{"id":"12192d15-aa87-4da8-bf10-ecb994bb5ec7","arxiv_id":"2507.17169","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct a symmetric monoidal functor from admissible ribbon bordisms labeled by cochains over a modular tensor category to linear cochain complexes, extending the DGGPR TQFT and preserving homotopy equivalences.","lead":"This paper builds a 3-dimensional topological quantum field theory whose outputs are cochain complexes (vector spaces with an extra squared-zero map) rather than plain vector spaces, upgrading an existing theory for non-semisimple modular tensor categories. It keeps track of homotopies, so the theory can be localized to a derived category-valued version, pointing toward the physically motivated 'derived TQFT'.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 9.3 is written only for non-admissible bordisms; the cutting step in Section 11 can leave the admissible subcategory for bordisms with empty outgoing boundary, so functoriality on those morphisms is not established.","rationale":"I read the paper in good faith and found the overall strategy plausible, but the most load-bearing point is not the full faithfulness of the separation functor, which appears fixable and is at least addressed in Appendix A. The specific weakness is the transition from Theorem 9.2 to the admissible main theorem 9.3. The authors themselves flag that they present only the non-admissible argument and leave the admissible case to the reader. The cutting arguments in Corollary 11.7 and Proposition 11.1 create pieces whose admissibility is not verified. For a connected admissible bordism with empty outgoing boundary, the admissibility condition is exactly a projective label, and the cutting can isolate that label into a collar piece, leaving an inadmissible complement. Since the cochain-valued theory is obtained by restricting the Lyubashenko theory, which is defined only on admissible bordisms, this is a real gap in the proof of the main theorem. The reader's verdict of conditional is appropriate; I do not see a reason to move it, and I am not claiming the theorem is false. The proposed test would settle whether the gap is repairable by a more careful choice of cut or whether a genuinely new argument is needed for admissible bordisms with empty outgoing boundary.","tokens_in":50220,"tokens_out":24426,"duration_ms":294089,"concrete_test":"Use a concrete nonsemisimple example, such as a small quantum group category, and take the admissible bordism M: S^2 with one P-marked point → ∅ realized by a 3-ball with a single coupon connected to the marked point and a small P-labeled loop. Apply the cutting procedure of Corollary 11.7, sliding the coupon into the incoming collar and cutting. Check whether the residual piece is admissible: if it has empty outgoing boundary and no projective-labeled edge, then the proof of Proposition 11.1 does not cover M, and Theorem 9.3 lacks a proof for this morphism. Then repeat while keeping the P-labeled loop in the residual piece; the test is whether every piece can simultaneously be admissible and have the form required by Lemmas 11.3-11.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 9.3 is the main theorem, but Section 9 explicitly says the admissible variant is obtained by repeating the non-admissible proof word-for-word with no additional argument. That claim is insecure. The proof of Proposition 11.1 for a connected bordism with nonempty boundary invokes Corollary 11.7, which drags a unique coupon into a boundary collar and cuts the bordism into pieces N0 and N1. Admissibility is never checked for these pieces. For a bordism M: Σ → ∅ with empty outgoing boundary, admissibility of M requires some edge labeled by a projective. If the unique coupon and its incident edges are isotoped into the incoming collar, the residual piece is a couponless bordism from Σ to ∅ with empty outgoing boundary and no projective-labeled edge, so it is not admissible. Since Z_A is only defined on admissible bordisms, the induction in Proposition 11.1 cannot be run on this decomposition inside Bord^adm_Ch(A). This gap is not merely cosmetic: it affects exactly the morphisms whose admissibility is forced by projectivity, which is the nonsemisimple case the paper is about. The construction may still be correct, but as written the proof of Theorem 9.3 does not cover such bordisms.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for a finite modular tensor category A, a symmetric monoidal functor Z^*_A from an admissible bordism category whose objects are surfaces marked by finite-length cochains over A to the symmetric monoidal category Ch(Vect), lifting the DGGPR theory Z_A. It identifies the state spaces with Hom complexes Hom^*_A(1, E^{⊗g} ⊗ x_I), shows that the theory preserves certain algebraic homotopy equivalences, and uses this to discuss a localization to an ∞-categorical TQFT with labels in the homotopy ∞-category K(A). The construction proceeds by a base change to Vect^Z via a separation functor, followed by an explicit differential on state spaces and graphical verification that bordisms induce cochain maps.","tokens_in":50472,"tokens_out":5360,"duration_ms":63491,"significance":"If the main theorem is fully established, the paper gives a genuinely useful cochain-level refinement of the DGGPR theory: it is parameter-free, it identifies state spaces with concrete Hom complexes, and it provides a plausible first step toward derived TQFTs for nonsemisimple modular tensor categories. The graphical calculus in Sections 10–11 is detailed and the paper is careful about many categorical foundations. The localization and derived-theory parts are appropriately hedged as approximations or work in progress, which is honest. However, as written, the admissible case of the main theorem is not actually proved: several load-bearing steps are explicitly deferred or asserted to be 'word-for-word' adaptations that they are not.","major_comments":[{"comment":"Theorem 9.3 is not obtained by repeating the non-admissible proof word-for-word, because admissibility of the pieces produced by cutting is not preserved. In Corollary 11.7 a unique coupon is dragged into a boundary collar and the bordism is cut into pieces N0 and N1; for a bordism M: Σ → ∅ with empty outgoing boundary, admissibility of M requires some projectively labeled edge. If the unique coupon and its incident edges are isotoped into the incoming collar, the residual piece is couponless, has empty outgoing boundary, and has no projective-labeled edge, so it is not admissible. Since Z_A is only defined on admissible bordisms, the induction in Proposition 11.1 cannot be run on this decomposition inside Bord^adm_{Ch(A)}. This gap affects exactly the morphisms whose admissibility is forced by projectivity, which is the nonsemisimple case; a separate argument for the empty outgoing boundary case is needed.","section":"§9.3, proof of Proposition 11.1 (admissible variant)"},{"comment":"The identification of state spaces for arbitrary markings is deferred. The proof establishes the cochain isomorphism only for positively marked surfaces and then states that 'a slightly more subtle analysis' handles general markings. This identification is part of the statement of Theorem 9.3, and because the differential for a general surface is defined by conjugation with the flip bordism (Definition 10.1), one must explicitly verify that the natural isomorphism of Theorem 8.5 intertwines the conjugated differential with the differential on Hom^*_A(1, E^{⊗g} ⊗ x_I). As written, the theorem is not proved for general markings.","section":"§12.3, Eq. (49)"},{"comment":"The definition of dΣ for surfaces with negative framings contains the assertion 'After dealing with various annoyances with orientations, one sees...' without a proof. The resulting formula, with the factor -(-1)^mi and the dualized differentials, is used in Lemmas 11.4 and 11.5, so this is not purely cosmetic. Please supply the missing verification, for example by tracking the flip bordism through the definition in (27)–(28).","section":"Definition 10.1"}],"minor_comments":[{"comment":"There are several typos that should be corrected: 'nonseimisimple' in Section 1.1, 'monoical' in Section 1.2, 'homomotopy' in Section 13.3, and 'representastions' in reference [26].","section":"Throughout"},{"comment":"The notation k(m) is used both for the one-dimensional graded vector space and for the corresponding object in k Str^red; this is convenient but can be confusing in formulas such as (27). A short glossary or a sentence fixing this convention would help.","section":"§10.1"},{"comment":"The proof of Proposition 6.3 is very dense. In particular, the decomposition over 'partially labeled monomial bordisms' and the claim that the functor ccat' preserves this decomposition would benefit from a brief justification that the relations (R0), (R5), (U4) respect the decomposition.","section":"Appendix A"},{"comment":"The phrase 'up to some approximation' is used repeatedly in describing the localized ∞-category. This is acceptable for a proposed construction, but it should be stated explicitly that the ∞-category Bord^adm_{K(A)} is not formally constructed in this paper and is only heuristically described.","section":"§13.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the central idea is promising. The main obstacle is the missing proof for the admissible case, especially the empty outgoing boundary situation; if the authors can supply that argument, I would be inclined to support publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does something real: it lifts the DGGPR universal Lyubashenko theory to a symmetric monoidal functor valued in Ch(Vect), with an explicit differential on state spaces, a d^2=0 computation done by graphical calculus (Lemma 10.3), and a homotopy preservation theorem (13.3) that produces an actual localization to an infinity-categorical target. That last result is the one people will cite. The base-change machinery is credited to Runkel–Szegedy–Watts, the theory is a lift of De Renzi et al., and the derived-TQFT material is honestly flagged as work in progress. No fitted parameters, no circularity.\n\nThe soft spot is where the paper tells you to look. Section 9 says the admissible case \"applies word-for-word\" from the non-admissible proof. That claim does not survive contact with the cutting argument. The proof of Proposition 11.1, via Corollary 11.7, drags the unique coupon into a boundary collar and cuts the bordism into pieces that Corollary 11.6 must handle. For an admissible M: Σ → ∅, admissibility is forced by the presence of a projective-labeled edge. If that edge is one of the coupon's incident edges, the residual couponless piece from Σ to ∅ has empty outgoing boundary and no projective edge — it is not admissible, and Z_A is not defined on it. The decomposition leaves the admissible subcategory, and the induction cannot run. This hits exactly the morphisms whose admissibility is forced by projectivity, which is the nonsemisimple case the paper exists for. The construction may be correct, but Theorem 9.3 is not proven for those bordisms as written.\n\nMinor items, in proportion: Definition 10.1 defers the orientation signs to \"various annoyances\" rather than proving them; Section 12.3 leaves general markings to \"a slightly more subtle analysis\"; and the coupon-sliding lemmas in Section 11 are pictorial, not fully formal. Each is fixable. The localization discussion in Section 13.3 calls itself vague, and it is, but it is explicitly a proposal.\n\nFor whom: people in low-dimensional TQFT and non-semisimple categories who want a concrete step toward a derived theory. Theorem 13.3 alone is worth the read. But the admissible theory — the one that matters for non-semisimple input — needs its proof gap closed before the main theorem is trustworthy as stated.\n\nRecommendation: yes, send it to peer review. A serious referee should ask the authors to fix the admissible-case argument, not reject the paper. The construction is substantial and probably right; it is just not fully proved yet.","headline":"New cochain-valued lift of the DGGPR theory with a real differential and homotopy localization — but the admissible-case proof has a genuine gap exactly where the paper says it follows word-for-word.","tokens_in":51047,"tokens_out":6757,"would_cite":true,"duration_ms":65659,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18M15","57K16","18N60"],"pacs":[],"model":"deepseek-v4-flash","headline":"A cochain-valued TQFT extending the universal Lyubashenko theory is constructed for nonsemisimple modular tensor categories, with state spaces given by Hom complexes and localization to dg vector spaces.","keywords":["cochain-valued TQFT","nonsemisimple modular tensor categories","Lyubashenko invariant","ribbon bordism category","separation functor","base change for TQFTs","homotopy equivalence","localization to dg vector spaces"],"falsifier":"Take a contractible bounded complex of projectives, label a knot in a closed 3-manifold by it, and compute the claimed alternating-sum invariant $\\sum_m (-1)^m Z_A(M, T_m)$; a nonzero result would disprove the theory. Likewise, Theorem 13.3 predicts that no bordism labeled by homotopy equivalences can produce a non-homotopy-equivalence chain map, so exhibiting such a bordism would settle the claim negatively.","tokens_in":49977,"feed_emoji":"🎀","tokens_out":9552,"duration_ms":94052,"temperature":0.7,"pith_summary":"The paper proves that the universal Lyubashenko TQFT, a 3-dimensional field theory built from any nonsemisimple modular tensor category, extends from vector spaces to the symmetric monoidal category of linear cochain complexes. On a genus $g$ surface with markings by cochains, the state space of the extended theory is the Hom complex $\\mathrm{Hom}^*_A(1, E^{\\otimes g} \\otimes x_I)$. The extension preserves algebraic homotopy equivalences, so it localizes to a theory valued in the $\\infty$-category of dg vector spaces. The authors offer this localized theory as the first concrete step toward a fully derived TQFT for derived quantum group representations, whose state spaces should be infinite-dimensional Hochschild-type cohomologies.","feed_headline":"Lyubashenko invariants lift to cochain-valued TQFTs","feed_subtitle":"State spaces become Hom complexes, and homotopy equivalences are preserved, so the theory localizes to dg vector spaces.","key_machinery":"The load-bearing mechanism is the separation functor $\\Delta$, which converts a bordism labeled by the Kelly product $S \\otimes A$, for semisimple symmetric $S$ and in particular $S = \\mathrm{Vect}^{\\mathbb{Z}}$, into a formal sum of an $S$-labeled string diagram and an $A$-labeled reduced bordism. The functor $\\Delta$ is well-defined only because the reduced concatenation functor $\\mathrm{ccat}^{\\mathrm{red}}$ is fully faithful (Proposition 6.3), proved by a dimension count on partially labeled monomial bordisms. On top of this base change, the paper defines an explicit differential $d_\\Sigma$ on every state space using string-diagram relations (V5) and (V6) in the graded-vector-space factor, with $d^2 = 0$ following from cancellation of paired terms; coupon-sliding moves then show every ribbon bordism intertwines the differentials, yielding the cochain-valued functor.","core_discovery":"For a finite modular tensor category $A$, Theorem 9.3 constructs a symmetric monoidal functor $Z^*_A : \\mathrm{Bord}^{\\mathrm{adm}}_{\\mathrm{Ch}(A)} \\to \\mathrm{Ch}(\\mathrm{Vect})$ extending the universal Lyubashenko theory of [11]. Its value on a genus $g$ marked surface $\\Sigma_{\\vec{x}}$ is naturally isomorphic to the Hom complex $\\mathrm{Hom}^*_A(1, E^{\\otimes g} \\otimes x_I)$, where $E$ is the canonical end and $x_I$ is the oriented tensor product of the marking objects. Theorem 13.3 shows that $Z^*_A$ sends algebraic homotopy equivalences between marked surfaces, namely product bordisms whose straight-line labels are homotopy equivalences in $\\mathrm{Ch}(A)$, to homotopy equivalences of cochain complexes. The functor therefore descends to a symmetric monoidal theory valued in the $\\infty$-category of dg vector spaces, with domain given up to approximation by ribbon bordisms labeled in the homotopy $\\infty$-category $\\mathcal{K}(A)$.","pith_inferences":["Not spelled out in the paper: if the localized state space on the twice-punctured sphere is the linear mapping space $\\mathrm{Maps}_{\\mathcal{K}(A)}$, then a linear $\\infty$-Yoneda argument would make this single surface determine the entire homotopy $\\infty$-category $\\mathcal{K}(A)$.","A testable consequence the authors only hint at: replacing a marking object by a projective resolution of the unit should change the state space materially, so one can check the proposed derived theory's prediction that unmarked-surface values come from resolutions of the unit.","The alternating-sum formula for knots suggests an Euler-characteristic interpretation of the cochain invariant; deriving surgery formulas for it from known surgeries on scalar Lyubashenko invariants would be a concrete extension.","A fully formal set of movie moves for sliding coupons through tubular neighborhoods would convert the differential-intertwining arguments into a checkable calculus and might expose hidden sign conventions."],"forward_implications":["On a genus $g$ marked surface, the state space is the Hom complex $\\mathrm{Hom}^*_A(1, E^{\\otimes g} \\otimes x_I)$, so mapping-class-group actions and gluing maps act on genuine cochain complexes rather than on vector spaces.","For a knot in a closed 3-manifold labeled by a bounded complex of projectives, the invariant is the alternating sum $\\sum_m (-1)^m Z_A(M, T_m)$ of renormalized Lyubashenko invariants.","Algebraic homotopy equivalences of labeling complexes are sent to homotopy equivalences of vector-space cochains, so the functor localizes to the symmetric monoidal $\\infty$-category of dg vector spaces.","The localized domain is, up to approximation, an $\\infty$-category of ribbon bordisms labeled by the homotopy $\\infty$-category $\\mathcal{K}(A)$, and the fiber over any marked surface is a cartesian power of $\\mathcal{K}(A)$.","Restricting to objects of $A$, viewed as cochains concentrated in degree zero, recovers the original universal Lyubashenko theory."],"supporting_citations":[{"why":"Supplies the base theory $Z_A$, the renormalized Lyubashenko invariant, and the admissible bordism category that the paper extends.","marker":"[11]"},{"why":"Provides the tautological TQFT method and the model for the base-change Theorem 7.3.","marker":"[29]"},{"why":"Provides ribbon graph evaluation, framed braid groups, and the anomalous bordism category formalism used throughout.","marker":"[32]"},{"why":"Identifies the localization $\\mathrm{Ch}(\\mathrm{Vect})[W^{-1}]$ with the $\\infty$-category of dg vector spaces and supplies the homotopy $\\infty$-category $\\mathcal{K}(A)$.","marker":"[22]"},{"why":"Gives the Dwyer-Kan localization result used to localize the symmetric monoidal fibrations and obtain the $\\infty$-categorical theory.","marker":"[18]"},{"why":"Supplies Lyubashenko's 3-manifold invariants and mapping class group actions that the universal Lyubashenko theory reprojects.","marker":"[24]"},{"why":"Provides the categorical foundations for ribbon categories, duals, and tensor products on which the constructions rely.","marker":"[13]"}],"fun_headline_variants":["Cochain-valued TQFTs preserve homotopy equivalences","Lyubashenko TQFT localizes to derived dg categories","Nonsemisimple tensor categories yield cochain TQFTs","Hom complexes as state spaces: a new TQFT"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction stands or falls on a technical 'no extra morphisms' claim about a reduced bordism category, proved by a dimension count; if that claim failed, the separation functor that carries the base change could not be built, and the paper also assumes without a fully formal proof that coupons can be slid through small tubular neighborhoods.","fun_headline_variants_meta":{"raw":{"variants":["Cochain-valued TQFTs preserve homotopy equivalences","Lyubashenko TQFT localizes to derived dg categories","Nonsemisimple tensor categories yield cochain TQFTs","Hom complexes as state spaces: a new TQFT"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000238,"raw_usage":{"total_tokens":1562,"prompt_tokens":1049,"completion_tokens":513,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":665,"completion_tokens_details":{"reasoning_tokens":438}},"tokens_in":665,"tokens_out":513,"duration_ms":6501,"temperature":1.0,"reasoning_tokens":438,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:54:51.863628+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a contractible bounded complex of projectives, label a knot in a closed 3-manifold by it, and compute the claimed alternating-sum invariant $\\sum_m (-1)^m Z_A(M, T_m)$; a nonzero result would disprove the theory. Likewise, Theorem 13.3 predicts that no bordism labeled by homotopy equivalences can produce a non-homotopy-equivalence chain map, so exhibiting such a bordism would settle the claim negatively.","supporting_citations":[{"cited_title":"De Renzi, A","cited_arxiv_id":null,"evidence_quote":"Supplies the base theory $Z_A$, the renormalized Lyubashenko invariant, and the admissible bordism category that the paper extends."},{"cited_title":"Runkel, L","cited_arxiv_id":null,"evidence_quote":"Provides the tautological TQFT method and the model for the base-change Theorem 7.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides ribbon graph evaluation, framed braid groups, and the anomalous bordism category formalism used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the localization $\\mathrm{Ch}(\\mathrm{Vect})[W^{-1}]$ with the $\\infty$-category of dg vector spaces and supplies the homotopy $\\infty$-category $\\mathcal{K}(A)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Dwyer-Kan localization result used to localize the symmetric monoidal fibrations and obtain the $\\infty$-categorical theory."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lyubashenko's 3-manifold invariants and mapping class group actions that the universal Lyubashenko theory reprojects."},{"cited_title":"Etingof, S","cited_arxiv_id":null,"evidence_quote":"Provides the categorical foundations for ribbon categories, duals, and tensor products on which the constructions rely."}],"review_version":1}