HEXAGONbeta

Supersymmetric Yang--Mills theories with several times reduce to ten dimensions

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

Contributed by Deep Bhattacharjee ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

Supersymmetric Yang--Mills theories in signatures with several times are built by restricting the gaugino to the kernel of kk mutually orthogonal null vectors and letting the supersymmetry close on translations contracted with their product. We prove that every theory of this kind, defined by five axioms that Nishino's construction satisfies, has residual dimension s+t−2ks+t-2k equal to ten (or to two, a degenerate case), and that its null vectors can be real only if k≤min⁡(s,t)k\le\min(s,t). The cubic fermion terms reduce to a Fierz identity on the residual spinors, and that identity forces the dimension of the spinors to be 2(d−2)2(d-2) on a dd-dimensional vector space, which happens only for d=3,4,6,10d=3,4,6,10. With Majorana--Weyl spinors the signature must therefore satisfy s−t∈{8,0,−8}s-t\in\{8,0,-8\}. There is no such theory in (9+8n)+1(9+8n)+1, (10+8n)+2(10+8n)+2, (11+8n)+3(11+8n)+3 or (12+8n)+4(12+8n)+4 for n≥1n\ge1, although it has been claimed that the existence proofs for the signatures (9+m)+(1+m)(9+m)+(1+m) carry over to these; the signatures 27+327+3 and 19+319+3 are among those excluded. For a single time the result does not depend on the construction: in (9+8n)+1(9+8n)+1 with n≥1n\ge1 every massless multiplet without tensor charges has at least 222+4n2^{2^{2+4n}} states, so a unitary theory with no tensor charges on its massless states has no massless vector multiplet.

Provenance statement

Deep Bhattacharjee set the task and directed the work. Claude (Anthropic), through Claude Code, chose the claim to be tested, checked the literature (every INSPIRE record citing the two papers that make the claim), developed the proofs, wrote the Python checks in explicit gamma matrices (34 checks, all passing) and the figure scripts, and drafted and revised the paper. Every statement is proved in the text; the numerical checks test the identities independently and no proof depends on them. Code and LaTeX source: Zenodo, doi:10.5281/zenodo.23558751 (release v1.0.0 of https://github.com/creelie/cs-grav).
AI provenance filesAI-PROVENANCE-HEXAGON.md · 3 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 22, 2026