{"id":"2ac38475-5656-405f-b23a-bdc25d632788","arxiv_id":"2501.09474","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Holomorphic, torsion-parallel vector fields on KT manifolds induce Killing and holomorphic vector fields on Hermitian-Einstein moduli spaces, which become toric or QKT fibrations under extra closure conditions.","lead":"The paper derives conditions under which symmetries of Kähler-with-torsion manifolds lift to symmetries of Hermitian-Einstein and instanton moduli spaces, and models those moduli spaces as torus or S3×S1 fibrations. It provides a general framework that subsumes and extends Witten's S3×S1 instanton construction used in AdS3/CFT2 dualities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"QKT principal-bundle claim in §6.5 is not derived from first principles: the curvature G is inferred (footnote 47) and freeness of the SU(2)×U(1) action is asserted, not proved.","rationale":"The main vector-field theorems in Sections 3–4 are internally coherent: the key lemma (3.20) follows from (3.25) and (2.61) with the usual interior-product convention, the Killing/holomorphy computations are consistent, and the examples are worked out in detail. The analytic regularity assumption (invertibility of O, Gauduchon metric, irreducible connections) is imported from Lübke–Teleman and is standard; it is a genuine condition but not the most likely point of failure. The QKT claim, by contrast, is a headline result and is explicitly built from an inferred curvature rather than a derived one. The reader's stated weakest assumption was invertibility, but the reader's rationale already identified the QKT inference as the reason for CONDITIONAL, so my concern partially agrees with the reader. Since the concern supports the existing CONDITIONAL verdict rather than moving it, no verdict adjustment is made. If the proposed computation of G were carried out and (6.54) verified, the QKT claim would be substantially strengthened.","tokens_in":67629,"tokens_out":20709,"duration_ms":207867,"concrete_test":"At a smooth point of M*_asd(S3×S1) with a fixed nonzero instanton number, compute the curvature of λ directly: use the moduli-space metric (2.51) and the explicit formula (6.37) for dF_{αLr} to obtain dλ^r + ½ε^r_st λ^s∧λ^t, then project onto the horizontal distribution and verify whether the Sp(1) component satisfies (6.54), (G^r_sp(1))_ab = ½δ^rs ω^s_ab. Simultaneously check that each α_Lr is nowhere vanishing (solve α_Lr = 0 on the smooth part) to confirm the action is free. If either check fails at a generic point, the QKT modelling claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 6.5 models M*_asd(S3×S1) as a principal SU(2)×U(1) bundle over a QKT base. The load-bearing step is the identification of the curvature G of the connection λ: equation (6.54) sets (G^r_sp(1))_ab = ½δ^rs ω^s_ab. This is not computed from the moduli-space metric or from the definition of λ in (6.46); footnote 47 explicitly states that G has not been obtained from a first-principles calculation and is instead inferred from representation theory and the HKT structure of M*_asd. If (6.54) is incorrect, the base need not be QKT and the headline structural claim fails. The same section also requires the SU(2)×U(1) action to be free, but the non-vanishing proof in §3.3.3 ends with the assertion that A*_HE 'is expected' to contain non-invariant connections, which is not a proof; moreover §6.5.2 concedes that the full so(4) action has fixed points, so global principal-bundle structure is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper investigates the geometry that symmetries of the underlying manifold induce on moduli spaces of Hermitian-Einstein connections M*_HE(M^{2n}) over compact KT manifolds and on instanton moduli spaces M*_asd(M^4) over KT, bi-KT, HKT, and bi-HKT four-manifolds. Section 2 reproduces and reorganizes the Lübke-Teleman construction of a strong KT structure on M*_HE, including the gauge-fixing operator O of (2.44), the connection curvature Θ of (2.61), and the torsion H of (2.63). Section 3 proves the central structural results: a holomorphic ∇̂-covariantly constant vector field X on M^{2n} induces a vector field α_X on M*_HE that is Killing and holomorphic (equations (3.34) and (3.35)), and if X^♭∧θ is a (1,1)-form, α_X is D̂-covariantly constant (equations (3.26)-(3.31)); the key technical input is the lemma Θ(a^h, a^h_X) = ι_X a^h of (3.20). Section 4 applies these results to M*_HE(S3×S3) and M*_HE(S3×T3), giving holomorphic T²-fibration models with curvature (F1,F1) and (F1,0), respectively. Section 5 adapts the analysis to instantons over KT and bi-KT four-manifolds, re-deriving Hitchin's argument that the metric and torsion of M*_asd do not depend on the choice of KT structure. Section 6 treats HKT and bi-HKT manifolds, computes the so(4)⊕so(2) symmetry action on M*_asd(S3×S1), and models the moduli space as S1×P with P an SU(2)-bundle over a QKT base (equations (6.51)-(6.55)); Section 6.6 covers S3/Zn × S1 and squashed metrics.","tokens_in":67875,"tokens_out":59411,"duration_ms":516996,"significance":"If the results hold, the paper is a substantial and systematic contribution to the geometry of gauge-theory moduli spaces. Its most solid part is Section 3: the lifting theorem for holomorphic ∇̂-covariantly constant vector fields, the Killing/holomorphic/parallel properties of the lifted fields, and the key lemma (3.20) are derived in detail and are internally coherent, and the paper re-derives the Lübke-Teleman, Hitchin, and Moraru-Verbitsky structures in a uniform notation with many helpful intermediate steps. Explicit credit is given to prior work, and the paper is honest about its limitations, notably footnote 47 on the QKT curvature and the explicitly conditional 'is expected' statement in §3.3.3. The derived structural predictions — the curvature identity F = (F1,F1) for M*_HE(S3×S3), the vanishing dF_{V0}=0 for M*_asd(S3×S1), and the so(4)⊕so(2) action on the bi-HKT structure — are concrete and checkable. The QKT modeling of M*_asd(S3×S1) is an interesting structural conjecture whose value depends on completing the consistency computation behind (6.54).","major_comments":[{"comment":"The modeling of M*_asd(S3×S1) as a principal SU(2)×U(1)-bundle over a QKT base uses (6.54) — the identification (G^r_sp(1))_ab = ½δ^{rs}ω^s_ab of the sp(1) component of the curvature of λ — as an essential input in (6.55). Footnote 47 concedes that G has not been obtained from a first-principles computation on the moduli space and is instead fixed by 'the restrictions imposed on G by the HKT structure.' The argument that determines the coefficient — comparing (6.50) with the second line of (6.16), decomposing G^r = G^r_sp(k−1) ⊕ G^r_sp(1), and solving the ansatz (6.53) for A^{rs} — is summarized in three sentences. The reader is not shown the quaternionic linear algebra that fixes A^{rs} = ½δ^{rs}, nor the verification that the sp(k−1) component indeed drops out of (6.16). Since (6.54) is the load-bearing step for the claim that the base is QKT (rather than some more general almost-quaternionic structure), the derivation should either be written out or the statement explicitly marked as conditional on (6.54).","section":"§6.5.1, eq. (6.54), footnote 47"},{"comment":"The freeness of the k-action on M*_HE, on which the principal-bundle models of Sections 4 and 6 depend, is not proved. The contradiction argument ends with 'It is expected that A*_HE contains non-invariant connections under the action of k', which is an unproven assertion; footnote 30 sketches a rescue for ASD instantons (ι_X F = 0 plus anti-self-duality implies F = 0) but does not justify the claim, and does not cover the general Hermitian-Einstein cases of Section 4. In addition, the inference 'the action... has no fixed points. Therefore, it is free' conflates local freeness with freeness: even if every non-zero infinitesimal generator is nowhere vanishing, finite stabilizers are not excluded, and a finite stabilizer would prevent M*_HE → M*_HE/K from being a genuine principal bundle. Section 6.5.2 itself concedes that the full so(4) action has fixed points, so the principal-bundle model rests precisely on the individual left and right su(2)⊕u(1) actions being free; that is what the incomplete argument must establish. For the instanton case with non-trivial instanton number the missing non-vanishing argument can be completed (for an ASD connection, ι_X F = 0 with X nowhere vanishing forces F = 0, since a non-zero ASD 2-form on a 4-manifold is non-degenerate), so this gap is repairable, but as written the argument is incomplete at a load-bearing point.","section":"§3.3.3 and footnote 30"},{"comment":"Equation (6.38) is inconsistent with the formula (6.24) it is said to follow from, and with equation (6.39). From (6.24), L_{α_X}Ω̂s(α1,α2) = −∫ L_Xω̂s ∧ ⟨a1^h∧a2^h⟩, using (6.26), (6.27), and (6.31), a direct computation gives L_{L1}ω̂2 = −ω̂3 (via Cartan's formula with dL1 = L2∧L3, dL2 = L3∧L1, dL3 = L1∧L2), hence L_{α_{L1}}Ω̂2 = +Ω̂3, whereas (6.38) states −Ω̂3. Moreover (6.38) and (6.39) cannot both hold under the paper's convention ω(X,Y) = g(X,IY) of §2.1.1: L_{α_{L1}}Ω̂2 = −Ω̂3 would imply L_{α_{L1}}Î2 = −Î3, contradicting (6.39). The downstream equations (6.39), (6.43), (6.50), and (6.54) appear consistent with the corrected sign, so the error is localized; nevertheless (6.38) as displayed is wrong, and the stated agreement with Witten [12] should be re-verified with the corrected sign.","section":"§6.3.2, eq. (6.38)"}],"minor_comments":[{"comment":"The sentence 'Viewing S3×S3 as the group manifold SU(3)×SU(3)' should read SU(2)×SU(2); the same typo appears in the surrounding discussion of the left-invariant vector fields.","section":"§4.1.1"},{"comment":"In the summary sentence 'for K = SU(2)×U(1) or S(3)×SO(2)', the symbol S(3) is a typo for SO(3).","section":"§6.5.1"},{"comment":"The characterization of 'X^♭∧θ is (1,1)' via X^♭∧θ = ι_IX^♭∧ι_Iθ is stated without comment; a one-sentence explanation that this is the (1,1)-condition would help, as the mixed vector/1-form notation (X∧θ^♭)_{ij} in (3.26)-(3.30) is otherwise hard to track.","section":"eq. (3.32) and (3.26)-(3.30)"},{"comment":"The symbol Ω is used both for the Hermitian form of the moduli space and for the frame connection of ∇̂ in (6.10)-(6.16); although the paper notes the clash, a distinct symbol for the frame connection would improve readability.","section":"§6.1.2 and §6.4"},{"comment":"The title line in the full text ('instant on connection moduli spaces') contains a spacing typo that should be corrected in the final version.","section":"Title line"}],"recommendation":"major_revision","confidential_remarks":"The main theorems of Sections 2-3 appear sound to me, and the examples in Sections 4-5 are, modulo the freeness gap in §3.3.3, well supported. The recommendation of major revision is driven by three localized but load-bearing issues: the inferred curvature (6.54) behind the QKT modeling, the incomplete freeness/non-vanishing argument, and the sign inconsistency in (6.38). All three are repairable within the manuscript's scope. On framing: a large fraction of the cited technical literature is the author's own previous work ([13], [26], [28], [35], [36], [37], [46], [74], [75]); the editor may wish to have the incremental novelty relative to [36] (which also treats ∇̂-covariantly constant vector fields and HKT geometry) and to [12] double-checked. Given the length of the component computations, a symbolic-verification appendix or companion file would materially increase confidence in the sign conventions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is worth your time if you work on KT/HKT geometry or instanton moduli spaces. The central lifting theorem in Section 3 is genuinely new: a holomorphic hat-nabla-parallel vector field on the base induces a Killing, holomorphic vector field on the moduli space, and the extra condition X^flat ∧ θ being (1,1) upgrades it to hat-D-parallel. The proof is detailed and largely self-contained, and it generalizes Witten's S3×S1 analysis to a much broader setting. The toric principal-bundle examples for S3×S3 and S3×T3 are concrete and useful.\n\nThe paper is weaker where it reaches for global structure. In Section 3.3.3 the proof that the induced action is free depends on the assertion that A*_HE contains non-invariant connections, which the author himself hedges as \"expected.\" For instantons with non-zero instanton number the vanishing argument goes through, but for general Hermitian-Einstein moduli spaces that is a gap. The bigger soft spot is the QKT modeling in Section 6.5. Equation (6.54), fixing the sp(1) part of the curvature, is not computed from the moduli-space metric; footnote 47 admits it is inferred from representation theory and consistency. That makes the QKT claim a well-motivated conjecture, not a theorem. The section also asserts freeness of the SU(2)×U(1) action rather than proving it; the earlier non-vanishing argument helps but is not fully carried out there.\n\nNone of this destroys the central lifting theorem. The analytic input from Lübke–Teleman is standard and the dependence is stated clearly. The citation pattern looks honest; self-citations are to relevant prior work, not to the main claim.\n\nWho is this for? People working on KT/HKT geometry, instanton moduli spaces, and AdS3/CFT2. It deserves a referee. Send it out, but require the author to either prove the non-vanishing lemma or state it as an assumption, and to either derive (6.54) or explicitly label the QKT section as a conjecture. As is, I would file it under \"promising, needs revision.\"","headline":"A serious, mostly solid generalization of symmetry lifting on Hermitian-Einstein and instanton moduli spaces, with the QKT principal-bundle claim in Section 6.5 the main soft spot.","tokens_in":664,"tokens_out":822,"would_cite":true,"duration_ms":45932,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C07","53C26","53C55","58D27"],"pacs":[],"model":"deepseek-v4-flash","headline":"Holomorphic parallel vector fields on KT manifolds lift to symmetries of their Hermitian-Einstein and instanton moduli spaces.","keywords":["Hermitian-Einstein connections","instanton moduli spaces","Kähler with torsion (KT) manifolds","HKT manifolds","covariantly constant vector fields","toric principal bundles","quaternionic-Kähler with torsion (QKT)","superconformal sigma models"],"falsifier":"One concrete test is to evaluate the curvature identity $\\Theta(a^h,a^h_X)=\\iota_X a^h$ on an explicit Hermitian-Einstein connection over $S^3\\times S^1$ or $S^3\\times S^3$; if it fails for any horizontal tangent vector $a^h$, the induced-field construction collapses.","tokens_in":67389,"feed_emoji":"📐","tokens_out":10848,"duration_ms":97605,"temperature":0.7,"pith_summary":"The paper tries to show that the symmetries of a geometry with torsion are inherited, in a precise sense, by the moduli space of Hermitian-Einstein connections built over it. Its central claim is that every holomorphic and $\\hat{\\nabla}$-covariantly constant vector field $X$ on a Kähler-with-torsion (KT) manifold $M^{2n}$ induces a vector field $\\alpha_X$ on the moduli space $\\mathscr{M}^*_{\\mathrm{HE}}(M^{2n})$ that is both Killing and holomorphic, preserving the metric, the Hermitian form, and the torsion. Under the extra condition that $X^\\flat\\wedge\\theta$ is a $(1,1)$-form, $\\alpha_X$ is covariantly constant for the torsion connection $\\hat{D}$ of the moduli space. These induced fields organise the moduli space, when orbits close, as a holomorphic toric principal bundle over a KT base, and they similarly explain the geometry of instanton moduli spaces, including a description of $\\mathscr{M}^*_{\\mathrm{asd}}(S^3\\times S^1)$ as a principal bundle with fibre $S^3\\times S^1$ over a quaternionic-Kähler-with-torsion (QKT) manifold. If correct, the results give a systematic way to construct new strong KT, bi-KT, HKT and QKT manifolds from gauge-theoretic moduli spaces and to understand the superconformal symmetries of two-dimensional $\\sigma$ models with such targets.","feed_headline":"KT symmetries survive on instanton moduli spaces","feed_subtitle":"Symmetries of torsion geometries persist on Hermitian-Einstein and instanton moduli spaces.","key_machinery":"The load-bearing mechanism is the horizontal lift of an induced vector field: for a holomorphic $\\hat{\\nabla}$-covariantly constant field $X$, the tangent vector $a^h_X=\\iota_X F$ on the space of connections is both tangent to the Hermitian-Einstein submanifold and horizontal for the gauge-fixing connection. The central identity is the curvature contraction $\\Theta(a^h,a^h_X)=\\iota_X a^h$, where $\\Theta$ is the curvature of the principal bundle of connections over the moduli space; it converts Lie derivatives on the moduli space into Lie derivatives of $X$ on the base, which is why $\\alpha_X$ inherits the Killing and parallel properties of $X$. This identity in turn rests on invertibility of the gauge-fixing operator $O=D^i_A D^A_i+\\theta^i D^A_i$, guaranteed for irreducible connections by a Gauduchon metric on the compact underlying manifold.","core_discovery":"On the smooth part of the moduli space of irreducible Hermitian-Einstein connections over a compact KT manifold, the paper establishes that a holomorphic and $\\hat{\\nabla}$-covariantly constant vector field $X$ on $M^{2n}$ lifts to a vector field $\\alpha_X$ whose horizontal representative is $\\iota_X F$. The lift is tangent and horizontal because of the Bianchi identity for the curvature and the Hermitian-Einstein equations, and the key identity $\\Theta(a^h,a^h_X)=\\iota_X a^h$ then implies that $\\alpha_X$ is Killing and holomorphic. If $X^\\flat\\wedge\\theta$ is a $(1,1)$-form, the same identity yields $d\\alpha_X^\\flat=\\iota_{\\alpha_X} H$, so $\\alpha_X$ is parallel with respect to the torsion connection $\\hat{D}$. The paper further shows that the existence of one such field forces a second, $Y=-IX$, and when orbits close the moduli space is locally a holomorphic principal $T^2$ fibration over a KT base, with the metric, Hermitian form and torsion decomposing accordingly. For instantons, the same mechanism models $\\mathscr{M}^*_{\\mathrm{asd}}(S^3\\times S^1)$ and its quotients as principal bundles with fibre $S^3\\times S^1$ over a QKT base, up to a discrete identification.","pith_inferences":["The mechanism suggests a general transfer principle: a suitably parallel Killing field on any geometric structure with torsion should lift to a parallel field on a moduli space of connections, provided the gauge-fixing operator is invertible; this may extend to other gauge-theoretic moduli spaces such as Higgs bundles or $G_2$-instanton moduli.","The condition $X^\\flat\\wedge\\theta\\in\\Lambda^{1,1}$ looks like a moment-map-type compatibility between the vector field and the Lee form; if made precise, the induced $\\hat{D}$-parallel fields would be the Hamiltonian generators of a torus action on the moduli space with respect to the Hermitian form $\\Omega$.","The QKT description of $\\mathscr{M}^*_{\\mathrm{asd}}(S^3\\times S^1)$ gives a concrete construction route for compact QKT manifolds: any Hermitian-Einstein bundle over $S^3\\times S^1$ should yield a QKT base, so new examples could be obtained by varying the rank and instanton number and computing the corresponding base metrics.","For $S^3\\times T^3$, the model predicts that the moduli space decomposes as $S^1$ times a circle bundle only when the vector field $V_3$ is used; the additional fields $V_1,V_2$ are Killing and holomorphic but not $\\hat{D}$-parallel, so a testable distinction is whether their associated flows close on the moduli space."],"forward_implications":["Every holomorphic and $\\hat{\\nabla}$-covariantly constant vector field on a KT manifold gives a Killing holomorphic vector field on the Hermitian-Einstein moduli space, so the moduli space inherits the torsion-parallel isometries of the base.","With closed orbits, the moduli space is locally a holomorphic principal $T^2$ fibration over a KT base; for $S^3\\times S^3$ the two curvature components are equal, while for $S^3\\times T^3$ one component vanishes and the moduli space is locally $S^1$ times a circle bundle.","The instanton moduli spaces $\\mathscr{M}^*_{\\mathrm{asd}}(S^3\\times S^1)$ and $\\mathscr{M}^*_{\\mathrm{asd}}(RP^3\\times S^1)$ are locally $S^1$ times a principal bundle with fibre $S^3$ or $RP^3$ over a QKT base, with the $u(1)$ component of the curvature zero.","The sigma model with target $\\mathscr{M}^*_{\\mathrm{asd}}(S^3\\times S^1)$ carries two copies of the large $N=4$ superconformal algebra, with the currents constructed from the induced vector fields and complex structures; this reproduces the known symmetry content required by the AdS$_3\\times S^3\\times S^1$ duality.","The moduli spaces themselves become a source of new strong KT, bi-KT, HKT and QKT manifolds, including low-dimensional QKT bases coming from the $S^3\\times S^1$ instanton moduli spaces."],"supporting_citations":[{"why":"Establishes that the moduli space of Hermitian-Einstein connections is a strong KT manifold and provides the analytic setup, including the Gauduchon metric and the invertible gauge-fixing operator, that the paper's constructions assume.","marker":"[19]"},{"why":"Provides the $S^3\\times S^1$ instanton-moduli symmetry construction and the prototype curvature-contraction formula that the paper generalises to all KT manifolds.","marker":"[12]"},{"why":"Proves that the metric and torsion induced on instanton moduli spaces are independent of the choice of complex structure on an oriented bi-KT manifold, a step reused in the bi-KT instanton analysis.","marker":"[23]"},{"why":"Establishes that instanton moduli spaces over HKT and bi-HKT manifolds inherit strong HKT and bi-HKT structures, the starting point for the $S^3\\times S^1$ analysis.","marker":"[25]"},{"why":"Supplies the description of the HKT structures on $S^3\\times S^1$ and the Lee-form identities used to identify the bi-HKT structure of the moduli space.","marker":"[13]"},{"why":"Provides the rigidity classification naming $S^3\\times S^3$ and $S^3\\times T^3$ as the only compact six-dimensional strong KT manifolds with $SU(3)$ holonomy, motivating the two main examples.","marker":"[31]"},{"why":"Defines quaternionic-Kähler manifolds with torsion (QKT), the structure used to describe the base of the $S^3\\times S^1$ instanton moduli bundle.","marker":"[28]"}],"fun_headline_variants":["Torsion symmetries lift to moduli spaces","Symmetries survive on Hermitian-Einstein moduli","KT symmetries persist on instanton moduli","Holomorphic vector fields act on moduli spaces","Moduli spaces inherit torsion symmetries"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The smooth construction assumes that the moduli space of irreducible Hermitian-Einstein connections is a manifold and that the gauge-fixing operator $O=D^i_A D^A_i+\\theta^i D^A_i$ has trivial kernel and is onto, which requires a compact underlying manifold with a Gauduchon metric and gauge group $U(r)$, $SU(r)$ or $PU(r)$; if this fails, the induced vector field $\\alpha_X$ and its Killing and parallel properties are not defined.","fun_headline_variants_meta":{"raw":{"variants":["Torsion symmetries lift to moduli spaces","Symmetries survive on Hermitian-Einstein moduli","KT symmetries persist on instanton moduli","Holomorphic vector fields act on moduli spaces","Moduli spaces inherit torsion symmetries"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000489,"raw_usage":{"total_tokens":2525,"prompt_tokens":1183,"completion_tokens":1342,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":799,"completion_tokens_details":{"reasoning_tokens":1269}},"tokens_in":799,"tokens_out":1342,"duration_ms":9943,"temperature":1.0,"reasoning_tokens":1269,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:58:52.940694+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete test is to evaluate the curvature identity $\\Theta(a^h,a^h_X)=\\iota_X a^h$ on an explicit Hermitian-Einstein connection over $S^3\\times S^1$ or $S^3\\times S^3$; if it fails for any horizontal tangent vector $a^h$, the induced-field construction collapses.","supporting_citations":[{"cited_title":"L¨ ubke and A","cited_arxiv_id":null,"evidence_quote":"Establishes that the moduli space of Hermitian-Einstein connections is a strong KT manifold and provides the analytic setup, including the Gauduchon metric and the invertible gauge-fixing operator, that the paper's constructions assume."},{"cited_title":"Scale and Conformal Inv ariance in 2d Sigma Models, with an Application to N=4 Supersymmetry,","cited_arxiv_id":null,"evidence_quote":"Supplies the description of the HKT structures on $S^3\\times S^1$ and the Lee-form identities used to identify the bi-HKT structure of the moduli space."}],"review_version":1}