Journal of High Energy Physics (Dec 2024)

Varieties of four-dimensional gauge theories

  • Ben Gripaios,
  • Khoi Le Nguyen Nguyen

DOI
https://doi.org/10.1007/JHEP12(2024)041
Journal volume & issue
Vol. 2024, no. 12
pp. 1 – 21

Abstract

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Abstract We use algebraic geometry to study the anomaly-free representations of an arbitrary gauge Lie algebra for 4-dimensional spacetime fermions. For irreducible representations, the problem reduces to studying the Lie algebras 𝔰𝔲 n for n ≥ 3. We show that there exist equivalence classes of such representations that are in bijection with the rational points on a projective variety that are dense in a region on the underlying real variety diffeomorphic to ℝ n−3. It follows that the chiral ones overwhelm the non-chiral ones for n ≥ 5. We present an efficient algorithm to find explicit anomaly-free irreducible representations and discuss the generalization to reducible representations.

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