[Paper Review] $L^2$-topology and Lagrangians in the space of connections over a Riemann surface

Tomasz Mrowka, Katrin Wehrheim|arXiv (Cornell University)|May 5, 2010

Geometric Analysis and Curvature Flows6 references4 citations

TL;DR

This paper establishes uniform local quasiconvexity and local pathwise connectedness of gauge orbits in the space of connections over a Riemann surface equipped with the $L^2$-topology, using a quantitative $L^2$-local slice theorem. The key contribution is a uniform bound on path length in gauge orbits, essential for proving compactness in Yang-Mills Floer theory with Lagrangian boundary conditions.

ABSTRACT

We examine the $L^2$-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl Lemma of harmonic analysis, and deduce local pathwise connectedness and local uniform quasiconvexity of the gauge orbits. Using these, we generalize compactness results for anti-self-dual instantons with Lagrangian boundary counditions to general gauge invariant Lagrangian submanifolds. This provides the foundation for the construction of instanton Floer homology for pairs of a $3$-manifold with boundary and a Lagrangian in the configuration space over the boundary.

Motivation & Objective

• To understand the $L^2$-topology on the space of connections over a Riemann surface and its quotient by the gauge group.
• To establish local connectivity and uniform local quasiconvexity of gauge orbits in the $L^2$-topology for connections in $L^p$-regularity with $p>2$.
• To provide geometric control over gauge-invariant Lagrangian submanifolds in the space of connections, crucial for Yang-Mills Floer theory on 3-manifolds with boundary.
• To extend compactness results for anti-self-dual instantons with Lagrangian boundary conditions beyond special cases like handlebody boundaries.
• To establish a quantitative foundation for future proofs of local convexity of gauge-invariant Lagrangians using curvature bounds and quasiconvexity.

Proposed method

• Uses a quantitative $L^2$-local slice theorem to analyze the geometry of gauge orbits in the space of $L^p$-connections with $p>2$.
• Applies the $L^2$-topology to the space of connections $\mathcal{A}^{0,p}(\Sigma)$ and studies the action of the gauge group $\mathcal{G}^{1,p}(\Sigma)$.
• Establishes local pathwise connectedness by constructing continuous paths in $L^2$-balls connecting nearby points in a gauge orbit.
• Imposes a uniform bound on the $L^2$-norm of the derivative of such paths, ensuring quasiconvexity with constant $C$ depending on the orbit.
• Uses the local Chern-Simons functional and energy estimates to control the behavior of instantons near reducible connections.
• Applies integration and decay estimates to show that the energy $\mathcal{E}(r)$ decays as $r^{2\beta}$ for small $r$, with $\beta>0$ depending on the Lagrangian geometry.

Experimental results

Research questions

• RQ1 Can the gauge orbits in the space of $L^p$-connections over a Riemann surface be shown to be locally pathwise connected under the $L^2$-topology?
• RQ2 Is there a uniform constant $C$ such that any two points in a sufficiently small $L^2$-ball of a gauge orbit can be joined by a path of $L^2$-length bounded by $C$ times their distance?
• RQ3 Can the local quasiconvexity of gauge orbits be used to prove compactness of moduli spaces of anti-self-dual instantons with general gauge-invariant Lagrangian boundary conditions?
• RQ4 What is the rate of decay of the energy $\mathcal{E}(r)$ of an instanton near a reducible connection, and how does it relate to curvature bounds?
• RQ5 Does the local Chern-Simons functional remain well-defined and continuous along short paths in the Lagrangian, enabling energy control?

Key findings

• The gauge orbit $\mathcal{G}^{1,p}(\Sigma)^*B$ is locally pathwise connected in the $L^2$-topology: for any $\varepsilon>0$, there exists $\delta>0$ such that any two points within $L^2$-distance $\delta$ can be joined by a continuous path in the $L^2$-ball of radius $\varepsilon$.
• The connecting path can be chosen to be smooth as a map into $\mathcal{A}^{0,p}(\Sigma)$, with $\|\partial_t A_t\|_{L^2} \leq C\|A_1 - A_0\|_{L^2}$, where $C$ depends on the orbit $[B]$.
• For irreducible connections, the constant $C$ in the quasiconvexity bound can be chosen uniformly on $L^p$-bounded subsets of the orbit.
• The energy $\mathcal{E}(r)$ of an instanton near a reducible connection decays as $\mathcal{E}(r) \leq C r^{2\beta}$ for small $r$, with $\beta = \pi^{-1}(1 + \pi C_{\mathcal{L}})^{-2} > 0$.
• The local Chern-Simons functional $\mathcal{C}\mathcal{S}(A(r,\cdot))$ is bounded by $\frac{\pi(1 + \pi C_{\mathcal{L}})^2}{4} \int_0^\pi r^2 \|F_\Xi(r,\phi)\|_{L^2}^2 d\phi$, which tends to zero as $r \to 0$.
• The energy of the extended instanton $\tilde{\Xi}$ is expressed as $\mathcal{E}(\rho) = -\mathcal{C}\mathcal{S}(A(\rho,\cdot)) + \mathcal{C}\mathcal{S}(A(\delta,\cdot))$, enabling energy control via the Chern-Simons functional.