Seiberg-Witten and Gromov invariants for self-dual harmonic 2-forms

This is the sequel to the author's previous paper which gives an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds. The main result of this paper asserts the following. Whenever the Seiberg-Witten invariants are defined over a closed minimal 4-manifold X, they are equivalent modulo 2 to "near-symplectic" Gromov invariants in the presence of certain self-dual harmonic 2-forms on X. A version for non-minimal 4-manifolds is also proved. A corollary to circle-valued Morse theory on 3-manifolds is also announced, recovering a result of Hutchings-Lee-Turaev about the 3-dimensional Seiberg-Witten invariants.Comment: 41 pages. Comments desired; to be submitte

G(2) holonomy, Taubes' construction of Seiberg-Witten invariants and superconducting vortices

Using a reformulation of topological N = 2 QFT\u2019s in M-theory setup, where QFT is realized via M5 branes wrapping co-associative cycles in a G2 manifold constructed from the space of self-dual 2-forms over a four-fold X, we show that superconducting vortices are mapped to M2 branes stretched between M5 branes. This setup provides a physical explanation of Taubes\u2019 construction of the Seiberg-Witten invariants when X is symplectic and the superconducting vortices are realized as pseudo-holomorphic curves. This setup is general enough to realize topological QFT\u2019s arising from N = 2 QFT\u2019s from all Gaiotto theories on arbitrary 4-manifolds. \ua9 2020, The Author(s)

Electrochemical Investigation of Azurin Thermodynamic and Adsorption Properties at Monolayer-Protected Cluster Film Assemblies – Evidence for a More Homogeneous Adsorption Interface

Thermodynamic and adsorption properties of protein monolayer electrochemistry (PME) are examined for Pseudomonas aeruginosa azurin (AZ) immobilized at an electrode modified with a networked film of monolayer-protected clusters (MPCs) to assess if nanoparticle films of this nature offer a more homogeneous adsorption interface compared to traditional self-assembled monolayer (SAM) modified electrodes. Specifically, electrochemistry is used to assess properties of surface coverage, formal potential, peak broadening, and electron transfer (ET) kinetics as a function of film thickness. The modification of a surface with dithiol-linked films of MPCs (Au225C675) provides a more uniform binding interface for AZ that results in voltammetry with less peak broadening (mV) compared to SAMs (\u3e120–130 mV). Improved homogeneity of the MPC interface for protein adsorption is confirmed by atomic force microscopy imaging that shows uniform coverage of the gold substrate topography and by electrochemical analysis of film properties during systematic desorption of AZ, which indicates a more homogeneous population of adsorbed protein at MPC films. These results suggest MPC film assemblies may be used in PME to provide greater molecular level control of the protein adsorption interface, a development with applications for strategies to study biological ET processes as well as the advancement of biosensor technologies

Distance Dependence of Electron Transfer Kinetics for Azurin Protein Adsorbed to Monolayer Protected Nanoparticle Film Assemblies

The distance dependence and kinetics of the heterogeneous electron transfer (ET) reaction for the redox protein azurin adsorbed to an electrode modified with a gold nanoparticle film are investigated using cyclic voltammetry. The nanoparticle films are comprised of nonaqueous nanoparticles, known as monolayer-protected clusters (MPCs), which are covalently networked with dithiol linkers. The MPC film assembly serves as an alternative adsorption platform to the traditional alkanethiolate self-assembled monolayer (SAM) modified electrodes that are commonly employed to study the ET kinetics of immobilized redox proteins, a strategy known as protein monolayer electrochemistry. Voltammetric analysis of the ET kinetics for azurin adsorbed to SAMs of increasing chain length results in quasi-reversible voltammetry with significant peak splitting. We observed rate constants (k°ET) of 12−20 s−1 for the protein at SAMs of shorter alkanethiolates that decays exponentially (β = 0.9/CH2 or 0.8/Å) at SAMs of longer alkanethiolates (9−11 methylene units) or an estimated distance of 1.23 nm and is representative of classical electronic tunneling behavior over increasing distance. Azurin adsorbed to the MPC film platforms of increasing thickness results in reversible voltammetry with very little voltammetric peaks splitting and nearly negligible decay of the ET rate over significant distances up to 20 nm. The apparent lack of distance dependence for heterogeneous ET reactions at MPC film assemblies is attributed to a two-step mechanism involving extremely fast electronic hopping through the MPC film architecture. These results suggest that MPC platforms may be used in protein monolayer electrochemistry to create adsorption platforms of higher architecture that can accommodate greater than monolayer protein coverage and increase the Faradaic signal, a finding with significant implications for amperometric biosensor design and development

Seiberg-Witten and Gromov invariants for self-dual harmonic 2-forms

For a closed oriented smooth 4-manifold X with b^2_+(X)>0, the Seiberg-Witten invariants are well-defined. Taubes' "SW=Gr" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This thesis describes well-defined counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic 2-forms, and it is shown that they recover the Seiberg-Witten invariants (modulo 2). This is an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds.The main results are the following. Given a suitable near-symplectic form w and tubular neighborhood N of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X-N, w) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form "near-symplectic" Gromov invariants as a map on the set of spin-c structures of X. They are furthermore equal to the Seiberg-Witten invariants with mod 2 coefficients, where w determines the "chamber" for defining the latter invariants when $b^2_+(X)=1$.In the final chapter, as a non sequitur, a new proof of the Fredholm index formula for punctured pseudoholomorphic curves is sketched. This generalizes Taubes' proof of the Riemann-Roch theorem for compact Riemann surfaces