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arxiv: 2605.04148 · v1 · submitted 2026-05-05 · ✦ hep-th · hep-ph

Heterotic Flux Vacua with a Small Superpotential

Evgeny I. Buchbinder , Andrei Constantin , Lucas T. Y. Leung , Andre Lukas , Burt Ovrut This is my paper

Pith reviewed 2026-05-08 17:56 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords heterotic string theoryflux compactificationCalabi-Yau manifoldmoduli stabilizationsuperpotentialNSNS fluxsupersymmetric vacuacomplex structure moduli
0
0 comments X p. Extension

The pith

Heterotic Calabi-Yau flux compactifications forbid supersymmetric minima with small or vanishing superpotential except at special points away from the large complex structure limit.

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

The paper investigates whether heterotic string compactifications on Calabi-Yau threefolds with NSNS three-form flux can achieve a small value for the flux-induced superpotential at supersymmetric vacua. Without the no-scale structure of type IIB, a large superpotential generates a tree-level scalar potential for all moduli, so controlled stabilization requires |W0| to be small enough to compete with non-perturbative effects. Using the four-dimensional effective theory and the special geometry of complex-structure moduli space, the authors derive two no-go theorems: supersymmetric vacua with exactly vanishing W0 exist only at singular loci, and no supersymmetric vacua with small W0 exist in the large complex structure limit. Explicit one- and two-parameter models computed with exact periods allow flux choices that produce |W0| of order unity, with some one-parameter cases moderately smaller, but these values remain only marginally suitable for moduli stabilization.

Core claim

Working within a four-dimensional effective field theory and exploiting the special geometry of Calabi-Yau complex structure moduli spaces, supersymmetric vacua with vanishing |W0| occur only at singular loci in moduli space, while no supersymmetric vacua with small |W0| exist in the large complex structure limit. Explicit analysis of one- and two-parameter models away from this limit identifies flux choices yielding |W0| of order unity, with some one-parameter models admitting values moderately below unity. These findings indicate that small heterotic flux superpotentials are highly constrained.

What carries the argument

The special geometry of the complex-structure moduli space, used to solve the complex-structure F-term equations and derive constraints on the superpotential.

If this is right

  • Supersymmetric flux vacua with exactly zero superpotential require singular points in the moduli space.
  • All supersymmetric minima in the large complex structure limit have large |W0|.
  • Away from the large complex structure limit, suitable flux choices in one- and two-parameter models can produce |W0| of order one or moderately below.
  • The |W0| values obtained are only marginally compatible with the requirements for non-perturbative moduli stabilization.

Where Pith is reading between the lines

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

  • Heterotic models may require additional mechanisms beyond flux plus non-perturbative effects to achieve reliable moduli stabilization.
  • The constraints could imply that viable supersymmetric heterotic vacua with stabilized moduli are rarer than in type IIB.
  • Further analysis of models with more moduli or including bundle moduli might reveal whether smaller |W0| becomes accessible in broader classes of compactifications.

Load-bearing premise

The four-dimensional effective field theory remains valid and the special geometry of the complex-structure moduli space can be used to solve the F-term equations even when |W0| is small and away from the large-complex-structure limit.

What would settle it

An explicit period computation in a one-parameter Calabi-Yau model that yields a supersymmetric minimum with |W0| much smaller than unity while remaining in the large complex structure regime, or a concrete counterexample to either no-go theorem.

Figures

Figures reproduced from arXiv: 2605.04148 by Andrei Constantin, Andre Lukas, Burt Ovrut, Evgeny I. Buchbinder, Lucas T. Y. Leung.

Figure 1
Figure 1. Figure 1: Plots of the values of U0, defined in Eq. (1.1), obtained from a systematic search for flux integers in the range |nA|, |mA| ≤ 10. The four panels correspond to the four CY manifolds in view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the values of W0 from Eq. (1.1) obtained from a systematic search for flux integers in the range |nA|, |mA| ≤ 10. The four panels correspond to the four CY manifolds in view at source ↗
Figure 3
Figure 3. Figure 3: The smallest value of |U0| we have been able to find within the search box is of order one and the details are provided in view at source ↗
Figure 3
Figure 3. Figure 3: The plots show the values of U0 and W0 from Eq. (2.16) for the two-parameter model defined in the text, with fluxes in the range |nA|, |mA| ≤ 5. 6 Conclusions In this work we have explored moduli stabilisation using fluxes in heterotic string theories. Flux in heterotic theories is quite different from type IIB flux. Firstly, the heterotic flux superpotential does not depend on the axio-dilaton and this im… view at source ↗
read the original abstract

We study heterotic Calabi--Yau compactifications with NSNS three-form flux in view of moduli stabilisation and investigate whether the value $|W_0|$ of the flux superpotential evaluated at supersymmetric minima can be small. Unlike in type IIB string theory, heterotic compactifications lack a no-scale structure, so that a non-vanishing flux superpotential generically induces a tree-level scalar potential for all moduli. Controlled moduli stabilisation therefore requires the flux superpotential to be sufficiently small in order to compete with non-perturbative effects. Working within a four-dimensional effective field theory and exploiting the special geometry of Calabi--Yau complex structure moduli spaces, we analyse the complex structure F-term equations and derive two no-go theorems: (1) supersymmetric vacua with vanishing $|W_0|$, which would lead to a vanishing tree-level scalar potential as in type IIB, occur only at singular loci in moduli space, and (2) no supersymmetric vacua with small $|W_0|$ exist in the large complex structure limit. Motivated by these results, we analyse explicit models away from the large complex structure regime using exact period expressions and identify one- and two-parameter examples in which suitable flux choices allow for values of $|W_0|$ of order unity, with some one-parameter models admitting values moderately below unity. Our findings show that small heterotic flux superpotentials are highly constrained, with the values of $|W_0|$ found in the examples studied being only marginally compatible with moduli stabilisation.

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

3 major / 2 minor

Summary. The manuscript derives two no-go theorems for supersymmetric complex-structure minima in heterotic Calabi-Yau compactifications with NSNS flux: (1) vanishing |W0| occurs only at singular loci in moduli space, and (2) no small-|W0| supersymmetric solutions exist in the large complex structure limit. It then constructs explicit one- and two-parameter models away from the LCS regime using exact period integrals, finding flux choices that yield |W0| of order unity (with some one-parameter cases moderately below unity), and concludes that small heterotic flux superpotentials are highly constrained and only marginally compatible with moduli stabilisation.

Significance. If the results hold, they provide concrete constraints on heterotic moduli stabilisation, underscoring the absence of no-scale structure and the resulting need for small |W0| to compete with non-perturbative effects. The explicit use of exact period expressions (rather than approximations) is a clear strength, furnishing falsifiable numerical examples that support the no-go theorems and quantify the limited range of achievable |W0|.

major comments (3)
  1. [F-term equations and no-go theorems] § on F-term equations and no-go theorems: the second no-go (no small |W0| in LCS limit) relies on the leading terms in the period expansion; it is not immediately clear whether sub-leading corrections in the exact periods could permit parametrically small |W0| while remaining in the LCS regime, which would affect the load-bearing claim that small |W0| is excluded there.
  2. [Explicit models section] Explicit models section: the chosen fluxes are stated to produce supersymmetric vacua with |W0| ~ O(1), but the manuscript does not report an explicit check of the scalar mass matrix or Hessian at these points to confirm they are genuine minima (as opposed to saddle points) once the full potential including non-perturbative terms is considered.
  3. [Discussion of 4D EFT validity] Discussion of 4D EFT validity: the central claim of marginal compatibility with moduli stabilisation rests on the assumption that the 4D effective theory and special geometry remain valid for the reported small |W0| values away from LCS; a quantitative estimate of the size of higher-order corrections (e.g., in the Kähler potential or flux-induced terms) would be needed to substantiate this.
minor comments (2)
  1. [Abstract] The abstract states that some models admit values 'moderately below unity' but does not quote the specific numerical values obtained; adding these numbers would improve clarity.
  2. [Explicit models] Notation for the periods and flux quanta is introduced without a dedicated table summarising the integer flux choices for each explicit model; a compact table would aid reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and indicate planned revisions.

read point-by-point responses

  1. Referee: [F-term equations and no-go theorems] § on F-term equations and no-go theorems: the second no-go (no small |W0| in LCS limit) relies on the leading terms in the period expansion; it is not immediately clear whether sub-leading corrections in the exact periods could permit parametrically small |W0| while remaining in the LCS regime, which would affect the load-bearing claim that small |W0| is excluded there.

    Authors: In the strict large complex structure limit the moduli are taken large, so that leading polynomial terms in the periods dominate while sub-leading corrections are suppressed by inverse powers of the moduli. These corrections are parametrically small and cannot cancel the leading O(1) piece to produce parametrically small |W0|. The no-go therefore continues to hold inside the LCS regime. We will add a clarifying sentence on this scaling in the revised manuscript. revision: yes

  2. Referee: [Explicit models section] Explicit models section: the chosen fluxes are stated to produce supersymmetric vacua with |W0| ~ O(1), but the manuscript does not report an explicit check of the scalar mass matrix or Hessian at these points to confirm they are genuine minima (as opposed to saddle points) once the full potential including non-perturbative terms is considered.

    Authors: Our analysis identifies points satisfying the complex-structure F-term equations with |W0| of order unity. We have not computed the full Hessian including non-perturbative Kähler-sector contributions, as this requires additional model-specific choices for gaugino condensation and Kähler moduli stabilization that lie outside the scope of the present work. The reported points are supersymmetric minima in the complex-structure sector at tree level. We will add a remark noting that a complete stability analysis of the full potential is left for future study. revision: yes

  3. Referee: [Discussion of 4D EFT validity] Discussion of 4D EFT validity: the central claim of marginal compatibility with moduli stabilisation rests on the assumption that the 4D effective theory and special geometry remain valid for the reported small |W0| values away from LCS; a quantitative estimate of the size of higher-order corrections (e.g., in the Kähler potential or flux-induced terms) would be needed to substantiate this.

    Authors: A fully model-independent quantitative bound on higher-order corrections is difficult without fixing the Kähler moduli and the specific Calabi-Yau geometry. In the explicit examples we employ exact periods and remain away from singular loci, so that the special-geometry description remains reliable. We will expand the discussion with a qualitative assessment of the regime of validity based on the distance from the LCS limit. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results consist of two no-go theorems on supersymmetric complex-structure minima derived directly from the F-term equations via special geometry of Calabi-Yau moduli spaces, plus explicit numerical solutions of one- and two-parameter models using exact period integrals away from the large-complex-structure limit. These steps rely on standard, externally established mathematical structures and direct computation rather than any fitted parameter renamed as a prediction, self-definitional closure, or load-bearing self-citation chain. The conclusion that small |W0| values are constrained follows from these independent derivations without reduction to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background assumptions of heterotic string compactification and special geometry rather than new free parameters or invented entities.

axioms (2)
  • domain assumption The four-dimensional effective theory obtained from heterotic Calabi-Yau compactification with NSNS flux is valid for analyzing supersymmetric minima.
    Invoked throughout the analysis of F-term equations and the tree-level potential.
  • standard math Special geometry governs the complex-structure moduli space and allows solution of the F-term equations.
    Used to derive both no-go theorems.

pith-pipeline@v0.9.0 · 5589 in / 1514 out tokens · 49121 ms · 2026-05-08T17:56:28.519198+00:00 · methodology

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