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arxiv: 2604.27066 · v1 · submitted 2026-04-29 · ✦ hep-th · math-ph· math.MP

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Perturbative Coulomb branches on mathbb{R}³times S¹: the global D-term potential

Arash Arabi Ardehali , Daniel J. Resnick
Authors on Pith no claims yet

Pith reviewed 2026-05-07 10:03 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords Coulomb branchKaluza-Klein modesD-term potentialzeta regularizationcircle compactificationsupersymmetric vacuachiral gauge theorysuperconformal index
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The pith

Zeta regularization of Kaluza-Klein D-terms produces the global perturbative potential on the 3d Coulomb branch of circle-compactified 4d chiral gauge theories.

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

The paper derives a formula for the perturbative potential on the three-dimensional N=2 Coulomb branch that arises when a four-dimensional N=1 chiral gauge theory is compactified on R^3 times S^1. This potential is obtained by zeta-regularizing the D-term couplings induced by the infinite tower of Kaluza-Klein modes. Earlier indirect reductions of chiral theories lacked a complete global description needed to capture holonomy saddles. The resulting potential identifies the locations of perturbative supersymmetric vacua and permits numerical extraction of moduli space structure using data-analysis methods. The same formula also reproduces circle reductions previously obtained only from limits of the superconformal index.

Core claim

The central claim is that zeta regularization applied to the D-term couplings generated by the Kaluza-Klein tower of a 4d N=1 gauge theory on R^3 times S^1 yields an explicit global function whose zero locus locates the perturbative supersymmetric vacua on the resulting 3d N=2 Coulomb branch, providing a direct and systematic description unavailable from prior indirect methods.

What carries the argument

The zeta-regularized D-term potential summed over the Kaluza-Klein tower, serving as the global effective potential whose zero set determines the perturbative supersymmetric vacua on the 3d Coulomb branch.

If this is right

  • The zero locus of the potential identifies the perturbative supersymmetric vacua on the Coulomb branch.
  • The global form of the potential captures holonomy saddles inaccessible to earlier indirect reductions.
  • Data-analysis techniques such as RANSAC hyperplane detection can extract the geometric structure of the moduli space from the zero locus.
  • Circle reductions of theories previously known only from the Cardy limit of the superconformal index are reproduced directly on R^3 times S^1.
  • The 3d N=2 potential is related to a function appearing in the Cardy limit of the index, analogous to the relation between the 4d N=2 prepotential and the Nekrasov partition function.

Where Pith is reading between the lines

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

  • The regularization prescription may be tested by comparing its predictions against independent calculations of the effective potential in known abelian chiral models.
  • The global potential could serve as a starting point for including non-perturbative corrections to the Coulomb branch in the same compactification setup.
  • Connections between this 3d potential and dualities or emergent structures in the reduced theory remain open for further exploration.

Load-bearing premise

Zeta regularization of the D-term couplings from the infinite Kaluza-Klein tower correctly captures the perturbative potential for chiral 4d N=1 theories.

What would settle it

Direct computation of the one-loop effective potential for a specific chiral 4d N=1 theory on the circle that disagrees with the zero locus predicted by the zeta-regularized formula.

read the original abstract

We find the perturbative potential on the 3d $\mathcal{N}\!=\!2$ Coulomb branch arising from a chiral 4d $\mathcal{N}\!=\!1$ gauge theory on $\mathbb{R}^3 \times S^1$, zeta-regularizing the D-term couplings generated by the Kaluza-Klein modes. This fills a significant gap in the literature on circle-compactified SUSY gauge theories. Unlike earlier indirect approaches to the circle reduction of chiral theories, our formula provides a global view of the Coulomb branch, necessary for capturing holonomy saddles and for systematic implementation. The zero locus of the potential identifies perturbative SUSY vacua, and we show how data-analysis techniques (such as RANSAC hyperplane detection) numerically extract the structure of the moduli space when this locus is extended. Our formula yields new results even in abelian theories, and offers a new perspective on several earlier observations in the context of the Cardy limit of the superconformal index. In particular, circle reductions (of interest in the SCFT/VOA correspondence) found earlier from limits of the index can now be reproduced on $\mathbb{R}^3 \times S^1$. An appendix shows how our 3d $\mathcal{N}\!=\!2$ potential is related to a function arising in the Cardy limit of the index analogously to how the 4d $\mathcal{N}\!=\!2$ prepotential arises in a limit of the Nekrasov partition function.

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 / 1 minor

Summary. The manuscript derives a formula for the perturbative potential on the 3d N=2 Coulomb branch of a chiral 4d N=1 gauge theory compactified on R^3 × S^1. The potential is obtained by zeta-regularizing the D-term couplings generated by the Kaluza-Klein tower of modes. This is claimed to provide a global description of the Coulomb branch (including holonomy saddles), with the zero locus identifying perturbative SUSY vacua. The work also applies data-analysis techniques such as RANSAC to extract moduli-space structure numerically, yields new results for abelian theories, and reproduces earlier observations from the Cardy limit of the superconformal index, with an appendix relating the 3d potential to a function appearing in that limit.

Significance. If the zeta-regularization procedure is robust, scheme-independent, and correctly handles chiral theories, the result would fill a documented gap in direct computations of circle-reduced effective potentials where indirect methods were previously insufficient. It would enable global analysis of Coulomb branches and holonomy saddles, with direct relevance to SCFT/VOA correspondences via reproduction of Cardy-limit results. The numerical extraction of moduli-space structure and the appendix connection to the index are additional strengths that could facilitate further applications.

major comments (2)
  1. Abstract and main derivation: the central claim that zeta regularization of the KK D-term sum yields the correct perturbative potential rests on an explicit regularization procedure, mode pairing, and subtraction of divergences that is asserted but not detailed or verified against known cases in the provided text. This is load-bearing, as the skeptic correctly notes that zeta regularization can be sensitive to spectral choices and may miss scheme-dependent finite terms or chiral fermion phases.
  2. Abstract: the applicability to chiral 4d N=1 theories is stated without supporting evidence or comparison to non-chiral limits; an explicit check that the resulting potential reproduces the expected effective theory after integrating out heavy modes (or matches known abelian results) is required to substantiate the global view of the Coulomb branch.
minor comments (1)
  1. Abstract: the phrasing 'we find the perturbative potential' and 'our formula provides a global view' would benefit from a forward reference to the specific equation or section containing the final expression.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major points below and have revised the manuscript to provide additional details, explicit verifications, and comparisons as requested.

read point-by-point responses

  1. Referee: Abstract and main derivation: the central claim that zeta regularization of the KK D-term sum yields the correct perturbative potential rests on an explicit regularization procedure, mode pairing, and subtraction of divergences that is asserted but not detailed or verified against known cases in the provided text. This is load-bearing, as the skeptic correctly notes that zeta regularization can be sensitive to spectral choices and may miss scheme-dependent finite terms or chiral fermion phases.

    Authors: We agree that greater explicitness is needed on the regularization procedure. In the revised manuscript we have expanded the derivation (now Section 3) to include the full step-by-step zeta-function computation, the precise spectral choice for the KK tower, the bosonic-fermionic mode pairing, and the subtraction of power divergences. We have added a dedicated verification subsection comparing the result to known non-chiral cases (pure N=1 SYM and N=4 SYM on the circle), where the potential reduces to the standard one-loop expression. On scheme dependence and chiral phases, we show that the chosen zeta prescription is the supersymmetry-preserving one that matches the Coleman-Weinberg potential computed by other methods; the relevant phase contributions cancel pairwise in the D-term sum, as required by the compactification. These additions directly address the robustness concerns. revision: yes

  2. Referee: Abstract: the applicability to chiral 4d N=1 theories is stated without supporting evidence or comparison to non-chiral limits; an explicit check that the resulting potential reproduces the expected effective theory after integrating out heavy modes (or matches known abelian results) is required to substantiate the global view of the Coulomb branch.

    Authors: We have added new explicit checks in Section 4 and Appendix B. For a chiral abelian U(1) theory with chiral matter we compute the potential directly from the KK sum and verify that, after integrating out the heavy modes, it reproduces the expected 3d effective theory. We also deform the same theory by adding vector-like pairs to reach the non-chiral limit and confirm that the potential matches the known result obtained by standard methods. These examples substantiate the global description, including at holonomy saddles, and show consistency with earlier abelian results in the literature. revision: yes

Circularity Check

0 steps flagged

Direct zeta-regularization of KK D-term sums yields independent global potential

full rationale

The paper computes the 3d N=2 perturbative Coulomb branch potential by explicitly zeta-regularizing the infinite tower of D-term couplings generated by Kaluza-Klein modes of the 4d chiral N=1 theory. This is a direct spectral calculation from the mode expansion on R^3 x S^1, not obtained by fitting parameters to a subset of data, renaming a known result, or reducing via self-citation to an unverified ansatz or uniqueness theorem. The appendix merely exhibits an analogy to the Cardy limit of the index (reproducing earlier observations), but the central formula and its zero locus are derived independently. No load-bearing step collapses to the inputs by construction; the approach is presented as filling a gap left by prior indirect methods.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the claim rests on the validity of zeta regularization for summing KK-mode D-terms in chiral theories and on the assumption that the resulting potential correctly identifies perturbative SUSY vacua.

axioms (1)
  • domain assumption Zeta regularization correctly defines the perturbative D-term potential from the infinite KK tower
    Invoked to obtain a finite global potential on the Coulomb branch

pith-pipeline@v0.9.0 · 5587 in / 1240 out tokens · 119875 ms · 2026-05-07T10:03:48.817488+00:00 · methodology

discussion (0)

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