Localization Techniques in Quantum Field Theories: January 16 – February 16, 2018

Organized by: Vasily Pestun and Maxime Zabzine

The program will be focusing on the development of localization techniques in quantum field theories and its applications. In particular we want to concentrate on the developments in the field since 2007.

The main idea of different localization formulas is that the specific finite dimensional integral can be evaluated exactly by summing up over fixed point contributions. The proof of the Berline-Vergne-Atiyah-Bott formula can be recasted  in terms of supergeometry and finite dimensional version of supersymmetry. This allows to discuss the possible generalizations of the Berline-Vergne-Atiyah-Bott formula to the infinite dimensional setup.

Since the discovery of localization formula in math there have been different attempts to implement it in infinite dimensional setting, in particular in the context of path integral. Localization of the path integral to various interesting finite-dimensional geometrical moduli spaces was pioneered by Witten (1982) and later was applied to two-dimensional topological sigma model, four dimensional topological gauge theory, two-dimensional Yang-Mills theory. Further development on supersymmetric localization is related to the calculation of Nekrasov’s partition function, or equivariant Donaldson-Witten theory based on earlier works on the equivariant integration of the hyper-K\”ahler quotients by Losev-Moore-Nekrasov-Shatashvili.

As for the non-topological supersymmetric theories, the localization technique, which captures the essential physical phenomena such as a $\beta$-function for running coupling constant, was developed in 2007 by the first organizer to compute the full partition function and the expectation value of supersymmetric Wilson loops on four dimensional sphere for N=2 supersymmetric field  theories. After this paper it has been explosion of the works  dealing with the localization of different supersymmetric theories in different dimensions. This explosion of exact results in quantum field theory led to many non-trivial developments in quantum field theory: the check of non-perturbative dualities, the further checks of AdS/CFT correspondence, the new look at the supersymmetric theories on the curved manifold and better analytical and algebraic understanding of the exact results (e.g., the  partition functions). This also led to the new mathematical results, e.g. the new ways of calculating the genus 0f Gromov-Witten invariants from Kahler potentials computed  by the partition functions of gauge theories on a two-sphere. The program will concentrate on these developments during last 10 years.

Program Application Deadline: October 30, 2017 (or when event is at maximum capacity). Applicants will be contacted soon after this date.