Classical ghosts - A literature study of the Batalin-Vilkovisky formalism

Abstract

Gauge symmetry appears in many physical theories and is often handled by just imposing gauge choices, i.e. removing the gauge symmetry as it consists of unphysical degrees of freedom. This approach may fail when symmetries do not close on-shell and obscures underlying structure of the theory. The Batalin-Vilkovisky formalism describes the connected structure of (higher) gauge symmetries, their corresponding (higher) Noether identities and equations of motion, by introducing a plethora of extra fields called ghosts, antifields and antighosts to represent these. This literature study starts with the introduction of graded structures, from graded vector spaces to Q-manifolds and their duals L∞-algebras, whereafter the physical phenomena and problems that motivates the BV formalism are explained. These phenomena and the formalism itself are then formulated in terms of the introduced mathematical framework. The extra fields are regarded as elements or sections of a symplectic Q-manifold. Central for the BV formalism is an action S that through the symplectic structure induces a differential which in turn gives the equations of motion, gauge transformation etc. when it acts upon the corresponding field. In the L∞-dual description the gauge symmetries etc. are formulated in terms of higher products that generalise the Lie bracket, while equivalence between physical theories corresponds to quasi-isomorphisms. After some minor examples and calculations the thesis ends by revisiting a BV formulation of 10D supersymmetric Yang-Mills, with the help of bosonic spinors and weight partition functions.