# Past Quantum Field Theory Seminar

We find a new duality for form factors of lightlike Wilson loops

in planar N=4 super-Yang-Mills theory. The duality maps a form factor

involving a lightlike polygonal super-Wilson loop together with external

on-shell states, to the same type of object but with the edges of the

Wilson loop and the external states swapping roles. This relation can

essentially be seen graphically in Lorentz harmonic chiral (LHC) superspace

where it is equivalent to planar graph duality.

In 1997, Maxim Kontsevich gave a universal formula for the

quantization of Poisson brackets. It can be viewed as a perturbative

expansion in a certain two-dimensional topological field theory. While the

formula is explicit, it is currently impossible to compute in all but the

simplest cases, not least because the values of the relevant Feynman

integrals are unknown. In forthcoming joint work with Peter Banks and Erik

Panzer, we use Francis Brown's approach to the periods of the moduli space

of genus zero curves to give an algorithm for the computation of these

integrals in terms of multiple zeta values. It allows us to calculate the

terms in the expansion on a computer for the first time, giving tantalizing

evidence for several open conjectures concerning the convergence and sum of

the series, and the action of the Grothendieck-Teichmuller group by gauge

transformations.

The "Weyl fermion" was discovered in a topological semimetal in

2015. Its mathematical characterisation turns out to involve deep and subtle

results in differential topology. I will outline this theory, and explain

some connections to Euler structures, torsion of manifolds,

and Seiberg-Witten invariants. I also propose interesting generalisations

with torsion topological charges arising from Kervaire semicharacteristics

and ``Quaternionic'' characteristic classes.

Quantization is the study of the interface between commutative and

noncommutative geometry. There are myriad approaches to it, mostly presented

as ad hoc recipes. I shall discuss the motivating ideas, and the relations

between some of the methods, especially the relation between 'deformation'

and 'geometric' quantization.

The talk will review the origins

of ambitwistor strings, and recent progress in extending them to a

wider variety of theories and loop amplitudes.

In this talk, we discuss an ongoing calculation of the

four-loop form factors in massless QCD. We begin by discussing our

novel approach to the calculation in detail. Of particular interest

are a new polynomial-time integration by parts reduction algorithm and

a new method to algebraically resolve the IR and UV singularities of

dimensionally-regulated bare perturbative scattering amplitudes.

Although not all integral topologies are linearly reducible for the

more non-trivial color structures, it is nevertheless feasible to

obtain accurate numerical results for the finite parts of the complete

four-loop form factors using publicly available sector decomposition

programs and bases of finite integrals. Finally, we present first

results for the four-loop gluon form factor Feynman diagrams which

contain three closed fermion loops.

The $\phi^4$ model in statistical physics describes the

continous phase transition in the liquid-vapour system, transition to

the superfluid phase in helium, etc. Experimentally measured values in

this model are critical exponents and universal amplitude ratios.

These values can also be calculated in the framework of the

renormalization group approach. It turns out that the obtained series

are divergent asymptotic series, but it is possible to perform Borel

resummation of such a series. To make this procedure more accurate we

need as much terms of the expansion as possible.

The results of the recent six loop analitical calculations of the

anomalous dimensions, beta function and critical exponents of the

$O(N)$ symmetric $\phi^4$ model will be presented. Different technical

aspects of these calculations (IBP method, R* operation and parametric

integration in Feynman representation) will be discussed. The

numerical estimations of critical exponents obtained with Borel

resummation procedure are compared with experimental values and

results of Monte-Carlo simulations.

The big phase space is an infinite dimensional manifold which is the arena

for topological quantum field theories and quantum cohomology (or

equivalently, dispersive integrable systems). tt*-geometry was introduced by

Cecotti and Vafa and is a way to introduce an Hermitian structure on what

would be naturally complex objects, and the theory has many links with

singularity theory, variation of Hodge structures, Higgs bundles, integrable

systems etc.. In this talk the two ideas will be combined to give a

tt*-geometry on the big phase space.

(joint work with Liana David)