Perturbative Holonomy Potentials and Vacua in Chiral Supersymmetric Gauge Theories

Arash Arabi Ardehali

REVIEW

Perturbative Holonomy Potentials and Vacua in Chiral Supersymmetric Gauge Theories

Arash Arabi Ardehali

High Energy Physics - Theory · Submitted 25 Oct 2026

Pith's verdict: A clear, technically solid derivation of global D-term control in chiral SUSY theories on R^3 x S^1, with explicit matching and nontrivial examples—worth serious referee time.

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arXiv ID2610.34240 (version 3)

Pith ID6BTGTLHG, Pith's permanent record of this paper and its review

CitationArash Arabi Ardehali. "Perturbative Holonomy Potentials and Vacua in Chiral Supersymmetric Gauge Theories." arXiv:2610.34240 (2026). https://pith.science/paper/6BTGTLHG

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Editor's note

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This paper takes a known closed-form D-term coefficient (from the author's companion paper) and turns it into a systematic procedure for classifying perturbative vacua across the full holonomy space. The main new thing is the global matching: outer patch zeros give gapped vacua or Coulomb branches directly from F^out = 0; inner patches get local FI/CS couplings via threshold subtraction, and the paper shows how condensates or further integrating-out can cancel the matched D-term. That is a genuine advance over the earlier indirect methods that only worked near trivial holonomy.

The examples are well-chosen. The abelian model with charges (1,5,-7,-8,9) at η=0 gives twelve isolated roots and two Coulomb intervals; the FI deformation enriches the vacuum structure and produces Higgs vacua at the endpoints. The nonabelian example reproduces the Gang–Yamazaki sector from the R-twisted Maruyoshi–Song reduction, and explicitly shows where wall zeros fail (pure SU(2)_1 has no SUSY vacuum) — that kind of honest reporting strengthens the paper.

What keeps this from being a standalone reference is heavy reliance on the companion paper [6] for the full derivation of the D-term coefficient and higher-rank examples. The core result (Eq. 2) is stated and used, but the derivation itself is only sketched. For a Letter that is fine, but the referee will need to check [6] to verify the regularization and the FI parameter matching. The perturbative control condition g^2_4 << ε^2 << 1 is clearly stated as a limit; the authors do not oversell the regime. The strong-coupling dynamics near walls is deliberately excluded, which is honest but means the framework is incomplete for many interesting applications.

The supplemental material on the wall theory at x=0 is speculative: it identifies a candidate SCFT via F-maximization and index, but the authors note the assumptions (no accidental symmetry, no monopole superpotential). That part is more exploratory and less tightly connected to the Letter's main claim.

Overall, the central claim — that the closed-form D-coefficient plus threshold matching gives global control of perturbative vacua — is well-supported by the derivation, the explicit matching formulas, and the worked examples. The paper is self-consistent and the limitations are openly stated.

Verdict: This deserves a serious referee. The reviewer will want to verify the matching scheme against [6], but the core logic is sound and the examples are concrete. I would recommend acceptance after the referee confirms the companion paper's derivations hold up.

I would bring this to reading group, would cite the results if I worked in this area, and the author shows clear, honest thinking.

Referee report

Significance

If the result holds, the paper provides a systematic and rigorous method to control perturbative D-terms globally on the holonomy space of chiral gauge theories on R^3 x S^1, going beyond earlier indirect approaches that were often limited to trivial holonomy. The explicit matching of outer and inner patches, combined with the closed-form coefficient, offers a practical tool for analyzing semiclassical vacua in a wide class of theories. The examples demonstrate nontrivial phenomena (such as vacua at non-zero holonomy despite vanishing D-coefficient at trivial holonomy) that underscore the need for a global perspective. This work is a significant contribution to the understanding of circle-compactified supersymmetric theories and will be of interest to researchers in supersymmetric gauge theory, compactification, and dualities.

Minor objections (4)

[Title and abstract] The title contains a typographical artifact: 'V acua' should be 'Vacua'. The same artifact appears in the abstract; the authors should correct these LaTeX-related spacing issues.
[Page 2, near Eq. (2)] The scheme dependence of the FI input is mentioned briefly ('The regulator and renormalized FI input specify the matching scheme') but no further explanation is given. The authors should briefly clarify what scheme (e.g., dimensional reduction with a specific subtraction) is used or state that scheme dependence cancels in the final results for physical vacua.
[Figure captions] The captions mention 'GPT-6 Astra assisted with the plotting code.' While this acknowledgment is acceptable in an acknowledgments section, including it in the figure caption is unusual. Consider moving this statement to the general acknowledgments.
[Supplemental Material, index expansion] In Eq. (30), the exponents are given as numerical approximations. It would be helpful to include a comment that these are exact linear combinations of the optimized R-charges, perhaps giving one example explicitly to illustrate the pattern.

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The paper

The paper claims

A closed-form one-loop-exact D-term coefficient globally controls perturbative vacua across the compact holonomy space in chiral supersymmetric gauge theories on R^3 × S^1.

The reading

This paper establishes a globally controlled perturbative description of D-term potentials in chiral supersymmetric gauge theories on $R^{3}$ × $S^{1}$. The key is a closed-form one-loop-exact D-term coefficient that includes four-dimensional Fayet–Iliopoulos parameters. This coefficient gives a uniformly controlled leading potential away from massless thresholds and fixes local FI and Chern–Simons couplings near thresholds. The framework allows lifting the known 3d Intriligator–Seiberg classification of semiclassical vacua to the entire compact holonomy space. Two examples demonstrate that true vacua can lie at nontrivial holonomy, even when the D-coefficient is nonzero at thresholds or vanishes at trivial holonomy.

What carries the argument

The closed-form one-loop exact D-term coefficient $F^{{out}}$_i(x) = η_i - (1/2) Σ_ρ ρ_i B_2(u_ρ), where B_2 is the second Bernoulli polynomial, η_i encodes the renormalized four-dimensional FI parameters, and u_ρ encodes the holonomy-dependent masses of charged KK modes. This formula, together with the Wilsonian inner/outer patch decomposition using modified Bernoulli functions K_2 and K_1, provides a uniform control of the leading potential away from thresholds and determines local FI and CS couplings near thresholds. Anomaly cancellation ensures $F^{{out}}$ is affine on each chamber, and its zeros, when isolated and off walls, give gapped vacua.

Extended reading notes

Core claim

The central discovery is that the perturbative D-term potential in chiral theories on $R^{3}$ × $S^{1}$ is governed by a one-loop-exact, closed-form function $F^{{out}}$(x) (the outer D-coefficient) that is affine on each chamber of the holonomy space. Its zeros reliably capture perturbative vacua in the weak-coupling regime, provided the hierarchy $g^{2}$_4 ≪ $ε^{2}$ ≪ 1 is maintained. The function is continuous, periodic under large gauge transformations, and Weyl-covariant. Its slopes are piecewise constant Chern–Simons levels that can jump at massless thresholds. By matching to inner patches via modified Bernoulli functions, local FI and CS couplings are fixed, and the classification of vacua—including gapped Maxwell–CS sectors, Coulomb branches, and Higgs branches—extends across the entire holonomy space, not just near trivial holonomy.

Load-bearing premise

The entire perturbative control rests on the hierarchy $g^{2}$_4 ≪ $ε^{2}$ ≪ 1, which requires taking the weak-coupling limit before the cutoff parameter, and assumes no additional quadratic superpotential masses for charged fields on the Coulomb slice and that the auxiliary-field expansion condition |Q·D| ≪ $M^{2}$ is satisfied.

Editorial extensions

If this is right

Perturbative topological vacua in any anomaly-free chiral theory on R^3 × S^1 can now be systematically enumerated by solving F^{out}(x)=0 on the Cartan torus, without needing the full kinetic matrix.
The global D-term control clarifies earlier works that only treated trivial holonomy or relied on indirect real-mass deformations.
Inner patches reveal that Coulomb branches can extend through walls via retained-field loops, and that Higgs branches can occur at walls even when the outer D-coefficient is nonzero.
The matching of local FI and CS couplings at walls provides a precise perturbative input for studying infrared dynamics near walls, such as supersymmetry enhancement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The one-loop exact D-term coefficient is argued to be exact due to supersymmetry non-renormalization; a direct two-loop test in a simple abelian model would confirm this claim.
The method can systematically revisit past studies of chiral gauge theories on R^3 × S^1 (e.g., Lee–Terning [24]) to identify vacuum sectors missed by restricting to trivial holonomy.
The framework may accelerate the search for 3d SCFTs from R-twisted circle reductions, as global D-term control reduces reliance on index calculations for diagnosing supersymmetry enhancement.

Original abstract

We present a Wilsonian framework, globally controlled across the holonomy space, for determining the perturbative data governing the vacua of chiral supersymmetric gauge theories on a circle. This allows lifting the Intriligator--Seiberg classification of semiclassical vacua in 3d $\mathcal{N}=2$ Chern--Simons-matter theories to $\mathbb{R}^3\times S^1$. The framework is showcased by finding a rich set of vacuum sectors in a simple U(1) model and by reproducing on flat space the R-twisted reduction (first diagnosed on $S^3\times S^1$) of an SU(2) Maruyoshi--Song theory to the Gang--Yamazaki theory.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-27: 1 request, 4,597 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.64 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.18 · compares with prior results 0.86 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We establish global control of perturbative D-term data governing semiclassical vacuum sectors in chiral supersymmetric gauge theories on $\mathbb{R}^3\times S^1$. A closed-form, one-loop-exact D-term coefficient, including 4d Fayet--Iliopoulos (FI) parameters, gives a uniformly controlled leading potential away from massless thresholds and fixes local FI and Chern--Simons couplings near them. This allows lifting the 3d Intriligator--Seiberg classification of semiclassical vacua to $\mathbb{R}^3\times S^1$ across the compact holonomy space. The framework is showcased by finding a remarkably rich set of vacuum sectors in a simple abelian model and reproducing on flat space the R-twisted reduction of an $\mathrm{SU}(2)$ Maruyoshi--Song theory to the Gang--Yamazaki theory, first found on $S^3\times S^1$. The examples demonstrate the stakes of global matching: true vacua may lie at nontrivial holonomy even with nonzero threshold values of the Coulomb-slice D-coefficient, and despite vanishing D-coefficient at trivial holonomy.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We establish global control of perturbative D-term data governing semiclassical vacuum sectors in chiral supersymmetric gauge theories on $\mathbb{R}^3\times S^1$.	contribution 0.93	0.95	0.60	asserted 1.00
s1	A closed-form, one-loop-exact D-term coefficient, including 4d Fayet--Iliopoulos (FI) parameters, gives a uniformly controlled leading potential away from massless thresholds and fixes local FI and Chern--Simons couplings near them.	result 0.58	0.92	0.23	asserted 1.00
s2	This allows lifting the 3d Intriligator--Seiberg classification of semiclassical vacua to $\mathbb{R}^3\times S^1$ across the compact holonomy space.	interpretation 0.53	0.84	0.30	asserted 0.98
s3	The framework is showcased by finding a remarkably rich set of vacuum sectors in a simple abelian model and reproducing on flat space the R-twisted reduction of an $\mathrm{SU}(2)$ Maruyoshi--Song theory to the Gang--Yamazaki theory, first found on $S^3\times S^1$.	result 0.76	0.93	0.10	asserted 1.00
s4	The examples demonstrate the stakes of global matching: true vacua may lie at nontrivial holonomy even with nonzero threshold values of the Coulomb-slice D-coefficient, and despite vanishing D-coefficient at trivial holonomy.	interpretation 0.90	0.82	0.36	hedged possibility 0.73

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The twelve isolated roots support gapped Maxwell– CS sectors. On the two open intervals, k out = ζ out = 0, so the holonomy and dual photon parametrize one-complexdimensional perturbative Coulomb branches. At x = 0, all five zero-mode chirals remain light, with k in = 0 and F in = η/R1. The classical Higgs locus 2π P i qi |ϕi | 2 = η/R1 modulo U(1) has generic complex dimension four. The Supplemental Material separately explores a candidate interacting superconformal field theory (SCFT) at its strongly-coupled origin for η = 0. The endpoints xL = 2/5 and xR = 3/7 retain x

**Figure 2.** FIG. 2. Continued outer coefficient of the nonabelian example.

Figures reproduced from arXiv:2610.34240 by the authors.

Discussion

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