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Residual-dimension obstructions for null-projected supersymmetric Yang--Mills theories

A mix of human-written and AI-generated textHuman understanding: all partsmath-ph — Mathematical Physicsmath.RT — Representation Theory

Contributed by Deep Bhattacharjee

Version 3 / Oct 22, 2026 / CC BY 4.0

Abstract

We prove necessary conditions for null-projected supersymmetric Yang--Mills theories with a Majorana--Weyl gaugino in signature (s,t)(s,t), under five explicit assumptions on the gaugino constraint, the transformation rules, the residual gauge field, the translation bracket and the absorption of the cubic gaugino terms. A real gaugino allows at most min⁡(s,t)\min(s,t) null vectors, the residual dimension must be two or ten, and s−ts-t must be 00 or ±8\pm8. The main step is a converse of the cubic gamma-matrix identity: if symmetric gamma matrices acting on NN-component spinors in dd dimensions satisfy it, then N=2(d−2)N=2(d-2). Consequently Nishino's construction does not transfer to the signatures (11+8n)+3(11+8n)+3 proposed by Rios, Marrani and Chester. In 27+327+3 a real null frame has at most three vectors against Nishino's ten, the translation bracket allows only two, and with two the residual spinors have 40964096 components where absorption needs 4848. For unitary single-time theories with vanishing tensor charges on a massless multiplet, we also exclude every finite collection of scalar, spinor and vector polarizations when there are more than sixteen real supercharges. Theories outside the five assumptions are not decided.

Provenance statement

Claude Code assisted the original preparation. OpenAI Codex audited the proofs of the second version and assisted with the helicity-span argument, the null-frame current lemma, scope corrections and editing. For the third version, ChatGPT-6.1 Sol Ultra (OpenAI) wrote an independent audit of the signature transfer proposed by Rios, Marrani and Chester, and Claude Code (Opus 5.5 with Ultracode, and Fable 5.1 at medium effort) assisted with Section 7.1, Proposition 7.1, Remarks 7.2 and 7.3, Appendix D, the program transfer_check.py, Figures 8 and 9, and the checks of every quotation and reference; a second independent review, supplied by the author, led to four corrections of statement in Section 7. The text is a mix of human writing and writing generated by these tools. The author has checked and understood every part and takes responsibility for the content. The gamma-matrix programs are supplementary numerical checks; no Lean formalisation is claimed.

Tools used

Anthropic
Claude CodeVersion Fable 5.1 Medium

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositSupersededOct 22, 2026
  2. v2Submitted by Deep BhattacharjeeSubmitted as a new versionSupersededOct 26, 2026
  3. v3Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 28, 2026