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The gaugino condensate from asymmetric four-torus with twists

Anber, Mohamed M.; Poppitz, Erich

The gaugino condensate from asymmetric four-torus with twists Thumbnail


Erich Poppitz


We calculate the gaugino condensate in SU(2) super Yang-Mills theory on an asymmetric four-torus 𝕋4 with ’t Hooft’s twisted boundary conditions. The 𝕋4 asymmetry is controlled by a dimensionless detuning parameter βˆ†, proportional to L3L4 βˆ’ L1L2, with Li denoting the 𝕋4 periods. We perform our calculations via a path integral on a 𝕋4. Its size is taken much smaller than the inverse strong scale Ξ› and the theory is well inside the semi-classical weak-coupling regime. The instanton background, constructed for βˆ† β‰ͺ 1 in [1], has fractional topological charge Q=1/2 and supports two gaugino zero modes, yielding a non-vanishing bilinear condensate, which we find to be βˆ†-independent. Further, the theory has a mixed discrete chiral/1-form center anomaly leading to double degeneracy of the energy eigenstates on any size torus with ’t Hooft twists. In particular, there are two vacua, ∣0〉 and ∣1〉, that are exchanged under chiral transformation. Using this information, the βˆ†-independence of the condensate, and assuming further that the semi-classical theory is continuously connected to the strongly-coupled large-𝕋4 regime, we determine the numerical coefficient of the gaugino condensate: γ€ˆ0|trλλ|0〉 =β€‰βˆ£γ€ˆ1|trλλ|1γ€‰βˆ£β€‰= 32Ο€2Ξ›3, a result equal to twice the known ℝ4 value. We discuss possible loopholes in the continuity approach that may lead to this discrepancy.


Anber, M. M., & Poppitz, E. (2022). The gaugino condensate from asymmetric four-torus with twists. Journal of High Energy Physics, 2023, Article 118 (2023).

Journal Article Type Article
Acceptance Date Dec 19, 2022
Online Publication Date Jan 23, 2022
Publication Date Jan 23, 2022
Deposit Date Oct 27, 2022
Publicly Available Date Apr 18, 2023
Journal Journal of High Energy Physics
Print ISSN 1126-6708
Publisher Scuola Internazionale Superiore di Studi Avanzati (SISSA)
Peer Reviewed Peer Reviewed
Volume 2023
Article Number 118 (2023)


Published Journal Article (Advance online version) (935 Kb)

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Copyright Statement
Advance online version This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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