Positivity of the universal pairing in 3 dimensions

Associated to a closed, oriented surface S is the complex vector space with basis the set of all compact, oriented 3-manifolds which it bounds. Gluing along S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3-manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2+1 dimensional TQFTs. The proof involves the construction of a suitable complexity function c on all closed 3-manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that c(AB) <= max(c(AA),c(BB)) for all A,B which bound S, with equality if and only if A=B. The complexity function c involves input from many aspects of 3-manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite volume hyperbolic 3-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3-manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3-manifolds due to Agol-Storm-Thurston.Comment: 83 pages, 21 figures; version 3: incorporates referee's comments and correction

Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications

We study a class of Legendrian surfaces in contact five-folds by encoding their wavefronts via planar combinatorial structures. We refer to these surfaces as Legendrian weaves, and to the combinatorial objects as N-graphs. First, we develop a diagrammatic calculus which encodes contact geometric operations on Legendrian surfaces as multi-colored planar combinatorics. Second, we present an algebraic-geometric characterization for the moduli space of microlocal constructible sheaves associated to these Legendrian surfaces. Then we use these N-graphs and the flag moduli description of these Legendrian invariants for several new applications to contact and symplectic topology. Applications include showing that any finite group can be realized as a subfactor of a 3-dimensional Lagrangian concordance monoid for a Legendrian surface in the 1-jet space of the two-sphere, a new construction of infinitely many exact Lagrangian fillings for Legendrian links in the standard contact three-sphere, and performing rational point counts over finite fields that distinguish Legendrian surfaces in the standard five-dimensional Darboux chart. In addition, the manuscript develops the notion of Legendrian mutation, studying microlocal monodromies and their transformations. The appendix illustrates the connection between our N-graph calculus for Lagrangian cobordisms and Elias-Khovanov-Williamson's Soergel Calculus.Comment: 114 Pages, 105 Figure

Solitons and Volume Preserving Flow

Solitons arise as solutions to non-linear partial differential equations. These equations are only analytically solvable in very few special cases. Other solutions must be found numerically. A useful technique for obtaining static solutions is gradient flow. Gradient flow evolution is in a direction which never increases energy, leading to solutions which are local minima. Here, a modified version of gradient flow referred to as volume preserving flow is introduced. This flow is constructed to evolve solutions towards local minima, while leaving a number of global quantities unaltered. Volume preserving gradient flow will be introduced and demonstrated in some simple models. Volume preserving flow will be used to investigate minimal surfaces in the context of double bubbles. Work will reproduce explicit results for double bubbles on the two-torus and construct a range of possible minimisers on the three-torus. Domain walls in a Wess-Zumino model with a triply degenerate vacuum will be used to represent the surfaces of the bubbles. Volume preserving flow will minimise the energy of the domain walls while maintaining the volumes of the space they contain. Global minima will represent minimal surfaces in the limit in which the domain wall thickness tends to zero. Numerical simulations of solitons in models which have conformal symmetry are problematic. Discretisation breaks the zero modes associated with changes of scale to negative modes. These lead to the collapse of solutions. Volume preserving flow provides a framework in which minimisation occurs orthogonally to these zero modes, maintaining a scale for the minimisation. Two such conformal models which permit Hopf solitons are the Nicole and AFZ models. They are comprised of the two components of the Skyrme-Faddeev model, taken to fractional powers to allow for solitons. Volume preserving flow will be used to find static solutions for a range of Hopf charges for each model. Comparisons will be made with the Skyrme-Faddeev model and general features of Hopf solitons will be discussed. A one parameter family of conformal Skyrme-Faddeev models will also be introduced. These models will be the set of linear combinations of the Nicole and AFZ models where the coefficients sum to one. Energy and topology transitions through this set of models will be investigated

Quantum Field Theory on Noncommutative Spaces

A pedagogical and self-contained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the Weyl-Wigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommutative Yang-Mills theory on infinite space and on the torus, Morita equivalences of noncommutative gauge theories, twisted reduced models, and an in-depth study of the gauge group of noncommutative Yang-Mills theory. Some of the more mathematical ideas and techniques of noncommutative geometry are also briefly explained.Comment: 111 pages LaTeX, 6 eps figures; Some material and discussions added for clarity, corrections done, references added; Final version to be published in Physics Report