Gluing theorem for instanton trajectories

Intuition Beginner

Compactness says a long sequence of instanton trajectories can break into pieces. Gluing is the reverse process. If one trajectory goes from $\alpha$ to $\gamma$ and another goes from $\gamma$ to $\beta$, gluing joins them into a single long trajectory from $\alpha$ to $\beta$.

The joined object is not immediately a solution. First, one cuts off the two pieces far out on their cylindrical ends and pastes them with a long neck. This produces an approximate instanton. Then analysis corrects the small error to make an actual instanton.

The gluing parameter is the neck length. As the neck gets longer, the glued trajectory looks more and more like the broken pair.

Visual Beginner

The long neck records how long the glued solution spends near the intermediate flat connection. Letting the neck length go to infinity recovers the broken trajectory.

Worked example Beginner

Suppose one trajectory runs from $\alpha$ to $\gamma$ and another from $\gamma$ to $\beta$. Choose a large number $T=20$. Cut the first trajectory after it has come very close to $\gamma$, cut the second trajectory before it leaves a neighborhood of $\gamma$, and insert a neck of length about $T$.

The result is almost a solution because each piece was already a solution and the cut happens where both pieces are close to the same flat connection. The only error is concentrated in the transition region. The gluing theorem says that, when $T$ is large enough, this small error can be corrected in one controlled way.

What this tells us: a broken trajectory is not just a limiting object. It is the boundary of a family of honest trajectories with a long neck.

Check your understanding Beginner

Formal definition Intermediate+

Let $\alpha ,\gamma ,\beta$ be nondegenerate irreducible flat connections on a closed oriented three-manifold $Y$. Let $$ A_1\in\mathcal M(\alpha,\gamma),\qquad A_2\in\mathcal M(\gamma,\beta) $$ be finite-energy ASD trajectories on $\mathbb{R}\times Y$ in temporal gauge, considered modulo translation. Here $\mathcal{M}(\alpha ,\gamma )$ denotes the moduli space of ASD gradient trajectories from $\alpha$ to $\gamma$ for the Chern-Simons functional 03.07.17.

A pre-gluing of $A_1$ and $A_2$ with gluing length $T\gg 0$ is a connection $$ A_1#T A_2 $$ formed by translating $A_1$ far left, translating $A_2$ far right, cutting both in a region where they are exponentially close to $\gamma$, and interpolating their connection forms using cutoff functions. The resulting connection is smooth and has the correct limits $\alpha$ and $\beta$, but its self-dual curvature is small rather than zero: $$ |(F{A_1#_T A_2})^+|\leq Ce^{-\delta T} $$ in a suitable weighted Sobolev norm.

The gluing problem is to find a correction $a_T$ such that $$ F^+_{A_1#_T A_2+a_T}=0 $$ after imposing a gauge-fixing condition. The linearized operator is the gauge-fixed ASD operator at the pre-glued connection. A right inverse with norm bounded independently of large $T$ is the main analytic input.

Key theorem with proof Intermediate+

Theorem (gluing for instanton trajectories). Let $A_1\in \mathcal{M}(\alpha ,\gamma )$ and $A_2\in \mathcal{M}(\gamma ,\beta )$ be regular finite-energy ASD trajectories, and assume the intermediate flat connection $\gamma$ is nondegenerate. For all sufficiently large gluing lengths $T$, there is a unique small correction $a_T$ in a fixed gauge such that $$ A_T=A_1#_T A_2+a_T $$ is an ASD trajectory from $\alpha$ to $\beta$. The map $(A_1,A_2,T)\mapsto A_T$ is smooth and parametrizes a neighborhood of the broken trajectory in the compactified moduli space.

Proof. Exponential decay at $\gamma$ implies that, after translating the two trajectories and cutting them in the neck region, their connection forms differ from $\gamma$ by terms of size $O(e^{-\delta T})$. The cutoff construction therefore produces a smooth connection $A_1{\#}_T{A}_2$ whose self-dual curvature satisfies $$ |(F_{A_1#_T A_2})^+|\leq Ce^{-\delta T}. $$

Write the ASD equation for a correction $a$ as $$ D_Ta+Q_T(a)=-(F_{A_1#_T A_2})^+, $$ where $D_T$ is the linearized gauge-fixed ASD operator and $Q_T(a)$ is quadratic in $a$. Regularity of $A_1$ and $A_2$ gives right inverses for their linearized operators. Cutting and pasting these inverses gives a right inverse $R_T$ for $D_T$ with operator norm bounded independently of large $T$.

The equation becomes a fixed-point problem $$ a=-R_T(F_{A_1#_T A_2})^+-R_TQ_T(a). $$ The first term is $O(e^{-\delta T})$, and the quadratic estimate for $Q_T$ makes the second term a contraction on a ball of comparable radius when $T$ is large. The contraction mapping theorem gives a unique small solution $a_T$. Smooth dependence follows from the implicit function theorem with the same uniform right inverse. The local parametrization statement follows because Uhlenbeck compactness 03.07.20 says every sufficiently nearby unbroken trajectory degenerates to the given broken pair as $T\to \infty$, and the slice theorem 03.07.18 fixes the remaining gauge ambiguity. $\square$

Bridge. Gluing builds toward 03.07.23 because the proof of ${\partial }^2=0$ identifies the boundary of a one-dimensional compactified moduli space with pairs of trajectories, and it appears again in 03.07.22 where orientations decide the signs of those boundary points. The foundational reason is that compactness gives broken limits while gluing supplies the reverse local chart; this is exactly what identifies analytic boundary strata with algebraic compositions. Putting these together, gluing generalises finite-dimensional Morse gluing, and the bridge is dual to the Uhlenbeck compactness statement in 03.07.20.

Exercises Intermediate+

Advanced results Master

The gluing construction has three analytic stages. First, choose gauges on the ends so that both trajectories converge exponentially to the same flat connection $\gamma$. Second, form the pre-glued connection using cutoffs over a neck of length $T$. Third, solve the nonlinear ASD equation by applying a uniform right inverse to the linearization and controlling the quadratic term.

Uniformity in $T$ is the core issue. The long neck can create small eigenvalues if the limiting Hessian at $\gamma$ has kernel. Nondegeneracy of $\gamma$ rules this out. The right inverse on the glued cylinder is constructed by cutting a test section into pieces, applying right inverses on the two original trajectories, and patching the results. Errors produced by the cutoffs are supported in the neck and decay exponentially with $T$.

Dimension bookkeeping matches the gluing picture. If $\mathcal{M}(\alpha ,\gamma )$ and $\mathcal{M}(\gamma ,\beta )$ have dimensions $d_1$ and $d_2$ after quotienting by translation, then the glued family has dimension $d_1+d_2+1$ near the broken stratum, where the additional parameter is the neck length. In particular, broken pairs of zero-dimensional moduli spaces appear as boundary points of one-dimensional moduli spaces.

For Floer homology, gluing supplies the local collar neighborhood of the compactified moduli space. Near a broken pair, the compactified moduli space is modeled on the half-open gluing parameter $[T_0,\infty ]$, where $T=\infty$ is the broken endpoint. After the reparametrization $q=e^{-T}$, this becomes a neighborhood of a boundary point with coordinate $q\in [0,\epsilon )$.

Synthesis. The foundational reason gluing works is that exponential decay makes the two halves exponentially close to the same flat connection along their ends. This is exactly what makes the pre-gluing error small enough for a uniform inverse and a contraction argument. The central insight identifies broken compactness limits with boundary points of unbroken moduli spaces, while the neck length is dual to the degeneration parameter in 03.07.20. Putting these together, gluing supplies the analytic collar needed for the algebraic differential in instanton Floer theory.

Full proof set Master

Proposition 1 (pre-gluing error is exponentially small). If $A_1$ and $A_2$ converge exponentially to the same nondegenerate flat connection $\gamma$, then the self-dual curvature of $A_1{\#}_T{A}_2$ is $O(e^{-\delta T})$ in the neck norm.

Proof. On the uncut portions, $A_1$ and $A_2$ are ASD, so the self-dual curvature vanishes. The only contribution comes from the cutoff interpolation in the neck. In the neck gauges, both connection forms differ from $\gamma$ by terms bounded by $C{e}^{-\delta T}$, and their derivatives satisfy the same type of bound on the overlap region. Differentiating the cutoff expression produces terms involving the difference of the two small end forms and their first derivatives. Hence every self-dual curvature term created by the cutoff is bounded by $C^{\prime }{e}^{-\delta T}$ in the chosen Sobolev norm. $\square$

Proposition 2 (contraction solves the correction equation). Suppose $D_T$ has a right inverse $R_T$ with $\Vert R_T\Vert \le M$, the pre-gluing error satisfies $\Vert e_T\Vert \le C{e}^{-\delta T}$, and the nonlinear term satisfies the quadratic estimate in Exercise 6. Then the correction equation has a unique small solution for large $T$.

Proof. Use the fixed-point map $$ \Phi(a)=-R_Te_T-R_TQ_T(a). $$ On the ball of radius $2MC{e}^{-\delta T}$, the quadratic estimate implies that $\Phi$ is a contraction for sufficiently large $T$. The same estimate, together with $\Vert R_T{e}_T\Vert \le MC{e}^{-\delta T}$, shows that $\Phi$ maps the ball to itself after increasing $T$ if needed. Banach's fixed-point theorem gives a unique fixed point in the ball. That fixed point is exactly a correction solving $D_{T}a+Q_T(a)=-e_T$. $\square$

Proposition 3 (gluing adds one dimension). Near a regular broken pair, the unparametrized glued moduli space has local dimension $d_1+d_2+1$, where $d_i$ are the dimensions of the two unparametrized component moduli spaces.

Proof. The gluing data consist of a point in the first component moduli space, a point in the second component moduli space, and the gluing length $T$ for the neck. For large finite $T$, the gluing theorem gives a unique nearby unbroken trajectory after gauge fixing. Conversely, compactness identifies every sufficiently nearby unbroken trajectory with such data. Thus the local parameter space is $\mathcal{M}(\alpha ,\gamma )\times \mathcal{M}(\gamma ,\beta )\times (T_0,\infty )$, whose dimension is $d_1+d_2+1$. $\square$

Connections Master

• Uhlenbeck compactness 03.07.20. Compactness identifies the broken trajectories that appear as limits; gluing constructs the local family of unbroken trajectories approaching each such limit.

• Spectral flow and Floer grading 03.07.19. The Fredholm index controlled by spectral flow determines the expected dimensions that make gluing add exactly one neck parameter.

• Orientations on instanton trajectory moduli 03.07.22. The gluing map must respect determinant-line orientations so that broken boundary points cancel with the correct signs.

• Instanton Floer homology 03.07.23. The proof that the Floer differential squares to zero uses compactness plus gluing to identify the boundary of one-dimensional moduli spaces with pairs counted by two successive differentials.

Historical & philosophical context Master

Taubes developed the gluing methods that made four-dimensional Yang-Mills moduli spaces analytically usable, constructing self-dual connections by patching approximate solutions and correcting them through nonlinear analysis ^{[Taubes 1982]}. These ideas became standard in Donaldson theory and in the cylindrical version used by Floer trajectories.

Donaldson's Floer tract presents gluing as the counterpart to Uhlenbeck compactness for instanton trajectories ^{[Donaldson 2002]}. Compactness describes the possible limits; gluing proves those limits are actual boundary points of the moduli spaces used to define the Floer differential.

Bibliography Master

@article{Taubes1982SelfDual,
  author = {Taubes, Clifford Henry},
  title = {Self-dual Yang-Mills connections over non-self-dual 4-manifolds},
  journal = {Journal of Differential Geometry},
  volume = {17},
  pages = {139--170},
  year = {1982}
}

@article{Taubes1984Indefinite,
  author = {Taubes, Clifford Henry},
  title = {Self-dual connections on 4-manifolds with indefinite intersection matrix},
  journal = {Journal of Differential Geometry},
  volume = {19},
  pages = {517--560},
  year = {1984}
}

@book{Donaldson2002FloerGluing,
  author = {Donaldson, Simon K.},
  title = {Floer Homology Groups in Yang-Mills Theory},
  series = {Cambridge Tracts in Mathematics},
  volume = {147},
  publisher = {Cambridge University Press},
  year = {2002}
}

@book{DonaldsonKronheimer1990Gluing,
  author = {Donaldson, Simon K. and Kronheimer, Peter B.},
  title = {The Geometry of Four-Manifolds},
  publisher = {Oxford University Press},
  year = {1990}
}