# Supersymmetric field theories on curved backgrounds and applications

**Project Level: **PhD, Masters, Honours

During the last four decades, supersymmetry has been at the forefront of theoretical and mathematical physics of fundamental interactions. It played a crucial role in constructing models aimed at the unification of all forces including quantum gravity, namely string theory. Supersymmetry has also led to several new developments in mathematical physics such as, for example, the study of conformal field theories (that play a fundamental role in string theory and in the description of phase transitions in statistical mechanics) and integrable systems.

The analysis of supersymmetric theories on curved backgrounds has made possible the exact computation of several important observables. For instance, quantum computations in (super)conformal field theories on a conformally flat background, can be extrapolated from the curved to the flat space limit. For rigid supersymmetric field theories on curved spacetimes localisation techniques (which can reduce an infinite dimensional path integral to a fine dimensional one) allow one to compute quantum observables, such as the partition function, indices, correlators or Wilson loops exactly. Remarkably, localisation has been developed also for calculations in supergravity theories and evaluation of the quantum entropy of black holes. One crucial requirement of localisation is that SUSY has to close off-shell. This leads to the natural use of covariant off-shell (superspace) techniques.

In quantum field theories, correlation functions may contain singularities at coincident points, the so-called contact terms. They arise as higher-derivative terms in quantum effective actions on some supersymmetric gravity and gauge multiplet backgrounds. For correlation functions of symmetry currents, contact terms can lead to so-called anomalies. These manifest the breakdown of a symmetry due to short distance quantum effects. Anomalies have had a predominant role in the non-perturbative study of quantum field theory and string theory and relate to the Atiyah-Singer index theory in mathematics. In supersymmetric theories, anomalies lie in supermultiplets. Despite their importance, for general supersymmetry and spacetimes, the structure of higher-derivative invariants associated to contact terms and anomalies is not understood. These find important applications in the study supersymmetric quantum field theories, holographic correspondences between gauge and quantum gravity and related topics.

This project aims at: i) classifying general off-shell supersymmetric curved backgrounds by mean of new mathematical techniques; ii) studying supersymmetric and superconformal field theories on curved manifolds and their applications to holographic correspondences; iii) classify the structure of supersymmetric anomalies in generality.

This project will support the research of the ARC Future Fellowship of Dr. Gabriele Tartaglino Mazzucchelli “Supersymmetry and Supergravity: New Approaches and Applications.” The student will largely benefit from a vibrant international collaboration on supersymmetry and related topics.