Analysis of gauged Witten equation

Gang Tian; Guangbo Xu

doi:10.1515/crelle-2015-0066

Article

Analysis of gauged Witten equation

Gang Tian and Guangbo Xu

Published/Copyright: November 4, 2015

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2018 Issue 740

Abstract

The gauged Witten equation was essentially introduced by Witten in his formulation of the gauged linear σ-model (GLSM), which explains the so-called Landau–Ginzburg/Calabi–Yau correspondence. This is the first paper in a series towards a mathematical construction of GLSM, in which we set up a proper framework for studying the gauged Witten equation and its perturbations. We also prove several analytical properties of solutions and moduli spaces of the perturbed gauged Witten equation. We prove that solutions have nice asymptotic behavior on cylindrical ends of the domain. Under a good perturbation scheme, the energies of solutions are shown to be uniformly bounded by a constant depending only on the topological type. We prove that the linearization of the perturbed gauged Witten equation is Fredholm, and we calculate its Fredholm index. Finally, we define a notion of stable solutions and prove a compactness theorem for the moduli space of solutions over a fixed domain curve.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1309359

Funding statement: G.T. is supported by NSF grant DMS-1309359 and an NSFC grant. G.X. is supported by AMS-Simons Travel Grant.

A Basic analytical results

A.1 Epsilon-regularity for Cauchy–Riemann equations

The Witten equation is an inhomogeneous Cauchy–Riemann equation. We first recall the ϵ-regularity result of [21] in the case of J-holomorphic curves with a continuous J. Let Y be a manifold of dimension 2⁢N and Y′⊂Y be a subset. Let h0 be a smooth Riemannian metric on Y which we used as a reference to define the norms on function spaces on Y. For any x∈Y and δ>0, we use Bδ⁢(x) to denote the open geodesic ball centered at x with radius δ.

Definition A.1 ([21, Definition 1.1]).

A continuous almost complex structure J on Y is said to be uniformly continuous on Y′ (with respect to h0), if the following is true:

∥J∥L∞⁢(Y′)<∞.
For any ϵ>0, there is a number δ>0 such that for any x∈Y′, there exists a C1-diffeomorphism ϕ:B⁢(x,δ)→B⁢(0,δ)⊂ℂN such that
(A.1)∥J-ϕ*⁢Js⁢t∥C0(Bδ(x))∩Y′)+∥h0-ϕ*⁢hs⁢t∥C0(Bδ(x)∩Y′<ϵ,
where Js⁢t is the standard complex structure and hs⁢t is the standard metric on ℂN.

For each ϵ>0, the largest δ for which (A.1) is true is called the modulus of uniform continuity, and is denoted by a function μJ⁢(ϵ).

Lemma A.2 ([21, Lemma 1.1]).

Let J* be a continuous almost complex structure on Y which is uniformly continuous on A⊂Y. For every p∈(2,+∞), there exist constants ϵ1=ϵ1⁢(μJ⁣*,A,h0)>0, ϵp>0, Cp=C⁢(p,μJ*,A,h0)<∞ with the following property. For any continuous almost complex structure J on Y with ∥J-J*∥L∞⁢(A)<ϵp and for any J-holomorphic map u∈C0∩L12⁢(B1,Y) such that u⁢(B1)⊂A and ∥d⁢u∥L2⁢(B1)<ϵ1, we have

(A.2)∥d⁢u∥Lp⁢(B1/2)≤𝐂p⁢∥d⁢u∥L2⁢(B1).

By Sobolev embedding L1p→C0,2p-1, (A.2) implies (using the same constant 𝐂p)

(A.3)diam⁡(u⁢(B1/2))≤𝐂p⁢∥d⁢u∥L2⁢(B1).

A consequence of Lemma A.2 is the following.

Lemma A.3.

Let (X,h0) be a Riemannian manifold of dimension 2⁢n and J be a continuous almost complex structure on X which is uniformly continuous on the whole (noncompact) manifold X with respect to h0. Then there exists ϵ2=ϵ2⁢(μJ,X,h0)>0 satisfying the following condition. Suppose ρ∈(0,1], ν∈C0⁢(Bρ×X,T⁢X) and u:Bρ→X satisfies the inhomogeneous equation

(A.4)∂⁡u∂⁡z¯+ν⁢(u)=0.

∥d⁢u∥L2⁢(Bρ)≤ϵ2,ρ⁢∥ν⁢(u)∥L∞⁢(Bρ)≤ϵp⁢ϵ2,

then

diam⁡(u⁢(Bρ/2))≤𝐂p⁢(∥d⁢u∥L2⁢(Bρ)+ρ⁢∥ν⁢(u)∥L∞⁢(Bρ))

and

∥d⁢u∥Lp⁢(Bρ/2)≤𝐂p⁢ρ2p-1⁢(∥d⁢u∥L2⁢(Bρ)+ρ⁢∥ν⁢(u)∥L∞⁢(Bρ)).

Proof.

Denote Y=ℂ×X. Let J~0=(J0,Js⁢t) be the product almost complex structure on Y, which is uniformly continuous on Y with respect to the product metric h~0=(h0,hs⁢t). We claim that

ϵ2=11+π⁢ϵ1⁢(μJ~0,ℂ×X,h~0)

satisfies the requirement of this lemma. Indeed, define κ=∥ν⁢(u)∥L∞⁢(Bρ) and

v~:B1→Bρ⁢κϵp×X, ⁢ν~:Bρ⁢κϵp×X→T⁢X,

w↦(ρ⁢κϵp⁢w,u⁢(ρ⁢w)); ⁢(w,x)↦ϵpκ⁢ν⁢(ϵpκ⁢w,x).

Then define an almost complex structure J~ν~ on Bρ⁢κϵp×X by

J~ν~⁢(∂s,X)=(∂t,J0⁢X+ν~),J~ν~⁢(∂t,X)=(-∂s,J0⁢X-J0⁢ν~).

Then (A.4) implies that v~ is holomorphic with respect to J~ν~. On the other hand, we have

∥J~ν~-J~0∥L∞⁢(v~⁢(B1))≤ϵp,∥d⁢v~∥L2⁢(B1)≤∥d⁢u∥L2⁢(Bρ)+π⁢ρ⁢κϵp≤ϵ1.

Then Lemma A.2 and (A.3) imply that

∥d⁢v∥Lp⁢(B1/2)≤𝐂p⁢(∥d⁢u∥L2⁢(Bρ)+ρ⁢∥ν⁢(u)∥L∞⁢(Bρ))

and

diam⁡(u⁢(Bρ/2))=diam⁡(v⁢(B1/2))≤𝐂p⁢(∥d⁢u∥L2⁢(Bρ)+ρ⁢∥ν⁢(u)∥L∞⁢(Bρ)).

The rescaling relation of Lp-norms implies that

∥d⁢u∥Lp⁢(Bρ/2)≤𝐂p⁢ρ2p-1⁢(∥d⁢u∥L2⁢(Bρ)+ρ⁢∥ν⁢(u)∥L∞⁢(Bρ)).

This finishes the proof of Lemma A.3. ∎

A.2 Mean value estimates

We quote several important mean value estimates for differential inequalities of the Laplace operator on the plane. Let Br be the radius r open disk in ℂ centered at the origin, with the standard coordinates (s,t). Let Δ=∂s2+∂t2.

Lemma A.4 ([36, p. 156]).

Let f:Br→R with f⁢(z)≥0 be a smooth function and

Δ⁢f≥-A-B⁢f2,

where A≥0, B>0. Then

∫Brf≤π16⁢B⟹f⁢(0)≤8π⁢r2⁢∫Brf+A⁢r24.

A.3 Hofer’s lemma

The following lemma is due to Hofer.

Lemma A.5 ([28, Lemma 4.6.4]).

Let (X,d) be a metric space, f:X→R be a non-negative continuous function. Suppose x∈X, δ>0 and the closed ball B¯2⁢δ⁢(x)⊂X is complete. Then there exist ξ∈X and ϵ∈(0,δ] such that

d⁢(x,ξ)<2⁢δ,supBϵ⁢(ξ)≤2⁢f⁢(ξ),ϵ⁢f⁢(ξ)≥δ⁢f⁢(x).

B Equivariant topology

Suppose G is a compact Lie group, N is a G-manifold and P→M is a principal G-bundle over a closed oriented manifold M, then any continuous section s of the associated bundle P×KN defines a cycle in the Borel construction NG, which represents an equivariant homology class

s*⁢[M]∈Hdim⁡MG⁢(N;ℤ).

In this current paper, we would like to define such an equivariant fundamental class for any solution (A,u) to the perturbed gauged Witten equation by using the section u. However, since the monodromy of the r-spin structure at the punctures could be nontrivial, the image of the section u is an equivariant cycle in X only in the orbifold sense. So the contribution from the cylindrical ends Uj should be weighted by a rational weight, and the fundamental class of a solution (A,u) should be a class

[A,u]∈H2G⁢(X~;ℤ⁢[r-1]).

We will carry this out explicitly in this subsection.

We first recall a general way of defining a rational fundamental class of an orbifold section of an associated bundle over an orbicurve. We assume that the reader is familiar with the notion of orbicurves (orbifold Riemann surfaces) and orbifold bundles over an orbicurve, so we will be sketchy when referring to such structures.

We assume that we have a compact Riemann surface Σ with several distinct punctures z1,…,zk. An orbifold chart near zj with local group Γj≃ℤrj is a holomorphic map

πj:𝔻→Σ

which maps 0 to zj and can be expressed as ζ↦ζrj in local coordinates. A collection of orbifold charts {πj}j=1k define an orbicurve structure. An equivalence relation can be defined among orbifold charts, and an equivalence class is called an orbicurve 𝒞.

Now suppose for each j, we have an injective homomorphism χj:ℤrj→G. An orbifold G-bundle over 𝒞 is a usual G-bundle over Σ*:=Σ∖{z1,…,zk}, together with a collection of “bundle charts”

(π~j,πj):(𝔻*×G,𝔻*)→(P|Σ*,Σ*),j=1,…,k,

where πj:𝔻*→Σ* extends to an orbifold chart near pj and π~j covers πj; moreover, π~j is invariant under the Γj-action on the left by γ⋅(ζ,k)=(γ⁢ζ,χj⁢(γ)⁢k). An equivalence class of orbifold bundle charts defines an orbifold G-bundle 𝒫→𝒞. As a topological space, 𝒫 is

𝒫:=P*∪(⋃j=1k𝔻×G)/∼

with the equivalence relation generated by p∼(ζ,k) if π~j⁢(ζ,k)=p.

Now if N is a G-manifold, we can have an “orbifold associated bundle” 𝒴:=𝒫×GN, which contains the usual associated bundle Y*:=P*×GN as a proper subset. Each bundle chart π~j induces a chart π~jN:𝔻*×N→Y* by

π~jN⁢(ζ,x)=[π~j⁢(ζ,1),x],

which is invariant under the Γj-action γ⁢(ζ,x)=(γ⁢ζ,γ⁢x).

Suppose we have a continuous section u:Σ*→Y*, identified with an equivariant map U:P*→N. Then the composition

U∘π~j:𝔻*×G→N

is again a G-equivariant map and invariant under the Γj-action. It can be viewed as a continuous section over the chart 𝔻*×N. If it extends continuous to the origin 0∈𝔻j for all j, then we have an orbifold section of 𝒴→𝒞.

Now we can define the rational fundamental class of a continuous orbifold section of 𝒴. First, we construct a CW complex out of the orbicurve. The complement Σ∖U is a surface with boundary, hence we can regard it as a CW complex in such a way that ∂⁡U is a subset of the 1-skeleton of Σ∖U. Then we take k copies of 2-cells 𝔻j and attach it to ∂⁡U by the rj-to-1 map ζj↦ζjrj. This CW complex is denoted by |𝒞|. Then, we see that the singular chain

[𝒞]:=[Σ∖U]+∑j=1k1rj⁢|𝔻j|

defines a rational homology class in H2⁢(|𝒞|;ℤ⁢[r-1]), if r is divisible by all rj.

Moreover, the class of orbibundle charts defines a continuous G-bundle |𝒫|→|𝒞| (in the usual sense); the orbifold section s defines a continuous section |s|:|𝒫|→N. Hence we obtained a continuous map (up to homotopy) |𝒞|→NG. The class [𝒞] is pushforwarded to

s*⁢[𝒞]∈H2⁢(NG;ℤ⁢[r-1])=H2G⁢(N;ℤ⁢[r-1]).

Acknowledgements

We would like to thank the Simons Center for Geometry and Physics for hospitality during our visit in 2013. We thank Kentaro Hori, David Morrison, Edward Witten for useful conversations on GLSM. The second author would like to thank Chris Woodward for helpful discussions. The revision of this paper was partially made during the second author’s visit to the Institute for Advanced Study and he would like to thank Helmut Hofer for hospitality.

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Received: 2014-11-05

Revised: 2015-07-09

Published Online: 2015-11-04

Published in Print: 2018-07-01

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