Student: | Jacob McNamara |
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School: | Harvard College |

E-mail: | jmcnamara (at) college [dot] harvard [dot] edu |

Project: | Application of the Yang-Mills Flow to the Atiyah-Floer Conjecture |

Mentor: | Chris Woodward |

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 [1] 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 L^{2} metric), and has been used in the proof of the well-known Donaldson-Uhlenbeck-Yau theorem. While Råde [2] 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 [3], as well as some of Duncan's thesis [1] 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 [1], as well as Råde [2] and Donaldson's book [4], 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 [1]. 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 [5]).

- 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.