nLab Nicolai map

Idea

The Nicolai map is a (non-local and non-linear) transformation of bosonic quantum fields to free fields whose existence characterizes the original bosonic quantum field theory as arising by integrating out the fermions of a globally supersymmetric field theory.

References

The original discussion:

• Hermann Nicolai, On a new characterization of scalar supersymmetric theories, Physics Letters B 89 3–4 (1980) Pages 341-346 [doi:10.1016/0370-2693(80)90138-0]

• Hermann Nicolai, Supersymmetry and functional integration measures, Nuclear Physics B

  176 2 (1980) 419-428 [doi:10.1016/0550-3213(80)90460-5]

• Hermann Nicolai, Supersymmetric functional integration measures, NATO Advanced Study Institute on Supersymmetry, Bonn (Aug 1984) 393-420 [cds:155731]

The infinitesimal version:

• Olaf Lechtenfeld, Konstruktion der Nicolai-Abbildung in supersymmetrischen Feldtheorien, PhD thesis, Bonn (1984) [pdf, pdf]

• K. Dietz, Olaf Lechtenfeld, Nicolai maps and stochastic observables from a coupling constant flow, Nuclear Physics B 255 (1985) 149-173 [doi:10.1016/0550-3213(85)90132-4]

• K. Dietz, Olaf Lechtenfeld, Ghost-free quantisation of non-abelian gauge theories via the Nicolai transformation of their supersymmetric extensions, Nuclear Physics B 259 2–3 (1985) 397-411 [doi:10.1016/0550-3213(85)90642-X]

• Olaf Lechtenfeld, Stochastic variables in ten dimensions?, Nuclear Physics B 274 3–4 (1986) 633-652 [doi:10.1016/0550-3213(86)90531-6]

• Rainald Flume, Olaf Lechtenfeld, On the stochastic structure of globally supersymmetric field theories, Physics Letters B 135 1–3 (1984) 91-95 [doi:10.1016/0370-2693(84)90459-3]

Construction in terms of a path-ordered integral:

• Olaf Lechtenfeld, Maximilian Rupprecht, Universal form of the Nicolai map, Phys. Rev. D 104 021701 (2021) [arXiv:2104.00012, doi:10.1103/PhysRevD.104.L021701]

• Olaf Lechtenfeld, Maximilian Rupprecht, Is the Nicolai map unique?, J. High Energ. Phys. 2022 139 (2022) [arXiv:2207.09471, doi:10.1007/JHEP09(2022)139]

On its non-perturbative nature:

• Olaf Lechtenfeld, Supersymmetric large-order perturbation with the Nicolai map, Physics Letters B

  835 (2022) 137507 [arXiv:2208.06420, doi:10.1016/j.physletb.2022.137507]

Review:

• Danny Birmingham, Matthias Blau, Mark Rakowski, George Thompson, section 3.2 of: Topological field theory, Physics Reports 209 4–5 (1991) 129-340 [doi:10.1016/0370-1573(91)90117-5]

• Olaf Lechtenfeld, The Nicolai-map approach to supersymmetry [arXiv:2309.00481]

On the Nicolai map for the $\mathbb{C}P^n$ sigma-model:

• Lorenzo Casarin, Olaf Lechtenfeld, Maximilian Rupprecht, Nicolai maps with four-fermion interactions [arXiv:2310.19946]

• Olaf Lechtenfeld: Nicolai maps for supersymmetric sigma models [arXiv:2409.18855]

For D=4 SYM and D=10 SYM in light cone gauge:

• Olaf Lechtenfeld, Nicolai maps for super Yang-Mills on the light cone [arXiv:2406.04406]

For super Yang-Mills theory in light cone gauge:

• Nipun Bhave, Saurabh Pant, Nicolai maps and uniqueness in the light-cone gauge [arXiv:2406.18238]

For supergravity:

• Federico Arrighi, Saurish Khandelwal, Olaf Lechtenfeld: Towards a Nicolai map for supergravity [arXiv:2509.17973]

• Ji-Seong Chae, Hun Jang, Junhyeok Lee: Perturbative Nicolai-Map Diagrammatics: Application to Poincaré Supergravity [arXiv:2605.29990]