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Volume 27, Issue 1, October 2019, Pages 170 - 184

Fredholm Property of Operators from 2D String Field Theory

Authors
Hai-Long Her
Department of Mathematics, Jinan University, Guangzhou, 510632, China,hailongher@jnu.edu.cn
Received 14 May 2019, Accepted 24 August 2019, Available Online 25 October 2019.
DOI
10.1080/14029251.2020.1683998How to use a DOI?
Keywords
Fredholm property; Floer homology; transversal non-degeneracy
Abstract

In a recent study of Landau-Ginzburg model of string field theory by Gaiotto, Moore and Witten, there appears a type of perturbed Cauchy-Riemann equation, i.e. the ζ-instanton equation. Solutions of ζ-instanton equation have degenerate asymptotics. This degeneracy is a severe restriction for obtaining the Fredholm property and constructing relevant homology theory. In this article, we study the Fredholm property of a sort of differential operators with degenerate asymptotics. As an application, we verify certain Fredholm property of the linearized operator of ζ-instanton equations.

Copyright
© 2020 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

It is well-known that critical points of a generic smooth function f : M → ℝ on a Riemannian manifold (M,g) determine the topology of this manifold. This is the subject of Morse theory ([14]). By studying the gradient flow lines between critical points of Morse function f, one can define the so-called Morse-Smale-Witten complex, its homology is called Morse homology ([20]). Symplectic Floer homology ([5, 6, 8, 11, 13, 17]), which was originally invented as a tool of proving the Arnold conjecture ([1]) on lower bound of number of non-degenerate Hamiltonian time 1-periodic orbits on a compact symplectic manifold (X, ω), can be considered as a generalization to infinite dimension of the Morse homology. The analogous role of Morse function on a Riemannian manifold is played by the symplectic action functional defined on loop space ℒX, critical points of action functional are 1-periodic solutions of Hamiltonian equation, gradient flow lines between critical points are played by Floer’s connecting trajectories which are solutions of some significant elliptic partial differential equations. As a result, for some classes of symplectic manifolds, Floer constructed such infinite dimensional version of MSW complex and verified the lower bound estimation by Arnold conjecture via its homology.

The PDE studied by Floer is the perturbed Cauchy-Riemann equation

∂¯J,Hu=∂u∂s+J∂u∂t+∇H=0,(1.1)
where J is an ω-compatible almost complex structure on X, H = Ht(·) is a time 1-periodic Hamiltonian function on X, the gradient ∇ H is determined by the associated metric ω(·,J·), and the map u : ℝ × S1 → X satisfying (1.1) is called a perturbed J-holomorphic cylinder. One key point in the construction of Floer homology is to verify that, at every solution u of equation (1.1), asymptotic to 1-periodic Hamiltonian orbits as s →±∞, the differential of ∂¯J,H, denoted by (see (2.5))
Du:W1,p(ℝ×S1,u*TX)→Lp(ℝ×S1,u*TX),
is a Fredholm operator. Recall that a bounded linear operator between Banach spaces is called a Fredholm operator if both its kernel and cokernel are of finite dimension. Floer showed that if solutions u satisfy some non-degenerate asymptotic conditions, then the Fredholm property holds. Based on this property, roughly, the index theorem can apply to calculate the dimension of moduli space of solutions of equation (1.1) (asymptotic to 1-periodic Hamiltonian orbits), then the differential of Floer complex can be defined.

In the recent study of Landau-Ginzburg(LG)-model for 2d string field theory ([9]), there also appears perturbed Cauchy-Riemann equation, i.e. the physicists’ so-called ζ-instanton equation ∂¯J,Hiju=0 with u : ℝ × ℝ → X (see next section for more details). Due to the time independence of Hamiltonian Hij, which is derived from the superpotential that is a J-holomorphic (time-independent) Morse function

W:(X,J)→ℂ,
however, solutions u of ζ-instanton equation only have degenerate asymptotics, which are called ζ-solitons (satisfying (2.8) and (2.9) below). Here degeneracy of ζ-solitons means that for each ζ-soliton there always exists an ℝ-family of solutions satisfying the ζ-soliton equation. Inspired by [9], some Floer-type homology theory is supposed to be constructed from moduli space of ζ-instantons. While Floer’s argument for non-degenerate 1-periodic solutions can not apply directly due to the degeneracy of asymptotics. In this article, we study operators related to those derived from ζ-instanton equation defined on some weighted Sobolev spaces. Under some weaker non-degeneracy conditions, we obtain the following main result

Theorem 1.1.

Assume that solutions γ±(t) of ζ-soliton equations (2.8) and (2.9) are transversally non-degenerate. For each solution u of ζ-instanton equation satisfying some asymptotic conditions (i.e. (2.10)–(2.12)), the map

Du:𝒱+,−⊕Wk,p,A(ℝ2;u*TX)→Wk−1,p,A(ℝ2;u*TX)
defined by (3.1) is Fredholm for any k ∈ ℝ,p > 1, and its index depends neither on k nor on p, which can be given by
Ind  Du=μ(γ+,γ−,u)+2,
where Wk,p,A(ℝ2;u*TX) are A-weighted Sobolev spaces (see subsection 3.1) and μ is the relative Maslov index.

This result will be proved in section 4 (Corollary 4.1). This is a first step of an ongoing proposal of constructing homology theory for LG-model. Roughly, based on such Fredholn property, we will study the Morse-Bott type moduli space of ζ-instantons, define Maslov-type index for ζ-instanton and calculate the dimension of moduli space, then the Floer complex can be constructed via moduli space of certain Floer-type trajectories connecting ζ-solitons. The compactness and transversality of moduli space of Floer trajectories are crucial properties to be verified such that all constructions go through. The relevant works will appear in another paper.

It turns out that one of essential steps of overcoming the (partial) degeneracy is to study certain analytic problem presented in the following sections. In the next section, to set the problem, we introduce more notations and give more detailed description of the motivation of this work.

We remark that after submitting this work, we were informed that Jiang [12] also independently studied some analytic problems related to LG-model with different setting and method.

2. Differential operators

2.1. Differential operators in Floer theory

Given a compact symplectic manifold (X,ω) and a time 1-periodic function Ht(·)= Ht+1(·), the Hamiltonian equation is

ddtx(t)=XHt,(2.1)
where XHt is the Hamiltonian vector field associated to Ht, which can be defined by ω(·, XHt) = dHt(·). The solutions of (2.1) generate a family of symplectomorphisms ϕt : X → X such that
ddtφt=XHt°φt,          φ0=Id.

We say an almost complex structure J on TX is compatible with the symplectic form ω if 〈·,·〉 = ω(·,J·) define a Riemannian metric on X. Denote by 𝒥 = 𝒥 (X, ω) the space of ω-compatible almost complex structures. For any J ∈ 𝒥 and smooth maps u : ℝ × S1 → X, the elliptic PDE Floer studied is the perturbed Cauchy-Riemann equation

∂u∂s+J(u)(∂u∂t−XHt)=0(2.2)
satisfying boundary conditions
lims→−∞u(s,t)=x+(t),          lims→+∞u(s,t)=x−(t),(2.3)
where x±(t) are two non-degenerate 1-periodic solutions of (2.1). A 1-periodic solution x(t) of (2.1) is non-degenerate if the following condition holds
det(dφ1(x(0))−Id)≠0.(2.4)

Denote by ℳ(x+,x−,H,J) the (unparameterized) moduli space of solutions of (2.2) and (2.3). For generic J and Ht, these spaces are smooth finite dimensional manifolds ([7, 18]). This is called transversality property which is very important in the construction of Floer homology and can be regarded as an analog to the Morse-Smale-Thom transversality in Morse theory.

Denote the left hand side of (2.2) by

∂¯J,Hu=∂u∂s+J(u)(∂u∂t−XHt).

Then ∂¯J,H can be thought of as a section of a Banach vector bundle ℰ → ℬ, where ℬ is a Banach manifold which is some W1,p-completion of the space of smooth maps C∞(ℝ × S1,X). The fiber over u ∈ ℬ is ℰu = Lp(ℝ × S1,u*TX). Then the moduli space ℳ (x+,x−,H,J) is just the zero set of the section ∂¯J,H:ℬ→ℰ. A key step to prove that for generic J and Ht the moduli space ℳ (x+,x−,H,J) is a smooth manifold is to show that the section ∂¯J,H is a Fredholm section. That is to say, at each u ∈ ℳ (x+,x−,H,J), the vertical differential Du=Du∂¯J,H of ∂¯J,H is a Fredholm operator. Explicitly,

Du:W1,p(ℝ×S1,u*TX)→Lp(ℝ×S1,u*TX)Duξ=∇sξ+J(u)(∇tξ−∇ξXHt)+∇ξJ(u)(∂tu−XHt(u)),(2.5)
where ∇ denotes a Levi-Civita connection with respect to the metric 〈·, ·〉 . Floer [6] proved that if x±(t) are non-degenerate, then Du is Fredholm operator. Moreover, the Fredholm index of Du is the difference of Conley-Zehnder indices associated with Hamiltonian 1-periodic solutions x± (see [4, 17]).

To simplify the formula (2.5) one can choose a unitary trivialization of the vector bundle u*TX → ℝ × S1. This is a smooth family of vector space isomorphisms

Ψ(s,t):(ℝ2n,ω0,J0)→(Tu(s,t)X,ω,J),
where ω0 and J0 are standard symplectic and complex structures on ℝ2n. Under this trivialization, it is equivalent to study an operator acting on vector-valued functions
L:W1,p(ℝ×S1,ℝ2n)→Lp(ℝ×S1,ℝ2n)Lξ=∂sξ+J0∂tξ+Sξ,
where 2n × 2n matrices S(s,t) are defined by
S=Ψ−1[∇sΨ+J(∇tΨ−∇ΨXHt)+∇ΨJ(u)(∂tu−XHt(u))].

We can define ψ±(t)=lims→∓∞Ψ(s,t) and

S±(t)=lims→∓∞S(s,t)=ψ±−1J(∇tψ±−∇ψ±XHt).

By [18] we know that both S± and J0S± are symmetric, and up to a compact perturbation one can assume S is symmetric for all s and t. Then one can associate to the symmetric matrix-valued function a symplectic matrix-valued function P : ℝ × ℝ → Sp(2n), where Sp(2n) := {Φ ∈ M2n×2n | ΦT J0Φ = J0} is the group of symplectic matrices, defined by

∂tP(s,t)−J0S(s,t)P(s,t)=0,         P(s,0)=Id.(2.6)

Thus we have symplectic paths P±(t)=lims→∓∞P(s,t). Then to show that operator Du is Fredholm is equivalent to verifying that under the non-degenerate conditions

det(P±(1)−Id)≠0,(2.7)
the operator L = ∂s + J0∂t + S is Fredholm.

Definition 2.1.

Symplectic paths P±(t) are called asymptotics of the operator L. We say the operator L has non-degenerate asymptotics if (2.7) hold, which is equivalent to that 1-periodic solutions x± are non-degenerate.

Denote by l± = ∂t − J0S±(t). l± can be regarded as an expression under the trivialization of the linearized operator of Hamiltonian equation (2.1) at solutions x±. Then the non-degenerate condition (2.7) is equivalent to that operators l± are invertible. Also we note that another condition which is equivalent to (2.7) is that matrices S± have no vanishing eigenvalues.

2.2. Differential operators in LG-model

Let (X,ω = dλ) be an exact symplectic manifold with boundary and with a compatible almost complex structure J, and W : X → ℂ be a Lefschetz fibration ([19]) which is a J-holomorphic Morse function. To this data physicists associate a Landau-Ginzburg (LG) model.

For any pair of distinct critical points xi, xj of W, denote the space of smooth paths connecting xi and xj by

𝒫Mi,j:={γ∈C∞(ℝ,X)|limt→−∞γ(t)=xi,      limt→+∞γ(t)=xj}

The action functional

𝒜W:𝒫Mi,j→ℝ
can be defined as
𝒜W(γ)=∫−∞+∞γ*λ+∫−∞+∞Re(ζij−1W°γ(t))dt,
where ζij is the phase such that iζij=W(xj)−W(xi)|W(xj)−W(xi)| is the unit in ℂ.

The critical curves of the action functional 𝒜W are solutions of physicists’ ζ-soliton equation

ddtγ(t)=J∇[Re(ζij−1W(γ(t)))],(2.8)
satisfying boundary conditions
limt→−∞γ(t)=xi,         limt→+∞γ(t)=xj.(2.9)
which can be considered as an equation for phase flow of autonomous Hamiltonian Hij:=Re(ζij−1W)a. The gradient is defined with respect to the associated Riemannian metric g(·,·)= ω(·,J·).

We generically assume that, to critical points xi and xj, the associated Lefschetz thimbles Li and Lj intersect transversally in the fiber W−1(z0) over a regular value z0 of W on the line segment in ℂ between W(xi) and W(xj). In this case, there will be a finite number of geometrically different critical curves of 𝒜W, one for each intersection point p ∈ Li ∩ Lj. Denote such a unparameterized curve by Γp which actually represents a ℝ-family of solutions of (2.8) and (2.9) because (2.8) is an autonomous Hamiltonian equation. Denote by 𝒫ij = ∪p∈Li∩Lj Γp.

Definition 2.2.

We say a solution γ(t) of (2.8) and (2.9) are transversally non-degenerate if for each intersection point p ∈ Li ∩ Lj the unparameterized curve Γp are isolated. This is equivalent to the assumption that the associated Lefschetz thimbles Li and Lj intersect transversally in a fiber W−1(z0) over a regular value of W.

For two distinct Γ+, Γ− ∈ 𝒫ij, the ζ-instanton equation is the perturbed Cauchy-Riemann equation

∂u∂s+J∂u∂t+∇Hij=0,(2.10)
satisfying
limt→−∞u(s,t)=xi,         limt→+∞u(s,t)=xj,(2.11)
lims→+∞u(s,t)=γ−(t)∈Γ−,         lims→−∞u(s,t)=γ+(t)∈Γ+.(2.12)

In [9], Physicists formulate an outline of construction of invariants via solutions of ζ-instanton equation (2.10) satisfying various boundary conditions. In particular, when solutions u of (2.10) satisfy boundary conditions (2.11) and (2.12), some sort of Morse-Floer type homology might be defined. It is a challenge for mathematicians to realize these constructions. One may expect that, by using Floer’s arguments on the study of the moduli space of solutions of (2.10), (2.11) and (2.12), similar construction will go through provided that some non-degenerate conditions hold. However, it is clear that Hij is time independent and (2.8) is an autonomous Hamiltonian equation which has no non-degenerate solutions. Therefore, the argument of Floer can not apply directly. Explicitly, in current LG-model setting, the differential Du of the equation (2.10) is

Du:W1,p(ℝ×ℝ,u*TX)→Lp(ℝ×ℝ,u*TX)Duξ=∇sξ+J(u)(∇tξ−∇ξXHij)+∇ξJ(u)(∂tu−XHij(u)),(2.13)
and, under a unitary trivialization Ψ of the vector bundle u*TX → ℝ2, the equivalently expressed operator L is of similar form
L:W1,p(ℝ2,ℝ2n)→Lp(ℝ2,ℝ2n)Lξ=∂sξ+J0∂tξ+Sξ,(2.14)
where S(s,t) are defined by
S=Ψ−1[∇sΨ+J(∇tΨ−∇ΨXHij)+∇ΨJ(∂tu−XHij)].(2.15)

Similarly, we can define ψ±(t)=lims→∓∞Ψ(s,t) and

S±(t)=lims→∓∞S(s,t)=ψ±−1J(∇tψ±−∇ψ±XHij).(2.16)

Since solutions of autonomous Hamiltonian equation (2.8) and (2.9) always occur in an ℝ-family, the asymptotics of L are not non-degenerate any more. Fortunately, if solutions γ(t) of equations (2.8) and (2.9) are still of some non-degeneracy in the direction normal to the subspace Tγ ⊕ JTγ, which we call transversal non-degeneracy, then some Fredhom property may also be obtained. In this paper, we will show that, working with some weighted Sobolev norm, we can do some perturbation to get non-degenerate asymptotics and prove the following result

Theorem 2.1 (Theorem 4.1).

Given a strandard symplectic space with the standard complex structure (ℝ2n,ω0,J0). Let S : ℝ2 → M2n×2n(ℝ) be a smooth bounded matrix-valued function such that S(s,t) are symmetric for all (s,t) ∈ ℝ2. Assume that the following conditions hold:

  1. (i)

    As s →∓ ∞,S(s,t) converges to S±(t) in the C0-topology such that for each t ∈ ℝ, the matrix J0S±(t) is symmetric;

  2. (ii)

    As t → ∓ ∞,S(s,t) converges to constant symmetric matrices in C0-topology.

  3. (iii)

    There are splittings for those two paths of matrices as

    S±(t):=(000S˜±(t)),
    where S˜±(t):=−J0P˙±(t)P±(t)−1 and P± are two non-degenerate smooth paths in symplectic group Sp(2n − 2) (see Definition 3.2).

    Then the operator L = ∂s + J0∂t + S between some weighted Sobolev spaces (see Definitions 3.1, 3.3 and (3.2)) is Fredholm.

As a result, Theorem 1.1 holds. That is, under the assumption that solutions of (2.8) and (2.9) are transversally non-degenerate and in the sense of weighted norm, the operator Du is Fredholm. This is the first step for succeeding works on constructing Morse-Floer type homology and even A∞-category in LG-model.

We remark that in [10] Haydys studied the moduli space of solutions of a modified version of ζ-instanton equation. After suitable modification of the ζ-soliton equation the solutions of the new equation turn to be non-degenerate, while the physical meaning of the new equation seems not so obvious. However, some new methods are introduced in [10] and the results are inspiring for our current research.

3. Operators on weighted spaces

3.1. Weighted spaces

In the setting of LG-model, the domain of pseudo-holomorphic maps u is ℝ2 with coordinates s and t. We can obtain a compactification ℝ2¯ of the ℝ2 by extending the coordinate s and t to ±∞ such that (s,±∞)=(s′,±∞) for ∀ s,s′ ∈ [−∞,+∞]. Topologically, ℝ2¯ is identified with a digon. Denote by ℓ± the 1-dimensional boundary component {(∓∞, t), t ∈ [−∞,+∞]}. Let E be a symplectic vector bundle of rank 2n over ℝ2 together with two families of symplectic vector spaces Et± which are symplectic-linearly identified with a standard symplectic space with symplectic splitting (ℝ2,ω0) ⊕ (ℝ2n−2,ω0) for all t ∈ ℝ. These give rise to a symplectic vector bundle Ē over the compactification ℝ2¯ such that fibres over the compactification boundary have fixed symplectic splittings and trivialization, that is we have symplectic linear identifications

E¯(+∞,t)≅E¯(−∞,t)≅(ℝ2,ω0)⊕(ℝ2n−2,ω0)
for each t ∈ [−∞,+∞].

We choose a particular smooth function F(t) > 0 such that F˙(t)=dFdt=e−t2 and F(0)= 1. Recall that for a matrix B, the exponential of B is eB=Id+B+12!B2+⋯+1n!Bn+⋯.

Definition 3.1.

Given a symplectic vector bundle E over ℝ2 and symplectic vector spaces E(±∞,t) as above. Let J be a compatible complex structure on E. For A > 0, an A-weighted (or (A,J)-twisted) Sobolev space of sections ξ of E, denoted by W k,p,A(E,J) or simply W k,p,A(E), satisfies the following:

  1. (1)

    ξ are locally in Wk,p(E);

  2. (2)

    as s → ±∞, eApF(t)Jξ(s,t)∈Wk,p(E);

  3. (3)

    the Banach norm on Wk,p,A(E) is defined as ‖ξ‖Wk,p,A=‖eApF(t)Jξ‖Wk,p. For instance, when k =1,

    ‖ξ‖W1,p,Ap=∫−∞+∞∫−∞+∞|eApF(t)Jξ|p+|∇s(eApF(t)Jξ)|p+|∇t(eApF(t)Jξ)|pdsdt.

  4. (4)

    ξ (s,±∞) are independent of s.

We remark that the method of studying operators in weighted Sobolev spaces is partially motivated by the work of [3], in which the problem of degenerate asymptotics also appears (for example, near the cylindrical ends asymptotic to the Reeb orbits) in the study of analytic foundation of symplectic field theory. In section 2 of [3], since the usual Fredholm theory cannot apply directly to the case of differential operators with degenerate asymptotics, a kind of d-weighted Sobolev space Lkp,d of sections of the bundle E over closed surface is constructed. While in the case of LG-model, since we consider Riemann surface with boundary, the appropriate weight function AF(t) is chosen to overcome degeneracy. The calculation in the next section shows that the Fredholm property can be also obtained in the sense of such weighted spaces.

3.2. Symplectic paths

Let P : (−∞,+∞) → Sp(2n) be a smooth path in the symplectic group, J0 be the standard complex structure on ℝ2n. Then S(t):=−J0P˙(t)P(t)−1 is a path of matrices which are symmetric. Consider an operator

l:Wk,p(ℝ,ℝ2n)→Wk−1,p(ℝ,ℝ2n)
defined by
l:=ddt−J0S(t).

Definition 3.2.

We say a 2n-order smooth path in the symplectic group

P:(−∞,+∞)→Sp(2n),       P(0)=Id
is (k, p)-non-degenerate or simply non-degenerate if the operator l is invertible. Denote by 𝒫k,p(2n) or simply 𝒫 (2n) the set of above defined non-degenerate 2n-order symplectic paths.

Let θ± be two sections of the restricted bundle Ē|ℓ± such that at each point (±∞,t) they are in the first summand of Ē(±∞,t) ≅ (ℝ2,ω0) ⊕ (ℝ2n−2,ω0), respectively, and θ−(.,±∞) = θ+(·,±∞).

Choose a smooth cut-off function ρ : ℝ2 → [0,1] such that ρ(s,t)= 0 near s =+∞, ρ(s)= 1 near s = −∞.

Definition 3.3.

Denote by 𝒱+,− the vector space generated by the sections

σ+(s,t)=ρ(s,t)θ+        and        σ−(s,t)=(1−ρ(s,t))θ−.

3.3. Linear operators

Let E be the symplectic vector bundle of rank 2n defined above. For two non-degenerate (2n − 2)-order symplectic paths P± ∈ 𝒫(2n − 2), denote by 𝒪(ℝ2,E,P±) the the set of bounded linear operators

D:𝒱+,−⊕Wk,p,A(E)→Wk−1,p,A(Λ0,1(E)),(3.1)
with k ≥ 1, p > 2, A > 0, such that under a suitable unitary trivialization Ψ(s,t) : (ℝ2n,J0) → (E,J) the following are satisfied:
  1. (1)

    the operator D is equivalent to a matrix-valued function

    L:𝒱+,−⊕Wk,p,A(ℝ2,ℝ2n,J0)→Wk−1,p,A(ℝ2,ℝ2n,J0)(3.2)
    L=∂∂s+J0∂∂t+S(s,t),(3.3)
    where S(s,t) extend continuously to the compactification ℝ2¯ such that, in the interior of ℝ2, they are locally in W k,p, and near a boundary S(s,t) − S(±∞,t) and S(s,t) − S(s,±∞) are in Wk−1,p,A(ℝ2,ℝ2n × ℝ2n);

  2. (2)

    those two paths of matrices S(±∞,t) split respectively in each E(±∞,t) = ℝ2 ⊕ ℝ2n−2 as

    S±(t):=S(+∞,t)=0⊕S˜±(t)=(000S˜±(t)),(3.4)
    where S˜±(t):=−J0P˙±(t)P±(t)−1 is a path of (2n − 2)×(2n − 2)-matrices which are symmetric with respect to the Euclidean structure determined by the symplectic and the complex structure.

Recall

l±:Wk,p(ℝ,ℝ2n)→Wk−1,p(ℝ,ℝ2n)l±=ddt−J0S±(t).(3.5)

Denote by

l˜±:=ddt−J0S˜±(t).

Since P± are non-degenerate (2n − 2)-order symplectic paths, by the Definition 3.2, the operator l˜± are invertible. Note that the differential operators we consider have degenerate asymptotics, since the matrices S vanish on the first summand of each E(±∞,t) = ℝ2 ⊕ ℝ2n−2, so the operator l± are not invertible. Hence, we can not directly apply to them the argument of Fredholm theory as in [10].

We remark that when we consider our actual example of ζ-instanton equation (2.10), let E = u*TX, then in some sense of weighted norm, its linearized operator Du is just as the one in (3.1) which after trivialization also looks like L in (3.3). It is clear that the Definition 2.2 of transversal non-degeneracy is equivalent to the following

Definition 3.4.

We say a solution γ(t) of (2.8) and (2.9) are transversally non-degenerate if, under a suitable trivialization, for the operator L in (3.3) the following hold

  1. (i)

    S(s,t) is independent of s and of the form S(t):=(000S˜(t)),

  2. (ii)

    the operator l˜=ddt−J0S˜(t) is invertible.

4. Fredholm property

Assumption 1. We assume that the following conditions hold:

  1. (1)

    S : ℝ2 → M2n×2n(ℝ) is C∞-bounded;

  2. (2)

    S(s,t) converges to S±(t) in the C0(ℝ)-topology as s → ∓ ∞;

  3. (3)

    The operators ddt−J0S±(t) are invertible;

4.1. Works by Haydys

Here we recall some conclusions drawn by Haydys in [10] to get Fredholm property for certain operators with non-degenerate asymptotics. Such operators appears when one considers the modified version of ζ-instanton equations. The following lemma is just the Lemma 2.28 in [10].

Lemma 4.1.

Let

A=∑|α|≤μfα(x)∂∂xα,      x∈ℝn
be a uniformly elliptic C∞ bounded differential operator of order μ, where fα takes values in the space of l × l-matrices. If A : W μ,2(ℝn;ℝl) → L2(ℝn;ℝl) is Fredholm, then A : W k+μ,p(ℝn;ℝl) → Wk,p(ℝn;ℝl) is Fredholm for all k ∈ ℝ,p > 1 and its index depends neither on k nor on p.

The following proposition is just the Lemma 2.29 in [10].

Proposition 4.1.

Assume that in addition to conditions (1)–(3) of Assumption 1 the following holds:

  1. (i)

    For each t ∈ ℝ the matrix J0S±(t) is symmetric;

  2. (ii)

    S(s,t) converges to constant matrices Σ± in the C0-topology as t → ∓ ∞. Moreover,

    Σ+=limt→−∞S+(t)=limt→−∞S−(t)    and       Σ−=limt→+∞S+(t)=limt→+∞S−(t)
    are symmetric matrices.

Then L : Wk,p(ℝ2;ℝ2n) → W k−1,p(ℝ2;ℝ2n) is Fredholm for any k ∈ ℝ,p > 1 and its index depends neither on k nor on p.

Proof.

We outline the key points of proof. First consider special case when k = 1, p = 2.

  1. 1.

    Since

    l±:W1,2(ℝ,ℝ2n)→L2(ℝ,ℝ2n)
    are isomorphisms, one can show that the s-independent operators
    L±:W1,2(ℝ2,ℝ2n)→L2(ℝ2,ℝ2n),L±=∂s+J0∂t+S±(t)
    are invertible.

  2. 2.

    Verify that the operators

    k±=dds+Σ±:W1,2(ℝ,ℝ2n)→L2(ℝ,ℝ2n)
    are isomorphisms, which implies the operators with constant coefficients
    J0K±:W1,2(ℝ2,ℝ2n)→L2(ℝ2,ℝ2n),J0K±=−∂t+J0(∂s+Σ±)
    and hence K± are invertible.

  3. 3.

    Verify that any limit operator L* = ∂s + J0∂t + S* (s,t) of L must be K± or

    L±τ:=∂s+J0∂t+S±(t+τ),
    each of which is also an isomorphism.

Then the proposition follows from a result in the theory of limit operators (Theorem 5.6 of [15]) and the Lemma 4.1.

4.2. Fredholm property

Note in our case, however, the condition (3) of Assumption 1 does not hold since our ζ-solitons are degenerate. So we can not apply the Proposition 4.1 directly.

Theorem 4.1.

Assume that in addition to conditions (1)–(2) of Assumption 1 the following conditions hold:

  1. (a)

    For each t ∈ ℝ, the matrix J0S±(t) is symmetric;

  2. (b)

    S(s,t) converges to constant matrices Σ± in the C0-topology as t → ∓ ∞. Moreover,

    Σ+=limt→−∞S+(t)=limt→−∞S−(t),       Σ−=limt→+∞S+(t)=limt→+∞S−(t)

  3. (c)

    P± ∈ 𝒫(2n − 2),D ∈ 𝒪(ℝ2,E,P±).

Then the operator L in (3.3) (and hence D) is Fredholm for any k ∈ ℝ,p > 1 and its index Ind L depends neither on k nor on p.

Proof.

First we fix k = 1, p = 2.

Consider the restriction of L to W1,2,A(ℝ2,ℝ2n). Note that the space W1,2,A(ℝ2,ℝ2n) is isomorphic to the space W1,2(ℝ2,ℝ2n), via a transformation that is identity away from the boundaries and given by eA2F(t)J0 near the boundaries ℓ±. Note that the matrix eA2F(t)J0 commutes with J0 and F˙(t)=e−t2, then

L(eA2F(t)J0ξ)=eA2F(t)J0∂sξ+J0[eA2F(t)J0J0(A2F˙(t))ξ+eA2F(t)J0∂tξ]+SeA2F(t)J0ξ=eA2F(t)J0[∂s+J0∂t+(eA2F(t)J0)−1SeA2F(t)J0−(A2e−t2)Id]ξ=eA2F(t)J0[∂s+J0∂t+(eA2F(t)J0)−1(S−(A2e−t2)Id)eA2F(t)J0]ξ=eA2F(t)J0[∂s+J0∂t+SA]ξ,
where denote by SA=(eA2F(t)J0)−1(S−(A2e−t2)Id)eA2F(t)J0. Since the matrix J0S±(t) is symmetric, we can as well assume that J0S is symmetric for all s and t. Hence it is clear that SA is symmetric and SA goes to S as A → 0. Using the isomorphism one can conjugate our restricted operator to an operator
L′:W1,2(ℝ2,ℝ2n)→L2(ℝ2,ℝ2n),
which looks like
L′:=∂s+J0∂t+SA.

Let S′(s,t)=S(s,t)−(A2e−t2)Id. Then

S±′(t):=S′(±∞,t)=S±(t)−(A2e−t2)Id=−J0P˙′±(t)P±′(t)−1,(4.1)
where S±(t)=(000S˜±(t)), P′± are 2n-order symplectic paths. Since from (3.4) S˜±(t):=−J0P˙±(t)P±(t)−1, the operator L has asymptotics (000P±(t)), then a calculation shows that, up to a similarity transformation, the new operator L′ has asymptotics
P±′(t)=e−A2F(t)(Id00P±(t)),
which are not degenerate anymore.

Let SA±(t)=lims→±∞SA(s,t). We consider the operators

lA,±:W1,2,A(ℝ,ℝ2n)→L2,A(ℝ,ℝ2n),lA±:=ddt−J0SA±(t).

Note that lA± are isomorphisms, and for each t ∈ ℝ, the matrix J0SA±(t) are symmetric. Since S(s,t) converges to Σ±, SA(s,t) also converges to constant symmetric matrices, denoted by ΣA± in the C0-topology as t → ∓ ∞. Thus, we can apply the Proposition 4.1 to L′. As a consequence, we obtain a Fredholm operator L′.

We remark that in our actual example, we consider bounded linear operators

Du:𝒱+,−⊕Wk,p,A(u*TX)→Wk−1,p,A(Λ0,1(u*TX))(4.2)
for a solution u of the ζ-instanton equation asymptotic to a pair of ζ-solitons γ±. The restriction of Du to Wk,p,A(ℝ2, u*TX) is conjugated to a Cauchy-Riemann operator
Du′:Wk,p(u*TX)→Wk−1,p(Λ0,1(u*TX)),(4.3)
which under trivialization is equivalently expressed by L′.The index of D′u or L′ can be given by relative Maslov index ([16])
μ(γ+,γ−;u)=μ(Λ0+,Λ0−)
which is the Robin-Salamon index for a pair of paths of Lagrangian subspaces Λ0±:ℝ→ℒag(ℝ2n). This will be explained in the next subsection.

Let us now consider the summand 𝒱+,−. The image of these sections have their support near a boundary and decay exponentially, hence L maps the new elements into L2,A(E). The space 𝒱+,− is of dimension 2. Thus the operator L is Fredholm. Its Fredholm index is

Ind  L=Ind(L|W1,p,A(E))+2=Ind  L′+2
the sum of the index of the operator restricted to W1,p,A(E) and the dimension of 𝒱+,−.

When we consider arbitrary k and p, the last statement of the Theorem follows from Lemma 4.1.

As an application, we consider the symplectic manifold (X,ω,J) with superpotential W and solutions of ζ-instanton equation. The following corollary is just our main result theorem 1.1

Corollary 4.1.

Assume that solutions γ±(t) of (2.8) and (2.9) are transversally non-degenerate. For each solution u of (2.10)–(2.12) the map

Du:𝒱+,−⊕Wk,p,A(ℝ2;u*TX)→Wk−1,p,A(ℝ2;u*TX)
defined by (2.13) is Fredholm for any k ∈ ℝ,p > 1, and its index depends neither on k nor on p, which can be given by
IndDu=μ(γ+,γ−,u)+2.

Proof.

The conclusions hold by applying the Theorem 4.1 to the case that E = u*TX and the matrix-valued function S(s,t) defined by (2.15). The space 𝒱+,− now is generated by Hamiltonian vector field along the solutions γ±(t) of (2.8) and (2.9). We can verify that J0S±(t) is symmetric by computation via (2.16). Note that matrices Σ∓ are the Hessians of the function Re(ξij−1W) at xi and xj, so they are symmetric.

4.3. Relative Maslov index

Now for our concrete example of symplectic manifold (X,ω,J) with a superpotential W, for a pair of solutions γ± of (2.8) and (2.9) and a solution u of (2.10)–(2.12), we study the computation of the D′u index of in (4.3) based on the method in [10]. Since the index depends neither on k nor on p, we can set k = 1, p = 2. With an arbitrary smooth curve γ(t) in X satisfying

limt→−∞γ(t)=xi,         limt→+∞γ(t)=xj,        and          limt→+∞γ˙(t)=0,(4.4)
we can associate a pair of Lagrangian subspaces Λ± in Tγ(0)X. Moreover, the kernel of the operator
𝒜:W1,2(γ*TX)→L2(γ*TX)𝒜  ξ=J0∂tξ+Sξ
can be identified with Λ+ ∩ Λ−. In particular, ker 𝒜 is nontrivial if and only if Λ+ ∩ Λ− ≠ {0}. In precise, the Lagrangian subspaces are defined as
Λ±={v∈Tγ(0)X|limt→∓∞ξv(t)=0},
where ξv is the solution of Cauchy problem 𝒜 ξv = 0, ξv(0)= v.

For a pair of curves γ± satisfying (4.4), let u : ℝ2 → X be any C1-map such that each curve γs(t) := u(s,t) also satisfies (4.4) and γs → γ± as s → ∓ ∞ in the C1-topology. Then as s goes from −∞ to +∞, we obtain two paths of Lagrangian subspaces Λ±(s). If for the two curves γ+ and γ−, the associated pair of Lagrangian subspaces Λ+(±∞) and Λ−(±∞) intersect transversally, respectively, then applying the argument of Robbin-Salamon [16] to the pair of paths of Lagrangian subspaces (Λ+(s),Λ−(s)) of Tγs(0)X, one can associate with the triple (γ+,γ−,u) an integer μ(γ+,γ−,u)= μ(Λ+,Λ−), called relative Maslov index (see section 2.8 of [10]). Moreover, in this non-degenerate case, the number μ(Λ+,Λ−) is a homotopy invariant (see Corollary 3.3 of [16]).

Although solutions of ζ-soliton equation (2.8) are degenerate, which implies the associated two pairs of Lagrangian subspaces (Λ+(+∞),Λ−(+∞)) and (Λ+(−∞),Λ−(−∞)) might not intersect transversally, the relative Maslov index μ(Λ+,Λ−) is still well-defined. Then the actual Fredholm index of D′u would be given by μ(γ+,γ−,u).

Proposition 4.2.

Assume that solutions γ±(t) of (2.8) and (2.9) are transversally non-degenerate. For each solution u of (2.10)–(2.12), the restriction of Du, defined by (4.2) and (2.13), to Wk,p,A(ℝ2;u*TX) is conjugate to a Fredholm operator D′u whose index can be given by the relative maslov index

IndDu′=μ(γ+,γ−,u).

Proof.

The argument is almost the same as the one in Proposition 2.32 in [10]. The point is by [2] the index of D′u (i.e. under a unitary trivialization the index of L′= ∂s +𝒜(s)) can be computed with the help of the spectral flow of 𝒜(s) and, under the identification Λ+(s0) ∩ Λ−(s0)= ker 𝒜(s0), the associated crossing forms Γ(Λ+,Λ−,s0) and Γ(𝒜, s0) coincide at each regular crossing s0. The difference of our case from the one in [10] is that the subspaces Λ+(s) and Λ−(s) are not trans-verse for s = ±∞, while the definition of relative Maslov index still does work for the case that Λ+(±∞) ∩ Λ−(±∞) ≠ 0 but with only regular crossings. The contributions at ±∞ to the index are 12signΓ(Λ+,Λ−,+∞)+12signΓ(Λ+,Λ−,−∞). So the index μ(γ+,γ−,u) generally might be a half integer. This is induced by the one-dimensional degeneracy of γ+ and γ−. We remark that, in actual computation, one would perturb slightly the ζ-solitons to get non-degenerate curves γ˜± satisfying (4.4), and calculate the index IndDũ at the perturbed map ũ which is close to ζ-instanton u. Then one can verify that IndDu˜=μ(γ˜+,γ˜−,u˜)−1.

Acknowledgments

The author would like to thank H.-L. Chang, X. Chen, H. Fan, A. Haydys, Y. He, J. Hu, W. Jiang, S. Li, C. Liu, Y. Long, Y. Ruan, S. Sun, G. Tian, G. Xu, B.-L. Wang and C. Zhu for helpful communications. This work was partially supported by projects No. 11671209 and No. 11271269 of National Science Foundation of China.

Footnotes

a

Generally, the functional 𝒜W and equation (2.8) can be defined for any phase ζ, while there might be no solution for general ζ since solutions to (2.8) and (2.9) project to the unique straight line of slope iζij in the complex W-plane.

References

[4]C.C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions of Hamiltonian equations, Commun. Pure Appl. Math, Vol. 37, 1984, pp. 207-253.
[9]D. Gaiotto, G.W. Moore, and E. Witten, Algebra of the Infrared: String Field Theoretic Structures in Massive N = (2, 2) Field Theory In Two Dimensions. Preprint, arXiv: 1506.04087
[17]D. Salamon, Lectures on Floer homology, Symplectic geometry and topology, Park City, UT, 1997, pp. 143-229. volume 7 of IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 1999.
[18]D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math, Vol. 45, 1992, pp. 1303-1360.
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TY  - JOUR
AU  - Hai-Long Her
PY  - 2019
DA  - 2019/10/25
TI  - Fredholm Property of Operators from 2D String Field Theory
JO  - Journal of Nonlinear Mathematical Physics
SP  - 170
EP  - 184
VL  - 27
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1683998
DO  - 10.1080/14029251.2020.1683998
ID  - Her2019
ER  -