The c-map on groups

Andrew Swann; Oscar Macia

REVIEW 2 major objections 4 minor 22 references

The c-map on groups

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that a simply-connected Kähler Lie group with exact Kähler form is projective special Kähler precisely when two symmetric matrix-valued one-forms satisfy an intrinsic system, and that the c-map of such homogeneous…

desk verdict A solid c-map homogeneity theorem, but the Section 7 example fails the paper's own equations (6.4)-(6.5), signaling a sign error that needs fixing before Proposition 6.2 is reliable. read the letter →

arxiv 1908.01736 v1 pith:EHA3RINS submitted 2019-08-05 math.DG

Oscar Macia , Andrew Swann This is my paper · ORCID

classification math.DG MSC 53C2653C3053C55

keywords projectivespecialKählermanifoldsquaternionicc-mapLiegroupsleft-invariantstructurestwistconstructiongeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Projective special Kähler (PSK) geometry is usually defined through an auxiliary cone, which makes it unclear when a Kähler group manifold should count as a homogeneous example. This paper starts with a left-invariant Kähler structure on a simply-connected Lie group and derives an intrinsic system of equations---two symmetric matrix-valued one-forms $p,q$ satisfying $p=Jq$ and the torsion-free condition---that is equivalent to the existence of a compatible PSK structure. The equivalence uses the assumption that the Kähler form is exact, $\omega_S=d\kappa$, which lets the cone data be rotated into forms that descend to the base. The paper then proves that the c-map of such a homogeneous PSK structure is always a left-invariant quaternionic Kähler metric on a Lie group of dimension $4n+4$. This gives a workable mathematical notion of homogeneity for the c-map and a first step toward understanding which homogeneous quaternionic Kähler metrics arise this way.

What carries the argument

The central device is a rotation of the two matrix-valued one-forms $u,v$ that encode the difference between the special and Levi-Civita connections on the cone. Writing $P=u\cos z+v\sin z$, $Q=-u\sin z+v\cos z$ and setting $z=2\tau$ with $\tau$ the fibre coordinate trivialized by the exactness of $\omega_S$, makes $P,Q$ basic: they descend to forms $p,q$ on $S$. On $S$ the tensor $p$ satisfies $p=Jq$, and the full PSK condition becomes an intrinsic system with no reference to the cone.

What would settle it

Run the intrinsic system on a concrete Kähler Lie group with a flat factor, such as a product $S_0\times\mathbb{R}$ with $S_0$ flat Kähler; Proposition 4.1 says no solution exists, so an explicit solution of (6.1)--(6.5) would refute the paper's classification.

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Extended reading notes

Core claim

On a simply-connected Kähler Lie group $S$ of real dimension $2n$ with exact Kähler form $\omega_S=d\kappa$, a compatible projective special Kähler structure exists if and only if there are matrix-valued one-forms $p,q\in\Omega^1(S,M_n(\mathbb{R}))$ with $p^T=p$, $q^T=q$, $p=Jq$, satisfying the torsion-free condition $p\wedge a+q\wedge b=0=p\wedge b-q\wedge a$ and the four equations (6.2)--(6.5), which express flatness of the special connection in terms of the Lie-group curvature blocks $M,\Lambda$, the one-form $\kappa$, and the one-forms $\mu,\lambda$ of the Levi-Civita connection. The structure is homogeneous exactly when $p,q$ can be chosen left-invariant. Under that choice, the twist construction of the c-map produces a coframe with constant coefficients, so the resulting quaternionic Kähler metric is left-invariant on a Lie group of dimension $4n+4$ (Theorem 8.1).

Load-bearing premise

The load-bearing assumption is that the Kähler form of $S$ is exact, $\omega_S=d\kappa$ with $\kappa$ a left-invariant one-form, together with simple connectivity (or passage to the universal cover); if exactness fails, the flat connection and trivialization used to rotate the cone forms $u,v$ into basic forms, and then into left-invariant $p,q$, need not exist.

Editorial extensions

If this is right

Every homogeneous PSK structure in this sense has a well-defined c-map image that is a homogeneous quaternionic Kähler manifold of dimension $4n+4$, so the c-map does not leave the class of group manifolds.
The intrinsic equations let one verify whether a given left-invariant Kähler metric is projective special Kähler by solving a concrete system on the Lie algebra, without building the special Kähler cone.
A Kähler group whose universal cover is a product of three or more non-flat Kähler factors can be PSK only for $\mathrm{CH}(1)^3$ with all factors of holomorphic sectional curvature $-1$; the two-factor case $\mathrm{CH}(1)^2$ is solved in §7 and yields a quaternionic Kähler image $\mathrm{SO}(3,4)/(\mathrm{SO}(3)\times\mathrm{SO}(4))$.
The de Rham decomposition of any PSK Kähler group covered by the theorem contains no flat Kähler factor.

Reading between the lines

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

If the exactness assumption can be removed through the general intrinsic equations of [21], the rotation argument might carry over to a wider class of homogeneous PSK structures; testing that is a natural next step.
The pair $(p,q)$ with $p=Jq$ is equivalent to a symmetric three-tensor on $S$; one could use such tensors to construct new examples by algebraic geometry or Lie algebra cohomology.
The rotational freedom $R_s(p,q)$ in Remark 6.3 suggests a circle-family of homogeneous PSK data with the same underlying metric; understanding this freedom may correspond to a moduli or gauge symmetry in the c-map image.
Should a homogeneous PSK structure with non-exact Kähler form be found in higher dimensions, the paper's setup predicts its c-map image might not be a Lie group with left-invariant metric; that would sharpen the role of exactness.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper is worth a referee's time, but it has a concrete bug in the worked example that must be fixed before publication, because it bears on the main equivalence, Proposition 6.2.

What is genuinely new: a left-invariant, intrinsic formulation of homogeneous projective special Kähler (PSK) structures on Lie groups; the no-flat-factor result (Prop 4.1); the product classification that yields CH(1)^3 as the only product of three or more factors (Prop 5.1); and the clean theorem that the c-map of an invariant PSK structure is necessarily a left-invariant quaternionic Kähler structure on a Lie group (Thm 8.1). The paper is also honest about the overlap with Mantegazza's independent work [21] and about the exactness assumption ωS=dκ, which is a genuine restriction outside real dimension four.

The soft spot is Section 7. The example is CH(1)×CH(1) with c1=√2, c2=2, and p,q as given. I substituted these into (6.4) and (6.5) with the paper's conclusion 2κ=-√2 b1-2b2. It fails. Moreover, no left-invariant κ with dκ=ωS can satisfy both equations for those p,q,λ. The (1,2) and (2,1) components of (6.5) actually give the same condition, not contradictory ones, but that condition requires c2=4 with the paper's κ, and even then dκ would not be ωS. The stress-test's specific component split is not exactly what happens, but its bottom line is right: the example is not self-consistent. Even the stated conclusion 2κ=-√2 b1-2b2 differentiates to -2ωS, not +2ωS. This points to a sign or matrix-ordering error in the derivation of (6.4)-(6.5) from U=V=0, or in the assignments for the example.

How much does this hurt? Theorem 8.1's proof does not actually use the signs of (6.4)-(6.5); it only uses that left-invariant p,q make η exp(2iτ) constant coefficient. So the c-map theorem is likely safe. But the intrinsic characterization and the example verification are central, and the exactness restriction remains an open limitation in higher dimensions.

Recommendation: send to a competent referee. The paper deserves serious review, not desk rejection. The referee should ask for a corrected derivation of (6.4)-(6.5), a recomputed example, and a paragraph reconciling with Mantegazza [21]. If the equations are fixed, this is a solid addition to the PSK and c-map literature.

Referee Report

2 major / 4 minor

Summary. The paper develops an intrinsic description of homogeneous projective special Kähler (PSK) structures on Lie groups with exact left-invariant Kähler form. Starting from the cone formulation of the c-map, the authors derive matrix equations (6.1)-(6.5) on the group S, define homogeneity via left-invariant one-forms p,q obtained after a rotation, analyze flat factors and products, and prove in Theorem 8.1 that the c-map sends such data to a left-invariant quaternionic Kähler structure on a Lie group of dimension 4n+4. A four-dimensional example, CH(1)xCH(1), is worked out to illustrate the equations, and the resulting quaternionic Kähler spaces are identified with known symmetric examples.

Significance. If the intrinsic system is correct, the paper gives a concrete algebraic tool for constructing homogeneous PSK examples and shows geometrically why the c-map preserves the left-invariant/group ansatz. It connects the supergravity c-map literature with the twist geometry of [20] and complements the concurrent work [21] by Mantegazza. The derivation is largely self-contained and the Theorem 8.1 formulation is attractive. However, the main intrinsic system (6.4)-(6.5) contains a sign error that invalidates the equations exactly as printed and makes the CH(1)xCH(1) example inconsistent; this must be fixed before the paper can be accepted.

major comments (2)

[§6, Eqs. (6.4)-(6.5), and §7] The CH(1)xCH(1) data in Section 7 do not solve the intrinsic system as printed. With mu=0, lambda=diag(-c1 b1, -c2 b2), p=(b2,b1;b1,0), q=(a2,a1;a1,0), dq=0, equation (6.5) reduces to p^lambda - lambda^p + 4 kappa^p = 0. Its (1,2) component is -c2 b1^b2 + 4 kappa^b1 = 0, so for c2=2 one needs 4 kappa^b1 = 2 b1^b2, whereas the displayed kappa=-(sqrt(2)/2)b1-b2 gives 4 kappa^b1 = 4 b1^b2. In fact, solving (6.4)-(6.5) as printed forces kappa=-(sqrt(2)/2)b1-(1/2)b2, and differentiating that kappa gives -omega_S, not omega_S. The last sentence of Section 7, claiming that the displayed 2 kappa = -sqrt(2)b1-2b2 satisfies d kappa = omega_S, is therefore false in two separate ways. The signs of the kappa terms in (6.4)-(6.5) appear to be reversed: replacing -4 kappa^q by +4 kappa^q and +4 kappa^p by -4 kappa^p makes the example consistent with 2 kappa = sqrt(2)b1+b2 and d kappa = omega_S. Since (6.4)-(6.5) are part of the equivalence in Proposition 6.2, this sign error is load-bearing and must be corrected.
[§6, Definition 6.1 and Proposition 6.2] The paper's intrinsic characterization is conditional on the exactness assumption omega_S = d kappa with kappa a left-invariant one-form. This assumption is what allows the flat connection on the circle bundle and the descent of P,Q to left-invariant p,q on S. The sentence 'all group manifold examples of projective special Kähler structures in the literature satisfy this definition' is asserted without proof or citation, and the paper itself notes that [21] treats the non-exact case. The authors should either prove this claim for the known examples or explicitly restrict Definition 6.1, Proposition 6.2, and Theorem 8.1 to the exact case with the scope stated as a limitation.

minor comments (4)

[§1, p.2] The phrase 'We find a certain of integrability conditions' should read 'We find a certain set of integrability conditions'.
[§7, p.14] The sentence 'As q+ and p+=Jq+ are span T*S1' is ungrammatical and should be rephrased, for example as 'Since q+ and p+=Jq+ span T*S1'.
[§8, p.15] 'a principle parameter' should be 'a principal parameter'.
[References [17], [18]] The text that the correctness of results in [18] is not clear is an unexplained caveat; it should either be expanded into a precise mathematical statement or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the intrinsic equations are derived from the standard cone formulation, not from the conclusions they are used to prove.

full rationale

The paper's main derivation starts from the standard projective special Kähler cone conditions (2.12)-(2.15), already summarized from [20], and obtains the intrinsic system (6.1)-(6.5) by an explicit rotation P = u cos z + v sin z, Q = -u sin z + v cos z. The relation p = Jq is inherited from u = Jv, and the torsion-free and curvature equations are rewritten, not assumed. Definition 6.1 defines 'homogeneous' as the choice of left-invariant p,q after the rotation; Proposition 6.2's 'if and only if p,q can be chosen left-invariant' is a restatement of that definition, not a derived prediction. Theorem 8.1 depends on the c-map twist theorem of [20], which is prior published work by the same authors; that citation is load-bearing for the twist being quaternionic Kähler, but the cited theorem is a general result proven there with assumptions that do not include the present homogeneity equations, so invoking it is normal mathematical dependency rather than circularity. The exactness assumption omega_S = d kappa is explicitly introduced and acknowledged as a limitation (with [21] treating the non-exact case); it restricts the scope but does not smuggle the conclusion into the input. Any apparent inconsistency in the Section 7 example (e.g., the (1,2) component of (6.4)-(6.5) forcing c2 = 0) is a potential correctness or typographical error, not an instance of a claimed result reducing to its own input by construction. The paper contains no fitted parameter renamed as a prediction and no uniqueness theorem invoked from the authors' prior work to forbid alternatives.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim is a differential-geometric theorem with no data fitting and no new physical entities. It relies on the Hodge condition, exactness of the Kähler form, simple-connectedness, and the prior twist construction from [20]. No free parameters are fitted to data; the constants c_i and r_i in the examples are determined by the geometric equations.

assumptions (5)

domain assumption S is a Lie group with a left-invariant Kähler structure, and (S,2ωS) is Hodge, meaning there is a circle bundle C0 -> S with connection φ satisfying dφ = 2π*ωS.
This is the starting setup in Section 2. It is standard for projective special Kähler manifolds, which are usually defined via a cone over a Hodge manifold.
domain assumption The Kähler form on S is exact, ωS = dκ for a left-invariant one-form κ.
Introduced in Section 6 before Definition 6.1. It is load-bearing because it makes the circle-bundle connection flat after a shift, enabling the descent from cone forms u,v to left-invariant forms p,q on S. The authors explicitly note that Mantegazza [21] treats the non-exact case, so this is a real restriction.
domain assumption The group S is simply-connected, or one passes to the universal cover.
Proposition 6.2 states the intrinsic characterization for simply-connected Kähler groups. Simple-connectedness is used to trivialize the flat circle bundle over the universal cover in Section 6.
domain assumption The twist construction of [20] sends a special Kähler cone with conic symmetry to a quaternionic Kähler manifold.
Theorem 8.1 relies directly on the general c-map and twist results from the authors' prior paper [20]. This is a nontrivial cited theorem rather than re-derived here.
standard math The universal cover of a Kähler Lie group admits a de Rham decomposition into Kähler factors.
Used in Sections 4 and 5 to analyze flat factors and product structures. The paper notes in Section 4 that the correctness of some related decomposition statements in [17,18] is not clear, though the proofs here appear to use only the standard de Rham decomposition.

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Pith. "Pith review of The c-map on groups." pith.science (2026). https://pith.science/paper/EHA3RINS

@misc{pith2026190801736,
  author       = {Pith},
  title        = {Pith review of: The c-map on groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHA3RINS}},
  note         = {Machine review of arXiv:1908.01736}
}

read the original abstract

We study the projective special Kaehler condition on groups, providing an intrinsic definition of homogeneous projective special Kaehler that includes the previously known examples. We give intrinsic defining equations that may be used without resorting to computations in the special cone, and emphasise certain associated integrability equations. The definition is shown to have the property that the image of such structures under the c-map is necessarily a left-invariant quaternionic Kaehler structure on a Lie group.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [21]

Mantegazza, Construction of projective special Kähler manifolds , preprint uploaded to arXiv 4th August, 2019

M. Mantegazza, Construction of projective special Kähler manifolds , preprint uploaded to arXiv 4th August, 2019

work page 2019

[2] [20]

Maciá and A

Ó. Maciá and A. F. Swann, Twist geometry of the c-map , Comm. Math. Phys. 336 (2015), no. 3, 1329–1357

work page 2015

[3] [1]

D. V. Alekseevsky, Classiﬁcation of quaternionic spaces with transitive solva ble group of motions , Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 315–362

work page 1975

[4] [2]

D. V. Alekseevsky, V. Cortés, and T. Mohaupt, Coniﬁcation of Kähler and hyper-Kähler manifolds, Comm. Math. Phys. 324 (2013), 637–655

work page 2013

[5] [3]

Alvarez-Gaumé and D

L. Alvarez-Gaumé and D. Z. Freedman, Geometrical structure and ultravi- olet ﬁniteness in the supersymmetric σ-model, Comm. Math. Phys. 80 (1981), no. 3, 443–451

work page 1981

[6] [4]

Bagger and E

J. Bagger and E. Witten, Matter couplings in N = 2 supergravity, Nucl. Phys. B 222 (1983), 1–10

work page 1983

[7] [5]

Castellani, R

L. Castellani, R. D’Auria, and S. Ferrara, Special geometry without special coordinates, Classical Quantum Gravity 7 (1990), no. 10, 1767–1790

work page 1990

[8] [6]

Cecotti, S

S. Cecotti, S. Ferrara, and L. Girardello, Geometry of type II superstrings and the moduli of superconformal ﬁeld theories , Internat. J. Modern Phys. A 4 (1989), no. 10, 2475–2529

work page 1989