{"id":"4bcd5d78-0626-463e-944b-94e93660c27b","arxiv_id":"2610.01909","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The torus one-point amplitudes of the four super Virasoro minimal strings are computed from the RNS worldsheet and match random matrix predictions.","lead":"This paper computes, for the first time, the complete torus one-point amplitude in all four versions of the super Virasoro minimal string, a class of exactly solvable 2D string theories dual to random matrices. The results match independent matrix model predictions, after fixing errors in previously published conformal block formulas and constructing new modular kernels.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The load-bearing element is the postulated odd-spin block FDV1 of §3.3: the 0A- constant -1/16 and 0B- zero both rest on the odd contribution I^-_1 = -1/32 built from FDV1, so a wrong identification shifts both.","rationale":"The reader's weakest_assumption identifies exactly the element I find load-bearing. The signatures of the paper — the momentum-independent 0A- = -1/16 and the vanishing 0B- = 0 — come from equal even and odd spin structure contributions (-1/32 each), and the odd contribution depends wholly on the postulated block FDV1; the reader correctly isolated this. The concern is genuine but disclosed: the authors state in §3.3 that they 'simply postulate' the block, and there is real supporting evidence in the modular covariance of ⟨DV⟩_1 (10^-13 spacelike, 5×10^-7 timelike) and the consistency of the DVS(11) kernel. Credit is also due for the fact that only one overall normalization is fitted (5.64), leaving three torus amplitudes and the sphere/mixed data as genuine predictions, and for the external anchoring through the [25] predictions and the recovered NS kernels of [36]. However, the modular-covariance checks use the same recursion-built blocks, so a systematic error in the identification of the level-3/2 descendant block with H^W_even would not be exposed; the definitive resolution is the first-principles derivation the authors defer, or an independent low-order computation of ⟨DV⟩_1. This does not change the reader's CONDITIONAL verdict: the concern is confirmable, disclosed, and consistent with the evidence presented, so I keep the verdict unchanged.","tokens_in":60476,"tokens_out":9802,"duration_ms":53397,"concrete_test":"Compute the low-order q-expansion (m = 0, 1, 2) of the ⟨~G_{-3/2}G_{-3/2}V⟩_1 block directly from the superconformal algebra — by Gram-matrix diagonalization or an OPE/free-field evaluation of the descendant one-point trace in the odd spin structure — without using the elliptic recursion of §3.2. Compare term-by-term with the postulation FDV1 = e^{iπ/4} P q^{P²/2} H^W_even (3.12). If any coefficient beyond numerical noise disagrees, the postulation fails and I^-_1 = -1/32 (hence A_{0A-} = -1/16 and A_{0B-} = 0) must be recomputed; agreement at the tested orders would materially strengthen the claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (5.36) has three amplitudes resting on the odd spin structure contribution I^-_1. From (5.39)-(5.40), N_NS C_T2/4 (I^-_3+I^-_4+I^-_2) = -1/32 and N_NS C_T2/4 I^-_1 = -1/32; summing with the signs of (5.34) gives A_{0A-} = -1/32 + (-1/32) = -1/16 and A_{0B-} = -1/32 - (-1/32) = 0. Thus the two most distinctive predictions — the exactly constant hard-edge amplitude and the exact vanishing — are both controlled by I^-_1 = ∫_{F0} d²τ (⟨V⟩_1⟨DV⟩_1 - ⟨DV⟩_1⟨V⟩_1) (5.33), in which ⟨DV⟩_1 is assembled from the new block FDV1 = e^{iπ/4} P q^{P²/2} H^W_even (3.12). In §3.3 the authors write: 'Carefully following this chain of logic would result in a first principles derivation of the block FDV1. However, at this point, we can simply postulate that it is given by the even elliptic block, modified to account for the additional P factors.' The identification rests on two pillars: the Ward-identity computation fixing ⟨DV⟩_1 = i∫dP ρ_R(P) P² C_even (qq)^{P²/2}[1+...], where the ellipsis is asserted to 'assemble into |H^W_even|²'; and numerical modular covariance (3.54) to ~10^-13 (spacelike) / 5×10^-7 (timelike) plus the DVS(11) kernel check. These are consistency checks on the same recursion-built ingredients, not an independent derivation. A wrong assignment of the descendant block would change I^-_1 and hence both flagged amplitudes; note the exact 0B- = 0 cancellation relies on |I^-_1| matching the even-sector -1/32 at the 10^-7 level. The missing element, disclosed by the authors, is the first-principles derivation they defer, or an independent low-order computation of the descendant block.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper develops the torus one-point super Virasoro block technology for N=1 super Liouville theory, correcting the recursion data of [35], adding a new odd-spin-structure block (FDV1) for a level-3/2 superdescendant, and constructing eight modular S-kernels by a bootstrap ansatz. It reports modular covariance checks of eight torus one-point functions in both spacelike and timelike super Liouville theory, then applies the machinery to the four super Virasoro minimal string theories 0A^+/-, 0B^+/-. The main numerical results are the torus one-point amplitudes in Eq. (5.36): A_{0A+}=(4h-1)/48, A_{0B+}=(4h-1)/24, A_{0A-}=-1/16, A_{0B-}=0, quoted as agreeing with the topological-recursion and intersection-theory predictions of [25] to about 10^-7. The paper also identifies a zero-momentum Ramond insertion in 0A^- and computes a mixed sphere amplitude equal to unity. One global vertex-operator normalization N_NS is fixed by matching the 0A+ torus amplitude to the intersection-theory result, as stated around Eq. (5.64).","tokens_in":61291,"tokens_out":6163,"duration_ms":75674,"significance":"If the central identification is correct, this is a significant step: it provides a complete RNS worldsheet computation of torus one-point amplitudes in all four super Virasoro minimal string theories, including the constant hard-edge amplitude of 0A^- and the vanishing 0B^- amplitude, and it supplies a full set of modular S-kernels for the N=1 torus. The numerical work is extensive and careful, with modular-covariance deviations at the 10^-13 level (spacelike) and 5x10^-7 level (timelike), kernel crossing checks at the 10^-11 to 10^-7 level, and amplitude comparisons at the 10^-7 level. The new odd-spin-structure block is the key innovation, but it is explicitly postulated rather than derived, and it controls the most distinctive predictions for 0A^- and 0B^-. The one fitted normalization is disclosed and is not, by itself, disqualifying; after fixing it, the functional forms, the vanishing amplitude, and the constant amplitude are genuine predictions.","major_comments":[{"comment":"The identification of FDV1 is the load-bearing element of the 0A^- and 0B^- torus amplitudes. In Eq. (3.45) the omitted terms are asserted to 'assemble into |H^W_even|^2', and shortly afterwards the block is 'simply postulate[d]' rather than derived. From Eqs. (5.39)-(5.40), I^-_1 = -1/32 is built from FDV1, and through Eq. (5.34) it produces both A_{0A-}=-1/16 and the exact cancellation A_{0B-}=0 in Eq. (5.36). The numerical modular-covariance check (3.54) and the DVS(11) kernel test check the same recursion-built object and therefore do not constitute an independent validation of the descendant block. I ask for a first-principles derivation of FDV1, or at least an independent low-order check (for example, direct evaluation of the first few q-coefficients from the superconformal algebra and the picture-changing procedure), before the claim of a complete worldsheet computation is made.","section":"3.3, Eqs. (3.12), (3.45), and (5.34)-(5.40)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The normalization fixing makes part of the comparison circular in a way that should be stated more prominently. Equation (5.64) fixes N_NS by matching the 0A+ torus amplitude, Eq. (5.65) fixes C_S2 by setting the 0A+/0B+ sphere three-point amplitude to unity, and Eq. (5.66) fixes N_R by requiring the 0B+ mixed sphere amplitude to be unity. Consequently the equality of the 0A^- Ramond-puncture amplitude to unity in Eq. (5.63) follows directly from the chosen normalizations, not from an independent worldsheet prediction. I recommend that the text explicitly separate the genuinely predictive comparisons (the P-dependence and relative factors in Eq. (5.36), and the vanishing 0B^- amplitude) from the normalization-fixing conventions used to set N_NS, C_S2, and N_R.","section":"5.4, Eqs. (5.63)-(5.66)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The eight S-kernels are obtained by positing bootstrap conditions rather than by solving the defining Moore-Seiberg consistency relations. The manuscript verifies the crossing transformations numerically and checks idempotency-type relations, which is valuable, but it does not prove that the ansatz (universal product structure of Eq. (4.3) plus the listed parity and residue conditions) has a unique solution. Since the abstract describes modular covariance as 'demonstrated' partly through these kernels, I suggest softening the claim to 'strongly supported by numerical checks' or adding an analytic uniqueness argument for the kernel bootstrap.","section":"4.2-4.4, Eqs. (4.3)-(4.10)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The abstract says the structure constants 'satisfy modular crossing', but the timelike verification in Section 3.4 is numerical at the 5x10^-7 level, and the kernels in Section 4 are bootstrap ansatze verified numerically. I suggest wording such as 'we provide strong numerical evidence for modular crossing' in the abstract to match the actual status.","section":"Abstract and Section 3.4","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The exchange of the 0A^- and 0B^- labels relative to [25] is important and easy to miss in a footnote. A small table comparing the labels used here, those of [25], and those of [2] for the semiclassical limit would make the conventions much clearer.","section":"Footnote 4 and Section 5.4","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence 'we have not derived the blocks from first principles as they did' is confusing because the paper does supply recursion relations and checks for these blocks. Clarify that the statement refers specifically to the absence of an independent derivation of the corrections to the [35] recursion data, not to the recursive representation itself.","section":"Section 3.2, paragraph before Eq. (3.23)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The timelike amplitudes use q^{P^2/2} \\bar q^{P^2/2} without the \\Vert\\cdot\\Vert^2 shorthand of Eq. (3.10). Repeating the definition of the no-complex-conjugation assembly rule directly in Appendix B, or pointing the reader back to Eq. (3.10), would prevent a natural misreading of the contour-shifted timelike integrals.","section":"Appendix B, Eq. (B.4)","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"This is a substantial paper with a clear central claim and unusually detailed numerical checks. The main issue is the unproved odd-spin-structure block FDV1, which is explicitly acknowledged by the authors as a postulate but is needed for the two most distinctive amplitude results. I would condition acceptance on either a derivation of FDV1 or an independent low-order verification, and on a more transparent separation between normalization-fixing conventions and genuine predictions in Section 5.4. The paper otherwise appears publishable in a leading hep-th journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"The paper does what it says: it fixes three bugs in the [35] torus one-point block recursion, identifies a genuinely new odd-spin-structure block for the level-3/2 superdescendant, constructs six new modular S-kernels, and completes the worldsheet torus one-point computation in all four super Virasoro minimal strings. The headline result — A_{0A+}=(4h-1)/48, A_{0B+}=(4h-1)/24, A_{0A-}=-1/16, A_{0B-}=0 — is verified numerically to roughly 10^-7 and matches the matrix-model predictions of [25]. That is a real closing of a gap in an actively developed solvable-string program.\n\nThe technical foundations feel solid. The recursion corrections are motivated by both Ward identities and modular covariance: the missing q^{P^2/2} factor is forced by the cusp behavior, and the f^I_{r,s} momentum factors are needed to get anything to transform correctly. Restarting the recursion at order 8 with 40-digit precision, checked at 10^-13 in the spacelike theory, is genuinely independent confirmation of the block data. The fact that the six new kernels reproduce the two known NS kernels of [36] is a good external anchor.\n\nThe soft spot is the one the authors flag themselves in section 3.3: the new block FDV1 is postulated, not derived. The 0A- constant amplitude and the 0B- vanishing both rest on the odd-spin contribution I^-_1 built from it, so a wrong identification would shift both. This is not an error, it is an honest gap, and the numerical modular covariance plus kernel crossing checks are consistency tests on the same ingredients, not an independent derivation. The asserted wording in (3.45) that the omitted terms \"assemble into |H^W_even|^2\" is the genuinely softest point. One normalization N_NS is fitted to match intersection theory; that is fine, because the functional forms and the constant and vanishing amplitudes are predictions, not fits.\n\nWho gets value: anyone working on solvable 2d strings, super Liouville CFT, or modular kernels of N=1 SCFT. It deserves a serious referee — the numerical verification burden is nontrivial but well documented. The main thing a referee should push on is the derivation gap for FDV1; if the deferred first-principles argument or an independent low-order computation does not appear, the claim stays conditional rather than proven. That does not change my recommendation: send it.","headline":"Careful, technically serious paper that completes the SVMS torus computation and builds reusable super-Liouville technology; the one soft spot — a postulated block — is disclosed and well-mitigated by numerics.","tokens_in":61704,"tokens_out":1964,"would_cite":true,"duration_ms":16725,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"The torus one-point amplitudes of all four super Virasoro minimal string theories, computed directly from the RNS worldsheet, agree with topological recursion on the spectral curves of the dual matrix integrals…","keywords":["super Liouville theory","torus one-point function","superconformal blocks","modular S-kernel","super Virasoro minimal string","matrix integral","topological recursion","Ramond puncture"],"falsifier":"A first-principles computation of the torus one-point function of $\\widetilde G_{-3/2}G_{-3/2}V$ that does not postulate the block would settle the claim: if the exact answer differs from the even elliptic block, the $0\\mathrm{A}^-$ torus amplitude would no longer be the constant $-1/16$ and the $0\\mathrm{B}^-$ amplitude would not vanish. A concrete check is to push the numerical modular-covariance test in the $\\tilde R$ sector toward the cusp, where the block's leading behaviour is governed by the postulated $q^{P^2/2}$ zero-mode factor and the $P^2$ prefactor; any mismatch there would expose the error.","tokens_in":61860,"feed_emoji":"🎻","tokens_out":7214,"duration_ms":40585,"temperature":0.7,"pith_summary":"This paper completes the worldsheet-side toolkit for $N=1$ super Liouville theory on the torus and uses it to test a conjectured string/matrix-model duality. It corrects the recursive representations of the torus one-point super Virasoro blocks, identifies a new block in the odd spin structure built on a distinguished superdescendant, and constructs modular crossing kernels for all four spin structure sectors. Applying the technology to the super Virasoro minimal string, it computes the NS-sector sphere three-point and torus one-point amplitudes in all four theories ($0\\mathrm{A}^{\\pm}$ and $0\\mathrm{B}^{\\pm}$) directly from the RNS worldsheet and finds they agree with topological recursion on the dual matrix integral spectral curves. The agreement matters because it confirms the super Virasoro minimal string duality, including the momentum-independent hard-edge amplitude of the $0\\mathrm{A}^-$ theory and the vanishing of the stable NS amplitudes of the $0\\mathrm{B}^-$ theory.","feed_headline":"Four superstring torus amplitudes match matrix integrals","feed_subtitle":"Worldsheet computation confirms the duality, plus a constant hard-edge amplitude and one trivial theory.","key_machinery":"The carrying object is the distinguished level three-half superdescendant $\\widetilde G_{-3/2}G_{-3/2}V_P$, whose torus one-point function is non-vanishing only in the odd spin structure ($\\tilde R$) sector and behaves like that of a Virasoro primary. Superconformal Ward identities fix the three-point functions of this descendant, and its block $F^{DV}_1$ is combined with the picture changing operator to regulate the ghost zero-mode ambiguity in the $\\tilde R$ sector. The technical apparatus consists of corrected Zamolodchikov-type elliptic recursion relations generating all torus one-point blocks, the eight modular $S$-kernels (six of them new) that express the blocks in the modular dual channel, and the structure constants and spectral densities of the spacelike and timelike super Liouville theories, whose modular covariance anchors the whole construction.","core_discovery":"The central claim is that the torus one-point amplitudes of the four super Virasoro minimal string theories, computed directly from the RNS worldsheet, equal the predictions of topological recursion on the spectral curves of the dual matrix integrals, with the simple values $A^{0\\mathrm{A}+}_{1,(1|0)}=(4h_{\\rm ext}-1)/48$, $A^{0\\mathrm{B}+}_{1,(1|0)}=(4h_{\\rm ext}-1)/24$, $A^{0\\mathrm{A}-}_{1,(1|0)}=-1/16$, and $A^{0\\mathrm{B}-}_{1,(1|0)}=0$, where $h_{\\rm ext}=\\frac18 Q^2+\\frac12 P_{\\rm ext}^2$. Along the way the paper establishes modular covariance of the torus one-point functions in both spacelike and timelike super Liouville theory and provides the missing modular crossing kernels. The computation hinges on a new object: the level three-half superdescendant $\\widetilde G_{-3/2}G_{-3/2}V_P$, whose torus one-point function transforms like that of a Virasoro primary and is non-vanishing only in the odd spin structure; its one-point function supplies the $\\tilde R$ contribution to the string amplitude through a careful treatment of the picture changing operator. The paper also identifies an exceptional zero-momentum Ramond insertion in the $0\\mathrm{A}^-$ theory despite the absence of generic Ramond states, interprets it as a Ramond puncture, and matches its mixed sphere three-point amplitude to the intersection-theory prediction, taking the value unity.","pith_inferences":["The block decomposition of the distinguished superdescendant is theory-independent, so $\\tilde R$-sector torus one-point functions of this operator in other $N=1$ SCFTs should follow by substituting their structure constants and spectral densities; the paper suggests this route but does not carry it out.","Summing a gas of the $0\\mathrm{A}^-$ Ramond punctures should deform the hard-edge spectral curve of the dual matrix integral, in parallel with turning on Ramond-Ramond flux in the $\\hat c=1$ Type 0A string; that sum is left as future work.","The kernel bootstrap strategy, fixed by analytic structure, parity, and idempotency rather than by solving Moore-Seiberg relations, should extend to $N=1$ fusion kernels for sphere four-point amplitudes and connect to the $U_q(\\mathrm{osp}(1|2))$ 6j-symbols; the paper notes that this map is not yet precise."],"forward_implications":["If the amplitudes are correct, the super Virasoro minimal string duality of [25] is confirmed from the worldsheet, including the momentum-independent constant $-1/16$ amplitude of the $0\\mathrm{A}^-$ theory with its hard-edge spectral curve.","The four string theories are cleanly distinguished by the GSO projection together with the relative sign of the total worldsheet supercharge, and the sign ambiguity in the timelike super Liouville structure constants is fixed by modular covariance.","The corrected torus blocks and the eight modular $S$-kernels are theory-independent $N=1$ kinematic data, so torus one-point functions of any $N=1$ SCFT can be assembled from them.","The exceptional zero-momentum Ramond insertion in $0\\mathrm{A}^-$ is a worldsheet candidate for the Ramond punctures of JT supergravity and its dual matrix integral."],"supporting_citations":[{"why":"Supplies the spectral curves, intersection-theory predictions, and the four-theory classification from 3d supergravity that the worldsheet computations must reproduce.","marker":"[25]"},{"why":"Provides the original recursive representations of the $N=1$ torus one-point blocks that this paper corrects and supplements with the new odd spin structure block.","marker":"[35]"},{"why":"Defines the bosonic Virasoro minimal string and its spectral curve, which governs the $0\\mathrm{A}^+$ and $0\\mathrm{B}^+$ amplitudes.","marker":"[3]"},{"why":"Establishes the JT supergravity/matrix integral duality, defines the Ramond punctures matched to the $0\\mathrm{A}^-$ zero-momentum insertion, and provides the semiclassical limit of the minus theories.","marker":"[2]"},{"why":"Bootstraps the timelike $N=1$ Liouville structure constants and records the relative sign ambiguity that this paper resolves by modular covariance.","marker":"[22]"},{"why":"Bootstraps timelike super Liouville structure constants and initiates the worldsheet study of the super Virasoro minimal string that this paper completes.","marker":"[23]"},{"why":"Constructs the NS-sector modular $S$-kernels that this paper's bootstrap reproduces and extends to the remaining six kernels.","marker":"[36]"},{"why":"Supplies the picture-changing operator treatment of the odd spin structure ghost correlator that resolves the $\\tilde R$-sector ambiguity.","marker":"[37]"}],"fun_headline_variants":["Super Virasoro torus amplitudes match matrix integrals","All four superstring torus amplitudes agree with recursion","Odd-spin superdescendant bridges worldsheet to matrix model","Torus bootstrap yields crossing kernels for super Liouville"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The odd spin structure block $F^{DV}_1$ that produces the non-vanishing correlator $\\langle \\widetilde G_{-3/2}G_{-3/2}V\\rangle_1$ is assumed rather than derived: the authors write that they 'can simply postulate that it is given by the even elliptic block, modified to account for the additional $P$ factors,' and the $\\tilde R$-sector contribution to the $0\\mathrm{A}^-$ and $0\\mathrm{B}^-$ torus amplitudes rests entirely on this postulate.","fun_headline_variants_meta":{"raw":{"variants":["Super Virasoro torus amplitudes match matrix integrals","All four superstring torus amplitudes agree with recursion","Odd-spin superdescendant bridges worldsheet to matrix model","Torus bootstrap yields crossing kernels for super Liouville"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"cost":0.000207788,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000207788,"upstream_inference_prompt_cost":0.000140868,"upstream_inference_completions_cost":6.692e-05},"total_tokens":1450,"prompt_tokens":1211,"completion_tokens":239,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":174}},"tokens_in":1211,"tokens_out":239,"cost_source":"openrouter_usage","duration_ms":2615,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":174,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-02T08:09:17.757730+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"A first-principles computation of the torus one-point function of $\\widetilde G_{-3/2}G_{-3/2}V$ that does not postulate the block would settle the claim: if the exact answer differs from the even elliptic block, the $0\\mathrm{A}^-$ torus amplitude would no longer be the constant $-1/16$ and the $0\\mathrm{B}^-$ amplitude would not vanish. A concrete check is to push the numerical modular-covariance test in the $\\tilde R$ sector toward the cusp, where the block's leading behaviour is governed by the postulated $q^{P^2/2}$ zero-mode factor and the $P^2$ prefactor; any mismatch there would expose the error.","supporting_citations":[{"cited_title":"The Super Virasoro Minimal String from 3d Supergravity","cited_arxiv_id":"2604.26038","evidence_quote":"Supplies the spectral curves, intersection-theory predictions, and the four-theory classification from 3d supergravity that the worldsheet computations must reproduce."},{"cited_title":"Recurrence relations for toric N=1 superconformal blocks","cited_arxiv_id":"1207.5740","evidence_quote":"Provides the original recursive representations of the $N=1$ torus one-point blocks that this paper corrects and supplements with the new odd spin structure block."},{"cited_title":"The Virasoro Minimal String","cited_arxiv_id":"2309.10846","evidence_quote":"Defines the bosonic Virasoro minimal string and its spectral curve, which governs the $0\\mathrm{A}^+$ and $0\\mathrm{B}^+$ amplitudes."},{"cited_title":"JT Gravity and the Ensembles of Random Matrix Theory","cited_arxiv_id":"1907.03363","evidence_quote":"Establishes the JT supergravity/matrix integral duality, defines the Ramond punctures matched to the $0\\mathrm{A}^-$ zero-momentum insertion, and provides the semiclassical limit of the minus theories."},{"cited_title":"On the three-point functions in timelike N=1 Liouville CFT","cited_arxiv_id":"2505.08890","evidence_quote":"Bootstraps the timelike $N=1$ Liouville structure constants and records the relative sign ambiguity that this paper resolves by modular covariance."},{"cited_title":"Towards the super Virasoro minimal string","cited_arxiv_id":"2505.08892","evidence_quote":"Bootstraps timelike super Liouville structure constants and initiates the worldsheet study of the super Virasoro minimal string that this paper completes."},{"cited_title":"S-move matrix in the NS sector of $N=1$ super Liouville field theory","cited_arxiv_id":"2310.03496","evidence_quote":"Constructs the NS-sector modular $S$-kernels that this paper's bootstrap reproduces and extends to the remaining six kernels."},{"cited_title":"Foundations of String Theory,https://github.com/xiyin137/stringbook","cited_arxiv_id":null,"evidence_quote":"Supplies the picture-changing operator treatment of the odd spin structure ghost correlator that resolves the $\\tilde R$-sector ambiguity."}],"review_version":1}