Open Access

Metamorphosis of fractional instantons on a twisted $T^{4}$ with a double-trace deformation: A numerical study

Department of Physics, University of Toronto, 60 St George Street, Toronto, Ontario M5S 1A7, Canada

^*Contact author: ben.dobozy@mail.utoronto.ca
^†Contact author: erich.poppitz@utoronto.ca

Phys. Rev. D 114, 065022 – Published 23 September, 2026

DOI: https://doi.org/10.1103/xnjb-vl84

Abstract

We use numerical minimization of the lattice action of trace-deformed Yang-Mills theory on $T^{4}$ with twisted boundary conditions to find the classical minimum action configurations of fractional topological charge. We vary the twists and ratios of torus periods to interpolate between different $R^{4 - k} \times T^{k}$ geometries. This allows us to see how the corresponding minimum action saddle point configurations—monopole-instantons ( $k = 1$ ), center vortices ( $k = 2$ ), and fractional instantons ( $k = 3$ , 4)—morph into each other. We also study how the transition between them depends on the presence of a deformation potential. In particular, we argue that the recent analytic picture of chains of monopole-instantons collimating their flux into center-vortex sheets, while technically relying on the deformation potential, also holds in pure Yang-Mills theory, for tori whose shape causes the Abelianization due to the deformation to align with the one due to the twists. Our results also indicate that with nonzero deformation potential, some transitions between different minimal-action fractional charge configurations may be discontinuous and involve level crossing.

Physics Subject Headings (PhySH)

Article Text

References (65)

G. ’t Hooft, A property of electric and magnetic flux in non-Abelian gauge theories, Nucl. Phys. B153, 141 (1979).
G. ’t Hooft, Aspects of quark confinement, Phys. Scr. 24, 841 (1981).
M. Unsal and L. G. Yaffe, Center-stabilized Yang-Mills theory: Confinement and large $N$ volume independence, Phys. Rev. D 78, 065035 (2008).
J. D. Bjorken, Elements of quantum chromodynamics, Prog. Math. Phys. 4, 423 (1979).
M. Luscher, Some analytic results concerning the mass spectrum of Yang-Mills gauge theories on a torus, Nucl. Phys. B219, 233 (1983).
A. Arabi Ardehali and D. J. Resnick, Perturbative Coulomb branches on $R^{3} \times S^{1}$ : The global $D$ -term potential, arXiv:2604.27066.
G. V. Dunne and M. Ünsal, New nonperturbative methods in quantum field theory: From large- $N$ orbifold equivalence to bions and resurgence, Annu. Rev. Nucl. Part. Sci. 66, 245 (2016).
E. Poppitz, Notes on confinement on $R^{3} \times S^{1}$ : From Yang-Mills, Super-Yang-Mills, and QCD(adj) to QCD(F), Symmetry 14, 180 (2022).
A. González-Arroyo, On the fractional instanton liquid picture of the Yang-Mills vacuum and confinement, arXiv:2302.12356.
M. M. Anber and E. Poppitz, Higher-order gaugino condensates on a twisted $T^{4}$ , J. High Energy Phys. 02 (2025) 114.
D. J. Gross, R. D. Pisarski, and L. G. Yaffe, QCD and instantons at finite temperature, Rev. Mod. Phys. 53, 43 (1981).
M. Garcia Perez et al. (RTN Collaboration), Instanton like contributions to the dynamics of Yang-Mills fields on the twisted torus, Phys. Lett. B 305, 366 (1993).
A. Gonzalez-Arroyo and P. Martinez, Investigating Yang-Mills theory and confinement as a function of the spatial volume, Nucl. Phys. B459, 337 (1996).
A. Gonzalez-Arroyo and A. Montero, Selfdual vortex—like configurations in SU(2) Yang-Mills theory, Phys. Lett. B 442, 273 (1998).
A. Montero, Vortex configurations in the large $N$ limit, Phys. Lett. B 483, 309 (2000).
P. van Baal, Some results for SU( $N$ ) gauge fields on the hypertorus, Commun. Math. Phys. 85, 529 (1982).
K.-M. Lee and P. Yi, Monopoles and instantons on partially compactified $D$ -branes, Phys. Rev. D 56, 3711 (1997).
T. C. Kraan and P. van Baal, Monopole constituents inside SU( $N$ ) calorons, Phys. Lett. B 435, 389 (1998).
Y. Tanizaki and M. Ünsal, Center vortex and confinement in Yang–Mills theory and QCD with anomaly-preserving compactifications, Prog. Thoer. Exp. Phys. 2022, 04A108 (2022).
M. M. Anber and E. Poppitz, Mass-deformed super Yang-Mills theory on $T^{4}$ : Sum over twisted sectors, $θ$ -angle, and $C P$ violation, J. High Energy Phys. 10 (2025) 182.
E. Poppitz and M. E. Shalchian T., String tensions in deformed Yang-Mills theory, J. High Energy Phys. 01 (2018) 029.
J. Greensite, An Introduction to the Confinement Problem (Springer, New York, 2020), Vol. 972.
D. R. Junior, G. Krein, L. E. Oxman, and B. R. Soares, Study of the emergence of a gluon mass scale from center vortices using a wave-functional formalism, Phys. Rev. Lett. 136, 111902 (2026).
M. M. Anber and E. Poppitz, The gaugino condensate from asymmetric four-torus with twists, J. High Energy Phys. 01 (2023) 118.
M. M. Anber and E. Poppitz, Multi-fractional instantons in SU( $N$ ) Yang-Mills theory on the twisted four-torus, J. High Energy Phys. 09 (2023) 095.
N. Dorey, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, The calculus of many instantons, Phys. Rep. 371, 231 (2002).
G. ’t Hooft, Some twisted selfdual solutions for the Yang-Mills equations on a hypertorus, Commun. Math. Phys. 81, 267 (1981).
F. D. Wandler, Numerical fractional instantons in SU(2): Center vortices, monopoles, and a sharp transition between them, arXiv:2406.07636.
Y. Hayashi and Y. Tanizaki, Unifying monopole and center vortex as the semiclassical confinement mechanism, Phys. Rev. Lett. 133, 171902 (2024).
C. Güvendik, T. Schaefer, and M. Ünsal, The metamorphosis of semi-classical mechanisms of confinement: From monopoles on $R^{3} \times S^{1}$ to center-vortices on $R^{2} \times T^{2}$ , J. High Energy Phys. 11 (2024) 163.
M. M. Anber and E. Poppitz, On the global structure of deformed Yang-Mills theory and QCD(adj) on $R^{3} \times S^{1}$ , J. High Energy Phys. 10 (2015) 051.
D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, Theta, time reversal, and temperature, J. High Energy Phys. 05 (2017) 091.
A. A. Cox, E. Poppitz, and F. D. Wandler, The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles, J. High Energy Phys. 10 (2021) 069.
M. M. Anber and L. Vincent-Genod, Classification of compactified $s u (N_{c})$ gauge theories with fermions in all representations, J. High Energy Phys. 12 (2017) 028.
Y. Hayashi, Y. Tanizaki, and H. Watanabe, Non-supersymmetric duality cascade of QCD(BF) via semiclassics on $R^{2} \times T^{2}$ with the baryon-’t Hooft flux, J. High Energy Phys. 07 (2024) 033.
Y. Hayashi and Y. Tanizaki, Semiclassics for the QCD vacuum structure through $T^{2}$ -compactification with the baryon-’t Hooft flux, J. High Energy Phys. 08 (2024) 001.
Y. Hayashi, Y. Tanizaki, and H. Watanabe, Semiclassical analysis of the bifundamental QCD on $R^{2} \times T^{2}$ with ’t Hooft flux, J. High Energy Phys. 10 (2023) 146.
Y. Tanizaki and M. Ünsal, Semiclassics with ’t hooft flux background for QCD with 2-index quarks, J. High Energy Phys. 08 (2022) 038.
Y. Hayashi, T. Misumi, and Y. Tanizaki, Monopole-vortex continuity of $N = 1$ super Yang-Mills theory on $R^{2} \times S^{1} \times S^{1}$ with ’t Hooft twist, J. High Energy Phys. 05 (2025) 194.
Y. Hayashi, T. Misumi, and Y. Tanizaki, On 2d perimeter law in $N = 1$ super Yang-Mills theory on $R^{2} \times S^{1} \times S^{1}$ with ’t Hooft twist from monopole-vortex continuity, Int. J. Mod. Phys. A 41, 2548006 (2026).
M. Nguyen and M. Ünsal, Refined instanton analysis of the 2D $C P^{N - 1}$ model: Mass gap, theta dependence, and mirror symmetry, J. High Energy Phys. 03 (2025) 162.
M. Nguyen and M. Ünsal, Self-dual monopole loops, instantons and confinement, arXiv:2509.09625.
M. Garcia Perez, A. Gonzalez-Arroyo, J. R. Snippe, and P. van Baal, Instantons from over—improved cooling, Nucl. Phys. B413, 535 (1994).
P. de Forcrand, M. Garcia Perez, and I.-O. Stamatescu, Improved cooling algorithm for gauge theories, Nucl. Phys. B, Proc. Suppl. 47, 777 (1996).
A. Montero, Numerical analysis of fractional charge solutions on the torus, J. High Energy Phys. 05 (2000) 022.
C. Bonati, M. Cardinali, and M. D’Elia, $θ$ dependence in trace deformed $S U (3)$ Yang-Mills theory: A lattice study, Phys. Rev. D 98, 054508 (2018).
A. Athenodorou, M. Cardinali, and M. D’Elia, Spectrum of trace deformed Yang-Mills theories, Phys. Rev. D 104, 074510 (2021).
C. Bonati, M. Cardinali, M. D’Elia, M. Giordano, and F. Mazziotti, Reconfinement, localization and thermal monopoles in $S U (3)$ trace-deformed Yang-Mills theory, Phys. Rev. D 103, 034506 (2021).
C. Bonati, M. Caselle, A. Negro, D. Panfalone, and L. Verzichelli, Effective string description of the reconfined phase in the trace deformed SU(2) Yang-Mills theory in ( $2 + 1$ ) dimensions, in Lattice 2024 (2025), p. 394, arXiv:2501.13684.
C. Bonati, M. Caselle, A. Negro, D. Panfalone, and L. Verzichelli, Confining flux tube in the trace deformed ( $2 + 1$ ) dimensional SU(2) gauge theory, arXiv:2606.20270.
M. Unsal, Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions, Phys. Rev. D 80, 065001 (2009).
E. Poppitz and F. D. Wandler, Gauge theory geography: Charting a path between semiclassical islands, J. High Energy Phys. 02 (2023) 014.
M. García Pérez, A. González-Arroyo, and M. Okawa, Spatial volume dependence for $2 + 1$ dimensional SU(N) Yang-Mills theory, J. High Energy Phys. 09 (2013) 003.
M. Garcia Perez, A. Gonzalez-Arroyo, A. Montero, and P. van Baal, Calorons on the lattice: A new perspective, J. High Energy Phys. 06 (1999) 001.
M. M. Anber, A. A. Cox, and E. Poppitz, On the moduli space of multi-fractional instantons on the twisted $T^{4}$ , J. High Energy Phys. 02 (2026) 169.
E. Poppitz, $D$ -branes and fractional instantons on a twisted $T^{4}$ : The moduli space as an $N = 2$ supersymmetric Higgs branch, J. High Energy Phys. 07 (2026) 240.
M. Unsal, Abelian duality, confinement, and chiral symmetry breaking in QCD(adj), Phys. Rev. Lett. 100, 032005 (2008).
E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B202, 253 (1982).
A. Gonzalez Arroyo and C. P. Korthals Altes, The spectrum of Yang-Mills theory in a small twisted box, Nucl. Phys. B311, 433 (1988).
M. Ünsal, Strongly coupled QFT dynamics via TQFT coupling, J. High Energy Phys. 11 (2021) 134.
M. Nguyen, T. Sulejmanpasic, and M. Ünsal, Phases of theories with ZN 1-form symmetry, and the roles of center vortices and magnetic monopoles, Phys. Rev. Lett. 134, 141902 (2025).
A. Gonzalez-Arroyo, Yang-Mills fields on the four-dimensional torus. Part 1: Classical theory, in Advanced Summer School on Nonperturbative Quantum Field Physics (1997), pp. 57–91, arXiv:hep-th/9807108.
T. C. Kraan and P. van Baal, Periodic instantons with nontrivial holonomy, Nucl. Phys. B533, 627 (1998).
S. Duane, A. Kennedy, B. J. Pendleton, and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195, 216 (1987).
T. Lippert, The hybrid Monte Carlo algorithm for quantum chromodynamics, Lect. Notes Phys. 508, 122 (1998).

Phys. Rev. D 114, 065022– Published 23 September, 2026

Vol. 114, Iss. 6 — 15 September 2026

Received 30 June 2026
Accepted 3 September 2026

DOI: https://doi.org/10.1103/xnjb-vl84

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physical Review D

Reuse & Permissions

Export citation

Export citation

Metamorphosis of fractional instantons on a twisted $T^{4}$ with a double-trace deformation: A numerical study

Abstract

Physics Subject Headings (PhySH)

Article Text

References (65)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Metamorphosis of fractional instantons on a twisted T4 with a double-trace deformation: A numerical study

Abstract

Physics Subject Headings (PhySH)

Article Text

References (65)

Metamorphosis of fractional instantons on a twisted $T^{4}$ with a double-trace deformation: A numerical study