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

Periodic Reporting for period 4 - HBQFTNCER (Holomorphic Blocks in Quantum Field Theory: New Constructions of Exact Results)

Reporting period: 2019-04-01 to 2021-02-28

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.
We have tested the holomorphic block factorisation proposal for a large variety of theories formulated on compact manifolds. Most of the results obtained in this research line of the project have been collected in an invited contribution to the review `Localization techniques in quantum field theories' J. Phys. A: Math. Theor. 50 440301.

The team has explored the potential application of the holomorphic blocks formalism in the context of the holographic study of the black hole entropy. This has led to the introduction of the gravitation blocks and their gluing rules providing a unifying entropy functional and an extremization principle for black holes and black strings with arbitrary rotation and generic electric and magnetic charges JHEP 12 (2019) 168.

We discovered new web of dualities involving holomorphic blocks in various dimensions and correlation functions in theories with Virasoro or deformed Virasoro symmetry. We combined ideas and technologies from topological strings and integrable systems obtaining new tools to study the modular properties of the blocks and their transformation properties under various dualities. Thanks to these results we provided explicit examples of dualities for defect theories which are induced by dualities in the bulk, JHEP 02 (2019) 176, JHEP 04 (2019) 138.

We formulated a new surprising correspondence between three dimensional supersymmetric gauge theories and free-field correlators published in JHEP 11 (2019) 081 and JHEP 01 (2020) 061. These works led us to construct a new family of four dimensional minimally supersymmetric theories, which we realised to be closely related to certain functions studied by the mathematicians in a different context. Building on these results we discovered a class of theories which enjoy a new type of infrared duality which we named 4d mirror duality JHEP 09 (2020) 047.

In the context of the geometric program, where lower dimensional theories are realised from compactifications of higher dimensional theories on Riemann surfaces, we introduced the fundamental building blocks for the construction of an important family of non-Lagrangian theories, known as E-string compactifications appeared in SciPost Phys. 8 (2020). More results have been in obtained by team members in JHEP 06 (2020) 159, JHEP 11 (2019) 053, JHEP 09 (2020) 161. In particular Zafrir has obtained a very important result providing the first (N=1) Lagrangian realisation for a N=3 SCFT, JHEP 01 (2021) 062.
The results obtained in this project have been published in top journals and presented at international conferences attracting considerable attention from the researcher community. Many of the ideas and results established in this project have been used by other research groups becoming standard tools.

The project has demonstrated the great potential of the approach based on the holomorphic blocks and of the use of the interplay between dualities and correspondences between supersymmetric gauge theories and exactly solvable systems.

Toward the end of the project we started exploring new very promising research directions, in particular we introduced the first class of theories enjoying mirror dualities in four dimensions. We are now undertaking a systematic study of these theories which we expect to soon produce very surprising results. There were many other new ideas that emerged during the course of the project which we plan to investigate in the coming months.

We are confident that building on the results obtained in the project we will continue making significant progress toward the understanding of strongly coupled phenomena in quantum field theories.
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