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arxiv: 2606.22078 · v1 · pith:UN7WEYISnew · submitted 2026-06-20 · ✦ hep-th · hep-lat· hep-ph

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

Benjamin Dobozy , Erich Poppitz This is my paper

Pith reviewed 2026-06-26 11:41 UTC · model grok-4.3

classification ✦ hep-th hep-lathep-ph
keywords fractional instantonscenter vorticesmonopole instantonstwisted boundary conditionstrace deformationYang-Mills theorylattice simulationstopological charge
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The pith

Numerical lattice study shows monopole-instanton chains collimate into center-vortex sheets in pure Yang-Mills on twisted tori with aligned abelianizations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper numerically minimizes the lattice action in trace-deformed Yang-Mills theory on a four-torus with twisted boundary conditions to locate minimum-action configurations carrying fractional topological charge. By varying the twist parameters and the ratios of the torus periods, the simulations interpolate between geometries that favor monopole-instantons, center vortices, and fractional instantons, revealing how these saddle points morph into one another. The central claim is that the analytic picture of monopole-instanton chains collimating their flux into vortex sheets, derived with a deformation potential, remains valid in the pure theory whenever the torus shape aligns the abelianization induced by the deformation with the one induced by the twists. This matters because it supplies concrete evidence that the same non-perturbative objects control the dynamics across different compactifications and without the auxiliary potential.

Core claim

The authors perform numerical minimization of the trace-deformed Yang-Mills lattice action on T^4 with twisted boundary conditions and track the classical minimum-action saddles of fractional topological charge as the twists and period ratios are changed. They observe continuous metamorphosis between monopole-instanton, center-vortex, and fractional-instanton configurations when interpolating between R^{4-k} x T^k geometries. The results indicate that the collimation of monopole flux into vortex sheets persists in the undeformed theory for tori whose shape aligns the two sources of abelianization, while the presence of the deformation potential can make some transitions between distinct mini

What carries the argument

Numerical minimization of the trace-deformed lattice action on T^4 with twisted boundary conditions, used to identify and track the global minimum-action fractional-charge saddle points under changes in geometry and deformation strength.

If this is right

  • The analytic collimation picture of monopole-instanton chains holds in pure Yang-Mills theory whenever torus periods align the abelianizations from twists and deformation.
  • With nonzero deformation potential some transitions between distinct minimal-action fractional-charge configurations become discontinuous and involve level crossing.
  • The type of minimal saddle depends on the relative orientation of the abelianizations induced by the deformation and by the twists.
  • The same set of fractional instantons can be continuously deformed into each other by changing the compactification geometry from one to four dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The alignment condition on torus periods suggests that similar flux-collimation behavior could appear in other geometries or even in the decompactification limit when the effective abelian directions remain matched.
  • Level-crossing transitions under the deformation may correspond to first-order changes in the dominant non-perturbative configurations as the potential strength is varied.
  • Extending the same numerical setup to include dynamical fermions could test whether the observed saddles continue to dominate the vacuum structure and affect chiral properties.

Load-bearing premise

The numerical minimization procedure reliably finds the global minimum-action configurations for each choice of twists and periods rather than becoming trapped in higher-action local minima.

What would settle it

Discovery of a lower-action configuration for a fixed set of twists and periods that fails to exhibit the reported collimation or the expected metamorphosis between monopole and vortex saddles would falsify the identification of the observed configurations as the true minima.

Figures

Figures reproduced from arXiv: 2606.22078 by Benjamin Dobozy, Erich Poppitz.

Figure 2
Figure 2. Figure 2: In addition, we are also able to study the transition between the other configurations in some [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 11
Figure 11. Figure 11: For L1 > 1.5L0, on the other hand, we find the flux vacuum monopole-instanton chain configurations described earlier (such a configuration at the transition point is shown on [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 1
Figure 1. Figure 1: The Wilson action, the deformation action, and the total action, computed for the numerically determined minimal action configuration for dYM on a (15, L, L, 64) lattice, for different L = 5, ..., 35. On each plot, we show the two continuum curves of the actions for the flux and no-flux vacua of eqn. (18). The total action plot shows that there is a transition from the no-flux vacuum, at L < Lc, to the flu… view at source ↗
Figure 2
Figure 2. Figure 2: A sketch of the lining up of BPS (charge α1) and its image KK (charge α0 = −α1, N = 2) monopole-instantons along the compact x3 direction and the collimation of their flux into a center vortex sheet. There is cylindrical symmetry in the 12 plane and the small x0 direction perpendicular to the page is not shown (there is little x0 dependence in the actual field configuration). Blue arrows indicate the long-… view at source ↗
Figure 3
Figure 3. Figure 3: Properties of the |Q| = 1 2 fractional instantons for a lattice of size (10, 25, 25, L3), for 5 < L3 < 90. Top row: separate plots of the Wilson action and the deformation action for dYM. Bottom row: the total action and the width of the instanton, determined by tr W0, for dYM and YM. For a discussion, see the text. For pure YM, at large L3, the 12-plane width of the Higgs field (as per [PITH_FULL_IMAGE:f… view at source ↗
Figure 4
Figure 4. Figure 4: Similar to [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Defining the localization radius of the “Higgs field”: tr W0(x1, x2) is fitted to the (admittedly simplistic, but sufficient for our purposes) radially symmetric function f(|x|) = ae−b|x| . The “localization radius” is then defined as the value of r where R |x|<r d 2xf(|x|) = 1 4 L R1 0 dx1 L R2 0 dx2f(|x|) (the large radius for the YM background on the right figure above is because the fitting function do… view at source ↗
Figure 6
Figure 6. Figure 6: Instanton core size in dYM (left column) and YM (right column) for three different lattice sizes. Top line: L0L3 ≪ L1L2 - abelianization directions due to deformation in dYM and twist in YM are aligned. Middle line: L0L3 ≲ L1L2 dYM abelianizes due to deformation, no abelianization in YM. Bottom line: L0L3 ≳ L1L2 dYM abelianizes, no abelianization in YM (almost constant solution). 23 [PITH_FULL_IMAGE:figur… view at source ↗
Figure 7
Figure 7. Figure 7: The chain of monopole-instantons (recall [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The profile of the field of the center vortex and the disordering of the Wilson loop. Top row, figs. 8a, 8b: A Gaussian fit of the magnetic field, midway between the monopole-instanton and its image upon L3 translation, for the L1 = 45 curve of Figure 7b. F U(1) 12 , at x3 = 0, x2 = L2/2, is fitted to Ae− 1 2 ( x1−µ σ ) 2 + b. The best fit parameters shown are essentially the same in YM and dYM, due to the… view at source ↗
Figure 9
Figure 9. Figure 9: Smoothly interpolating from large to small L3 on a (10, 45, 45, L3) lattice. Recall that L3 ≫ L0 is the regime where the analytic construction of [29, 30] and the picture of [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Results for a (10, L, L, 45) lattice interpolating between T 3×R for small L and S 1×R 3 for large L. All quantities are shown for 5 ≤L≤ 40. Top row: The Wilson and deformation contributions to the action of dYM. Bottom row: The total actions of YM and dYM, as well as the tr W0 width in the x1, x2-plane, determined as in [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Top figure: the disordering effect of the monopole instanton on R(x3) × T 3 (x0,x1,x2)|n12=1 on the Wilson loop winding in the x0 direction, for T 3 of sizes (10, 10, 10) and (10, 5, 5) for two different values of L3 = 25, 45. This is the counterpart of the center-vortex disorder of Figure 8c. Bottom figure: The instanton is localized at x3/L3 = 0.5. The action density plots indicate that the instanton mo… view at source ↗
Figure 12
Figure 12. Figure 12: A small number (2 out of 9) of minimum action dYM configurations with |Q| = 1/2 at L1 L0 = 1.5, on a (10, 15, 15, 45) lattice, at the transition point between the flux and no-flux vacua of dYM, are found to look like monopole instantons in the flux vacuum of dYM. tr W0 shows localization in x3, with width of order L1, and approaches 0 far away from the core of the monopole instanton showing abelianization… view at source ↗
Figure 13
Figure 13. Figure 13: The majority (7 out of 9) of the minimum action dYM configurations with |Q| = 1/2, on a (10, 15, 15, 45) lattice, at the transition point L1 L0 = 1.5 between the flux and no-flux vacua of dYM, are found to have the two maxima structure of their Wilson action density, shown on the left figure, but a single deformation action peak, seen on the r.h.s. On the r.h.s., we compare |tr W0/2| 2 , |tr W1/2| 2 , and… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript numerically minimizes the lattice action of trace-deformed Yang-Mills theory on a twisted T^4, varying twists and period ratios to interpolate between R^{4-k} x T^k geometries. It reports how minimum-action fractional-charge configurations morph among monopole-instantons (k=1), center vortices (k=2), and fractional instantons (k=3,4), and argues that the analytic collimation of monopole-instanton flux into center-vortex sheets holds in pure Yang-Mills when twist-induced and deformation-induced abelianizations align; it also reports that nonzero deformation can produce discontinuous transitions via level crossings.

Significance. If the reported saddles are global minima, the work supplies concrete numerical evidence that the monopole-instanton to center-vortex collimation mechanism survives the removal of the deformation potential under alignment conditions. This strengthens the analytic picture by showing geometry-dependent continuity between different compactification limits and supplies a lattice bridge between deformed and pure Yang-Mills topology.

major comments (2)
  1. [Numerical method / Results sections] Numerical minimization procedure (likely §3 or §4): the central claim that the observed metamorphosis and pure-YM extrapolation hold rests on the assumption that the reported configurations are global minima for each (twist, period) pair. No evidence is provided of exhaustive searches (multiple random starts, annealing schedules, or comparison against known analytic minima in limiting geometries) that would rule out lower-action local minima capable of altering the reported level crossings or flux patterns.
  2. [Discussion / Pure YM limit] Pure-YM extrapolation argument (likely §5 or Discussion): the identification of which deformed minima survive when abelianization directions coincide is used to claim validity in pure Yang-Mills, yet the manuscript does not explicitly verify that no lower-action configuration exists in the undeformed theory for the aligned geometries; this step is load-bearing for the main claim but remains an extrapolation without direct confirmation.
minor comments (2)
  1. [Figures] Figure captions and axis labels should explicitly state the deformation parameter value and the precise twist vector for each panel to allow direct comparison with the alignment condition discussed in the text.
  2. [Throughout] Notation for the fractional charge Q and the periods L_i should be introduced once with a clear definition before being used in multiple sections; occasional re-use of symbols for different quantities reduces readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our numerical study. We address each major point below, clarifying our numerical procedures and the scope of the pure-YM extrapolation argument.

read point-by-point responses

  1. Referee: [Numerical method / Results sections] Numerical minimization procedure (likely §3 or §4): the central claim that the observed metamorphosis and pure-YM extrapolation hold rests on the assumption that the reported configurations are global minima for each (twist, period) pair. No evidence is provided of exhaustive searches (multiple random starts, annealing schedules, or comparison against known analytic minima in limiting geometries) that would rule out lower-action local minima capable of altering the reported level crossings or flux patterns.

    Authors: We agree that explicit documentation of global-minimum verification strengthens the results. Our minimizations were initialized from multiple classes of configurations (random SU(N) links, small perturbations of analytic monopole and vortex solutions in the appropriate limits, and twisted instanton-like fields), with the same minimum-action saddles recovered in each case for the reported geometries. Convergence was monitored via action stabilization and topological charge. In the revised manuscript we will add a dedicated paragraph in §3 describing these initialization protocols and the consistency checks performed across random seeds. revision: yes

  2. Referee: [Discussion / Pure YM limit] Pure-YM extrapolation argument (likely §5 or Discussion): the identification of which deformed minima survive when abelianization directions coincide is used to claim validity in pure Yang-Mills, yet the manuscript does not explicitly verify that no lower-action configuration exists in the undeformed theory for the aligned geometries; this step is load-bearing for the main claim but remains an extrapolation without direct confirmation.

    Authors: The central claim is conditional: when the twist-induced and deformation-induced abelianizations align, the deformed-theory minima reproduce the expected monopole-to-vortex collimation. Because the deformation supplies the potential that stabilizes the abelianization, direct minimization in the strictly undeformed theory lies outside the scope of the present lattice setup. We will expand the discussion section to emphasize that the argument is an extrapolation resting on alignment and continuity of the saddles, and to state explicitly that a fully undeformed confirmation would require a separate method (e.g., cooling or gradient flow without the double-trace term). revision: partial

Circularity Check

0 steps flagged

No significant circularity; numerical study is self-contained

full rationale

The paper conducts a numerical lattice minimization of the trace-deformed Yang-Mills action on twisted T^4, varying twists and periods to observe how monopole-instanton, center-vortex, and fractional-instanton saddles morph. The claim that the prior analytic picture of flux collimation holds in pure Yang-Mills when deformation and twist abelianizations align is an extrapolation from the computed minimum-action configurations, not a mathematical derivation that reduces any output to fitted inputs or self-citations by construction. No self-definitional steps, fitted predictions renamed as results, or load-bearing self-citation chains appear; the work relies on direct computation rather than tautological re-derivation of its own assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be extracted beyond the standard lattice regularization and the double-trace deformation term whose functional form is not specified here.

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discussion (0)

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Reference graph

Works this paper leans on

63 extracted references · 26 linked inside Pith

  1. [1]

    ’t Hooft,A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories,Nucl

    G. ’t Hooft,A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories,Nucl. Phys. B 153(1979) 141–160

    1979

  2. [2]

    ’t Hooft,Aspects of Quark Confinement,Phys

    G. ’t Hooft,Aspects of Quark Confinement,Phys. Scripta24(1981) 841–846

    1981

  3. [3]

    Unsal and L

    M. Unsal and L. G. Yaffe,Center-stabilized Yang-Mills theory: Confinement and large N volume independence,Phys. Rev. D78(2008) 065035, [arXiv:0803.0344]

    Pith/arXiv arXiv 2008

  4. [4]

    J. D. Bjorken,Elements of Quantum Chromodynamics,Prog. Math. Phys.4(12, 1979) 423–561

    1979

  5. [5]

    Luscher,Some Analytic Results Concerning the Mass Spectrum of Yang-Mills Gauge Theories on a Torus,Nucl

    M. Luscher,Some Analytic Results Concerning the Mass Spectrum of Yang-Mills Gauge Theories on a Torus,Nucl. Phys. B219(1983) 233–261

    1983

  6. [6]

    Arabi Ardehali and D

    A. Arabi Ardehali and D. J. Resnick,Perturbative Coulomb branches onR3 ×S 1: the global D-term potential,arXiv:2604.27066

    Pith/arXiv arXiv

  7. [7]

    G. V. Dunne and M. Ünsal,New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence,Ann. Rev. Nucl. Part. Sci.66(2016) 245–272, [arXiv:1601.03414]

    Pith/arXiv arXiv 2016

  8. [8]

    Poppitz,Notes on Confinement onR3 ×S 1: From Yang-Mills, Super-Yang-Mills, and QCD (adj) to QCD(F),Symmetry14(2022), no

    E. Poppitz,Notes on Confinement onR3 ×S 1: From Yang-Mills, Super-Yang-Mills, and QCD (adj) to QCD(F),Symmetry14(2022), no. 1 180, [arXiv:2111.10423]

    arXiv 2022

  9. [9]

    González-Arroyo,On the fractional instanton liquid picture of the Yang-Mills vacuum and Confinement,arXiv:2302.12356

    A. González-Arroyo,On the fractional instanton liquid picture of the Yang-Mills vacuum and Confinement,arXiv:2302.12356

    arXiv

  10. [10]

    M. M. Anber and E. Poppitz,Higher-order gaugino condensates on a twistedT4,JHEP02(2025) 114, [arXiv:2408.16058]

    arXiv 2025

  11. [11]

    D. J. Gross, R. D. Pisarski, and L. G. Yaffe,QCD and Instantons at Finite Temperature,Rev. Mod. Phys.53(1981) 43. [12]RTNCollaboration, M. Garcia Perez et al.,Instanton like contributions to the dynamics of Yang-Mills fields on the twisted torus,Phys. Lett. B305(1993) 366–374, [hep-lat/9302007]

    Pith/arXiv arXiv 1981

  12. [12]

    Gonzalez-Arroyo and P

    A. Gonzalez-Arroyo and P. Martinez,Investigating Yang-Mills theory and confinement as a function of the spatial volume,Nucl. Phys. B459(1996) 337–354, [hep-lat/9507001]. 32The only advantage for consideringSU(2)is that all quantities entering Eqns. (35)-(37) are explicitly worked out in [51]. 36

    Pith/arXiv arXiv 1996

  13. [13]

    Gonzalez-Arroyo and A

    A. Gonzalez-Arroyo and A. Montero,Selfdual vortex - like configurations in SU(2) Yang-Mills theory, Phys. Lett. B442(1998) 273–278, [hep-th/9809037]

    Pith/arXiv arXiv 1998

  14. [14]

    Montero,Vortex configurations in the large N limit,Phys

    A. Montero,Vortex configurations in the large N limit,Phys. Lett. B483(2000) 309–314, [hep-lat/0004002]

    Pith/arXiv arXiv 2000

  15. [15]

    van Baal,Some Results for SU(N) Gauge Fields on the Hypertorus,Commun

    P. van Baal,Some Results for SU(N) Gauge Fields on the Hypertorus,Commun. Math. Phys.85 (1982) 529

    1982

  16. [16]

    Lee and P

    K.-M. Lee and P. Yi,Monopoles and instantons on partially compactified D-branes,Phys. Rev. D56 (1997) 3711–3717, [hep-th/9702107]

    Pith/arXiv arXiv 1997

  17. [17]

    T. C. Kraan and P. van Baal,Monopole constituents inside SU(n) calorons,Phys. Lett. B435(1998) 389–395, [hep-th/9806034]

    Pith/arXiv arXiv 1998

  18. [18]

    Tanizaki and M

    Y. Tanizaki and M. Ünsal,Center vortex and confinement in Yang–Mills theory and QCD with anomaly-preserving compactifications,PTEP2022(2022), no. 4 04A108, [arXiv:2201.06166]

    arXiv 2022

  19. [19]

    M. M. Anber and E. Poppitz,Mass-deformed super Yang-Mills theory onT4: sum over twisted sectors,θ-angle, and CP violation,JHEP10(2025) 182, [arXiv:2509.00157]

    arXiv 2025

  20. [20]

    Poppitz and M

    E. Poppitz and M. E. Shalchian T.,String tensions in deformed Yang-Mills theory,JHEP01(2018) 029, [arXiv:1708.08821]

    Pith/arXiv arXiv 2018

  21. [21]

    Greensite,An introduction to the confinement problem, vol

    J. Greensite,An introduction to the confinement problem, vol. 972. Springer, 2020

    2020

  22. [22]

    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(2026), no. 11 111902, [arXiv:2510.19103]

    arXiv 2026

  23. [23]

    M. M. Anber and E. Poppitz,The gaugino condensate from asymmetric four-torus with twists,JHEP 01(2023) 118, [arXiv:2210.13568]

    arXiv 2023

  24. [24]

    M. M. Anber and E. Poppitz,Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted four-torus,JHEP09(2023) 095, [arXiv:2307.04795]

    arXiv 2023

  25. [25]

    Dorey, T

    N. Dorey, T. J. Hollowood, V. V. Khoze, and M. P. Mattis,The Calculus of many instantons,Phys. Rept.371(2002) 231–459, [hep-th/0206063]

    Pith/arXiv arXiv 2002

  26. [26]

    ’t Hooft,Some Twisted Selfdual Solutions for the Yang-Mills Equations on a Hypertorus, Commun

    G. ’t Hooft,Some Twisted Selfdual Solutions for the Yang-Mills Equations on a Hypertorus, Commun. Math. Phys.81(1981) 267–275

    1981

  27. [27]

    F. D. Wandler,Numerical fractional instantons in SU(2): center vortices, monopoles, and a sharp transition between them,arXiv:2406.07636

    arXiv

  28. [28]

    Hayashi and Y

    Y. Hayashi and Y. Tanizaki,Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism,Phys. Rev. Lett.133(2024), no. 17 171902, [arXiv:2405.12402]

    arXiv 2024

  29. [29]

    Güvendik, T

    C. Güvendik, T. Schaefer, and M. Ünsal,The metamorphosis of semi-classical mechanisms of confinement: from monopoles onR3 ×S 1 to center-vortices onR2 ×T 2,JHEP11(2024) 163, [arXiv:2405.13696]. 37

    arXiv 2024

  30. [30]

    M. M. Anber and E. Poppitz,On the global structure of deformed Yang-Mills theory and QCD(adj) onR 3 ×S 1,JHEP10(2015) 051, [arXiv:1508.00910]

    Pith/arXiv arXiv 2015

  31. [31]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg,Theta, Time Reversal, and Temperature, JHEP05(2017) 091, [arXiv:1703.00501]

    Pith/arXiv arXiv 2017

  32. [32]

    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,JHEP10(2021) 069, [arXiv:2106.11442]

    arXiv 2021

  33. [33]

    M. M. Anber and L. Vincent-Genod,Classification of compactifiedsu(Nc)gauge theories with fermions in all representations,JHEP12(2017) 028, [arXiv:1704.08277]

    Pith/arXiv arXiv 2017

  34. [34]

    Hayashi, Y

    Y. Hayashi, Y. Tanizaki, and H. Watanabe,Non-supersymmetric duality cascade of QCD(BF) via semiclassics onR 2 ×T 2 with the baryon-’t Hooft flux,JHEP07(2024) 033, [arXiv:2404.16803]

    arXiv 2024

  35. [35]

    Hayashi and Y

    Y. Hayashi and Y. Tanizaki,Semiclassics for the QCD vacuum structure through T2-compactification with the baryon-’t Hooft flux,JHEP08(2024) 001, [arXiv:2402.04320]

    arXiv 2024

  36. [36]

    Hayashi, Y

    Y. Hayashi, Y. Tanizaki, and H. Watanabe,Semiclassical analysis of the bifundamental QCD onR 2 ×T 2 with ’t Hooft flux,JHEP10(2023) 146, [arXiv:2307.13954]

    arXiv 2023

  37. [37]

    Tanizaki and M

    Y. Tanizaki and M. Ünsal,Semiclassics with ’t Hooft flux background for QCD with 2-index quarks, JHEP08(2022) 038, [arXiv:2205.11339]

    arXiv 2022

  38. [38]

    Hayashi, T

    Y. Hayashi, T. Misumi, and Y. Tanizaki,Monopole-vortex continuity ofN= 1 super Yang-Mills theory onR 2 ×S 1 ×S 1 with ’t Hooft twist,JHEP05(2025) 194, [arXiv:2410.21392]

    arXiv 2025

  39. [39]

    Hayashi, T

    Y. Hayashi, T. Misumi, and Y. Tanizaki,On 2d perimeter law inN=1 super Yang-Mills theory on R2 ×S 1 ×S 1 with ’tt Hooft twist from Monopole-Vortex continuity,Int. J. Mod. Phys. A41(2026), no. 07 2548006

    2026

  40. [40]

    Nguyen and M

    M. Nguyen and M. Ünsal,Refined instanton analysis of the 2DCPN−1 model: mass gap, theta dependence, and mirror symmetry,JHEP03(2025) 162, [arXiv:2309.12178]

    arXiv 2025

  41. [41]

    Nguyen and M

    M. Nguyen and M. Ünsal,Self-dual monopole loops, instantons and confinement,arXiv:2509.09625

    arXiv

  42. [42]

    Garcia Perez, A

    M. Garcia Perez, A. Gonzalez-Arroyo, J. R. Snippe, and P. van Baal,Instantons from over - improved cooling,Nucl. Phys. B413(1994) 535–552, [hep-lat/9309009]

    Pith/arXiv arXiv 1994

  43. [43]

    de Forcrand, M

    P. de Forcrand, M. Garcia Perez, and I.-O. Stamatescu,Improved cooling algorithm for gauge theories,Nucl. Phys. B Proc. Suppl.47(1996) 777–780, [hep-lat/9509064]

    Pith/arXiv arXiv 1996

  44. [44]

    Montero,Numerical analysis of fractional charge solutions on the torus,JHEP05(2000) 022, [hep-lat/0004009]

    A. Montero,Numerical analysis of fractional charge solutions on the torus,JHEP05(2000) 022, [hep-lat/0004009]

    Pith/arXiv arXiv 2000

  45. [45]

    Bonati, M

    C. Bonati, M. Cardinali, and M. D’Elia,θdependence in trace deformedSU(3)Yang-Mills theory: a lattice study,Phys. Rev. D98(2018), no. 5 054508, [arXiv:1807.06558]

    Pith/arXiv arXiv 2018

  46. [46]

    Athenodorou, M

    A. Athenodorou, M. Cardinali, and M. D’Elia,Spectrum of trace deformed Yang-Mills theories,Phys. Rev. D104(2021), no. 7 074510, [arXiv:2010.03618]. 38

    arXiv 2021

  47. [47]

    Bonati, M

    C. Bonati, M. Cardinali, M. D’Elia, M. Giordano, and F. Mazziotti,Reconfinement, localization and thermal monopoles inSU(3)trace-deformed Yang-Mills theory,Phys. Rev. D103(2021), no. 3 034506, [arXiv:2012.13246]

    arXiv 2021

  48. [48]

    Bonati, M

    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,PoS LATTICE2024(2025) 394, [arXiv:2501.13684]

    arXiv 2025

  49. [49]

    Unsal,Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions,Phys

    M. Unsal,Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions,Phys. Rev. D80(2009) 065001, [arXiv:0709.3269]

    Pith/arXiv arXiv 2009

  50. [50]

    Poppitz and F

    E. Poppitz and F. D. Wandler,Gauge theory geography: charting a path between semiclassical islands, JHEP02(2023) 014, [arXiv:2211.10347]

    arXiv 2023

  51. [51]

    García Pérez, A

    M. García Pérez, A. González-Arroyo, and M. Okawa,Spatial volume dependence for 2+1 dimensional SU(N) Yang-Mills theory,JHEP09(2013) 003, [arXiv:1307.5254]

    Pith/arXiv arXiv 2013

  52. [52]

    Garcia Perez, A

    M. Garcia Perez, A. Gonzalez-Arroyo, A. Montero, and P. van Baal,Calorons on the lattice: A New perspective,JHEP06(1999) 001, [hep-lat/9903022]

    Pith/arXiv arXiv 1999

  53. [53]

    M. M. Anber, A. A. Cox, and E. Poppitz,On the moduli space of multi-fractional instantons on the twistedT 4,JHEP02(2026) 169, [arXiv:2504.06344]

    arXiv 2026

  54. [54]

    Poppitz,D-branes and fractional instantons on a twisted four torus: the moduli space as an N=2 supersymmetric Higgs branch,arXiv:2604.21980

    E. Poppitz,D-branes and fractional instantons on a twisted four torus: the moduli space as an N=2 supersymmetric Higgs branch,arXiv:2604.21980

    Pith/arXiv arXiv

  55. [55]

    Unsal,Abelian duality, confinement, and chiral symmetry breaking in QCD(adj),Phys

    M. Unsal,Abelian duality, confinement, and chiral symmetry breaking in QCD(adj),Phys. Rev. Lett. 100(2008) 032005, [arXiv:0708.1772]

    Pith/arXiv arXiv 2008

  56. [56]

    Witten,Constraints on Supersymmetry Breaking,Nucl

    E. Witten,Constraints on Supersymmetry Breaking,Nucl. Phys. B202(1982) 253

    1982

  57. [57]

    Gonzalez Arroyo and C

    A. Gonzalez Arroyo and C. P. Korthals Altes,The Spectrum of Yang-Mills Theory in a Small Twisted Box,Nucl. Phys. B311(1988) 433–449

    1988

  58. [58]

    Ünsal,Strongly coupled QFT dynamics via TQFT coupling,JHEP11(2021) 134, [arXiv:2007.03880]

    M. Ünsal,Strongly coupled QFT dynamics via TQFT coupling,JHEP11(2021) 134, [arXiv:2007.03880]

    arXiv 2021

  59. [59]

    Nguyen, T

    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(2025), no. 14 141902, [arXiv:2401.04800]

    arXiv 2025

  60. [60]

    Gonzalez-Arroyo,Yang-Mills fields on the four-dimensional torus

    A. Gonzalez-Arroyo,Yang-Mills fields on the four-dimensional torus. Part 1.: Classical theory, in Advanced Summer School on Nonperturbative Quantum Field Physics, pp. 57–91, 6, 1997. hep-th/9807108

    Pith/arXiv arXiv 1997

  61. [61]

    T. C. Kraan and P. van Baal,Periodic instantons with nontrivial holonomy,Nucl. Phys. B533 (1998) 627–659, [hep-th/9805168]

    Pith/arXiv arXiv 1998

  62. [62]

    Duane, A

    S. Duane, A. Kennedy, B. J. Pendleton, and D. Roweth,Hybrid monte carlo,Physics Letters B195 (1987), no. 2 216–222. 39

    1987

  63. [63]

    Lippert,The Hybrid Monte Carlo algorithm for quantum chromodynamics,Lect

    T. Lippert,The Hybrid Monte Carlo algorithm for quantum chromodynamics,Lect. Notes Phys.508 (1998) 122, [hep-lat/9712019]. 40

    Pith/arXiv arXiv 1998