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Holomorphic Blocks in Quantum Field Theory: New Constructions of Exact Results

Periodic Reporting for period 3 - HBQFTNCER (Holomorphic Blocks in Quantum Field Theory: New Constructions of Exact Results)

Reporting period: 2017-10-01 to 2019-03-31

Quantum field theory (QFT) plays a prominent role in our understanding of many areas of physics from elementary particles to statistical systems, from condensed matter physics to cosmology.

The standard approach to QFT relies on perturbation theory. This method loses effectiveness once the interactions become strong, limiting our quantitative understanding to the weakly coupled regimes. It is then a central challenge in theoretical physics to develop non-perturbative techniques to study the dynamics of strongly coupled quantum fields which could allow us to understand quantitatively processes like confinement.

It is believed that strongly coupled phases could be described by a new set of emergent degrees of freedom in terms of which the theory is weakly coupled and easier to describe. Identifying the emergent degrees of freedom is still a formidable problem. For theories with a high degree of symmetry, such as supersymmetric or conformal field theories, sometimes it is possible to realise this paradigm in the context of dualities. In infrared dualities we can describe the strongly coupled phase of a gauge theory in terms of a different weakly coupled gauge theory. In holographic dualities the emergent fields live in a space with one extra dimension and the dual theory is a gravity theory.

Since dualities have the potential of opening up new non-perturbative windows on strongly coupled phases they have almost monopolised the attention of a large part of the string theory community in the last 25 years. In recent years a major breakthrough has been the application of the localisation technique to the path integral of theories defined on compact manifolds, which has allowed us to obtain an unprecedented amount of exact results (valid for all values of the coupling) for a large number of protected observables. This has allowed us to test previously conjectured dualities, to discover many new ones and to learn much more about strongly coupled QFTs.

This project aims to establish new exact methods for the study of supersymmetric QFTs integrating the localisation technique with the idea of the holomorphic block decomposition which allows us to establish new connections between supersymmetric gauge theories and low dimensional exactly solvable systems such as conformal field theories, topological quantum filed theories and spin chains.
Some of the results obtained in the first half of this project are described below.

We have tested the holomorphic block factorisation proposal for a large variety of theories formulated on compact manifolds, including topologically twisted theories. Most of the results obtained in this research line of the project have been collected in an invited contribution in the review `Localization techniques in quantum field theories' J. Phys. A: Math. Theor. 50 440301, Pestun V and Zabzine M (ed).

We discovered new web of dualities involving holomorphic blocks in various dimensions and correlation functions in theories with Virasoro or deformed Virasoro symmetry.

We explored the relation between holomorphic blocks, topological strings and integrable systems obtaining new tools to study the modular properties of the blocks and their transformation properties under various dualities.

We explored the possibility of finding new IR conformal field theories with monopole operators appearing in the super-potential. This led us to the discovery of a new class of dualities for three dimensional theories which we tested by matching partition functions obtained via localizations. This has allowed us to obtain the correct description of the monopole deformed domain wall theories duals to conformal field theory kernels.
The results obtained so far have been published in top journals and have attracted the interest of many researchers from physics and mathematics. These results have confirmed the great potential of the approach based on the holomorphic blocks and constitute stepping stones to tackle the challenging problems we have planned to attack in the second half of the project.