Topology and Geometry

Publisher Summary Contact structures on manifolds and Legendrian and transversal knots in them are very natural objects, born over a century ago, in the work of Huygens, Hamilton, and Jacobi on geometric optics and work of Lie on partial differential equations. This chapter discusses Legendrian and transversal knots in dimension three where their theory is most fully developed and where they are most intimately tied to topology. In this dimension a predominately topological and combinatorial approach may be used to their study. The chapter then examines higher dimensional Legendrian knots. The chapter also explains the concept of Lagrangian protection. Analogous to the definition of finite type invariants of topological knots, finite type invariants of Legendrian knots can be defined. It is then possible to get a global contact isotopy if there is a fixed over twisted disk in the complement of both Legendrian knot. A complete understanding of Legendrian unknots in over twisted contact structures must wait for future work.

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Legendrian and Transversal Knots

Finding the exact, quantum corrected metric on the hypermultiplet moduli space in Type II string compactifications on Calabi-Yau threefolds still poses an outstanding open problem. We address this issue by relating the quaternionic-Kahler metric on the hypermultiplet moduli space to the complex contact geometry on its twistor space. In this framework, Euclidean D-brane instanton contributions are captured by contact transformations between different locally flat patches. We derive those by recasting the previously known A-type D2-instanton corrections in the language of contact geometry, covariantizing the result under electro-magnetic duality, and using mirror symmetry. As a result, we are able to express the effects of all D-instantons in Type II compactifications concisely as a sum of dilogarithm functions. We conclude with some comments on the relation to microscopic degeneracies of four-dimensional BPS black holes and to the wall-crossing formula of Kontsevich and Soibelman, and on the form of the yet unknown NS5-brane instanton contributions.

https://dspace.library.uu.nl/bitstream/handle/1874/42645/jhep032009044.pdf;sequence=1

D-instantons and twistors

These are lecture notes from the Clay Mathematics Institute summer school ``Floer Homology, Gauge Theory, and Low Dimensional Topology'' Alfred Renyi Institute; wwwclaymathorg/programs/summer_school/2004/ The main goal of these notes is to sketch a proof of Giroux correspondence between open book decompositions of three manifolds and contact structures, and then discuss various applications of this correspondence

Lectures on open book decompositions and contact structures

We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface Σ). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.

/pdf/non-abelian-localization-for-chern-simons-theory-11gm00ss9r.pdf

Non-Abelian localization for Chern-Simons theory

This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol 2 After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a detailed exposition of the original proof of the Lutz-Martinet theorem The text ends with a guide to the literature

Hansjörg Geiges

Papers

Contact geometry