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<p>We derive the explicit formula for fractional BPS lumps (or fractional instantons) in the <i>ℂP</i><sup><i>N</i> − 1</sup> nonlinear sigma model on a two-dimensional torus under various shift-clock twisted boundary conditions. After regularizing the <i>ℂP</i><sup><i>N</i> − 1</sup> model by an <i>N</i>-component Abelian-Higgs model, those twisted boundary conditions introduce nontrivial ’t Hooft fluxes <i>p</i>/<i>N</i> for the U(<CitationRef CitationID="CR1">1</CitationRef>) gauge field, and the topological charge becomes fractionalized as <i>k</i> + <i>p</i>/<i>N</i> ∈ <i>ℤ</i> + <i>p</i>/<i>N</i>. The moduli space is globally determined as the <i>ℂP</i><sup><i>Nk</i> + <i>p</i> − 1</sup>-fiber bundle on a 2-torus, which is a Kähler manifold of complex dimension <i>Nk</i> + <i>p</i> as predicted by the index theorem. We present two different parametrizations of the moduli space: one of them immediately identifies the small-lump singularity appearing in the <i>ℂP</i><sup><i>N</i> − 1</sup> limit, while the other makes the modular invariance manifest.</p><p>We also discuss the implications of our finding for the 4d SU(<i>N</i>) Yang-Mills theory on the 4-torus with ’t Hooft twists. By tuning the aspect ratio of the 4-torus, fractional instantons in the <i>ℂP</i><sup><i>N</i> − 1</sup> model with a non-Fubini-Study metric are obtained through the dimensional reduction of 4d Yang-Mills theory, whose moduli space coincides with the one obtained for the standard <i>ℂP</i><sup><i>N</i> − 1</sup> model as complex manifolds.</p>

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Fractional instantons in 2d ℂPN − 1 model and 4d Yang-Mills theory with ’t Hooft twists

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  • 发表日期：

    2026年3月19日

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  • Yui Hayashi,
  • Tatsuhiro Misumi,
  • Muneto Nitta,
  • Keisuke Ohashi,
  • Yuya Tanizaki

摘要

We derive the explicit formula for fractional BPS lumps (or fractional instantons) in the ℂPN − 1 nonlinear sigma model on a two-dimensional torus under various shift-clock twisted boundary conditions. After regularizing the ℂPN − 1 model by an N-component Abelian-Higgs model, those twisted boundary conditions introduce nontrivial ’t Hooft fluxes p/N for the U(1) gauge field, and the topological charge becomes fractionalized as k + p/N ∈ ℤ + p/N. The moduli space is globally determined as the ℂPNk + p − 1-fiber bundle on a 2-torus, which is a Kähler manifold of complex dimension Nk + p as predicted by the index theorem. We present two different parametrizations of the moduli space: one of them immediately identifies the small-lump singularity appearing in the ℂPN − 1 limit, while the other makes the modular invariance manifest.

We also discuss the implications of our finding for the 4d SU(N) Yang-Mills theory on the 4-torus with ’t Hooft twists. By tuning the aspect ratio of the 4-torus, fractional instantons in the ℂPN − 1 model with a non-Fubini-Study metric are obtained through the dimensional reduction of 4d Yang-Mills theory, whose moduli space coincides with the one obtained for the standard ℂPN − 1 model as complex manifolds.