Localisation with on-shell supersymmetry algebras via the Batalin-Vilkovisky formalism: Localisation as gauge fixing

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).

💡 Research Summary

The paper presents a unified perspective on supersymmetric localisation and gauge fixing by exploiting the Batalin‑Vilkovisky (BV) formalism. The authors observe that both global (super)symmetries and gauge symmetries can be treated on an equal footing within BV: each is encoded by a nilpotent differential (Q for global supersymmetry, Q_BV for gauge symmetry) acting on an extended field space that includes ghosts, antighosts, and auxiliary (Nakanishi‑Lautrup) fields. In the standard localisation framework one deforms the action by a Q‑exact term S → S + t Q Ψ_loc, where Ψ_loc is a fermionic functional of ghost number zero. Gauge fixing, on the other hand, proceeds via a Q_BV‑exact term S → S + Q_BV Ψ_gf, with Ψ_gf carrying ghost number –1. The key insight of the work is that when the gauge‑fixing condition is not invariant under the global supersymmetry, the two deformations can be merged into a single fermionic functional Ψ that contains both the localisation and gauge‑fixing pieces together with mixed terms. This combined fermion implements simultaneously the restriction to a Lagrangian submanifold (gauge fixing) and the localisation onto supersymmetric configurations.

A central technical step is the promotion of the ordinary supersymmetry parameter ε (which has ghost number zero) to a global ghost ε of ghost number +1, thereby aligning the grading of Q and Q_BV. The authors introduce a “global trivial pair” ( σ̄, β ) analogous to the usual gauge trivial pair ( c̄, b ). The localisation fermion then takes the schematic form Ψ_loc ∼ σ̄ V, where V is the usual localisation potential (e.g. a positive‑definite functional of the supersymmetry variation of the fields). This construction makes the global supersymmetry behave like a BRST symmetry within the BV algebra, allowing the standard BV machinery (master equation, BV Laplacian, antibracket) to be applied unchanged.

The most novel contribution is the extension of this framework to on‑shell supersymmetry algebras, i.e. algebras that close only upon use of the equations of motion. Traditional localisation requires auxiliary fields to achieve off‑shell closure; the BV approach sidesteps this by working directly with the on‑shell algebra in the extended field space. The master action S_BV is built to satisfy the classical (and, in the examples, the quantum) master equation even when the supersymmetry transformations close only on‑shell. Consequently, the Q‑exact deformation remains nilpotent and the localisation argument goes through without auxiliary fields.

Two concrete field‑theoretic examples are worked out in detail. First, the d = 1, N = 2 supersymmetric particle (the supersymmetric quantum mechanics model) is treated. The authors construct the BV action with global ghosts, write down the localisation fermion, and show that the deformed path integral localises onto the supersymmetric (BPS) trajectories, reproducing the Witten index. Both the “Q‑localisation” and the “R_ξ‑gauge” interpretations are presented, illustrating the equivalence of the BV localisation fermion with a familiar gauge‑fixing term.

Second, the d = 3, N = 2 supersymmetric Yang–Mills theory placed on a Seifert three‑manifold is examined. After a brief review of Seifert geometry, the authors introduce the BV action including the global supersymmetry ghosts and the appropriate trivial pairs. They formulate the localisation fermion for the on‑shell closed superalgebra and demonstrate that the resulting deformed theory reduces to an integral over the moduli space of supersymmetric connections, matching known results obtained by conventional off‑shell localisation (which typically requires a set of auxiliary fields). The paper also discusses the R_ξ‑gauge version of the BV localisation, showing how the gauge‑fixing function can be chosen to respect the Seifert fibration structure.

Beyond these examples, the authors outline several promising extensions. The BV‑localisation framework naturally accommodates higher‑form gauge symmetries and higher‑categorical algebras, suggesting applicability to topological field theories, twisted supersymmetric theories, and theories with background supergravity. They also propose the use of “fake” or “evanescent” supersymmetries to supersymmetrise non‑supersymmetric models (e.g. Chern–Simons theory) without introducing auxiliary fields, thereby enabling localisation calculations in a broader class of theories.

In summary, the paper establishes that the BV formalism provides a powerful, homological algebraic setting in which gauge fixing and supersymmetric localisation are unified as special cases of restricting the path integral to a Lagrangian submanifold defined by a fermionic functional. By promoting global supersymmetry parameters to ghosts and by employing trivial pairs for both gauge and global sectors, the authors achieve localisation for on‑shell supersymmetry algebras without auxiliary fields. The explicit examples of the supersymmetric particle and three‑dimensional N = 2 Yang–Mills theory illustrate the practicality of the method and open the door to applying on‑shell localisation to a wide variety of quantum field theories.