a family of exact solutions
[TABLE]
The -equivariance can be checked at each step of the construction so that the family only depends on 151515Lemma 4.12 is necessary to have the symmetry.. The only thing we want to emphasize is, one needs to choose a right inverse to which respects the symmetry and which satisfies that, for any ,
[TABLE]
i.e., the infinitesimal normalization condition. ∎
Proposition 4.13 implies the following corollary which is analogous to Corollary 4.4.
Corollary 4.14**.**
For , define
[TABLE]
let be the section induced from the map ; define by with image . Then for sufficiently small, the tuple is a local chart (with boundary) of around .
Proof.
The local surjectivity of follows from Item (1) of Proposition 4.13; the local injectivity follows from the lemma below whose proof is left to the reader. ∎
Lemma 4.15**.**
There is a neighborhood of satisfying the following conditions.
- (1)
For any , there exists a unique such that
[TABLE] 2. (2)
For any soliton solution representing a point in whose soliton component is , there exists a unique such that
[TABLE]
Corollary 4.16**.**
When the perturbation is small enough, is the disjoint union of and , both of which are compact.
Proof.
For each and , is represented by a soliton solution. The -equivariance of implies that it is a BPS soliton solution. Hence the image of is disjoint from , so is closed. On the other hand, is also closed since a sequence of -invariant solutions cannot converge to a non-invariant solution. Therefore the corollary holds. ∎
5. Technical Details of Gluing
In this section, we provide the details about certain notions used in Section 4. Meanwhile, we prove Lemma 4.5 and Lemma 4.6 in the gluing construction.
5.1. Proof of Lemma 4.5
Recall the approximate solution is obtained by pregluing the singular solution on the long cylinder . We use the simplified notation
[TABLE]
which won’t cause confusion within this proof.
The approximate solution fails to be a solution to the augmented vortex equation on the infinite cylinder . To estimate the failure, recall that we denoted
[TABLE]
Notice that by our definition which still satisfy . Hence we only have to estimate the first term. Let be the map that includes variables from the obstruction space . Since the volume form decays like , one has
[TABLE]
Here is the projection of the obstruction onto .
Now we estimate . Similar to the case of gluing pseudoholomorphic curves, it is nonzero only in the long cylinder . We only estimate over and the case for the other half of the long cylinder is the same. To estimate the failure over this long cylinder, first introduce the “constant” solution
[TABLE]
(It actually depends on and .) Then we can write
[TABLE]
which is very close to . We write the differential of the exponential map on as
[TABLE]
for sufficiently small, are sufficiently close to the parallel transport hence has bounded norm. Then there exists such that
[TABLE]
[TABLE]
On the other hand we write . Then over , for some , one has
[TABLE]
There is a similar estimate on the other half by replacing with the -twist of . Notice that , , and decay exponentially, in a rate faster than for some .
5.2. The inhomogeneous term
From now on to the end of this section, we prove Lemma 4.6. Here we give a more explicit characterization of the gradient vector field . The details can be found in [29].
Fact 5.1**.**
The following facts are true.
- (1)
Over the cylindrical end , there is a function depending on satisfying: 1) the correspondence is affine linear; 2) with respect to the trivialization ,
[TABLE]
3) if for some and , then , and the derivative is a bounded operator from to . 2. (2)
There is also a smooth function and only depends on the restriction of to a fixed compact subset of (see **[29, Definition 2.15]**). 3. (3)
For , over the cylindrical end we have
[TABLE]
Here at , depends smoothly on and . 4. (4)
The derivative of in is supported in . 5. (5)
For any , the set of zeroes of is
[TABLE]
Here is the perturbed superpotential defined by (3.2) (in our situation depends on ) and is viewed as an element of the group of R-symmetries acting on .
5.2.1. The linearized operator
Now we look at the linearization of . In three components,
[TABLE]
We denote by the first component and by the second and the third components. We look more carefully at over the cylindrical end . Here the first factor parametrizes deformations of , i.e. the perturbation of the superpotential. We write
[TABLE]
Here the first summand is
[TABLE]
The second and the third summands of (5.2) are
[TABLE]
[TABLE]
Here is a vector field along supported in (which may have nonzero limit at infinity) which appears because changing the gauge field will change the asymptotic limit of solutions. On the other hand, the linearization of reads
[TABLE]
For the approximate solution , we will use the same type of symbols by replacing the subscripts by . For example, the derivative the Witten equation will be decomposed as
[TABLE]
5.3. Augmentation of solitons
By definition, a soliton is only a solution to the Floer equation over the cylinder. However, to be glued with solutions over the principal component which has gauge fields, we need to regard a soliton as a map together with a flat connection. This is what we called an augmentation.
5.3.1. Some de Rham and Dolbeault theory on the cylinder
To add a gauge field to the soliton component, we need some more preparations. For , , let be the Banach space of functions on the cylinder such that
[TABLE]
It is easy to check that the “de Rham” operator
[TABLE]
is a bijection. Hence it has an inverse
[TABLE]
Similarly, the “Dolbeult” operator
[TABLE]
is an isomorphism. Its inverse is denoted by .
To prepare for the gluing construction, we need to introduce a new norm depending on a parameter . Indeed, the space is weighted by the function
[TABLE]
We define and define the corresponding norm by . When are clear from the context, we also abbreviate the norm by .
Then we want to define this type of norm on . Choose a cut-off function which is monotonic and changes from to [math] over . Define . Then we can decompose
[TABLE]
Then we define the norm over by this decomposition, namely
[TABLE]
Denote the space equipped with this norm by .
5.3.2. Augmenting soliton solutions
Recall that for fixed , given and , solitons connecting and live in a Banach manifold which depends on the parameters , and the value of (which depends on ).
The augmentations of solitons live in a bigger Banach manifold. Let be the set of pairs , where is a -map from to , and . Moreover, we require the following asymptotic condition on . Namely, is approximate to (resp. ) at (resp. ) in the -fashion. Namely, for , one can write
[TABLE]
[TABLE]
We define a bundle by the direct sum
[TABLE]
Notice that norm on is independent of but the norm on depends on . Now we define a family of sections of . Recall that the soliton equation defines a section which reads
[TABLE]
The augmented soliton equation, which includes the variable , is defined as follows. For , define
[TABLE]
Then the norm of the correspondence is independent of . Then define
[TABLE]
It is easy to to see that the only solution for is , and for , reduces to (5.8).
Then consider the linearization of . For fixed , denote this linearization at a soliton by
[TABLE]
By the expression (5.10), we can write
[TABLE]
Here is the operator of (5.5), and is the linearization of the original soliton equation at . The off-diagonal term
[TABLE]
is the derivative of with respect to the variation of the gauge filed . Among them , the norms on the domain and the target of are independent of .
Lemma 5.2**.**
The norm of is uniformly bounded for all .
Proof.
This is because the norms on is un-weighted while the norm on is weighted by the function . ∎
5.4. Right inverses along the soliton solution
To construct a right inverse for approximate solutions, we only need to consider the case for fixed value of . Hence we suppress the use of and in the notations. Given a soliton solution (which is already augmented in the above fashion) , we can write the linearized operator as
[TABLE]
Here is the linearization of the gauged Witten equation (including the gauge fixing) for the solution over the principal component. is the linearization of the operator . The off-diagonal term is the derivative of the term in the direction of variations of variables on the principal component, via the variation of the real parameter . More precisely,
[TABLE]
Notice that it is of rank one. Moreover, it only depends on the restriction of over a fixed compact subset of (see Item (2) of Fact 5.1).
When we include the obstruction spaces, we also denote the linearized operator by and the same symbols of the components , and .
Now we construct a right inverse of . Notice that before augmenting the soliton , we have chosen a right inverse (independent of )
[TABLE]
to the linearization of the soliton equation plus the obstruction . On the other hand, for , using the right inverse (5.6) we obtain
[TABLE]
Then define
[TABLE]
It is easy to see that this is a bounded right inverse to . The equivariance still holds since , and are equivariant. Moreover, since the first component above still lies in the image of , the infinitesimal normalization condition (4.15) still holds for . It then follows that
[TABLE]
is a bounded right inverse of .
We need the following technical results.
Lemma 5.3**.**
The norm of the operator is independent of . Moreover, there exist and such that for and sufficiently large,
[TABLE]
Proof.
only depends on the parameter through the variation of in the gradient term . Since is in the space where the norm is not shifted, the norm of is hence independent of . The second part of this lemma follows from the exponential decay of towards critical points. The details are omitted. ∎
Lemma 5.4**.**
* is uniformly bounded with respect to the norm for all .*
Proof.
The operator is independent of , hence its norm is uniformly bounded. By the above lemma, the norm of is independent of . Hence the norm of is uniformly bounded for all if we can show that has its norm uniformly bounded. Indeed, the norms on the domains and the targets of and are not shifted hence their norms are uniformly bounded. Moreover, is uniformly bounded since it is a translation invariant operator. By Lemma 5.2, the norm of is also uniformly bounded. Hence by the expression of ((5.12)), the norm of is uniformly bounded. ∎
5.5. Proof of Lemma 4.6
The values of the constants are reset within this subsection.
5.5.1. The auxiliary operations
Recall the definition of in (4.12). One has and is very close to . Therefore one can define the parallel transport
[TABLE]
[TABLE]
Similarly we define
[TABLE]
[TABLE]
Choose and two cut-off functions such that when , ; when , ; when , ; when , (see the picture). Moreover, we can have
[TABLE]
Abbreviate them by and respectively.
[math]2T$$(2+\hbar)T$$\chi_{\infty}$$-(2+\hbar)T$$-2T[math]
We define
[TABLE]
as follows. Take and . Denote where is the function associated to given in Fact 5.1 and is its derivative in direction. Therefore over , . We also recall that from , we can decompose (see (5.7))
[TABLE]
By (5.9) we defined such that
[TABLE]
Then we define
[TABLE]
[TABLE]
and
[TABLE]
Recall the parallel transport maps and . We define
[TABLE]
[TABLE]
and
[TABLE]
It is an infinitesimal deformation of the approximate solution. Moreover, one can extend to an operator (which is denoted by the same symbol)
[TABLE]
On the other hand, we define cut and paste as linear maps
[TABLE]
such that is a left inverse of . More precisely, let where , , . Define where
[TABLE]
[TABLE]
Here by abuse of notation, we regard and as operators on and (indeed identically) as “parallel transport.” On the other hand, if , , then define
[TABLE]
5.5.2. The approximate right inverse and estimates
Recall that at the soliton solution , we have a linear operator written as in (5.11), and we have constructed a right inverse written as in (5.13). Recall that the relation between and is . We define the approximate right inverse to by
[TABLE]
Lemma 5.5**.**
For sufficiently large, is bounded by a constant independent of .
Proof.
We prove this lemma by showing the three factors of are all uniformly bounded. First, for , by the definition (5.17) (5.18) and the definition of the weighted Sobolev norm on (see Subsection 5.3), comparing the weight functions, we have . Second, by Lemma 5.4 the norm of is independent of .
Lastly we show that is uniformly bounded. We only provide the detail of the estimate of where . For other variables the estimates are similar and even simpler. By definition,
[TABLE]
We abbreviate . Let be the weight function on which is equal to over the cylindrical end. Then for some constant which is independent of , we have
[TABLE]
By (5.7), (5.9), and the definition of the norm on the cylinder, one has
[TABLE]
On the other hand, the map
[TABLE]
is uniformly bounded. Combining with (5.14), we have
[TABLE]
for some constant independent of . ∎
Lemma 5.6**.**
There exist such that for sufficiently large,
[TABLE]
The proof of Lemma 5.6 occupies the remaining of this section. Together with Lemma 5.5 it implies that for sufficiently large (equivalently, sufficiently small), there is a right inverse to having the same image of , which has a uniformly bounded norm, independent of , and is equivariant with respect to the symmetry group of the soliton solution. This finishes the proof of Lemma 4.6.
5.5.3. Proof of Lemma 5.6
Take . Denote and
[TABLE]
[TABLE]
Using the notations of (5.16),
[TABLE]
Note that by the definitions of and , as well as the property of ,
[TABLE]
Then Lemma 5.6 follows from the three lemmata below. Here labels the converging rate (exponentially) of the solution towards critical points.
Lemma 5.7**.**
For some constant , we have
[TABLE]
Lemma 5.8**.**
For some constant , we have
[TABLE]
Lemma 5.9**.**
.
Now we start to prove the three lemmata, 5.7, 5.8 and 5.9. Let
[TABLE]
be the natural projections. The proofs are all straightforward calculations.
Proof of Lemma 5.7.
Using the notations specified in Subsection 5.2 (see (5.2) and nearby discussions), we have
[TABLE]
On the other hand, since over , , we have
[TABLE]
Using this fact, we see that over ,
[TABLE]
Here we used the fact that which follows from Item (2) of Fact 5.1. One sees that (5.20) is equal to (5.19) over , where and . Now we estimate each part of (5.20). By (5.14), we have
[TABLE]
By Lemma 5.3, for some , we have
[TABLE]
Over the support of , the distance between and is controlled by . Hence for some ,
[TABLE]
Similarly, for some ,
[TABLE]
Lastly, on the support of , by (5.4),
[TABLE]
Since the distance between and is exponentially small, and is exponentially small on the support of , we see that for some ,
[TABLE]
Therefore, by (5.21)–(5.24), for an appropriate , we have
[TABLE]
Now we estimate . By definition, over ,
[TABLE]
On the other hand, using the fact that over , , one obtains
[TABLE]
Here , are certain bilinear functions. One sees that (5.28) is equal to (5.27) over where . Then for certain and , we have
[TABLE]
(5.26) and (5.29) conclude this Lemma. ∎
Proof of Lemma 5.8.
Over , one has
[TABLE]
Over , by the definition of ,
[TABLE]
Here is the characteristic function of . Note that in the last equality we used the fact that over ,
[TABLE]
Hence (5.31) is equal to (5.30) over , where and . Then for some we have
[TABLE]
In deriving the last inequality, we used Lemma 5.3, (5.14), and the fact that and are exponentially closed over .
On the other hand, over , by (5.22),
[TABLE]
Using a -version of (5.32), i.e., over , , we see that over ,
[TABLE]
The last equality follows from the fact that the supports of and are disjoint. (5.34) is equal to (5.33) over where . Then for some ,
[TABLE]
Proof of Lemma 5.9.
By definition (see (5.16))
[TABLE]
Since is supported in , is also supported in 161616Here we used Item (2) of Fact 5.1.. Hence after applying , vanishes. On the other hand, by (5.16)
[TABLE]
Since is supported in , by the definition of ,
[TABLE]
On the other hand, by the definition of (see (5.10)),
[TABLE]
By (5.15), it is supported in , hence vanishes after composing with . Therefore . ∎
6. Constructing an Atlas
The local charts constructed in Section 4 don’t have coordinate changes among them. In this section we construct from these local charts an atlas which satisfies [31, Definition A.14]. This section is organized as follows. In Subsection 6.1 we construct the charts for the lower stratum and state the main propositions. In Subsection 6.2 we construct the transition functions between these charts. In Subsection 6.3 we complete the atlas construction on the whole moduli space , which finishes the proof of Theorem 3.4. In Subsection 6.4 we determine the orientation on the boundary component corresponding to BPS soliton solutions.
6.1. The sum charts
In Section 4, for each we have constructed a local chart for . We first want to construct a system of charts whose footprints cover , the moduli space of non-BPS soliton solutions, and which have coordinate changes among them. By Corollary 4.16, we can do the same thing independently for , the moduli space of BPS soliton solutions.
6.1.1. The thickened moduli
By the compactness of , one can choose finitely many such that the union of the footprints of the charts contains . Let be the automorphism group of the representative of . Define
[TABLE]
This set has a natural partial order given by inclusion of sets. For each , define
[TABLE]
We still need to consider a thickened moduli first. Let be the moduli space of configuration of points of indexed by elements in . There is a universal curve with a forgetful map . Consider the stratum of corresponding to configurations in which all points lie in a rational component attached to , and let be a small open neighborhood of this stratum and be the corresponding universal curve. Let denote a point of and be the preimage of under , which is a curve having at most two components and -tuple of markings. Similar as before, we can lift the fibre bundle to a family over . We can still define the Banach manifolds in a similar fashion as defining previously.
By forgetting all markings but only the -th marking, one also obtains a map . Hence one obtains a linear map
[TABLE]
Define the thickened moduli space by (compare to (4.5))
[TABLE]
6.1.2. Local charts
Choose . In Section 4 we constructed a family for and that satisfy the normalization condition for . Consider
[TABLE]
This is a -invariant subset of . Since each is close to for all (up to translation on the rational component), there exists exactly one -orbit of points on the rational component such that satisfies the normalization condition with respect to . By adding these markings, and using the inclusion , we obtain a -invariant subset together with a covering map .
The construction of the local chart is based on the following proposition.
Proposition 6.1**.**
- (1)
There exists a -invariant open subset containing which is homeomorphic to a topological manifold. 2. (2)
There is a family of right inverses for all to the defining equation of , such that the -components of the images of vanish for all . 3. (3)
The map defined by
[TABLE]
is -invariant and is transversal to .
Proof.
The fact that a neighborhood of in is a topological manifold follows from transversality and the gluing construction of Section 4. Moreover, since we have used a right inverse to the defining equation of to construct the local chart , and , we can use the same right inverse to construct the local chart of .
For the transversality of , denote . Let , be the intersections with the lower stratum. Then there is a local homeomorphism which is given by “pregluing correction.” Then under this homeomorphism, is a topological submanifold. does not coincide with , but every point in the former can be corrected in a unique way to a point in the latter, by moving the marked points. This shows that is a topological submanifold with a canonical germ of tubular neighborhoods. Hence is transversal. ∎
This proposition is proved in the next subsection. From it one obtains a local chart of . Shrink the footprint to such that the union of still cover . Define and let be the subset defined in the same way as via (6.1) but replacing with . Then there exists a continuous family of -invariant open subsets which as converge to . Since is transverse,
[TABLE]
is a topological manifold with a -action. Let the quotient orbifold be , on which we have an orbibundle
[TABLE]
There is a natural section , and a natural map , whose image is denoted by .
Corollary 6.2**.**
For sufficiently small , the tuple is a local chart of whose footprint contains an open neighborhood of .
Proof.
Same as Corollary 4.4. ∎
6.2. The coordinate changes
For each and each , fix some satisfying Corollary 6.2 and denote the chart by . In this section we would like to construct various coordinate changes among these charts.
First, consider the following subsets of (see (6.1))
[TABLE]
and the corresponding subsets of , denoted by
[TABLE]
For each element of , we can upgrade them by adding markings for all , such that together with satisfies the normalization condition with respect to . The way of adding is not unique, but different choices differ by an action of . For sufficiently small, denote by the -neighborhood of inside . Then this operation can be extended to all members of , and we obtain a covering
[TABLE]
Here is the direct product of for all . There is also a -action on the total space . By the inclusion , is naturally included into the thickened moduli space and is contained in the locus of . Moreover, choosing , if is small enough, then it is included in and the inclusion is -equivariant. Hence (see Remark 2.5 and the diagram (2.1)) it induces maps
[TABLE]
Lemma 6.3**.**
* is an orbibundle embedding.*
Proof.
It is obvious that and are injective. Hence by definition, we only need to construct tubular neighborhoods to fulfill the locally flatness condition of . By Item (2) of Proposition 6.1, there is a family of right inverses for all to the linearization of . Consider the “augmented” equation
[TABLE]
One also has a family of right inverses to the linearizations of this equation. Indeed, the derivative of restricted to the direction of varying the parameter plus the direction of reparametrizing the soliton is an isomorphism onto , so it has a unique right inverse.
Equip every obstruction space a -invariant norm. Notice that over the embedding image of in , the -component of the values of the projection , denoted by , is zero. Hence for any , there exists a neighborhood of the embedding image of over which . For each , forgetting markings labeled by we obtain a parameter ; write . Then the family satisfy
[TABLE]
where the right hand side is linear in and hence its norm is controlled by . Hence by the implicit function theorem, using the right inverses we have discussed (we use different Banach manifolds in the lower stratum and the top stratum), for sufficiently small, one can correct it to satisfying and . This gives a (germ of) tubular neighborhood of the embedding image of , and parametrizes the normal directions. ∎
Everything in the cocycle condition (see Definition 2.4) is straightforward to check. The same construction can be done for BPS-solitons. We omit the details. The set indexing charts for is denoted by . Define . We summarize what we have obtained so far.
Proposition 6.4**.**
- (1)
For each , there is a local chart of . The footprints of these charts cover . 2. (2)
Whenever , there is a coordinate change whose footprint contains . 3. (3)
These coordinate changes satisfy the cocycle condition.
6.3. The virtual orbifold atlas
The charts constructed so far cover a neighborhood of the lower stratum. We need to construct more charts to cover the remaining of the moduli space. First, there is a precompact open neighborhoods such that
[TABLE]
Then for all , one can construct a local chart around as in Subsection 4.1. We can choose finitely many , such that
[TABLE]
We obtain an open cover of by the open sets for . Abbreviate . We can shrink them to precompact open subsets such that all still cover . Then define
[TABLE]
We still use to denote the partial order on defined by inclusions of subsets. Define
[TABLE]
We have constructed charts for all whose footprints cover . Using the same method as in [31] it is much easier to construct charts for all whose footprints cover . We have also constructed coordinate changes for all and whose footprints contain . It is also much easier to construct coordinate changes for all other relations in , whose footprints cover .
So far we have construction a collection of charts and a collection of coordinate changes that satisfy the cocycle condition. However to make these data a virtual orbifold atlas, their footprints have to satisfy the overlapping condition of Definition 2.4. This can be done by properly shrinking the charts. The specific method of shrinking we use here is modified from a similar one in [20].
Order the set as such that if then . For each choose precompact shrinkings
[TABLE]
Then for , define
[TABLE]
Lemma 6.5**.**
The collection is an open cover of and satisfies the overlapping condition, namely
[TABLE]
Proof.
We first prove the overlapping condition. Suppose with . If does not hold, then there is some . Then
[TABLE]
This contradicts because
[TABLE]
Therefore . So the overlapping condition holds. To prove that all cover , take an arbitrary . Since all cover , there is some such that
[TABLE]
Let be the largest number satisfying the above relation. We claim that . If it is not the case, then there is some such that , and there is some such that contains and . Then
[TABLE]
which contradicts the maximality of . This proves the covering property.∎
Therefore we can shrink the charts so that the shrunk footprints are . To simplify the notations, we still denote the shrunk charts by and shrunk footprints by . Then by Definition 2.4, we have constructed a virtual orbifold atlas
[TABLE]
on the moduli space . Therefore we finish the proof of Theorem 3.4 except for the part about orientations.
6.4. Orientation of the boundary
In this last part, we settle the signs appeared in the wall-crossing formula Theorem 3.3. As is well-known, choosing (independently) orientations on the unstable manifolds and of the negative gradient flow line of the real part of determines orientations on the moduli space and . These orientations can be extended in a continuous way to all . It then induces an orientation on all charts over the uncompactified moduli space . So the first two components of (3.6) are oriented as
[TABLE]
Now we determine the induced orientation on the third component . Let be represented by a soliton solution . To study orientation, it suffices to assume that is regular and is maximally transverse. Here being maximally transverse means that the linearization of the equation
[TABLE]
at is surjective (see [31, Section 5] for more details).
Since we are only considering the index zero case, the orientation on the zero dimensional moduli space determines a sign
[TABLE]
On the other hand, the element in the determinant line of the linearization of the gradient flow equation provides a sign
[TABLE]
Then the counting appearing in Theorem 3.3 is defined as
[TABLE]
Recall that in Corollary 4.14, we constructed a family of objects
[TABLE]
which converges as to the soliton solution such that this family gives a continuous map which is a homeomorphism onto its image. Here the construction depends on choosing a concrete BPS soliton as part of a representative of the point . On the other hand, if we replace by a reparametrization defined as
[TABLE]
then fixing the family as varies near [math] also gives a local homeomorphism into (which is a one-dimensional manifold with boundary near ). It is easy to see that the direction in which increases is the same as the direction in which decreases. Therefore, to evaluate the boundary contribution, we need to compare this direction with the canonical orientation class in (where is the linearization of the gauged Witten equation, including the gauge fixing, at ).
Now we are ready to examine the orientation on . Write the linearized operator
[TABLE]
at the BPS soliton solution in the block form as
[TABLE]
Here we use the type of notations similar to (5.11). Notice that and are the directional derivatives in the direction of increasing the value of , hence are of rank one; is the derivative of the parameter in (5.8), hence is also of rank one. Since both and are Fredholm and oriented, there is an naturally induced orientation of the following deformed operator
[TABLE]
Let be the operator represented by the last row above. Introduce the symbol which means “canonically isomorphic” (up to homotopy). Then one has
[TABLE]
Here the first isomorphism is from the coherence of the orientation in the sense of Floer–Hofer [10], and the second isomorphism is from the deformation of operators described above. At , a positive element of is
[TABLE]
By [31, Lemma 6.1], the canonical identification sends to
[TABLE]
By the calculation we did in the last part of [31, Section 6], we know that the third sign is equal to . On the other hand, as shown previously, is the direction opposite to there the gluing parameter increases (this is exactly the opposite to the sign appeared in Picard–Lefschetz formula, see [31, Section 6]). Therefore, the third oriented component of is
[TABLE]
where is equipped with the product orientation (though the moduli space is not a product). This finishes the proof of Theorem 3.4 and hence Theorem 3.3.
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