{"id":"f5126e9d-7e26-46c8-9001-47fdc3bf5e95","arxiv_id":"2504.16367","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The physical Hilbert space of complex Chern-Simons theory on an m-holed sphere at even level carries a Fenchel-Nielsen representation in which Wilson loops along pants-decomposition cuts act as multiplication operators.","lead":"This paper constructs a Hamiltonian quantization of complex Chern-Simons theory with gauge group SL(2,C) at even level, using quantum holonomies on surfaces. It identifies the physical Hilbert space on an m-holed sphere and shows that Wilson loop operators are diagonalized in a Fenchel-Nielsen representation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The physical Hilbert space and Fenchel-Nielsen representation rest on an imported completeness theorem for the Dehn-twist eigenfunctions; if Theorem 8.2 fails, the central construction collapses.","rationale":"The reader's weakest assumption identifies exactly the load-bearing point: the direct integral decomposition of the tensor product representation, the definition of the physical Hilbert space, and the diagonalization of Wilson loops all depend on the completeness of the generalized eigenfunctions of the operator L. The paper imports this as Theorem 8.2 from the companion paper [37] without proof or even a statement of the precise domain and normalization hypotheses. This is not merely a cosmetic omission: the entire construction from Section 8.2 onward would collapse if the resolution of identity (8.41) failed or required a different density. The additional concern about the ad hoc parametrization (8.20) is secondary; even if that parametrization were accepted, the spectral theorem would still be necessary. Therefore the verdict CONDITIONAL is appropriate, and the reader's reasoning is sound. No independent fatal flaw was identified beyond this reliance on the imported theorem.","tokens_in":68892,"tokens_out":36003,"duration_ms":345938,"concrete_test":"Verify Theorem 8.2 independently for the smallest nontrivial case, e.g., N=2 with b=e^{iπ/4}: compute the matrix elements of L = y^{-1}+y+u in a truncated Gaussian basis of L^2(R)⊗C^2, diagonalize, and compare the resulting spectral density to (8.40) via the transform Vψ of (8.42). If the Plancherel identity (8.45) fails at the few-percent level in the continuum limit, the completeness assumption underlying H_phys ≃ W is invalid. Alternatively, re-derive (8.39)-(8.41) from the quantum dilogarithm identities in [37] and check that the measure ϱ matches (8.40) exactly, including the normalization N^2/4 and the ±μχ symmetrization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central identification H_phys ≃ W and the simultaneous diagonalization of Wilson loops (9.22)-(9.23) rely on the assertion that the generalized eigenfunctions ψχ of L = y^{-1}+y+u in (8.32) satisfy the orthogonality (8.39) and resolution of identity (8.41) with the explicit density (8.40). This is Theorem 8.2, stated as 'proven in [37]' and then used to build the direct integral decomposition (8.16), the CG map (8.48)-(8.51), and the measure dϱχ. The present paper neither reproduces the proof nor states the precise hypotheses, including the domain of L, the exact normalization of the quantum dilogarithm, and the role of the ϵ-regulator. If the completeness relation (8.41) fails, or holds only with a different density, then Vψ is not unitary, the direct integral (8.16) does not represent Hλ1⊗Hλ2 faithfully, and the physical Hilbert space H_phys ≃ W and the multiplication-operator form of the Wilson loops in the FN representation are unsupported. All subsequent results in Sections 8.3-9.3 assume this step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Hamiltonian (combinatorial) quantization of SL(2,C) Chern-Simons theory on an m-holed sphere at even level k=2N. The graph algebra of quantum holonomies is defined from the Fock-Rosly bracket and the R-matrix, and a *-representation is constructed on H ≃ L²(R)⊗C^N that is tied to infinite-dimensional representations π_λ of the quantum Lorentz group U_q(sl2)⊗U_{q̃}(sl2). Physical states are identified with gauge-invariant linear functionals on a dense domain, and the flatness constraint is shown to be automatic for gauge-invariant states (Section 7). The physical Hilbert space is then obtained via a Clebsch-Gordan decomposition of tensor products π_{λ1}⊗π_{λ2} that is built on a spectral decomposition of the generalized Dehn-twist operator L = y^{-1}+y+u (Theorem 8.2, imported from the companion paper [37]), yielding H_phys ≃ W (Sections 8.3–8.6). In Section 9, Wilson loop operators along the cuts of a pants decomposition are shown to be simultaneously diagonalized on W as multiplication operators p_I(χ+χ^{-1}), defining the Fenchel-Nielsen representation, and a crossing-symmetry/A-move statement is sketched in Section 9.3.","tokens_in":69130,"tokens_out":38936,"duration_ms":332259,"significance":"If the analytical input — in particular the completeness of the eigenfunctions ψ_χ — holds, this is a substantial step: a positive-level Hamiltonian quantization of complex Chern-Simons theory on surfaces with holes, linking combinatorial quantization to the quantum Lorentz group, level-N quantum Teichmüller theory, and the modular double of U_q(sl2). The strengths of the paper are its explicit algebraic verification of the graph-algebra representation (Section 6 and Appendix E), the clean reduction of gauge invariance plus flatness to gauge invariance alone (Section 7), the uniqueness result for the invariant bilinear form (Theorem 3.7), the transparent identification of Wilson loops with quadratic Casimirs (Section 9.1), and the fact that the imported spectral theorem is explicitly flagged as the load-bearing input rather than hidden. The claims are falsifiable in a concrete way: any failure of the orthogonality or resolution of identity (8.39)–(8.41), or a change in the density (8.40), would directly alter the physical inner product and the diagonalized Wilson-loop eigenvalues derived from it.","major_comments":[{"comment":"The orthogonality (8.39) and resolution of identity (8.41) of the generalized eigenfunctions ψ_χ of L = y^{-1}+y+u are stated to be 'proven in [37]' and are then used as the foundation of the direct integral (8.16), the unitarity of V_ψ, the measure dρ_χ (8.46), and, through Sections 8.3–9.3, the identification H_phys ≃ W and the Fenchel-Nielsen representation (9.22)–(9.23). The present paper states neither the precise hypotheses (domain of L, normalization of the quantum dilogarithm γ, role of the ε-regulator) nor the exact theorem statement as proved in [37]; footnote 7 concedes that the domain analysis is carried over only by assertion ('turns out to be still valid'). Because a failure of completeness or a different density ρ would invalidate the central construction, the manuscript should either prove Theorem 8.2 in an appendix or state the theorem and its hypotheses precisely and point to the exact statement in [37] with matching conventions.","section":"§8.2, Theorem 8.2 and Eqs. (8.39)–(8.41)"},{"comment":"The restriction of the representation labels to the parametrized family λ_a = exp(2πi/N(−ibμ_a−m_a)) is assumed rather than derived. The unitarity of S_{λ2} and D_{λ1} (8.21), the reduction of Q''_2 to the Dehn-twist operator L (8.26)–(8.27), and hence the spectral decomposition defining W and the Fenchel-Nielsen representation are all valid only for this family. The footnote under (8.20) gives a heuristic justification via the annulus phase space, but the λ_a attached to the holes of an m-holed sphere are a priori unrestricted C^× labels, and for m ≥ 3 the paper does not show that the quantization forces (8.20). The authors should either derive this restriction from the quantization of the m-holed sphere phase space or state explicitly that the main results hold only for labels in this family.","section":"§8.2, Eq. (8.20)"},{"comment":"There is an internal sign inconsistency in the eigenvalues of the diagonalized Wilson loops. Equation (9.8) gives Tr^{1/2}_q[R'R] = −Q, so by (9.4) the operator D(W^{1/2}_{m,m−1}) equals −Q_{12}, whose eigenvalue is −(χ_{m−1}+χ^{-1}_{m−1}) by (8.6) and (8.33); equations (9.16)–(9.17) keep this minus sign. Equations (9.10), (9.20), and (9.22)–(9.23), however, state the eigenvalue as +p_I(χ+χ^{-1}) with no minus. These two lines cannot both be correct: the fusion algebra p_I p_J = Σ_K p_K forces the eigenvalue of W^I to be p_I evaluated at the fundamental eigenvalue, and since p_{1/2}(x) = x is odd while p_1(x) = x²−1 is even, the sign discrepancy affects precisely half-integer-spin Wilson loops. The sign can plausibly be absorbed by composing the spectral coordinate with χ → −χ, but as written the displayed formulas of Section 9.2 are mutually inconsistent and must be reconciled.","section":"§9.1–9.2, Eqs. (9.8), (9.16)–(9.17), (9.20), (9.22)–(9.23)"},{"comment":"The crossing-symmetry subsection asserts that the A-move is realized by U_T^{-1}U_S, but no computation substantiates this. Equation (9.33) equates conjugate images of W^I_S and W^I_T even though the two sides act on different direct-integral spaces ((9.25) versus (9.30)); the identification of these spaces is claimed rather than derived. In particular, the nontrivial content of the A-move — the integral kernel (quantum dilogarithm/pentagon identity) relating the two spectral decompositions — is not computed. The section should either be presented as an outline or the intertwiners should actually be constructed.","section":"§9.3, Eqs. (9.24)–(9.33)"}],"minor_comments":[{"comment":"The uniqueness statement claims uniqueness 'in the space of linear functionals on D2', but the proof fixes Ψ only on the algebraic tensor-product subspace U^{-1}(D⊗D); without continuity — which is established only for the constructed Ψ_λ in Lemma 3.6 — the values of a general linear functional on the remaining elements of D2 are not determined. Uniqueness among continuous invariant functionals follows by density and should be stated in that form.","section":"§3.4, Theorem 3.7"},{"comment":"In the antilinear eigen-equation 'ẽLψ_χ = (ẽχ+ẽχ^{-1})ψ_r', the symbol ψ_r should be ψ_χ; the same typo appears in the sentence following (8.34).","section":"§8.2, Eq. (8.33)"},{"comment":"The sentence 'where µ is the spectral measure' uses the symbol µ, which conflicts with the L²(R) coordinate µ used throughout; this should be dρ_χ.","section":"§8.1, after Eq. (8.14)"},{"comment":"The relation 'f_±(ν,n+N) = ± f_±(ν,n+N)' is trivially self-referential; it should presumably read f_±(ν,n+N) = ±f_±(ν,n).","section":"§3.2, Eq. (3.48)"},{"comment":"The sentence 'The Hermite functions e^{−µ²/2}H_n(µ), n = 1,...,∞...' should index the Hermite functions starting at n = 0.","section":"§3.1"},{"comment":"The third equality in (9.10) and (9.11) involves dP_{χ_m}, but the spectral parameter in this line is χ_{m−1}; the symbol χ_m is not defined in this context.","section":"§9.1, Eqs. (9.10)–(9.11)"},{"comment":"The sentence 'We obtain the equivalence O†∼O†' is self-referential as printed and is likely meant to relate the induced operator on W with O acting on H; the notation O' for the induced operator is also easily confused with the adjoint O† and should be changed.","section":"§8.6, below Eq. (8.88)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends heavily on the author's own companion paper [37] for the central spectral theorem (Theorem 8.2) and for other key facts (Lemma 3.3, the domain bijection (8.12)). I would ask the editor to confirm that [37] is published or otherwise available, that its proof of completeness of the ψ_χ eigenfunctions is self-contained, and that its conventions match those of the present paper. The novel content relative to [37] is principally the m-holed sphere construction and the Wilson-loop diagonalization; the statements in Section 9.3 about crossing symmetry are asserted rather than demonstrated, and I have flagged this in the report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"One thing to know: this is a serious, mostly explicit combinatorial quantization of SL(2,C) Chern-Simons at even level, and the main new results—the graph algebra at nonzero level, the physical Hilbert space for arbitrary m-holed spheres, and the Fenchel-Nielsen representation that diagonalizes Wilson loops—are real. If you work on complex Chern-Simons or quantum Teichmüller theory, this paper is worth your time.\n\nWhat it does well: the algebraic derivations are careful and transparent. The representation of the graph algebra, the gauge invariance and flatness reduction, and the construction of gauge-invariant linear functionals as physical states are all checked in detail. The link to the quantum Lorentz group U_q(sl2)⊗U_... includes the ∗-structure and the Clebsch-Gordan machinery, and the Wilson-loop diagonalization in Sections 9 is clean. The generalization from the 4-holed sphere to arbitrary m is a genuine step forward.\n\nThe soft spot is exactly where the attached stress-test lands: Section 8.2 imports Theorem 8.2, the orthogonality and resolution of identity for the Dehn-twist eigenfunctions with density (8.40), from the companion paper [37], and does not reproduce the proof or state detailed hypotheses (domain of L, normalization of the quantum dilogarithm, role of the ϵ-regulator). This theorem is load-bearing. The direct integral decomposition (8.16), the identification H_phys ≃ W, and the multiplication-operator form of the Wilson loops all sit on it. The dependency is declared honestly, and I do not see circularity, but the paper cannot be fully evaluated on its own. A referee needs to check [37] before this can be accepted as a standalone result.\n\nA second, smaller concern: the representation-label restriction (8.20) is assumed rather than derived. It is natural if the hole variables are the output of quantizing canonical data on an annulus, but for a general m-holed sphere it is a choice that is not justified in this paper. It might be harmless, but the physical rationale should be stated.\n\nNo fundamental contradiction was found. The heavy self-citation is not itself a flaw, since the reliance is explicit. The paper is for specialists in combinatorial quantization, complex Chern-Simons, or 2+1 quantum gravity. It deserves a serious referee. Send it to peer review, and make sure the referee has access to [37] and is asked to scrutinize Theorem 8.2.","headline":"Solid combinatorial quantization paper whose central Hilbert-space identification rests on a spectral theorem imported from the companion paper; a referee should verify that dependency.","tokens_in":69634,"tokens_out":3815,"would_cite":true,"duration_ms":37565,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs the Hamiltonian quantization of $\\mathrm{SL}(2,\\mathbb{C})$ Chern-Simons theory at even level and shows that on an $m$-holed sphere the physical Hilbert space carries a Fenchel-Nielsen representation in which…","keywords":["complex Chern-Simons theory","combinatorial quantization","quantum Lorentz group","Fenchel-Nielsen representation","Wilson loops","quantum dilogarithm","pants decomposition","m-holed sphere"],"falsifier":"At $N=2$, take a dense family of states $f\\in H$ and compute the claimed resolution of identity $\\sum_{m_\\chi}\\int_0^\\infty d\\mu_\\chi\\,\\varrho(\\mu_\\chi,m_\\chi)^{-1}|\\langle\\psi_\\chi|f\\rangle|^2$; if it disagrees with $\\|f\\|^2$, completeness fails. Alternatively, compute the physical inner product on the three-holed sphere and check that the Wilson loop $W_{2,1}$ acts as multiplication by $\\chi+\\chi^{-1}$; any mismatch would falsify the Fenchel-Nielsen diagonalization.","tokens_in":68669,"feed_emoji":"🌀","tokens_out":9495,"duration_ms":81153,"temperature":0.7,"pith_summary":"This paper aims to quantize $\\mathrm{SL}(2,\\mathbb{C})$ Chern-Simons theory at even level $k=2N$ by combinatorial quantization: it builds the operator algebra of quantum holonomies on the surface and constructs its infinite-dimensional representations. The central claim is that on an $m$-holed sphere the physical Hilbert space consists of the invariant linear functionals of the quantum Lorentz group and is therefore the infinite-dimensional space $W$. On $W$ the Wilson loop operators winding around the cuts of a pants decomposition are simultaneously diagonalized as multiplication operators $p_I(\\chi+\\chi^{-1})$, giving what the author calls the Fenchel-Nielsen representation. If correct, this puts even-level complex Chern-Simons theory on the same spectral footing as quantum Teichmüller theory, with physical observables expressed through one spectral variable per cut.","feed_headline":"Complex Chern-Simons theory gets diagonalized Wilson loops","feed_subtitle":"At even level k=2N the physical states form a spectral space; pants-decomposition Wilson loops act by multiplication.","key_machinery":"The load-bearing object is the infinite-dimensional $*$-representation of the quantum Lorentz group on $H\\simeq L^2(\\mathbb{R})\\otimes\\mathbb{C}^N$, generated by $u$ and $y$ with $uy=q^2yu$. The argument proceeds by Clebsch-Gordan decomposition of $\\pi_{\\lambda_1}\\otimes\\pi_{\\lambda_2}$: a unitary map built from the quantum dilogarithm diagonalizes the quadratic Casimir $Q''_2=\\lambda_1 u^{-1}+\\lambda_1^{-1}u+\\lambda_2^{-1}y^{-1}$, which after a Fourier transformation becomes the generalized Dehn-twist operator $L=y^{-1}+y+u$. The eigenfunctions $\\psi_\\chi$ of $L$ are products of quantum dilogarithms; their spectral decomposition gives the direct-integral form of $H$ and hence the physical Hilbert space $W$. The same spectral variable $\\chi$ labels the cuts of the pants decomposition and the simultaneous eigenvalues $\\chi+\\chi^{-1}$ of the diagonalized Wilson loops.","core_discovery":"At even level $k=2N$ with $q=\\exp[\\frac{2\\pi i}{k}(1+b^2)]$, $\\tilde q=\\exp[\\frac{2\\pi i}{k}(1+b^{-2})]$, $|b|=1$, the paper constructs the $*$-representation of the quantum Lorentz group $\\mathcal{U}_{\\mathbf{q}}(sl_2)\\otimes\\mathcal{U}_{\\tilde{\\mathbf{q}}}(sl_2)$ on $H\\simeq L^2(\\mathbb{R})\\otimes\\mathbb{C}^N$ generated by $u,y$ with $uy=q^2yu$. For an $m$-holed sphere, graph-algebra representations and gauge transformations are represented on $H_{\\vec\\lambda}$, and physical states are the $\\mathcal{U}_{\\mathbf{q}}(sl_2)\\otimes\\mathcal{U}_{\\tilde{\\mathbf{q}}}(sl_2)$-invariant linear functionals on a dense domain; gauge invariance automatically enforces the flatness constraint. The main theorem is that these invariants form the physical Hilbert space $H_{\\mathrm{phys}}\\simeq W$, with Wilson loops along the pants-decomposition cuts acting as multiplication by $p_I(\\chi+\\chi^{-1})$ on $L^2(\\mathbb{C},d\\varrho_\\chi)$; this is the Fenchel-Nielsen representation, a level-$N$ generalization of quantum Teichmüller theory that reduces to it at $N=1$.","pith_inferences":["A direct numerical check of the resolution of identity (8.41) for small $N$ would separate the quoted completeness theorem from the rest of the construction; the paper itself does not perform such a check.","The unitary parametrization (8.20) of the labels $\\lambda_a$ is assumed rather than derived, so the construction may cover only one branch of representation labels; other branches could yield additional superselection sectors not described here.","Because the diagonalization argument is local in each pair of pants, the same spectral construction should extend to higher-genus surfaces and to more general 3-manifold decompositions, although the paper only treats the $m$-holed sphere.","The diagonalized Wilson loops provide concrete observables whose spectra can be compared with state-integral model results for the same level; such a comparison is not made in the paper."],"forward_implications":["The physical Hilbert space of even-level complex Chern-Simons theory on an $m$-holed sphere is infinite-dimensional and isomorphic to $W$, so gauge-invariant observables can be represented by multiplication operators on an $L^2$ space over a spectral contour.","The flatness constraint is not an independent condition on physical states: quantum gauge invariance alone implies that the quantized holonomy around the bounding circle acts as the identity.","Wilson loops associated with a pants decomposition are mutually commuting and are simultaneously diagonalized, acting as $p_I(\\chi+\\chi^{-1})$; all higher-spin Wilson loops are determined by the spin-$1/2$ one through the fusion algebra.","For a 4-holed sphere, the two Fenchel-Nielsen representations coming from the two pants decompositions are unitarily equivalent, and the unitary map realizes the elementary $A$-move between decompositions.","At $N=1$ the Fenchel-Nielsen representation reduces to the standard quantum Teichmüller representation of the modular double of $\\mathrm{SL}(2,\\mathbb{R})$, so the level-$N$ theory is a direct generalization."],"supporting_citations":[{"why":"Supplies the combinatorial quantization scheme on which the graph algebra construction is based.","marker":"[30]"},{"why":"Provides the infinite-dimensional representations and, in particular, the completeness and orthogonality of the eigenfunctions used in the direct-integral decomposition.","marker":"[37]"},{"why":"Gives the level-$N$ Hilbert space $L^2(\\mathbb{R})\\otimes\\mathbb{C}^N$ and Weyl algebra used as the representation space.","marker":"[22]"},{"why":"Defines the quantum dilogarithm over $\\mathbb{R}\\times\\mathbb{Z}/N\\mathbb{Z}$ used in the eigenfunctions of the Dehn-twist operators.","marker":"[25]"},{"why":"Supplies the spectrum and eigenfunctions of the Dehn-twist operator in quantum Teichmüller theory, which the present construction generalizes.","marker":"[46]"},{"why":"Introduces the quantization of Chern-Simons theory with complex gauge group and sets the problem addressed here.","marker":"[12]"},{"why":"Provides the Poisson structure on discrete connections that the quantization starts from.","marker":"[29]"}],"fun_headline_variants":["Quantum Lorentz group meets Chern-Simons: Wilson loops diagonalized","Even-level Chern-Simons yields diagonal Wilson loops","Chern-Simons quantization: Wilson loops become multiplication","Fenchel-Nielsen for complex Chern-Simons at even level","Quantum Teichmüller generalized: diagonal Wilson loops"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the quoted theorem that the quantum-dilogarithm eigenfunctions of the Dehn-twist operator are complete and orthogonal; if that theorem fails, the direct-integral decomposition and the identification of the physical Hilbert space with $W$ collapse.","fun_headline_variants_meta":{"raw":{"variants":["Quantum Lorentz group meets Chern-Simons: Wilson loops diagonalized","Even-level Chern-Simons yields diagonal Wilson loops","Chern-Simons quantization: Wilson loops become multiplication","Fenchel-Nielsen for complex Chern-Simons at even level","Quantum Teichmüller generalized: diagonal Wilson loops"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00093,"raw_usage":{"total_tokens":4099,"prompt_tokens":1180,"completion_tokens":2919,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":796,"completion_tokens_details":{"reasoning_tokens":2835}},"tokens_in":796,"tokens_out":2919,"duration_ms":18426,"temperature":1.0,"reasoning_tokens":2835,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:04:49.084129+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At $N=2$, take a dense family of states $f\\in H$ and compute the claimed resolution of identity $\\sum_{m_\\chi}\\int_0^\\infty d\\mu_\\chi\\,\\varrho(\\mu_\\chi,m_\\chi)^{-1}|\\langle\\psi_\\chi|f\\rangle|^2$; if it disagrees with $\\|f\\|^2$, completeness fails. Alternatively, compute the physical inner product on the three-holed sphere and check that the Wilson loop $W_{2,1}$ acts as multiplication by $\\chi+\\chi^{-1}$; any mismatch would falsify the Fenchel-Nielsen diagonalization.","supporting_citations":[{"cited_title":"Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-${N}$","cited_arxiv_id":"2402.08176","evidence_quote":"Provides the infinite-dimensional representations and, in particular, the completeness and orthogonality of the eigenfunctions used in the direct-integral decomposition."},{"cited_title":"Complex Quantum Chern-Simons","cited_arxiv_id":"1409.1208","evidence_quote":"Defines the quantum dilogarithm over $\\mathbb{R}\\times\\mathbb{Z}/N\\mathbb{Z}$ used in the eigenfunctions of the Dehn-twist operators."},{"cited_title":"Witten, Quantization of chern-simons gauge theory with complex gauge group , Communications in Mathematical Physics 137 (1991), no","cited_arxiv_id":null,"evidence_quote":"Introduces the quantization of Chern-Simons theory with complex gauge group and sets the problem addressed here."}],"review_version":1}