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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - QUASIFT (Quantum Algebraic Structures In Field Theories)

Teaser

Summary

- What is the problem/issue being addressed?

The current universal framework to address quantum physical systems
with infinitely many interacting degrees of freedom is Quantum Field
Theory. The traditional method of Feynman diagrams or perturbation
theory works well when the interaction is weak, but breaks when the
interaction is strong. The problem is to develop new mathematical
formalisms to address computations and predictions in Quantum Field
Theory.

- Why is it important for society?

Quantum field theory is the current paradigm of the theoretical
physics to describe the space-time and matter at the most fundamental
level. The current model of elementary particles and subatomic
physics, the Standard Model, is a Quantum Field Theory. Besides
nuclear physics, Quantum Field Theory describes a range of phenomena
in condensed matter physics such as strongly correlated organic
conductors, carbon nanotubes, high temperature superconductors,
quantum Hall systems, Josephon junction arrays, superfluid helium, and
many others. All these phenomena, at the cutting edge of the
research, might find groundbreaking technological applications. It is
necessary to have a good theoretical control of these phenomena, and
developing the necessary mathematical formalisms of Quantum Field
Theories is the most promising way to progress our theoretical
understanding of these physical systems.

- What are the overall objectives?

The overall objective of this project is to find novel algebraic
structures hidden in the mathematical formalism of quantum field
theory which allow more effective computations then perturbation
theory, or computations which are totally not accessible with
perturbation theory. Strongly coupled dynamical systems which have
exact solution are usually called integrable systems, and in this
project we are aiming to find mathematical structures analogous to the
structures of integrable systems in quantum field theories.

Work performed

The theoretical work performed from the beginning of the project
resulted in 25 publications or preprints in preparation to the
publication in which the results have been reported.

The main results achieved so far are centered around the two topics:
the supersymmetric localization and the integrable systems of
multiplicative Higgs bundles. We connected the results in these two
domains, we have built the novel mathematical formalism of the latter,
and we have profound connection to the formal mathematical theory
called geometric Langlands program originating in the number theory.
Many of the quantities exactly computable in low-dimensional quantum
field theories such at two-dimensional Toda theory and its massive
deformations have been linked to the supersymmetry protected
observables in higher dimensional gauge theories, such as
four-dimensional supersymmetric Yang-Mills.

Final results

All results reported in the 25 publications (except the two review
articles among them) are novel and beyond the state of the art, with
the main progress outlined above in the work performed section.

We expect to obtain novel results until the end of the project in two
domains: (i) generalization of integrable system of multiplicative
Higgs bundles for a reductive Lie group G to the supergroup. In the
case when G is the supergroup PSL(4|4) this will allow us to address
with proper mathematical formalism the maximally supersymmetric
Yang-Mills theory in four dimensions in the large N limit with
conjectured Yangian symmetry and (ii) developing the computational
connection between integrable systems approach and conformal bootstrap
approach to non-perturbative dynamics of quantum field theories.