pith. sign in

arxiv: 2505.00638 · v4 · submitted 2025-05-01 · ✦ hep-th

Only Flat Spacetime is Full BPS in Four Dimensional N=3 and N=4 Supergravity

Abhinava Bhattacharjee , Subramanya Hegde , Bindusar Sahoo This is my paper

Pith reviewed 2026-05-22 17:05 UTC · model grok-4.3

classification ✦ hep-th
keywords supergravityhigher derivativesBPS solutionssupersymmetric vacuaN=3 supergravityN=4 supergravityWeyl multipletflat spacetime
0
0 comments X

The pith

Only flat spacetime is fully supersymmetric in N=3 and N=4 higher derivative supergravity

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

The authors investigate fully supersymmetric solutions in N=3 and N=4 higher derivative Poincaré supergravity theories constructed via the conformal supergravity framework. These theories incorporate the standard Weyl multiplet along with a set of terms related to the Weyl square by supersymmetry. Working in the superconformal formalism, they demonstrate that flat spacetime is the only solution that preserves the full supersymmetry. This stands in contrast to N=2 theories in the same class, which also permit the Bertotti-Robinson geometry as a fully supersymmetric solution. The analysis reveals why the vacuum structure becomes more restricted with higher numbers of supersymmetries.

Core claim

Flat spacetime is the only fully supersymmetric solution in the considered class of N=3 and N=4 Poincaré supergravity theories. These theories are obtained in the conformal supergravity framework using the standard Weyl multiplet and supersymmetric terms associated with the Weyl square. In the superconformal formalism, no other solutions, such as curved geometries, satisfy the conditions for full BPS preservation. N=2 theories in this class allow both flat spacetime and the Bertotti-Robinson AdS2 x S2 geometry.

What carries the argument

Superconformal formalism for determining fully supersymmetric solutions in higher derivative theories based on the Weyl multiplet

Load-bearing premise

The theories under consideration are those derived within conformal supergravity using the standard Weyl multiplet and including supersymmetric terms related to the Weyl square term

What would settle it

The existence of a non-flat, stationary or curved fully supersymmetric solution in one of these N=3 or N=4 higher derivative theories would falsify the claim that only flat spacetime works

read the original abstract

We investigate the fully supersymmetric solutions in a class of N=3 and N=4 higher derivative Poincar\'e supergravity theories. These class of theories are obtained within the framework of conformal supergravity using the standard Weyl multiplet and contains a set of terms related to the Weyl square term by supersymmetry. We work in the superconformal formalism and show that flat spacetime is the only fully supersymmetric solution in these theories. However, in N=2 Poincar\'e supergravity theories that falls into the same class, two fully supersymmetric stationary solutions exist: Bertotti-Robinson geometry ($AdS_2\times S^2$) and flat spacetime, which are known in the literature. We discuss the reason behind the richer vacua in N=2 supergravity compared to its N=3 and N=4 cousins.

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

1 major / 2 minor

Summary. The manuscript investigates fully supersymmetric solutions in a class of N=3 and N=4 higher-derivative Poincaré supergravity theories constructed within conformal supergravity using the standard Weyl multiplet and supersymmetric completions of the Weyl-squared term. Working in the superconformal formalism, the authors impose the full set of Killing spinor equations and conclude that flat spacetime is the only solution compatible with all variations vanishing. They contrast this with N=2 theories in the same class, where both flat spacetime and the Bertotti-Robinson geometry (AdS₂ × S²) are known to exist, attributing the difference to the off-shell multiplet structure.

Significance. If the central derivation is correct, the result clarifies how the number of supercharges and the structure of the off-shell multiplets constrain the space of fully BPS backgrounds in higher-derivative supergravities. This distinction between N=2 and N≥3 cases is potentially useful for classifying supersymmetric vacua and for effective descriptions arising from string compactifications. The use of the superconformal formalism to derive algebraic constraints on curvature and auxiliary fields is a methodological strength.

major comments (1)
  1. [Section 3 (Killing spinor analysis)] The manuscript states that the additional supercharges in N=3 and N=4 impose stricter algebraic constraints than in N=2, but the explicit expansion of the Killing spinor variations (or the resulting conditions on the Weyl tensor and auxiliary fields) is not provided in sufficient detail to verify that only the flat solution survives. A concrete example of the constraint equations for at least one non-flat ansatz would strengthen the claim.
minor comments (2)
  1. [Abstract] The abstract summarizes the result without indicating the key technical step (imposition of the full Killing spinor equations in the superconformal frame); a single sentence on this point would improve readability.
  2. [Section 2 (Theory construction)] Notation for the supersymmetrized Weyl-squared invariants and their companions should be introduced with explicit reference to the relevant superconformal Lagrangian terms to avoid ambiguity when comparing N=2, N=3, and N=4 cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment of its significance, and their recommendation for minor revision. We address the major comment below.

read point-by-point responses

  1. Referee: [Section 3 (Killing spinor analysis)] The manuscript states that the additional supercharges in N=3 and N=4 impose stricter algebraic constraints than in N=2, but the explicit expansion of the Killing spinor variations (or the resulting conditions on the Weyl tensor and auxiliary fields) is not provided in sufficient detail to verify that only the flat solution survives. A concrete example of the constraint equations for at least one non-flat ansatz would strengthen the claim.

    Authors: We agree that additional explicit detail would improve verifiability. In the revised manuscript we will expand Section 3 to display the full set of Killing spinor variations for the N=3 and N=4 theories, derive the resulting algebraic constraints on the Weyl tensor and auxiliary fields, and include a concrete calculation showing that the Bertotti-Robinson ansatz violates at least one of these constraints once the extra supercharges are imposed (while remaining consistent in the N=2 truncation). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from Killing spinor constraints on defined class

full rationale

The paper constructs the class of N=3 and N=4 higher-derivative Poincaré theories by embedding into conformal supergravity with the standard Weyl multiplet and supersymmetrizing the Weyl-squared term plus companions. It then imposes the complete set of Killing spinor equations in the superconformal formalism. For N=3 and N=4 the extra supercharges produce stricter algebraic constraints on curvature and auxiliary fields than in N=2, forcing the only solution with all variations vanishing to be flat space. This step is independent of any fitted parameters, self-definitional loops, or load-bearing self-citations; the contrast with the known Bertotti-Robinson solution in N=2 is attributed to off-shell multiplet differences, which is an external structural fact. The derivation is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions from conformal and Poincaré supergravity; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption The higher derivative terms are related to the Weyl square term by supersymmetry
    This defines the specific class of theories studied.
  • domain assumption Fully supersymmetric solutions preserve all supercharges
    Standard definition used to identify full BPS solutions.

pith-pipeline@v0.9.0 · 5686 in / 1272 out tokens · 153381 ms · 2026-05-22T17:05:36.535045+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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Supersymmetry and Attractors in N = 4 Supergravity

    hep-th 2026-03 unverdicted novelty 4.0

    Numerical confirmation of the attractor mechanism and 1/4 supersymmetry preservation for generic dyonic extremal black holes in pure N=4 supergravity.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · cited by 1 Pith paper · 27 internal anchors

  1. [1]

    Unity of Superstring Dualities

    C.M. Hull and P.K. Townsend,Unity of superstring dualities,Nucl. Phys. B438(1995) 109 [hep-th/9410167]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  2. [2]

    Microscopic Origin of the Bekenstein-Hawking Entropy

    A. Strominger and C. Vafa,Microscopic origin of the Bekenstein-Hawking entropy,Phys. Lett. B379(1996) 99 [hep-th/9601029]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  3. [3]

    Black Hole Entropy, Special Geometry and Strings

    T. Mohaupt,Black hole entropy, special geometry and strings,Fortsch.Phys.49(2001) 3 [hep-th/0007195]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  4. [4]

    Freedman and A

    D.Z. Freedman and A. Van Proeyen,Supergravity, Cambridge Univ. Press, Cambridge, UK (5, 2012), 10.1017/CBO9781139026833

    work page doi:10.1017/cbo9781139026833 2012

  5. [5]

    Tod,All Metrics Admitting Supercovariantly Constant Spinors,Phys

    K.P. Tod,All Metrics Admitting Supercovariantly Constant Spinors,Phys. Lett. B121(1983) 241

    work page 1983

  6. [6]

    Kowalski-Glikman, POSITIVE ENERGY THEOREM AND VACUUM STATES FOR THE EINSTEIN-MAXWELL SYSTEM, Phys

    J. Kowalski-Glikman, POSITIVE ENERGY THEOREM AND VACUUM STATES FOR THE EINSTEIN-MAXWELL SYSTEM, Phys. Lett. B150(1985) 125

    work page 1985

  7. [7]

    Corrections to macroscopic supersymmetric black-hole entropy

    G. Lopes Cardoso, B. de Wit and T. Mohaupt, Corrections to macroscopic supersymmetric black hole entropy,Phys. Lett. B451(1999) 309 [hep-th/9812082]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  8. [8]

    Stationary BPS Solutions in N=2 Supergravity with R^2-Interactions

    G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Stationary BPS solutions in N=2 supergravity with R**2 interactions,JHEP12(2000) 019 [hep-th/0009234]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  9. [9]

    Supersymmetry and Attractors

    S. Ferrara and R. Kallosh,Supersymmetry and attractors,Phys. Rev. D54(1996) 1514 [hep-th/9602136]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  10. [10]

    All supersymmetric solutions of minimal supergravity in five dimensions

    J.P. Gauntlett, J.B. Gutowski, C.M. Hull, S. Pakis and H.S. Reall, All supersymmetric solutions of minimal supergravity in five- dimensions,Class. Quant. Grav. 20(2003) 4587 [hep-th/0209114]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  11. [11]

    All supersymmetric solutions of minimal supergravity in six dimensions

    J.B. Gutowski, D. Martelli and H.S. Reall, All Supersymmetric solutions of minimal supergravity in six- dimensions,Class. Quant. Grav. 20(2003) 5049 [hep-th/0306235]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  12. [12]

    Supergravity vacua and lorentzian Lie groups

    A. Chamseddine, J.M. Figueroa-O’Farrill and W. Sabra, Supergravity vacua and Lorentzian Lie groups,hep-th/0306278

    work page internal anchor Pith review Pith/arXiv arXiv

  13. [13]

    Maximally supersymmetric solutions of ten- and eleven-dimensional supergravities

    J.M. Figueroa-O’Farrill and G. Papadopoulos, Maximally supersymmetric solutions of ten-dimensional and eleven-dimensional supergravities, JHEP03(2003) 048 [hep-th/0211089]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  14. [14]

    Tod,More on supercovariantly constant spinors,Class

    K.P. Tod,More on supercovariantly constant spinors,Class. Quant. Grav.12(1995) 1801. – 23 –

    work page 1995

  15. [15]

    Stationary Axion/Dilaton Solutions and Supersymmetry

    E. Bergshoeff, R. Kallosh and T. Ortin, Stationary axion / dilaton solutions and supersymmetry,Nucl. Phys. B478(1996) 156 [hep-th/9605059]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  16. [16]

    Supergravity Vacua Today

    N. Alonso-Alberca and T. Ortin,Supergravity vacua today, in Spanish Relativity Meeting on Gravitation and Cosmology (ERE 2002), 10, 2002 [gr-qc/0210039]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  17. [17]

    All the supersymmetric configurations of N=4,d=4 supergravity

    J. Bellorin and T. Ortin,All the supersymmetric configurations of N=4, d=4 supergravity, Nucl. Phys. B726(2005) 171 [hep-th/0506056]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  18. [18]

    Chamseddine,N=4 Supergravity Coupled to N=4 Matter,Nucl

    A.H. Chamseddine,N=4 Supergravity Coupled to N=4 Matter,Nucl. Phys. B185(1981) 403

    work page 1981

  19. [19]

    Y. Chen, S. Murthy and G.J. Turiaci,Gravitational index of the heterotic string,JHEP09 (2024) 041 [2402.03297]

    work page arXiv 2024

  20. [20]

    Classification of maximally supersymmetric backgrounds in supergravity theories

    J. Louis and S. Lust, Classification of maximally supersymmetric backgrounds in supergravity theories,JHEP02 (2017) 085 [1607.08249]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  21. [21]

    Dabholkar and J.A

    A. Dabholkar and J.A. Harvey,Nonrenormalization of the Superstring Tension,Phys. Rev. Lett.63(1989) 478

    work page 1989

  22. [22]

    String-Corrected Black Holes

    V. Hubeny, A. Maloney and M. Rangamani,String-corrected black holes,JHEP05(2005) 035 [hep-th/0411272]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  23. [23]

    A Stringy Cloak for a Classical Singularity

    A. Dabholkar, R. Kallosh and A. Maloney,A Stringy cloak for a classical singularity,JHEP12 (2004) 059 [hep-th/0410076]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  24. [24]

    How Does a Fundamental String Stretch its Horizon?

    A. Sen,How does a fundamental string stretch its horizon?,JHEP05(2005) 059 [hep-th/0411255]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  25. [25]

    Fradkin and A.A

    E.S. Fradkin and A.A. Tseytlin,CONFORMAL SUPERGRAVITY,Phys. Rept.119(1985) 233

    work page 1985

  26. [26]

    de Roo, J.W

    M. de Roo, J.W. van Holten, B. de Wit and A. Van Proeyen, Chiral Superfields inN= 2Supergravity, Nucl. Phys. B173(1980) 175

    work page 1980

  27. [27]

    de Wit, R

    B. de Wit, R. Philippe and A. Van Proeyen, The Improved Tensor Multiplet inN= 2Supergravity, Nucl. Phys. B219(1983) 143

    work page 1983

  28. [28]

    N=2 Supergravity Lagrangians with Vector-Tensor Multiplets

    P. Claus, B. de Wit, M. Faux, B. Kleijn, R. Siebelink and P. Termonia, N=2 supergravity Lagrangians with vector tensor multiplets,Nucl. Phys. B512(1998) 148 [hep-th/9710212]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  29. [29]

    Off-shell N=2 tensor supermultiplets

    B. de Wit and F. Saueressig,Off-shell N=2 tensor supermultiplets,JHEP09(2006) 062 [hep-th/0606148]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  30. [30]

    Construction of all N=4 conformal supergravities

    D. Butter, F. Ciceri, B. de Wit and B. Sahoo,Construction of all N=4 conformal supergravities, Phys. Rev. Lett.118(2017) 081602 [1609.09083]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  31. [31]

    Butter, F

    D. Butter, F. Ciceri and B. Sahoo,N= 4conformal supergravity: the complete actions, JHEP 01(2020) 029 [1910.11874]

    work page arXiv 2020

  32. [32]

    Hegde, M

    S. Hegde, M. Mishra, D. Mukherjee and B. Sahoo, Higher derivative invariants in four dimensionalN= 3 Poincaré supergravity,JHEP02 (2023) 145 [2211.06628]. – 24 –

    work page arXiv 2023

  33. [33]

    Hegde and B

    S. Hegde and B. Sahoo, New higher derivative action for tensor multiplet inN= 2 conformal supergravity in four dimensions, JHEP01(2020) 070 [1911.09585]

    work page arXiv 2020

  34. [34]

    Kaku and P.K

    M. Kaku and P.K. Townsend, POINCARE SUPERGRAVITY AS BROKEN SUPERCONFORMAL GRAVITY,Phys. Lett. B76(1978) 54

    work page 1978

  35. [35]

    de Wit, J.W

    B. de Wit, J.W. van Holten and A. Van Proeyen, Transformation Rules of N=2 Supergravity Multiplets,Nucl. Phys. B167(1980) 186

    work page 1980

  36. [36]

    Kaku, P.K

    M. Kaku, P.K. Townsend and P. van Nieuwenhuizen,Properties of Conformal Supergravity, Phys. Rev. D17(1978) 3179

    work page 1978

  37. [37]

    Bergshoeff, M

    E. Bergshoeff, M. de Roo and B. de Wit,Extended Conformal Supergravity,Nucl. Phys. B182 (1981) 173

    work page 1981

  38. [38]

    de Roo,Matter Coupling in N=4 Supergravity,Nucl

    M. de Roo,Matter Coupling in N=4 Supergravity,Nucl. Phys. B255(1985) 515

    work page 1985

  39. [39]

    Towards the full N=4 conformal supergravity action

    F. Ciceri and B. Sahoo,Towards the fullN= 4conformal supergravity action, JHEP01(2016) 059 [1510.04999]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  40. [40]

    Ciceri and B

    F. Ciceri and B. Sahoo,N=4 supergravity higher-derivative invariants,Manuscript under preparation

    work page

  41. [41]

    The $\mathcal{N}=3$ Weyl Multiplet in Four Dimensions

    J. van Muiden and A. Van Proeyen,TheN= 3 Weyl multiplet in four dimensions,JHEP01 (2019) 167 [1702.06442]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  42. [42]

    Comment on "The N = 3 Weyl Multiplet in Four Dimensions"

    S. Hegde and B. Sahoo,Comment on “TheN=3 Weyl multiplet in four dimensions”,Phys. Lett. B791(2019) 92 [1810.05089]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  43. [43]

    Hegde, M

    S. Hegde, M. Mishra and B. Sahoo,N = 3 conformal supergravity in four dimensions,JHEP04 (2022) 001 [2104.07453]

    work page arXiv 2022

  44. [44]

    Aikot and B

    A. Aikot and B. Sahoo, Scalar-Tensor multiplet in four dimensional N=2 conformal supergravity,2412.16527

    work page arXiv

  45. [45]

    Ciceri, A

    F. Ciceri, A. Kleinschmidt, S. Murugesan and B. Sahoo, Torus reduction of maximal conformal supergravity,JHEP12(2024) 151 [2408.06026]

    work page arXiv 2024

  46. [46]

    Adhikari and B

    S. Adhikari and B. Sahoo, SU(2)×SU(2) dilaton Weyl multiplets for maximal conformal supergravity in four, five, and six dimensions, JHEP02(2025) 059 [2411.16322]

    work page arXiv 2025

  47. [47]

    Adhikari, A

    S. Adhikari, A. Aikot, M. Mishra and B. Sahoo, Dilaton Weyl multiplets forN= 3 conformal supergravity in four dimensions,JHEP04(2025) 062 [2412.14874]

    work page arXiv 2025

  48. [48]

    Adhikari, A

    S. Adhikari, A. Aikot, M. Mishra and B. Sahoo, Variant dilaton Weyl multiplet for N=3 conformal supergravity in four dimensions,Phys. Rev. D111(2025) 106024 [2502.13683]

    work page arXiv 2025

  49. [49]

    Heydeman, X

    M. Heydeman, X. Shi and G.J. Turiaci,Can black holes preserveN >4supersymmetry?, 2504.20146

    work page arXiv

  50. [50]

    Black hole partition functions and duality

    G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Black hole partition functions and duality,JHEP03(2006) 074 [hep-th/0601108]. – 25 –

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  51. [51]

    From Special Geometry to Black Hole Partition Functions

    T. Mohaupt,From Special Geometry to Black Hole Partition Functions,Springer Proc. Phys. 134(2010) 165 [0812.4239]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  52. [52]

    EXTREMAL BLACK HOLES AND ELEMENTARY STRING STATES

    A. Sen,Extremal black holes and elementary string states,Mod. Phys. Lett. A10(1995) 2081 [hep-th/9504147]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  53. [53]

    Precision Counting of Small Black Holes

    A. Dabholkar, F. Denef, G.W. Moore and B. Pioline,Precision counting of small black holes, JHEP10(2005) 096 [hep-th/0507014]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  54. [54]

    Y. Chen, J. Maldacena and E. Witten,On the black hole/string transition,JHEP01(2023) 103 [2109.08563]. – 26 –

    work page arXiv 2023