Localisation with on-shell supersymmetry algebras via the Batalin-Vilkovisky formalism: Localisation as gauge fixing
Leron Borsten, Dimitri Kanakaris, Hyungrok Kim
TL;DR
This work unifies localisation and gauge fixing within the Batalin–Vilkovisky formalism, showing that localisation with respect to global (super)symmetries can be treated on the same footing as gauge fixing. By introducing localising and global-ghost structures and employing on-shell BV techniques, the authors demonstrate genuine on-shell localisation for quantum field theories, eliminating the need for auxiliary fields in several cases. They develop a comprehensive BV framework for both off-shell and on-shell algebras, including bivector-type open algebras, and validate the approach through explicit examples: a $d=1$, $\mathcal{N}=2$ superparticle and a $d=3$, $\mathcal{N}=2$ SYM theory on Seifert manifolds, computing indices and partition functions. The results offer a robust, generalizable method for localisation beyond traditional off-shell constructions, with potential applications to higher symmetries, twisted theories, and supergravity contexts.
Abstract
The Batalin-Vilkovisky formalism provides a powerful technique to deal with gauge and global (super)symmetries that may only hold on shell. We argue that, since global (super)symmetries and gauge symmetries appear on an equal footing in the Batalin-Vilkovisky formalism, similarly localisation with respect to global (super)symmetries appears on an equal footing with gauge fixing of gauge symmetries; in general, when the gauge-fixing condition is not invariant under the global symmetries, localisation (with respect to a localising fermion) and gauge fixing (with respect to a gauge-fixing fermion) combine into a single operation. Furthermore, this perspective enables supersymmetric localisation using only on-shell supermultiplets, dispensing with auxiliary fields, extending an insight first discovered by Losev and Lysov arXiv:2312.13999. We provide the first examples of on-shell localisation for quantum field theories (together with a companion paper by Arvanitakis arXiv:2511.00144).