Tag - Topological field theory

Eric Samperton: Topological quantum computation is hyperbolic

12th October, 2023

We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The diagrams can even be arranged to have additional nice properties, such as being alternating with minimal crossing number. Moreover, the reduction is polynomially uniform in the self-braiding exponent of the colouring object. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.

Ethan Ackelsberg: Polynomial Patterns in Finite Fields: a Dynamical Point of View

18th April, 2023

In its dynamical formulation, the Furstenberg-Sárközy theorem states that for any invertible measure-preserving system (X,μ,T), any set A⊆X with μ(A)>0, and any integer polynomial P with P(0)=0,

c(A)=limN−M→∞ 1/(N−M) ∑n=MN−1 μ(A∩TP(n)A)>0.

The limit c(A) obtains the 'correct' value μ(A)2 when T is totally ergodic. In fact, when T is totally ergodic, one has an ergodic theorem for polynomial actions: for any integer polynomial P and any f∈L2(μ),

limN−M→∞ 1/(N−M) ∑n=MN−1 TP(n)f= ∫X f dμ,

where the limit is taken in L2(μ). We will explain that the correct notion of total ergodicity for polynomial actions of more general rings depends on the dynamical behavior of actions along finite index ideals. From this point of view, the action of a large finite field on itself is asymptotically totally ergodic, since the index of the only proper ideal {0} grows with the size of the field. Guided by ergodic-theoretic results about polynomial (multiple) recurrence in totally ergodic systems, we then obtain several new results about polynomial configurations in large subsets of finite fields.

Marcy Robertson: Expansions, completions and automorphisms of welded tangled foams

22nd April, 2021

Welded tangles are knotted surfaces in ℝ4. Bar-Natan and Dancso described a class of welded tangles which have 'foamed vertices' where one allows surfaces to merge and split. The resulting welded tangled foams carry an algebraic structure, similar to the planar algebras of Jones, called a circuit algebra. In joint work with Dancso and Halacheva we provide a one-to-one correspondence between circuit algebras and a form of rigid tensor category called 'wheeled props'. This is a higher-dimensional version of the well-known algebraic classification of planar algebras as certain pivotal categories.

This classification allows us to connect these 'welded tangled foams' to the Kashiwara-Vergne conjecture in Lie theory. In work in progress, we show that the group of homotopy automorphisms of the (rational completion of) the wheeled prop of welded foams is isomorphic to the group of symmetries KV, which acts on the solutions to the Kashiwara-Vergne conjecture. Moreover, we explain how this approach illuminates the close relationship between the group KV and the pro-unipotent Grothendieck–Teichmueller group.

Dan Freed: Complex Chern-Simons invariants of 3-manifolds and abelianization

28th April, 2017

A hyperbolic 3-manifold M carries a flat PSL2(ℂ)-connection whose Chern-Simons invariant has been much studied since the early 1980s. For example, its real part is the volume of M. Explicit formulas in terms of a triangulation involve the dilogarithm. In joint work with Andy Neitzke we use 3-dimensional spectral networks to abelianize the computation of complex Chern-Simons invariants. The locality of the classical Chern-Simons invariant, expressed in the language of topological field theory, plays an important role.