||jmcnamara (at) college [dot] harvard [dot] edu
||Application of the Yang-Mills Flow to the Atiyah-Floer Conjecture
The Atiyah-Floer Conjecture asserts the equality of two versions of Floer homology, both defined for a principal bundle Q over a compact three manifold Y. The two versions are the instanton Floer homology, which is defined by counting Yang-Mills instantons, or connections of minimal energy, on the cylinder Y × ℝ, and the symplectic Floer homology of the moduli space of flat connections associated to a Heegaard splitting of Y, which counts pseudoholomorphic curves. One program to prove this conjecture aims to establish a bijection between instantons and pseudoholomorphic curves by "stretching" Y along the splitting, with the expectation that each instanton will converge to a unique pseudoholomorphic curve. While it has been established by Duncan  and others in a number of cases that convergence holds, it is not known if this function from instantons to pseudoholomorphic curves is bijective.
In studying this limiting process and attempting to show bijectivity, a key analytic tool, which is not currently understood, is the Yang-Mills heat flow. This is the gradient flow of the Yang-Mills energy (in the L2 metric), and has been used in the proof of the well-known Donaldson-Uhlenbeck-Yau theorem. While Råde  has shown long time existence and convergence of the Yang-Mills flow in dimensions two and three, the four dimensional case is not yet understood in general. This project will attempt to establish good analytical properties of the Yang-Mills flow on four manifolds (and specifically those with tubular ends), and perhaps use these results to study the neck-stretching limit for a tubular neighborhood of the instantons.
- Before Arrival:
- Read Atiyah's paper , as well as some of Duncan's thesis  for background. Familiarized myself with the notions of instanton and symplectic Floer homology. Worked to understand Atiyah's heuristic that the instanton equation should formally converge to the equation for a pseudoholomorphic curve.
- Week 1:
- Discussed current state of knowledge with Professor Woodward, and considered possible specific project questions. Conducted literature review, and read more of Duncan's thesis , as well as Råde  and Donaldson's book , which provide many useful results on the Yang-Mills flow and on instantons and gauge theory in general. Concluded that a good first goal would be to establish the analytical properties of the Yang-Mills flow on four manifolds with tubular ends. Prepared first presentation on the Yang-Mills flow.
- Week 2:
- Began examining the Yang-Mills flow on Y × ℝ. Separated the total curvature into electric and magnetic components, and studied various consequences of Yang-Mills flow equation. Discovered issue with temporal gauge (not preserved under flow). Found two limiting cases: If magnetic field vanishes, then we obtain the heat equation for each component of the connection. If electric field varies slowly in physical time, obtain a slightly perturbed version of the three-dimensional Yang-Mills flow. Hope to control rate of flow of energy to ends of manifold. Suggests studying perturbed Y-M flow could be very useful.
- Week 3:
- Studied case of X = Y × Σ, where we shrink Σ. Found expression for the Laplacian of the component of the curvature in the S direction in the case of a pseudoholomorphic strip. Began examining convergence properties of components of connection in the S direction. Formulated a number of possible frameworks of proof of injectivity. Possible idea: Transversality of the deformation operators for maps contradicts a sequence of non-minimal Yang-Mills fields converging to holomorphic strip.
- Week 4:
- Investigated issues of continuity of the limit of the Yang-Mills heat flow at infinity, in the context of studying a path of connections connecting two instantons on separate components of the index one moduli space. Potentially get a sequence of large-energy connections near two instantons converging to the same limit at infinity, which could lead eventually to a contradiction, and thus a proof of injectivity. Thought that exploiting transversality may not work due to Cauchy-Riemann equation for the relevant components of curvature, but not necessarily: maybe rescaling?
- Week 5:
- Began considering heat flow on map side, the harmonic map flow. From solution of Yang-Mills flow on four-manifold, obtain solution to something like harmonic map flow in the space of maps into the moduli space of flat connections. Not quite though, since Laplacian is not the Levi-Cevita Laplacian, but slight perturbation. Map relating flows is the Narasimhan-Seshadri map from connections with small curvature to flat connections, constructed in Duncan's thesis . Looked into literature on the Harmonic map flow, which has similar regularity properties to the Yang-Mills flow.
- Week 6:
- Investigated injectivity of NS map from instantons to holomorphic curves directly. Two instantons, for a FIXED metric, which map to the same holomorphic curve, are thus related by a complex gauge transformation w. Since both instantons satisfy the instanton equation F+ = 0, we obtain an equation that the gauge transformation must satisfy. This should imply the subharmonicity of h = w*w, which could potentially imply that h = 1, or that the two instantons are actually related by a real gauge transformation (see ).
- Duncan, David L. 2013. Compactness Results for the Quilted Atiyah-Floer Conjecture.
- Råde, Johan. 1992. On the Yang-Mills heat equation in two and
three dimensions. Berlin.
- Atiyah, Michael. 1988. New Invariants of 3- and 4-Dimensional Manifolds. AMS.
- Donaldson, Simon. 2004. Floer Homology Groups in Yang-Mills Theory.Cambridge University Press.
- Jarvis, Stuart and Norbury, Paul. 1998. Degenerating metrics and instantons on the four-sphere. Journal of Geometry and Physics.