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

Understanding the nonperturbative dynamics of four-dimensional non-abelian gauge theories from first principles is a challenging problem lacking a complete solution. While many approaches exist, none of them can account for all interesting aspects of the physics. This paper is devoted to one particular direction of research: the study of the weak-coupling semiclassical dynamics of $SU(N)$ gauge theories on compact spaces of the form ${\mathbb{R}}^{4-k}\times{\mathbb{T}}^{k}_{*}$. Here ${\mathbb{T}}^{k}_{*}$ denotes a $k$ dimensional torus with $k=1,2,3,4$. The subscript $*$ indicates the inclusion of at least one of ’t Hooft twisted boundary conditions [tHooft:1979rtg, tHooft:1981sps] or a double-trace deformation [Unsal:2008ch]. We use $\Lambda$ to denote the strong coupling scale of the gauge theory. We take Euclidean signature in all ${\mathbb{R}}^{4-k}\times{\mathbb{T}}^{k}_{*}$ cases, except on a few occasions where the use of Minkowski signature is explicitly stated.

If the size of the torus $L_{{\mathbb{T}}^{k}}$ is small,¹¹1For simplicity, the discussion here only refers to the overall size of the ${\mathbb{T}}^{k}$, denoted by $L_{{\mathbb{T}}^{k}}$, tacitly assuming that all periods of the torus are of the same order. Later, we relax this assumption, considering different ratios of periods of ${\mathbb{T}}^{k}$. such that $L_{{\mathbb{T}}^{k}}N\Lambda\ll 1$, one can show that the ${\mathbb{R}}^{4-k}\times{\mathbb{T}}^{k}_{*}$ theory abelianizes: the gauge group is Higgsed, by the boundary conditions or the deformation, $SU(N)\rightarrow U(1)^{N-1}$ or $SU(N)\rightarrow{\mathbb{Z}}_{N}$, at a high scale ${1\over L_{{\mathbb{T}}^{k}}N}\gg\Lambda$, ensuring that the theory is weakly coupled. Let us be slightly more precise: for $k=1,2$, the long-distance ${\mathbb{R}}^{3}$ or ${\mathbb{R}}^{2}$ theory is weakly coupled because of the abelianization. For $k=3,4$, the theory on ${\mathbb{R}}\times{\mathbb{T}}^{3}$ or ${\mathbb{T}}^{4}$ is weakly coupled at small $L_{{\mathbb{T}}^{k}}$ even without the nontrivial ’t Hooft twists or deformation—this is the “femtouniverse” of [Bjorken:1979hv, Luscher:1982ma]. However, now the small-$L_{{\mathbb{T}}^{k}}$ and large-$L_{{\mathbb{T}}^{k}}$ theories are believed to not be continuously connected in the sense discussed below. The reason for our cautious statement is that the continuous classical vacuum degeneracy in compactifications without ’t Hooft twists or deformations complicates²²2For recent work on this problem in theories with supersymmetry, see [ArabiArdehali:2026kvt]. the semiclassical study of the ground state properties of the femtouniverse—a task that has not yet been completed, to the best of our knowledge. For this reason, we only study ${\mathbb{R}}^{4-k}\times{\mathbb{T}}^{k}_{*}$ theories with ’t Hooft twists and/or deformation.

The weak coupling at small $L_{{\mathbb{T}}^{k}}$ permits the use of theoretically controlled semiclassical methods to study ground state properties, $\theta$-angle dependence, symmetry realization, spectra, etc.; various aspects are reviewed in [Dunne:2016nmc, Poppitz:2021cxe, Gonzalez-Arroyo:2023kqv]. Remarkably, it is found that the properties of the ground state of the pure gauge theory, obtained analytically at small $L_{{\mathbb{T}}^{k}}$, evolve continuously into the ones seen in the strongly-coupled $L_{{\mathbb{T}}^{k}}\rightarrow\infty$, or ${\mathbb{R}}^{4}$, limit. Demonstrating this continuity, known as “adiabatic continuity,” requires, at the current state of the art, the use of numerical simulations to leave the weakly-coupled small-$L_{{\mathbb{T}}^{k}}$ regime; for certain quantities in supersymmetric theories, one can appeal to nonrenormalization theorems (see [Anber:2024mco]) to argue for small to large volume continuity. Here, we shall not review further details of the semiclassical dynamics nor discuss different ideas to connect the semiclassical and the strongly coupled limits; for this, we recommend the reviews cited above.

The purpose of this paper is to study the relation between the nontrivial saddles in the path integral on ${\mathbb{R}}^{4-k}\times{\mathbb{T}}^{k}_{*}$ for different $k$. These are finite action instantons contributing to the nonperturbative dynamics. We begin by listing the kind of finite action configurations that give the leading semiclassical contribution to the nonperturbative dynamics. We use the antisymmetric twist tensor “$n_{\mu\nu}$,” defined (mod $N$) in [tHooft:1979rtg], to denote the imposition of nontrivial twisted boundary conditions in the ${\mathbb{T}}^{k}$ (without, at this point, being explicit about the particular choice made). Likewise, we use “def.” to denote the addition of a double-trace deformation [Unsal:2008ch]. The upshot is that, for the different values of $k$, the pattern of gauge symmetry breaking, the kinds of minimal action saddles, their “quantum numbers” (e.g. topological charge $Q$), as well their localization properties, are found to be as follows:

In each case, we have assumed that the choice of $n_{\mu\nu}$ twists is such that the minimal $Q={1\over N}$; this choice shall be made more explicit in the body of the paper. Let us now discuss various aspects of (1.1)-(1.1):

1. 1.

   For all values of $k$, the minimum action semiclassical objects have topological charge $1/N$. For $k=1$, this has been known since the classification of finite action configurations on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ of [Gross:1980br], which implies that $Q=1/N$ at the center-symmetric point [Unsal:2008ch]. For $k=2,3$, the existence of $Q={1/N}$ fractional instantons on ${\mathbb{R}}\times{\mathbb{T}}^{3}_{n_{\mu\nu}}$ and ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{n_{\mu\nu}}$ has been realized already in [RTN:1993ilw, Gonzalez-Arroyo:1995ynx, Gonzalez-Arroyo:1998hjb, Montero:2000pb].³³3See Appendix B for a classification argument of finite action configurations on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{n_{12}}$ (and, by a simple extension, to ${\mathbb{R}}\times{\mathbb{T}}^{3}_{n_{12}}$). While the discussion of this semi-infinite volume limit classification seems absent in the literature, it appears to be known to many and we include it for completeness. For $k=4$, ’t Hooft argued [tHooft:1979rtg, tHooft:1981sps] that $Q=1/N$ for some choices of twists, later studied in great detail in [vanBaal:1982ag]. These instantons are exactly self-dual (or, for $k=1$ with no supersymmetry, approximately self-dual). Thus, their semiclassical contribution to the path integral scales as $\exp(-{8\pi^{2}\over g^{2}N})$, where $g^{2}$ is the gauge theory coupling at a scale of order $L_{{\mathbb{T}}^{k}}$. As indicated, these finite action configurations are localized in the noncompact ${\mathbb{R}}^{4-k}$ and have size of order $L_{{\mathbb{T}}^{k}}$.

2. 2.

   Notice that for $k=2,3,4$, there is a single, up to moduli, minimal action configuration with $Q={1/N}$. For $k=1$, on the other hand, there are $N$ different objects of fractional topological charge, distinguished by their $U(1)^{N-1}$ “magnetic” charges. These are labelled by the simple roots, $\alpha_{1},...,\alpha_{N-1}$, of $SU(N)$, along with the affine root $\alpha_{0}$$=$$-(\alpha_{1}$$+$$\alpha_{2}$$+$$...$$+$$\alpha_{N-1})$. These monopole-instantons are the $N$ “constituents” of the BPST $Q=1$ instanton [Lee:1997vp, Kraan:1998sn].

3. 3.

   In each of the cases with $k<4$, a dilute gas of the relevant $Q=1/N$ instantons disorders large Wilson loops in the noncompact ${\mathbb{R}}^{3}$ (case (1.1)) or ${\mathbb{R}}^{2}$ (case (1.1)), leading to area law and confinement. For $k=1$, see [Unsal:2008ch]. For $k=2$, confinement by center vortices is shown in [Tanizaki:2022ngt]. For $k=3$, Refs. [RTN:1993ilw, Gonzalez-Arroyo:1995ynx] showed that tunnelling due to fractional instantons causes the correlator of two winding Wilson loops, separated in ${\mathbb{R}}$, to obey the area law (this is recently reviewed in Section 6 of [Anber:2025vjo]). In all cases, the string tensions in the pure gauge theory, computed in the leading semiclassical approximation (for the case (1.1), see [Poppitz:2017ivi]) with exponential-only accuracy, are proportional to $\exp(-{8\pi^{2}\over g^{2}N})$

4. 4.

   The ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}_{\text{def.}}$ and ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{n_{\mu\nu}}$ cases, (1.1) and (1.1), are examples of semiclassically tractable mechanisms of confinement due to monopole-instantons and center vortices, respectively. These mechanisms have been extensively discussed in the lattice community, in the strong coupling regime of a large ${\mathbb{T}}^{4}$. There, monopoles and center vortices are identified after appropriate gauge fixing of lattice configurations belonging to an ensemble generated via Monte Carlo techniques. It is observed that removing the configurations containing center vortices/monopoles from the ensemble destroys the area law. There are many important details; for these and references, we recommend Greensite’s monograph [Greensite:2011zz].⁴⁴4For more recent work discussing newer developments on the use of effective models of ensembles of unoriented center vortices, see Ref. [Junior:2025gxg], also containing an updated list of references.

5. 5.

   We will not dwell much on the $k=4$ case in this Introduction. This is because the study of $Q=1/N$ configurations on ${\mathbb{T}}^{4}_{n_{\mu\nu}}$ is really the subject of most of the rest of the paper. Here, we only mention that fractional instantons on ${\mathbb{T}}^{4}_{n_{\mu\nu}}$ have been used to calculate the gaugino condensate in super-Yang-Mills theory [Anber:2022qsz, Anber:2023sjn, Anber:2024mco], showing complete agreement (owing to the supersymmetric nonrenormalization theorems) with the ${\mathbb{R}}^{4}$ results reviewed in [Dorey:2002ik].

6. 6.

   Our final remark is that there are few known analytic solutions with $Q=1/N$ in the geometries (1.1)-(1.1). In fact, there are only two kinds: the $N$ BPS monopole instantons on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ [Lee:1997vp] and ’t Hooft’s constant field strength fractional instantons on ${\mathbb{T}}^{4}$ [tHooft:1981nnx]. Most of what is known about the saddle points mentioned in (1.1)-(1.1) has been learned by numerically minimizing lattice actions with ’t Hooft twists, largely due to the long-term efforts of the Madrid group [Gonzalez-Arroyo:2023kqv]. In its use of numerical tools, this paper is not an exception.

After a further look at the various semiclassical objects appearing in (1.1)-(1.1) it should perhaps not come as a surprise that they can be related to each other. This is easiest to contemplate for the $k=2,3,4$ cases, where the only “quantum number” is the topological charge $Q=1/N$. Thus, at least naively (however, see Section 6), one imagines compactifying a center vortex, a 2d sheet wrapped on ${\mathbb{T}}^{2}$, as in (1.1), on an ${\mathbb{S}}^{1}$ orthogonal to the sheet, obtaining (1.1); this can then be further compactified to obtain (1.1). Conversely, one could begin with a ${\mathbb{T}}^{4}$ with appropriately chosen $n_{\mu\nu}$ twists (giving the minimal $Q=1/N$) and then take limits of its periods such that it approximates any one of the cases with $k=1,2,3$. This may be especially clear if one wants to obtain the $k=2,3$ cases, where the same twist in the ${\mathbb{T}}^{2}$ or ${\mathbb{T}}^{3}$ appearing in (1.1) or (1.1) can be imposed on the original ${\mathbb{T}}^{4}$—so that one obtains the desired ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{n_{\mu\nu}}$ or ${\mathbb{R}}\times{\mathbb{T}}^{3}_{n_{\mu\nu}}$ in the limit.⁵⁵5A twist involving a ${\mathbb{T}}^{4}$ direction taken to infinity will not be relevant and serves only to select the desired boundary condition, one of the many possible, at ${\mathbb{R}}^{2}$ or ${\mathbb{R}}$ infinity; see Appendix B.

Obtaining ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}_{\text{def.}}$ from (1.1)-(1.1), however, is bound to be a little trickier. First of all, as mentioned above, now there are $N$ monopole-instantons with different $U(1)^{N-1}$ magnetic charges, as per (1.1), instead of a single $Q=1/N$ fractional instanton; we call this the “multiplicity problem.” Second, if there is no deformation on the ${\mathbb{T}}^{4}$ one starts with, there is no way to strictly obtain ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}_{\text{def.}}$, a setup where abelianization only occurs because of the deformation.⁶⁶6As explained in Section 4, at finite ${\mathbb{T}}^{4}$ periods, abelianization around a localized fractional instanton can occur without deformation potential, due to the twists alone, provided the torus periods satisfy particular relations. In some cases, this abelianization is aligned with the one due to the deformation (however, this does not occur in the strict ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ limit; see [Wandler:2024hsq]). These issues were tackled by the authors of [Hayashi:2024yjc, Guvendik:2024umd], who proposed replacing ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}_{\text{def.}}$ by ${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}\times{\mathbb{S}}^{1}_{\text{def.}}$ and realized that adding a ’t Hooft twist in the thus-obtained two-torus ${\mathbb{S}}^{1}\times{\mathbb{S}}^{1}_{\text{def.}}$ solves the “multiplicity problem.” This is because an ${\mathbb{S}}^{1}$-period translation in the presence of a ’t Hooft twist is a center symmetry transformation in the orthogonal ${\mathbb{S}}^{1}_{\text{def.}}$ direction. This center symmetry cyclically permutes the $N$ different monopole-instantons [Anber:2015wha], thus causing them to alternate on the covering space of the ${\mathbb{S}}^{1}$, in effect creating a single object involving a monopole-instanton and all its $N$ images under the ${\mathbb{S}}^{1}$ translation. Refs. [Hayashi:2024yjc, Guvendik:2024umd] then used analytic tools familiar from ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ studies of deformed Yang-Mills theory [Unsal:2008ch] and classical electrodynamics to show that this single object acts like a center vortex: it localizes in the noncompact ${\mathbb{R}}^{2}$, is wrapped around ${\mathbb{S}}^{1}\times{\mathbb{S}}^{1}_{\text{def.}}$, and disorders the Wilson loops surrounding it. The resulting picture is shown, for $N=2$, on our Figure 2, referring the reader to Section 5.1 for details.

The monopole-instanton/center-vortex continuity is an interesting new development, showing, in a theoretically controlled semiclassical framework, that these two confinement mechanisms are related. We note, however, that the relation between center vortices and monopole-instantons in the strong coupling regime has appeared earlier in the lattice literature—see [Greensite:2011zz], in particular Ch. 8 there, where pictures of gauge-fixed lattice configurations appear, identical to our Fig. 2.

However, it is satisfying to have a setup where a controllable analytic argument is available: the semiclassical limit on small compact spaces gives a clear sense as to why these configurations dominate the path integral—in contrast with the phenomenological models of the strong coupling regime. Furthermore, weak coupling semiclassics on ${\mathbb{R}}^{4-k}\times{\mathbb{T}}^{k}_{*}$ allows one to treat properties and theories which are either very challenging or simply intractable at the current level of development of lattice techniques. These include the multibranched structure and associated $\theta$-dependence of the vacuum⁷⁷7Required by consistency with various generalized anomalies [Gaiotto:2017yup] (these are described in the old-fashioned language close to the one used here in [Cox:2021vsa]). and the study of theories with massless fermions in various representations, including chiral gauge theories and general supersymmetric theories, as in e.g. [Anber:2017pak, Hayashi:2024gxv, Hayashi:2024qkm, Hayashi:2023wwi, Tanizaki:2022plm]. As a concrete application of the recent developments, the picture of [Hayashi:2024yjc, Guvendik:2024umd] was used to explain some puzzles regarding confinement in supersymmetric theories [Hayashi:2024psa, Hayashi:2025mgk]. A more speculative remark [Guvendik:2024umd] is that viewing the monopole-instantons as BPST instanton constituents—recalling Comment 2. after Eqns. (1.1-1.1)—suggests that the continuity is a hint that both monopole-instantons and center vortices may lurk inside the gas of four dimensional instantons, in a way waiting to be made more concrete; see [Nguyen:2023rww, Nguyen:2025voy] for related developments.

In conclusion of this overview, we hope to have conveyed the idea that the study of how the different nonperturbative saddles appearing in (1.1)-(1.1) morph into each other is of interest, as it sheds light on the relation between the confinement mechanisms operating in the various geometries—and perhaps, ultimately, in the ${\mathbb{R}}^{4}$ limit. We now continue to discuss in more detail the scope and results of this paper.

In this paper, we further study the relation—or “metamorphosis”—between the various minimal action instantons in the geometries (1.1)-(1.1). We focus on $SU(2)$ Yang-Mills theory with a double-trace deformation potential, which we abbreviate as dYM, the deformed Yang-Mills theory of Ünsal and Yaffe [Unsal:2008ch]. We use numerical minimization of the lattice action of dYM on ${\mathbb{T}}^{4}$, subject to twisted boundary conditions. To motivate the use of numerics, we note that the analytic tools used in [Hayashi:2024yjc, Guvendik:2024umd] to study the monopole-instanton/center-vortex continuity apply in the limit where $L_{{\mathbb{S}}^{1}}\gg L_{{\mathbb{S}}^{1}_{\text{def}}}$ (see Section 5.1). One of our goals is to use numerical methods to study the construction away from this limit. These also allow us to explore the core of the solution and the shape of the center vortex obtained from the monopole-instanton chain of Figure 2. In addition, we are also able to study the transition between the other configurations in some detail, e.g. between (1.1) and (1.1), as alluded to in Section 1.1.

The continuity between different fractionally charged semiclassical configurations on ${\mathbb{T}}^{4}$ was previously studied using cooling (see [GarciaPerez:1993lic, deForcrand:1995qq, Montero:2000mv]) to find the lattice action minima [Wandler:2024hsq], albeit in pure gauge theory, without the addition of a deformation potential. An important technical point is that the addition of the nonlocal deformation potential requires the use of a different method to find the minima of the action. We reduce the action using a gradient flow method, similar to the Hamiltonian evolution phase of hybrid Monte Carlo, as described in Appendix A. dYM has been the subject of Monte Carlo simulations, e.g. [Bonati:2018rfg, Athenodorou:2020clr, Bonati:2020lal, Bonati:2025hik], but to the best of our knowledge, ours is the first study devoted to finding classical minimal action instanton configurations in dYM.

The main result of Section 3 is our Figure 1. It presents numerical evidence that either the “flux” or “no-flux” state is the global energy minimum in dYM, for any shape of ${\mathbb{T}}^{3}_{(L_{0},L_{1},L_{2})|_{n_{12}=1}}$, with a level crossing occurring at a critical value $\left({L_{1}/L_{0}}\right)_{c}\simeq 1.5$. For $L_{1}/L_{0}$ larger than the critical value, the “flux” vacuum is the minimum energy state, while for smaller values of the ratio, it is the “no-flux” vacuum. This level crossing has implications on the transition between center vortices on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, as per (1.1), and fractional instantons on ${\mathbb{R}}\times{\mathbb{T}}^{3}$, (1.1), in dYM, the subject of Section 6.

Here, we only give a brief description of the results. For $L_{1}<1.5L_{0}$, we find the ${\mathbb{R}}\times{\mathbb{T}}^{3}_{n_{12}}$ fractional instantons studied long ago in [RTN:1993ilw, Gonzalez-Arroyo:1995ynx] (seen in the low-$L_{1}$ region of Fig. 10, as discussed there). These fractional instantons, as per our Eqn. (1.1), are well localized in $L_{3}$. As is also well known, they disorder the $\,{\rm tr}\,W_{0}$ Wilson loop, which approaches values $\pm 2$ away from the solution. Both these features are seen on Fig. 11.

For $L_{1}>1.5L_{0}$, 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 Fig. 12).

The new feature specific¹⁰¹⁰10We recall that along the same path in pure YM theory, for the lattices we study, the transition proceeds, instead, via the maximally delocalized constant-field strength instanton [tHooft:1981nnx], which has minimal action when $L_{1}L_{2}=L_{0}L_{3}$, studied in [Wandler:2024hsq]. to dYM is the transition region near $L_{1}\simeq 1.5L_{0}$. Here, both the Wilson and deformation action of the minimum action configurations have a peak (as a function of $L_{1}$), seen on Fig. 10. For the critical value of $L_{1}$, our action minimization algorithm finds two kinds of configurations, of roughly the same total action. These are either monopole-instantons in the flux vacuum, localized in $x_{3}$ and shown on Fig. 12, or the delocalized (in $x_{3}$) configurations shown on Fig. 13 and described there in detail. This apparent degeneracy is consistent with a level crossing transition, however, we do not have a detailed understanding of the delocalized configurations; their study requires a more fine-grained resolution of the transition region, which is beyond our scope here.

The study of this paper further confirms that the set of minimal action configurations listed in (1.1)-(1.1) are related to each other in intricate ways, forming a rich set of interconnected saddle points related by changing the twists and ratios of periods of ${\mathbb{T}}^{4}$. In this regard, our study could be furthered by considering in more detail the “critical” region of the transition between the monopole-instanton and fractional instanton configurations of (1.1) and (1.1), studied in Section 6 and associated with a transition between the flux and no-flux vacua of dYM. This, however, requires a significantly more fine-grained study of the transition and the associated computer resources. Additionally, the gradient flow technique used to generate minimum action configurations could in principle be used to place upper bounds on the height of the energy barriers between vacua near the critical point. By giving configurations in one vacuum progressively stronger “kicks” or fictitious kinetic energy, until the lattice settles into the other vacuum, it could be possible to trace the path through field space the lattice takes. By computing the energy of these intermediate configurations it would be possible to place an upper bound on the height of the energy barrier.

A more general question—interesting from both mathematics and physics points of view—concerns the nature and moduli of classical self-dual instantons of (fractional) charge $Q=r/N$, $\forall r\in\mathbb{N}$, in $SU(N)$ YM theory on a twisted ${\mathbb{T}}^{4}$. For general $r>1$, the moduli space of such field configurations is only understood locally, as in [Anber:2025yub, Poppitz:2026gfa], using a combination of numerical and analytic tools. On the numerical side, it would be interesting to study whether one could make further progress, first for $N=2$, for any $r>1$, by generalizing the methods of [GarciaPerez:1993lic, deForcrand:1995qq].