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Matrix Models Recursions in Topological Strings and Beyond

Final Report Summary - MMRTSB (Matrix models recursions in topological strings and beyond)

A central challenge in theoretical physics is the understanding of the strongly coupled dynamics of quantum fields. An example of strongly coupled quantum field theory is quantum chromodynamics (QCD), the gauge theory that describes the interaction of quarks and gluons. The standard approach to quantum field theory relies on perturbation theory that, however, loses effectiveness once the interactions become of order one. To overcome this problem theoretical physicists have devoted a great deal of energy to develop non-perturbative or exact methods which turn out to be particularly powerful for theories with extended supersymmetry.

This effort to grasp strongly coupled physics has produced at the same time results of great interest in mathematics. In particular the study of theories with two (N = 2) supersymmetry has been a fertile ground of interaction between mathematicians and physicists.

In recent years, the development of the technique of localisation has made it possible to make significant progress in the study of supersymmetric theories since it allows to reduce the path integral of a theory formulated on a curved manifold to a finite-dimensional matrix integrals. This has made it possible to discover and test new dualities and correspondences between theories. In particular, two new correspondences between field theories obtained via two different reductions of a six-dimensional (6D) 'mother theory' describing the interaction of N M5 branes have been proposed. The first one is the celebrated Alday, Gaiotto and Tachikawa (AGT) correspondence relating 4D N = 2 generalised quiver theories to 2D conformal field theories. The second correspondence instead, proposed by Dimofte and Gukov, relates 3D N = 2 gauge theories to analytically continued Chern-Simons theories on 3 manifolds and thus to the theory of knot invariants.

This project has focused on the study of 3D supersymmetric gauge theories on compact manifolds with the goal of exploring their relation to topological string theory and to knot invariants. Our results can be summarised as follows. A direct study of the matrix integral obtained via localisation has allowed us to find exact expressions for partition function of a large class of theories. We have been able to test conjectured mirror dualities between theories and to construct partition functions for theories with no known Lagrangian description. The main results of this project have been the discovery of a surprising property of partition functions defined on three manifolds admitting a decomposition in solid tori, namely we have shown that partition functions can be expressed in terms of more fundamental building blocks that we named 'holomorphic blocks' corresponding to solid tori partition functions. We provided a detailed explanation of the relation of 3D N = 2 theories to topological strings. This result was achieved via the geometric engineering of holomorphic blocks as open topological string amplitudes which can be computed with matrix model recursive methods. We developed a complete formalism to define holomorphic blocks and gave a complete characterisation of holomorphic blocks globally in parameter space observing that they are subject to Stokes phenomena. We concluded our project by addressing the relation of holomorphic blocks to knot theory, and showed that for theories arising from the compactification of M5 branes on a 3-manifold M, the blocks correspond to a basis of wavefunctions in analytically continued Chern-Simons theory on M.

The block factorisation discovered in this project has been observed by other research groups in the case of 2D theories on the two-sphere and there are indications that it may appear also in higher dimensional theories. This is an interesting research direction that we will further develop in the future.