Compute the continuum determinant for the single-root \(SU(N>2)\) saddle

Determine the regulated continuum fluctuation determinant for the off-diagonal adjoint multiplets in the single-root charge-\(1/2\) \(SU(N>2)\) Eguchi–Hanson instanton, and use it together with the large-size region and any additional same-sector contributions to determine the complete boundary-sector-projected gaugino bilinear condensate.

Background

For SU(N>2)SU(N>2), embedding the Eguchi–Hanson SU(2)SU(2) instanton into one color block gives the correct two adjoint zero modes, but its noncentral boundary holonomy produces continuum fluctuations in the off-diagonal color sectors. Renormalization-group invariance requires these fluctuations to supply powers of the regulator scale and coupling that are absent from the zero-mode measure.

The paper writes the determinant in terms of an unknown finite spectral function. The resulting isolated saddle has apparent dependence on the insertion point and the Eguchi–Hanson resolution parameter that is incompatible with the Ward identities if treated as complete. The authors therefore leave unresolved the determinant, the strongly coupled large-modulus region, possible additional saddles, and possible moduli-space boundary terms needed to obtain the full condensate.

References

The unknown continuum determinant, the noncompact region t\gtrsim\Lambda{-1} (where t parametrizes the radial-size modulus) which extends well into the strongly-coupled regime, additional configurations in the same boundary sector, or a moduli-space surface term must be understood before the single-root sector can be completed. Thus, the zero-mode calculation does not determine the SU(N>2) bilinear condensate.

— Gaugino condensates in $\mathcal{N}=1$ super-Yang-Mills Theory on Eguchi-Hanson space  (2610.00486 - Anber, 30 Sep 2026) in Introduction, discussion following Eq. (single-root formal amplitude); Section 5.1, discussion following Eq. (single-root holomorphic scaling)

To interpret the EH amplitude as a condensate in a specific chiral vacuum, one would need additional data, such as a bra explicitly prepared in that vacuum or an infinitesimal chiral-symmetry-breaking source.

— Gaugino condensates in $\mathcal{N}=1$ super-Yang-Mills Theory on Eguchi-Hanson space  (2610.00486 - Anber, 30 Sep 2026) in Section “EH-prepared states and boundary-conditioned observables”; final paragraph

Consequently, the factor 1/6 constitutes a genuine clustering puzzle only if the two-block family is shown to give the complete separated four-point function, the EH boundary state is shown to select a definite vacuum, and an independent calculation in that same state and holonomy sector yields a different bilinear condensate.

— Gaugino condensates in $\mathcal{N}=1$ super-Yang-Mills Theory on Eguchi-Hanson space  (2610.00486 - Anber, 30 Sep 2026) in Section 5.2, discussion following Eq. (main-SU4-final-four-bilinear-result); concluding discussion