pith. machine review for the scientific record. sign in

arxiv: 2512.03146 · v2 · submitted 2025-12-02 · ✦ hep-th

Recognition: 2 theorem links

· Lean Theorem

Homotopy transfer for massive Kaluza-Klein modes

Camille Eloy , Olaf Hohm , Camilla Lavino , Henning Samtleben , Yehudi Simon
Authors on Pith no claims yet

Pith reviewed 2026-05-17 01:45 UTC · model grok-4.3

classification ✦ hep-th
keywords Kaluza-Kleinmassive modesgauge invariancehomotopy transferL-infinityHiggs mechanismtorusperturbation theory
0
0 comments X

The pith

A perturbative algorithm produces new fields for massive Kaluza-Klein modes that stay invariant under broken gauge transformations but transform under unbroken ones.

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

This paper develops techniques for handling massive Kaluza-Klein modes in perturbation theory by explicitly showing how the Higgs mechanism gives them mass. It introduces an algorithm that constructs new fields invariant under the gauge transformations of the higher modes, which get broken during compactification. These fields, however, still transform covariantly under the gauge transformations of the zero modes that remain unbroken. The approach relies on mapping gauge-redundant fields to gauge-invariant ones through a homotopy transfer procedure in the algebraic formulation of the theory. A sympathetic reader would care because this provides a systematic way to include massive modes in calculations without dealing with their gauge redundancies, which is crucial for consistent perturbative expansions in compactified gravity theories.

Core claim

The central claim is that homotopy transfer applied to the L∞ algebra description of gravity yields an algorithm in perturbation theory for new fields associated with massive Kaluza-Klein modes. These fields are gauge invariant under all higher-mode gauge transformations that are broken by the compactification but transform covariantly under the zero-mode gauge transformations that remain unbroken. This is illustrated for Kaluza-Klein theory on a torus as a proof of concept, with the intention to apply it to generalized Scherk-Schwarz backgrounds in exceptional field theory.

What carries the argument

the homotopy transfer map that converts gauge-redundant gravity fields into gauge-invariant fields while preserving covariance under unbroken symmetries

If this is right

  • The algorithm allows treatment of massive Kaluza-Klein modes to arbitrary order in perturbation theory.
  • It displays the Higgs mechanism that renders the higher modes massive.
  • The resulting fields can be used for consistent expansions in generalized Scherk-Schwarz reductions.
  • These techniques provide a way to handle broken gauge symmetries in compactified field theories systematically.

Where Pith is reading between the lines

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

  • If the method generalizes, it could be used to derive effective interactions among massive modes in string compactifications.
  • The construction might offer a new perspective on how gauge symmetries are realized in effective theories with massive fields from compactification.
  • Explicit computations in the torus case could verify the algorithm by checking invariance properties order by order.

Load-bearing premise

The homotopy transfer procedure applied to the algebraic formulation of gravity produces fields with the desired gauge invariance properties for the Kaluza-Klein modes.

What would settle it

Computing the gauge transformation of the new fields under a higher-mode transformation at second order in the perturbative expansion and finding a non-zero variation would disprove the algorithm's correctness.

Figures

Figures reproduced from arXiv: 2512.03146 by Camilla Lavino, Camille Eloy, Henning Samtleben, Olaf Hohm, Yehudi Simon.

Figure 1
Figure 1. Figure 1: Homotopy transfer from X to X˚. Let us now define the homotopy data for Kaluza-Klein theory on a torus. The projector is 12 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Proca chain complex This should be homotopy equivalent to the two-term complex for strict Proca, where pro￾jection and inclusion are only non-trivial in degrees zero and one: p0(A) = Aµ − 1 m ∂µφ , ι0(Aµ) = Aµ 0 ! , p1(E) = E µ , ι1(E µ ) = Eµ 1 m ∂νEν ! . (A.6) It is straightforward to verify the chain map conditions ∂¯ ◦ p = p ◦ ∂ and ∂ ◦ ι = ι ◦ ∂¯. In particular, this requires the non-trivial inclusion… view at source ↗
read the original abstract

We develop techniques to treat massive Kaluza-Klein modes to arbitrary order in perturbation theory. The Higgs mechanism that renders the higher Kaluza-Klein modes massive is displayed. To this end we give an algorithm in perturbation theory that yields new fields with the following characteristics: they are gauge invariant under all higher-mode gauge transformations, which are broken, but they transform covariantly under the zero-mode gauge transformations, which are unbroken. We employ the formulation of field theory in terms of $L_{\infty}$ algebras together with their homotopy transfer, which here maps the gauge redundant fields of gravity to gauge invariant fields. We illustrate these results, as a proof of concept, for Kaluza-Klein theory on a torus. In an accompanying paper these results will be applied to a large class of generalized Scherk-Schwarz backgrounds in exceptional field theory.

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 paper develops a perturbative algorithm using L∞ algebras and homotopy transfer to construct new fields for massive Kaluza-Klein modes. These fields are gauge invariant under the broken higher-mode gauge transformations but transform covariantly under the unbroken zero-mode gauge transformations. The approach displays the Higgs mechanism for mass generation and is illustrated as a proof of concept for Kaluza-Klein theory on a torus, with intended application to generalized Scherk-Schwarz backgrounds in exceptional field theory.

Significance. If the homotopy transfer preserves the stated gauge properties to all orders, the method supplies a systematic, gauge-redundancy-free description of massive modes in compactifications. This could facilitate the construction of effective theories and the analysis of residual symmetries in string theory and supergravity reductions. The algorithmic character and the explicit torus illustration are strengths that make the result potentially reusable.

major comments (2)
  1. [Torus example] § on the torus illustration (proof-of-concept example): the manuscript does not provide an explicit check that the homotopy operator commutes with the zero-mode gauge action on the dgLa at second order or higher. Without this verification, it remains unclear whether covariance under unbroken transformations is preserved beyond leading order, which is load-bearing for the central claim.
  2. [Perturbative algorithm] Algorithm section: the recursive definition of the transferred fields is given, but the paper does not derive or display the explicit second-order correction term that would confirm the absence of non-covariant contributions under zero-mode transformations.
minor comments (2)
  1. [Introduction] The L∞ bracket notation and the precise definition of the homotopy operator should be recalled briefly for readers who may not have the preceding literature at hand.
  2. [Main text] A short table summarizing the gauge transformation properties of the original versus transferred fields at each perturbative order would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript to incorporate explicit verifications at second order.

read point-by-point responses

  1. Referee: [Torus example] § on the torus illustration (proof-of-concept example): the manuscript does not provide an explicit check that the homotopy operator commutes with the zero-mode gauge action on the dgLa at second order or higher. Without this verification, it remains unclear whether covariance under unbroken transformations is preserved beyond leading order, which is load-bearing for the central claim.

    Authors: We agree that an explicit verification strengthens the torus illustration. While the general homotopy transfer construction in the L∞ algebra is designed to be equivariant under the unbroken zero-mode action (by choice of a compatible homotopy operator), we have now added an explicit computation at second order in the revised torus example section. This confirms that the homotopy operator commutes with the zero-mode gauge action to that order, with no non-covariant terms appearing. revision: yes

  2. Referee: [Perturbative algorithm] Algorithm section: the recursive definition of the transferred fields is given, but the paper does not derive or display the explicit second-order correction term that would confirm the absence of non-covariant contributions under zero-mode transformations.

    Authors: We have derived the explicit second-order correction to the transferred fields and included it in the revised algorithm section. The resulting expression contains only terms that transform covariantly under zero-mode gauge transformations, as required. This makes the recursive procedure more concrete while remaining consistent with the general L∞ homotopy transfer. revision: yes

Circularity Check

0 steps flagged

Homotopy transfer derivation is self-contained with no reduction to inputs

full rationale

The paper develops a perturbative algorithm using L∞ algebra homotopy transfer to construct fields invariant under broken higher-mode gauge transformations while covariant under unbroken zero-mode ones for Kaluza-Klein modes on a torus. This follows directly from the standard properties of homotopy transfer in the L∞ formulation of gravity, applied as a proof of concept without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to unverified assumptions. The gauge properties are derived from the algebraic structure and the explicit mapping, which remains independent of the target result and is illustrated for the torus case before generalization.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests primarily on the standard properties of L∞ algebras for encoding gauge symmetries in gravity and the applicability of homotopy transfer to map redundant fields to invariant ones in the KK setting.

axioms (2)
  • standard math L∞ algebras correctly capture the gauge structure and higher interactions in gravitational field theories
    Invoked as the foundational framework for the homotopy transfer procedure.
  • domain assumption Homotopy transfer maps gauge-redundant fields to gauge-invariant ones while preserving covariance under unbroken symmetries in KK compactifications
    This is the key mapping assumed to hold for the massive modes.

pith-pipeline@v0.9.0 · 5444 in / 1373 out tokens · 85119 ms · 2026-05-17T01:45:18.858682+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 14 internal anchors

  1. [1]

    The Large N Limit of Superconformal Field Theories and Supergravity

    J. M. Maldacena, “The largeNlimit of superconformal field theories and supergravity,” Int. J. Theor. Phys.38(1999) 1113–1133,arXiv:hep-th/9711200 [hep-th]. [Adv. Theor. Math. Phys.2,231(1998)]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  2. [2]

    Gauge Theory Correlators from Non-Critical String Theory

    S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,”Phys. Lett.B428(1998) 105–114,arXiv:hep-th/9802109 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  3. [3]

    Anti De Sitter Space And Holography

    E. Witten, “Anti-de Sitter space and holography,”Adv. Theor. Math. Phys.2(1998) 253–291,arXiv:hep-th/9802150 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  4. [4]

    On Kaluza-Klein theory,

    A. Salam and J. A. Strathdee, “On Kaluza-Klein theory,”Annals Phys.141(1982) 316–352

    work page 1982

  5. [5]

    Kac-Moody symmetries of Kaluza-Klein theories,

    L. Dolan and M. Duff, “Kac-Moody symmetries of Kaluza-Klein theories,”Phys. Rev. Lett.52(1984) 14

    work page 1984

  6. [6]

    The infinite-dimensional gauge structure of Kaluza-Klein theories.D= 1 + 4,

    C. Aulakh and D. Sahdev, “The infinite-dimensional gauge structure of Kaluza-Klein theories.D= 1 + 4,”Phys.Lett.B164(1985) 293

    work page 1985

  7. [7]

    Explicit construction of massive spin two fields in Kaluza-Klein theory,

    Y. Cho and S. Zoh, “Explicit construction of massive spin two fields in Kaluza-Klein theory,”Phys.Rev.D46(1992) 2290–2294. 34

    work page 1992

  8. [8]

    On Kaluza-Klein States from Large Extra Dimensions

    T. Han, J. D. Lykken, and R.-J. Zhang, “On Kaluza-Klein states from large extra dimensions,”Phys. Rev. D59(1999) 105006,arXiv:hep-ph/9811350

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  9. [9]

    Quantum Gravity and Extra Dimensions at High-Energy Colliders

    G. F. Giudice, R. Rattazzi, and J. D. Wells, “Quantum gravity and extra dimensions at high-energy colliders,”Nucl. Phys. B544(1999) 3–38,arXiv:hep-ph/9811291

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  10. [10]

    The spectrum of the eleven-dimensional supergravity compactified on the round seven sphere,

    E. Sezgin, “The spectrum of the eleven-dimensional supergravity compactified on the round seven sphere,”Phys. Lett. B138(1984) 57–62

    work page 1984

  11. [11]

    The fluctuating seven sphere in eleven-dimensional supergravity,

    B. Biran, A. Casher, F. Englert, M. Rooman, and P. Spindel, “The fluctuating seven sphere in eleven-dimensional supergravity,”Phys. Lett.134B(1984) 179

    work page 1984

  12. [12]

    The spectrum of theS 5 compactification of the chiral N= 2, D= 10 supergravity and the unitary supermultiplets ofU(2,2/4),

    M. G¨ unaydin and N. Marcus, “The spectrum of theS 5 compactification of the chiral N= 2, D= 10 supergravity and the unitary supermultiplets ofU(2,2/4),”Class. Quant. Grav.2(1985) L11

    work page 1985

  13. [13]

    The mass spectrum of chiral ten-dimensionalN= 2 supergravity onS 5,

    H. J. Kim, L. J. Romans, and P. van Nieuwenhuizen, “The mass spectrum of chiral ten-dimensionalN= 2 supergravity onS 5,”Phys. Rev.D32(1985) 389

    work page 1985

  14. [14]

    Exceptional Form of D=11 Supergravity

    O. Hohm and H. Samtleben, “Exceptional form ofD= 11 supergravity,”Phys. Rev. Lett. 111(2013) 231601,arXiv:1308.1673

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  15. [15]

    Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB

    O. Hohm and H. Samtleben, “Exceptional field theory I: E 6(6) covariant form of M-theory and type IIB,”Phys. Rev.D89(2014) 066016,arXiv:1312.0614

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  16. [16]

    Consistent Kaluza-Klein Truncations via Exceptional Field Theory

    O. Hohm and H. Samtleben, “Consistent Kaluza-Klein truncations via exceptional field theory,”JHEP1501(2015) 131,arXiv:1410.8145 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  17. [17]

    E$_{6(6)}$ Exceptional Field Theory: Review and Embedding of Type IIB

    A. Baguet, O. Hohm, and H. Samtleben, “E 6(6) exceptional field theory: Review and embedding of type IIB,”PoSCORFU2014(2015) 133,arXiv:1506.01065 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  18. [18]

    Kaluza-Klein spectrometry for supergravity,

    E. Malek and H. Samtleben, “Kaluza-Klein spectrometry for supergravity,”Phys. Rev. Lett.124(2020) 101601,arXiv:1911.12640 [hep-th]

    work page arXiv 2020

  19. [19]

    Kaluza-Klein spectrometry from exceptional field theory,

    E. Malek and H. Samtleben, “Kaluza-Klein spectrometry from exceptional field theory,” Phys. Rev. D102(2020) 106016,arXiv:2009.03347 [hep-th]

    work page arXiv 2020

  20. [20]

    Cubic and higher-order supergravity couplings for AdS vacua using Exceptional Field Theory,

    B. Duboeuf, E. Malek, and H. Samtleben, “Cubic and higher-order supergravity couplings for AdS vacua using Exceptional Field Theory,”JHEP05(2024) 214, arXiv:2311.00742 [hep-th]

    work page arXiv 2024

  21. [21]

    Kaluza-Klein Holography

    K. Skenderis and M. Taylor, “Kaluza-Klein holography,”JHEP05(2006) 057, arXiv:hep-th/0603016

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  22. [22]

    Closed String Field Theory: Quantum Action and the BV Master Equation

    B. Zwiebach, “Closed string field theory: Quantum action and the B-V master equation,” Nucl. Phys.B390(1993) 33–152,arXiv:hep-th/9206084 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  23. [23]

    Homotopy Lie Superalgebra in Yang-Mills Theory

    A. M. Zeitlin, “Homotopy Lie Superalgebra in Yang-Mills Theory,”JHEP09(2007) 068, arXiv:0708.1773 [hep-th]. 35

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  24. [24]

    $L_{\infty}$ Algebras and Field Theory

    O. Hohm and B. Zwiebach, “L ∞ algebras and field theory,”Fortsch. Phys.65no. 3-4, (2017) 1700014,arXiv:1701.08824 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  25. [25]

    L∞-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism,

    B. Jurˇ co, L. Raspollini, C. S¨ amann, and M. Wolf, “L∞-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism,”Fortsch. Phys.67no. 7, (2019) 1900025,arXiv:1809.09899 [hep-th]

    work page arXiv 2019

  26. [26]

    On the perturbation lemma, and deformations

    M. Crainic, “On the perturbation lemma, and deformations,”arXiv:math/0403266

    work page internal anchor Pith review Pith/arXiv arXiv

  27. [27]

    Classical algebraic structures in string theory effective actions,

    H. Erbin, C. Maccaferri, M. Schnabl, and J. Voˇ smera, “Classical algebraic structures in string theory effective actions,”JHEP11(2020) 123,arXiv:2006.16270 [hep-th]

    work page arXiv 2020

  28. [28]

    Gauge-invariant operators of open bosonic string field theory in the low-energy limit,

    D. Koyama, Y. Okawa, and N. Suzuki, “Gauge-invariant operators of open bosonic string field theory in the low-energy limit,”arXiv:2006.16710 [hep-th]

    work page arXiv 2006

  29. [29]

    Homotopy transfer and effective field theory I: Tree-level,

    A. S. Arvanitakis, O. Hohm, C. Hull, and V. Lekeu, “Homotopy transfer and effective field theory I: Tree-level,”Fortsch. Phys.70no. 2-3, (2022) 2200003,arXiv:2007.07942 [hep-th]

    work page arXiv 2022

  30. [30]

    Homotopy transfer and effective field theory II: Strings and double field theory,

    A. S. Arvanitakis, O. Hohm, C. Hull, and V. Lekeu, “Homotopy transfer and effective field theory II: Strings and double field theory,”Fortsch. Phys.70no. 2-3, (2022) 2200004,arXiv:2106.08343 [hep-th]

    work page arXiv 2022

  31. [31]

    Yang-Mills kinematic algebra via homotopy transfer from a worldline operator algebra,

    R. Bonezzi, C. Chiaffrino, O. Hohm, and M. F. Kallimani, “Yang-Mills kinematic algebra via homotopy transfer from a worldline operator algebra,”Phys. Rev. D112no. 10, (2025) 105006,arXiv:2508.17933 [hep-th]

    work page arXiv 2025

  32. [32]

    Gauge invariant perturbation theory via homotopy transfer,

    C. Chiaffrino, O. Hohm, and A. F. Pinto, “Gauge invariant perturbation theory via homotopy transfer,”JHEP05(2021) 236,arXiv:2012.12249 [hep-th]

    work page arXiv 2021

  33. [33]

    Holography as homotopy,

    C. Chiaffrino, T. Ersoy, and O. Hohm, “Holography as homotopy,”JHEP09(2024) 161, arXiv:2307.08094 [hep-th]

    work page arXiv 2024

  34. [34]

    Gauge Invariant Cosmological Perturbations,

    J. M. Bardeen, “Gauge Invariant Cosmological Perturbations,”Phys. Rev. D22(1980) 1882–1905

    work page 1980

  35. [35]

    Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions,

    V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, “Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions,”Phys. Rept.215(1992) 203–333

    work page 1992

  36. [36]

    Tree-level scattering amplitudes via homotopy transfer,

    R. Bonezzi, C. Chiaffrino, F. Diaz-Jaramillo, and O. Hohm, “Tree-level scattering amplitudes via homotopy transfer,”arXiv:2312.09306 [hep-th]

    work page arXiv

  37. [37]

    C. Eloy, O. Hohm, C. Lavino, H. Samtleben, and Y. Simon. work in progress. 36

    work page