{"id":"3c53752b-5b1d-41d0-8e5f-ca43d7627e64","arxiv_id":"2412.16587","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A constructive enumeration of all local minimal-model CFTs, organized by Jones index, yields selection rules for RG flows that recover known results and predict new ones.","lead":"This paper maps out all possible local (not necessarily modular invariant) quantum field theories with central charge c<1, organizing them into trees labeled by a Jones index that measures how much locality fails. It then uses this map to derive new constraints on which relevant perturbations can flow between minimal models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The RG preservation principle in Section 3 is stated as an expectation and carries the new selection rules; one concrete check would close the gap.","rationale":"The reader's weakest_assumption is exactly the preservation of the DHR superselection category along massless RG flows, introduced in Section 3 with 'we expect.' My independent reading confirms that this is the single load-bearing step for the RG selection rules: while the Section 2 classification is a substantial standalone result that matches the KL classification where applicable (modulo the undisclosed computer search and the then-unpublished E6 fusion rules), the novelty of the paper lies in deriving selection rules, no-go results, and the operator-mapping (eqs. 3.10-3.17) from this principle. The principle is neither proven nor empirically calibrated, and the paper explicitly carves out massive flows, so all the new predictions sit on the assumption that a massless flow preserves the full superselection structure at and above the perturbed node. I do not find an internal inconsistency in the classification itself, and I credit the independent support from matching KL results, the explicit index computations, and the cross-checks against known flows (e.g. tricritical to Ising). The verification step I propose is the sharpest available empirical test of the principle, and it is feasible with existing TBA and conformal perturbation theory results. Therefore the reader's CONDITIONAL verdict is appropriate; I would not strengthen to REJECT because the principle is physically plausible and the classification stands independently, but ACCEPT would overstate the confidence in the new selection rules.","tokens_in":37769,"tokens_out":1603,"duration_ms":14617,"concrete_test":"Test the preservation principle on the best-understood nontrivial case where IR data is known independently: the tricritical Ising (m=4, node 3/4 category su(2)_2^even or the Zamolodchikov flow from m=4 to m=3). Compute, in the known massive/massless TBA or conformal perturbation theory data, whether every UV primary in the chosen node has power-law (not exponential) correlators at long distance in the IR theory, and verify the DHR category of the IR fixed point matches the category of the UV perturbed node. If for the least relevant (1,3) perturbation all such correlators are massless and the category matches, the principle survives its sharpest available test; if any charged UV operator becomes massive while the flow is still described as massless, the principle needs qualification before the selection rules can be accepted.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's new physics claims derive from the principle in Section 3: under a massless relevant RG flow, the DHR superselection category at and above the perturbed node is preserved (pp. 26-27, explicitly 'we expect'). The classification of local minimal models in Section 2 is largely independent support, but the RG selection rules, including the Zamolodchikov-flow derivation and item (d)'s no-go, all rely on this unproven transfer principle. The text explicitly contrasts with massive flows and assumes no interpolating UV field acquires exponentially decaying correlators. That assumption is not derived from any known theorem, and it is exactly where a counterexample would break the central claim: if the sector category can change even when all UV fields remain massless, then the matching of categories between UV and IR (e.g. su(2)_{m-1} in eq. 3.2, and the claimed uniqueness of the IR model) is not established. This is a real soft spot, not merely a gap in rigor, because the principle is the load-bearing step that converts the classification into selection rules.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a finer classification of local unitary c<1 two-dimensional CFTs than the usual ADE classification of modular-invariant partition functions. Starting from a chiral algebra and imposing only T-invariance and OPE closure, the authors construct trees of local submodels inside each modular-invariant completion. Each node carries a global Jones index computed from the partition function via Eq. (2.19), and the nodes are organized by inclusion relations. The classification is compared with the Kawahigashi-Longo algebraic classification where the models are parity-symmetric. The second part uses the trees to constrain massless RG flows: the authors argue that the DHR superselection category at and above the perturbed node is preserved, yielding selection rules that recover the Zamolodchikov flow and produce new predictions, including a no-go statement for flows triggered by the (2,1) field.","tokens_in":38094,"tokens_out":4929,"duration_ms":45717,"significance":"If the claims hold, the paper provides a concrete, field-theoretic counterpart to the algebraic classification of local conformal nets: explicit spectra, inclusion trees, and Jones indices for all local minimal models, not only modular-invariant ones. The index formula (2.19) is simple and, on the examples checked, agrees with the KL classification and with Jones index constraints. The RG selection rules are falsifiable and give a unified explanation of known flows as well as new predictions (for example, the impossibility of a massless IR fixed point for the (2,1) perturbation, item (d) in Section 3). The paper is therefore potentially important for connecting bootstrap methods, algebraic QFT, and RG analysis in 2D CFT.","major_comments":[{"comment":"The preservation principle for massless RG flows is the load-bearing assumption of the paper: 'Under these conditions, we expect' preservation of global and relative Jones indices, completion structure, and DHR categories at or above the perturbed node. Not only is this introduced as an expectation rather than a derivation, but the paper also explicitly excludes massive flows. Items (b), (c), and (d) and the category matching in Eq. (3.2) all depend on this principle. The text gives no concrete criterion for when a sector remains massless and no proof that the DHR category cannot change in a massless flow. Since this is exactly where a counterexample would break the central new claims, the authors should either prove the principle from known results (e.g., from the transportability of DHR sectors) or test it in a non-trivial solvable case such as the tricritical Ising to Ising flow, stating clearly the regime of validity.","section":"Section 3, pp. 26-27"},{"comment":"The exhaustiveness of the classification rests on the statement 'A computer search shows there are no additional ones,' with no description of the algorithm, the search space, or the verification steps. The same issue arises in Sections 2.6 and 2.7, where numbers of submodels are quoted without proof. Since the paper's central claim is that the trees are complete, this is not a presentation issue: a reader cannot verify that all closed OPE subalgebras have been found. The authors should provide the search code as supplemental material or give a mathematical argument reducing the search to a finite check that can be inspected.","section":"Section 2.5, paragraph after Eq. (2.31)"},{"comment":"The submodels of the E6 modular invariants are built using fusion rules from Ref. [51], which was unpublished at the time of writing. The manuscript thanks the authors for communicating the results 'prior to the publication.' Now that [51] is available as arXiv:2502.14295, the authors should confirm that all fusion rules used here agree with the published version and state this explicitly. Until then, the E6 part of the classification and the RG statements in Section 3.2 are not independently verifiable from the present manuscript alone.","section":"Section 2.7, E6 examples"}],"minor_comments":[{"comment":"The global index in row 4 of Table 3 appears to have a typo: it contains sin^{-4}(π/(m+1)) twice, whereas the analogous entry in Table 2 has sin^{-4}(π/m) sin^{-4}(π/(m+1)). The category name '(A4n, Dn+2)' should presumably be '(A4n, D2n+2)'.","section":"Table 3, Section 2.6.2"},{"comment":"The caption says 'for m = 4n + 1', but the section is about the m = 4n - 1 case; the figure label should be corrected.","section":"Figure 11 caption"},{"comment":"There are several typos in the sets, for example 'ss= 1, 2, 3, . . . ,11, 12' and 's= 1, 12' in a context where s is otherwise summed over a range; please clean up the notation so that the field content is unambiguous.","section":"Section 2.7, list of non-parity-symmetric models"},{"comment":"Minor typos include 'completations', 'alegbra', 'tricrital', and 'form mU V = 13' for 'from mUV = 13'; these should be corrected in a final pass.","section":"Section 3.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is in good shape for a journal publication if the two central gaps are addressed: the unproven RG preservation principle and the undisclosed computer search for exhaustiveness. The former is the more serious issue because the novel physics claims in Section 3 are conditional on it. The authors may also want to clarify the status of Ref. [51] in the final version. I see no grounds for rejection: the classification part is valuable and the index computations appear sound wherever they can be checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The classification part of this paper is the real content, and it is good. The authors construct, for each unitary minimal model, all local OPE-closed subalgebras of the ADE modular invariants, producing a tree of theories with explicit spectra and Jones indices. They reproduce the Kawahigashi-Longo classification where the comparison is meaningful, and in addition find non-parity-symmetric models and track inclusion relations, which KL did not do. The index formula (2.19) is a clean, checkable statement. The m=3,4,5 examples are worked out in detail and are convincing. This is a genuinely useful contribution to the finer classification of c<1 local CFTs.\n\nThe RG section is a different matter. The selection rules all rest on the principle in Section 3 that a massless relevant flow preserves the DHR superselection category at and above the perturbed node. The paper states this as 'we expect', and it is not derived from anything more basic. The Zamolodchikov and D-series flows are consistent with it, and the no-go for the (2,1) perturbation is interesting, but these are predictions from a conjecture, not consequences of a theorem. The authors do flag the massive caveat, but the massless assumption is doing a lot of work. A concrete check, e.g. a non-perturbative argument or a lattice analog for one nontrivial flow, would close a real gap.\n\nTwo smaller issues. The exhaustiveness of the enumerations relies on a computer search that is not described and whose code is not released; for a classification claim, that matters. And the E6 fusion rules were borrowed from an unpublished manuscript [51], though that paper now exists on the arXiv, so this is mostly a timing issue, not a correctness one.\n\nThe complaint that the framework is circular does not hold up. The index formula and the modular-invariance-as-completeness equivalence come from the companion paper [6]; they are imported, not derived here, but that is a normal dependency, and the classification stands on its own.\n\nWho is this for? Anyone working on two-dimensional CFT, generalized symmetries, or RG flows between minimal models. It deserves a serious referee. My recommendation: send it to review, and insist the authors state the RG principle as a conjecture, provide the search code or at least a detailed description, and confirm the E6 fusions are now published. With those changes, the classification part is publishable as is.","headline":"The classification half of this paper is solid and worth engaging; the RG selection rules rest on an honestly flagged conjecture that needs sharper support.","tokens_in":38456,"tokens_out":2358,"would_cite":true,"duration_ms":21061,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T40","46L37"],"pacs":[],"model":"deepseek-v4-flash","headline":"All local c<1 minimal CFTs organize into OPE-closed trees; massless RG flows preserve the superselection category above the perturbed node, and this fixes which flows can exist.","keywords":["minimal models","modular invariance","Haag duality","superselection sectors","global index","renormalization group flows","selection rules","two-dimensional conformal field theory"],"falsifier":"Produce a massless RG flow between local unitary minimal models in which the neutral algebra's global index or superselection category is not reproduced at the IR fixed point. A concrete test is a lattice or bootstrap realization of the $(2,1)$ perturbation of the tricritical Ising model that ends at a non-trivial local CFT, which would contradict the paper's no-go; equivalently, compute $\\mu$ at both endpoints of the $(1,3)$ flow and check that the values match.","tokens_in":37545,"feed_emoji":"🔄","tokens_out":11324,"duration_ms":89441,"temperature":0.7,"pith_summary":"The paper claims that the set of local unitary minimal CFTs (central charge $c<1$) is not exhausted by the modular-invariant A-D-E list: if one drops S-invariance and keeps only locality plus closure of the operator algebra, every such theory is a node of a tree whose bottom is the stress-tensor algebra and whose top is a modular invariant completion. Each node carries a global index $\\mu$ computed from its torus partition function, measuring how much Haag duality is violated, i.e. how incomplete the theory is. The paper then uses the tree for RG flows: when a relevant scalar perturbation generates the neutral algebra (stress tensor plus perturbation), the whole tree structure above that node must be preserved along a massless flow, in particular the category of superselection sectors. Matching this category between a UV and an IR node produces selection rules, recovering the known $(1,3)$ flow between consecutive minimal models and excluding, for instance, a local fixed point for the $(2,1)$ perturbation.","feed_headline":"Superselection rules fix which minimal-model RG flows exist","feed_subtitle":"Massless flows preserve the symmetry category above the perturbed node, forbidding some endpoints.","key_machinery":"The machine doing the work is the tree of closed OPE subalgebras of complete minimal models, each node labelled by its local primary spectrum and by the global index $\\mu$ of equation (2.19). The index is the size of the inclusion $\\mathcal{A}(R_1\\cup R_2)\\subseteq \\widehat{\\mathcal{A}}(R_1\\cup R_2)$ for two intervals, i.e. the amount of Haag duality violation, and it equals the total quantum dimension of the node's superselection sector category. The tree encodes, for every node, all possible completions and their relative indices; the preservation rule states that along a massless flow only the internal structure of the neutral algebra may change, while every branch above it is a spectator.","core_discovery":"The central discovery is a constructive classification plus a preservation rule. Starting from a chiral algebra, the paper adds primary fields one at a time, imposing T-invariance (locality) and closure of the OPE; the resulting local algebras form a tree for each label $m$. There are only finitely many nodes for each $m$ (eight bosonic submodels of the diagonal series, further submodels for the D and E series), and the global index of a node is $$\\mu = \\left(\\frac{\\sum_{r,s} d_{r,s}^2}{\\sum_{r',s';r'',s''} d_{r',s'} M_{r',s';r'',s''} d_{r'',s''}}\\right)^2,$$ where $d_{r,s}$ are the quantum dimensions of the Virasoro representations and $M$ is the coupling matrix of the partition function; modular invariant nodes have $\\mu=1$. Acting as a generalized symmetry, the category of superselection sectors at and above a perturbed node cannot change under a massless RG flow. Consequently the global and relative indices and the inclusion pattern of all completions are preserved, and candidate IR endpoints are selected by matching the category of the UV neutral algebra.","pith_inferences":["If the preservation rule holds, it gives a practical consistency test for any proposed minimal-model flow: compute the global index (2.19) of the UV neutral algebra and require the IR node to have the same index and the same set of completions; this could be checked in lattice or tensor-network realizations.","The same tree construction should extend to rational CFTs with $c>1$, where complete models are coset constructions; enumerating their OPE-closed subalgebras would organize flows between coset models by the same category-preservation rule.","The paper's partially-massless conjecture implies that whenever some fields become massive while the stress tensor stays massless, the massive fields must be charged under an exact internal symmetry of the full theory; this is a testable 'naturalness' diagnostic in perturbative examples."],"forward_implications":["The finer classification predicts, for every $m$, a fixed finite set of local bosonic $c<1$ CFTs: eight submodels of the diagonal A series, plus D- and E-series submodels, together with their primary spectra and inclusion relations.","The $(1,3)$ perturbation from $m+1$ to $m$ carries the category $\\mathrm{su}(2)_{m-1}$, so the flow is allowed for every $m$, not only in the perturbative limit.","The $(1,2)$ perturbation preserves $\\mathrm{su}(2)^{\\mathrm{even}}_{m-1}$; for odd $m$ the IR can be a non-diagonal submodel, and in general non-diagonal models flow only to non-diagonal models.","The $(2,1)$ perturbation has a superselection category that appears in no smaller minimal model, so a massless flow to a local CFT is excluded and any endpoint must contain massive sectors.","Because every UV operator sits in a class of the preserved category, the rule assigns non-perturbative selection rules and field-class maps that match the known perturbative operator mixing."],"supporting_citations":[{"why":"Supplies the A-D-E modular invariant partition functions that form the top nodes of the trees.","marker":"[5]"},{"why":"Establishes that modular invariance is completeness and provides the partition-function formula for the global index used in (2.19).","marker":"[6]"},{"why":"Supplies the Virasoro fusion rules and OPE data used to close the operator algebras.","marker":"[4]"},{"why":"Proves the relation between global indices of a theory and its extensions and connects them to superselection sectors.","marker":"[9]"},{"why":"Classifies local conformal nets with c<1, providing the baseline list of superselection categories the paper matches.","marker":"[10]"},{"why":"Classifies possible superselection categories and two-dimensional local nets, used as the standard table for comparisons.","marker":"[11]"},{"why":"Contains the original calculation of the (1,3) flow between consecutive minimal models that the selection rule reproduces.","marker":"[19]"},{"why":"Gives the UV-to-IR field mapping via domain walls that the category argument recovers.","marker":"[27]"},{"why":"Studies RG flows of non-diagonal minimal models and establishes the non-diagonal-to-non-diagonal pattern.","marker":"[29]"}],"fun_headline_variants":["Minimal model RG flows pinned by superselection categories","RG flows of minimal models obey superselection rules","Tree of local CFTs fixes allowed RG endpoints","Superselection categories forbid some minimal-model RG endpoints","Haag duality violation sets RG flow selection rules"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire RG section rests on the assumption that a massless flow maps the UV theory completely onto the IR theory, so that the category of superselection sectors at and above the perturbed node is exactly preserved; this is stated as an expectation rather than proven, and massive flows are explicitly set aside.","fun_headline_variants_meta":{"raw":{"variants":["Minimal model RG flows pinned by superselection categories","RG flows of minimal models obey superselection rules","Tree of local CFTs fixes allowed RG endpoints","Superselection categories forbid some minimal-model RG endpoints","Haag duality violation sets RG flow selection rules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000324,"raw_usage":{"total_tokens":1872,"prompt_tokens":1051,"completion_tokens":821,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":745}},"tokens_in":667,"tokens_out":821,"duration_ms":20907,"temperature":1.0,"reasoning_tokens":745,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:26:30.431707+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce a massless RG flow between local unitary minimal models in which the neutral algebra's global index or superselection category is not reproduced at the IR fixed point. A concrete test is a lattice or bootstrap realization of the $(2,1)$ perturbation of the tricritical Ising model that ends at a non-trivial local CFT, which would contradict the paper's no-go; equivalently, compute $\\mu$ at both endpoints of the $(1,3)$ flow and check that the values match.","supporting_citations":[{"cited_title":"The ADE Classification of Minimal and A1(1) Conformal Invariant Theories,","cited_arxiv_id":null,"evidence_quote":"Supplies the A-D-E modular invariant partition functions that form the top nodes of the trees."},{"cited_title":"Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory,","cited_arxiv_id":null,"evidence_quote":"Contains the original calculation of the (1,3) flow between consecutive minimal models that the selection rule reproduces."},{"cited_title":"RG flows of non-diagonal minimal models perturbed by $\\phi_{1,3}$","cited_arxiv_id":"hep-th/9110018","evidence_quote":"Studies RG flows of non-diagonal minimal models and establishes the non-diagonal-to-non-diagonal pattern."}],"review_version":1}