{"id":"8b60ec4e-eb71-43e7-9a14-2369bb5bb749","arxiv_id":"1908.01736","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Kähler groups with exact Kähler form, homogeneous projective special Kähler structures are equivalent to left-invariant matrix one-forms satisfying explicit algebraic-differential equations, and their c-map images are always homogeneous quaternionic Kähler groups.","lead":"This paper gives a direct way to describe projective special Kähler geometry on Lie groups, and proves that the c-map construction always turns such homogeneous structures into quaternionic Kähler groups. It also determines which product groups can carry these structures, including a full analysis of the four-dimensional example.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Section 7 example appears not to satisfy the paper's own equations (6.4)-(6.5); the (1,2) component forces c2=0, so the intrinsic characterization is not self-consistent as written.","rationale":"I read the paper in good faith: the cone construction and the idea of rotating u,v to obtain basic P,Q are plausible, and the c-map homogeneity theorem would be a nice structural result. The exactness assumption is explicitly part of the hypotheses and is acknowledged as a limitation, so by itself it does not invalidate the conditional claims. However, the internal consistency of the main equations is the load-bearing point, and the paper's own worked example appears to contradict them. My computation of the (1,2) and (2,1) components of (6.4)-(6.5) is elementary; unless I have misread the convention for lambda or the matrix entries, no kappa can solve both equations for the stated p,q. This is a stronger concern than the exactness issue, because it attacks Proposition 6.2 directly. The reader did flag a concrete sign error in the Section 7 example but selected exactness as the weakest assumption; I partly agree with the reader's overall conditional verdict but would put the sign error at the centre. A corrected sign convention may well restore the theorems, so I do not recommend rejection; I recommend that the paper be accepted only after the intrinsic equations are re-derived and the example is recomputed.","tokens_in":13323,"tokens_out":30580,"duration_ms":272358,"concrete_test":"Re-derive (6.4)-(6.5) symbolically from U=V=0 using u=P cos z - Q sin z, v=P sin z + Q cos z, z=2 tau, phi=d tau + 2 kappa, tracking matrix multiplication and exterior signs, and then substitute the Section 7 ansatz into the corrected equations. If an uncancelled c2 a2^b2 term persists in any component, the equations as stated cannot admit the claimed solution.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim rests on the intrinsic system (6.1)-(6.5). In Section 7 this system is applied to CH(1)xCH(1), with mu=0, lambda=diag(-c1 b1, -c2 b2), p=(b2,b1;b1,0), q=(a2,a1;a1,0), db1=c1 a1^b1, db2=c2 a2^b2, and c1=sqrt(2), c2=2. Substituting into (6.5), the (1,2) component is 4 kappa ^ b1 = 0, while the (2,1) component is -c2 b1^b2 + 4 kappa ^ b1 = 0; together these force c2=0, contradicting c2=2. Likewise, the (1,2) component of (6.4) contains an uncancelled c2 a2^b2 term, since no term involving kappa can produce a2^b2. The paper's stated conclusion 2 kappa = -sqrt(2) b1 - 2 b2 does not satisfy either equation. This is not a typo in one number: it indicates 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. Because Proposition 6.2 is exactly the equivalence used in Theorem 8.1, the central claim is not fully established as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13610,"tokens_out":22863,"duration_ms":193965,"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":[{"comment":"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.","section":"§6, Eqs. (6.4)-(6.5), and §7"},{"comment":"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.","section":"§6, Definition 6.1 and Proposition 6.2"}],"minor_comments":[{"comment":"The phrase 'We find a certain of integrability conditions' should read 'We find a certain set of integrability conditions'.","section":"§1, p.2"},{"comment":"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'.","section":"§7, p.14"},{"comment":"'a principle parameter' should be 'a principal parameter'.","section":"§8, p.15"},{"comment":"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.","section":"References [17], [18]"}],"recommendation":"major_revision","confidential_remarks":"The sign error in (6.4)-(6.5) is serious but appears repairable; the corrected signs make the Section 7 example work. The paper is within the scope of the journal and the underlying c-map construction is valuable, so I recommend major revision rather than rejection. The claim that all literature examples satisfy Definition 6.1 should be checked carefully against [14] and [21] before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe 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.\n\nWhat 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.\n\nThe 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.\n\nHow 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":14149,"tokens_out":20670,"would_cite":true,"duration_ms":161804,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C26","53C30","53C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["projective special Kähler manifolds","quaternionic Kähler manifolds","c-map","Lie groups","left-invariant structures","twist construction","Kähler groups","special geometry"],"falsifier":"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.","tokens_in":13095,"feed_emoji":"🌀","tokens_out":15173,"duration_ms":126111,"temperature":0.7,"pith_summary":"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.","feed_headline":"C-map images of homogeneous special Kähler groups are Lie groups","feed_subtitle":"Left-invariant equations on the group decide the projective special Kähler condition, and the c-map stays homogeneous.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the twist construction for the c-map and the conic special Kähler conditions that the paper specializes to invariant data.","marker":"[20]"},{"why":"Defines projective special Kähler manifolds through the auxiliary cone, the framework the paper's intrinsic equations replace.","marker":"[16]"},{"why":"Provides the general non-exact intrinsic equations for projective special Kähler, and the four-dimensional homogeneous classification that matches the two cases solved here.","marker":"[21]"},{"why":"Connects the c-map to homogeneous quaternionic Kähler spaces and supplies the corrected classification context used for examples.","marker":"[13]"},{"why":"Introduces the local c-map from projective special Kähler to quaternionic Kähler manifolds, the map whose behavior on groups is the paper's target.","marker":"[15]"}],"fun_headline_variants":["C-map sends homogeneous special Kähler groups to Lie groups","Homogeneous special Kähler groups stay homogeneous under c-map","C-map preserves left-invariant special Kähler structures on Lie groups","Intrinsic equations characterize homogeneous special Kähler groups","C-map yields Lie groups from homogeneous special Kähler groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["C-map sends homogeneous special Kähler groups to Lie groups","Homogeneous special Kähler groups stay homogeneous under c-map","C-map preserves left-invariant special Kähler structures on Lie groups","Intrinsic equations characterize homogeneous special Kähler groups","C-map yields Lie groups from homogeneous special Kähler groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000454,"raw_usage":{"total_tokens":2228,"prompt_tokens":837,"completion_tokens":1391,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":453,"completion_tokens_details":{"reasoning_tokens":1305}},"tokens_in":453,"tokens_out":1391,"duration_ms":10198,"temperature":1.0,"reasoning_tokens":1305,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:04:44.527465+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Maciá and A","cited_arxiv_id":null,"evidence_quote":"Supplies the twist construction for the c-map and the conic special Kähler conditions that the paper specializes to invariant data."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines projective special Kähler manifolds through the auxiliary cone, the framework the paper's intrinsic equations replace."},{"cited_title":"Mantegazza, Construction of projective special Kähler manifolds , preprint uploaded to arXiv 4th August, 2019","cited_arxiv_id":null,"evidence_quote":"Provides the general non-exact intrinsic equations for projective special Kähler, and the four-dimensional homogeneous classification that matches the two cases solved here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Connects the c-map to homogeneous quaternionic Kähler spaces and supplies the corrected classification context used for examples."},{"cited_title":"Ferrara and S","cited_arxiv_id":null,"evidence_quote":"Introduces the local c-map from projective special Kähler to quaternionic Kähler manifolds, the map whose behavior on groups is the paper's target."}],"review_version":1}