Introdução à Supersimetria

Objectives

Supersymmetry (SUSY) is a profound symmetry with far-reaching implications in field theory, geometry, and number theory. It enables exact computations of partition functions, facilitates the derivation of index theorems, and reveals interesting connections to modular forms. As a result, SUSY serves as a powerful tool for uncovering exact results and exposing deep mathematical relationships. In this course, we will provide a gentle introduction to supersymmetry, focusing on supersymmetric quantum mechanics and exploring some of its various applications, particularly in classical index theorems.

Syllabus

Super vector space, Lie superalgebra. Grassmann calculus. Fermionic harmonic oscillator. Representation of the partition function and of the supertrace of a fermionic system using path integrals. Supersymmetric quantum mechanics (SUSY QM). Properties of supersymmetry. The Witten index and its representation using a path integral. The supersymmetric harmonic oscillator. Elliptic differential operators on compact manifolds. The De Rham complex. SUSY QM on a compact, orientable, Riemannian manifold without boundary. The supercharge operator Q and De Rham cohomology. The corresponding Witten index and the Euler characteristic of the complex defined by the operator Q. The Chern–Gauss–Bonnet theorem. SUSY QM on a compact spin manifold of even dimension, without boundary. The Dirac operator and the corresponding Witten index. The Atiyah–Singer index theorem. The Atiyah–Patodi–Singer index theorem. The eta invariant and pseudo-modular forms.

Prerequisites

Basic knowledge of quantum mechanics and of Riemannian geometry.

Cross Competence Component

The course unit fosters the development of transversal skills in critical thinking, creativity, and problem - solving strategies, both in class, through independent work, and across the various assessment components.

Ethical Principles

All member of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.