Pith. sign in

REVIEW 3 major objections 4 minor 58 references

The alpha-states of the Hurwitz worldsheet are labeled by Young diagrams, and their Hartle–Hawking weights are the Poissonized Plancherel measure.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 23:38 UTC pith:N2NI32IS

load-bearing objection The α-state identification is exact and well-grounded; the CLT/pseudorandomness claim is plausible but rests on a sketched asymptotic argument that needs either proof or clearer delegation. the 3 major comments →

arxiv 2607.15336 v1 pith:N2NI32IS submitted 2026-07-16 hep-th

The α-states of a string worldsheet

Suzanne Bintanja , Elliott Gesteau This is my paper
classification hep-th
keywords baby universe field theoryHurwitz worldsheetalpha-statessymmetric group representationsPlancherel measureKerov's central limit theorempseudorandomnesstopological string theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to show that the baby universe field theory of the Hurwitz worldsheet—a topological string theory obtained as a limit of string theory on AdS3—has a basis of alpha-states labeled by Young diagrams. The expectation value of a string insertion in an alpha-state is the normalized character of the symmetric group on the corresponding irreducible representation, and the probability of finding an alpha-state in the Hartle–Hawking state is the Poissonized Plancherel measure. Because string amplitudes can be rewritten as moments in this ensemble, the paper derives that in the weak-coupling limit (large p), normalized vertex-operator expectation values become independent Gaussian random variables—pseudorandom. This gives a physical interpretation of Kerov's central limit theorem and, in the dual Hurwitz TQFT, means that cluster-decomposing states become pseudorandom in the large-N limit.

Core claim

The paper establishes that the baby-universe Hilbert space of the Hurwitz worldsheet has an orthonormal basis |R⟩ labeled by all irreducible representations R of symmetric groups S_N (equivalently, Young diagrams with any number of boxes). In these states, the insertion operator Ẑ(μ) has expectation value equal to the normalized character |μ| χ_R(μ)/dim R (Eq. 11), and the Hartle–Hawking state decomposes over alpha-states with probabilities given by the Poissonized Plancherel measure e^{-p} p^{|R|} (dim R/|R|!)² (Eq. 12). The paper then uses the genus expansion of the worldsheet to show that, as p→∞ (g_s→0), the joint moments of these normalized characters converge to those of independent Ga

What carries the argument

The central identity is the Burnside-type character formula expressing Hurwitz numbers of the sphere as sums over symmetric-group irreps: H^•N_{μ1,...,μk} = Σ_R (dim R)^2/N! ∏ (|μ_i| χ_R(μ_i)/dim R). This formula converts gravitational path-integral amplitudes into sums over representation-theoretic data, making the alpha-states readable as normalized characters and the Hartle–Hawking overlaps as the Poissonized Plancherel measure. The subsequent Gaussian limit relies on the large-p suppression of connected worldsheets, which leaves only pairwise disconnected pairings to dominate the moments.

Load-bearing premise

The Gaussian limit follows only if the large-p amplitude is dominated by disconnected worldsheets that pair insertions and every other covering is suppressed by a positive power of p; if connected covers contribute at the same order for some winding profiles, the pseudorandomness conclusion for those profiles would not follow.

What would settle it

Evaluate the subleading-in-p term in the three-point amplitude for distinct single-cycle classes μ1, μ2, μ3. If the ratio to the Gaussian-predicted leading term does not vanish as p→∞—that is, if a connected three-sheeted cover contributes at the same order as the disconnected pairings—the derivation collapses. More broadly, scan the genus expansion for any connected covering whose contribution is not suppressed by a power of p relative to the disconnected pairings.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The baby universe Hilbert space of the Hurwitz worldsheet is exactly the direct sum of Hurwitz TQFT Hilbert spaces for all N, with alpha-states as cluster-decomposing states.
  • String amplitudes in the Hartle–Hawking state are moments of normalized characters sampled from the Poissonized Plancherel measure.
  • As g_s→0, vertex-operator expectation values in alpha-states become statistically indistinguishable from independent Gaussians; they are pseudorandom by any fixed number of measurements.
  • Restricting to N-sheeted covers gives the microcanonical ensemble with the ordinary Plancherel measure, and the thermodynamic-limit equivalence of ensembles follows.
  • The worldsheet provides a physical derivation of Kerov's central limit theorem in the grand-canonical ensemble, connecting string perturbation theory to asymptotic representation theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the same character-sum structure appears in other worldsheet theories, their alpha-states may also be labeled by representation-theoretic data, and their weak-coupling ensembles would inherit limit theorems from asymptotic representation theory.
  • The power-law (rather than exponentially suppressed) corrections in g_s suggest that subleading genus terms encode non-Gaussian deviations of the alpha-state ensemble; these could serve as a diagnostic of connected topology-change effects.
  • The identification of alpha-states with normalized characters implies that the worldsheet's baby universe Hilbert space carries a commutative algebra isomorphic to the algebra of class functions on all symmetric groups; one could test whether the operators Ẑ(μ) generate it exactly.
  • A converse program becomes plausible: given a BUFT with an ensemble of alpha-states whose moments are known, one might reconstruct the underlying worldsheet or TQFT from representation-theoretic data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper identifies the α-states of the baby-universe field theory associated with the Hurwitz worldsheet. Starting from the Hurwitz-number class-algebra formula (Burnside, Eq. (9)), it rewrites the grand-canonical worldsheet amplitude as a weighted sum over all Young diagrams (Eq. (10)). This is interpreted as the BUFT moment formula: the baby-universe Hilbert space has an orthonormal basis |R⟩ labeled by irreducible representations of symmetric groups, with expectation values ⟨R|Ẑ(μ)|R⟩ = |μ| χ_R(μ)/dim R (Eq. (11)) and Hartle–Hawking weights given by the Poissonized Plancherel measure (Eq. (12)). The Letter then uses an asserted dominance of disconnected worldsheets to compute the large-p moments of these α-state expectation values, identifies them with Gaussian joint moments (Eqs. (15)–(16)), and concludes that the ensemble becomes pseudorandom and that this is a physical interpretation of Kerov's central limit theorem. A dual Hurwitz-TQFT interpretation in terms of cluster-decomposing states is outlined and deferred to forthcoming work [41].

Significance. The exact α-state identification is a clean and elegant result: if Eqs. (10)–(12) hold — and they follow from Burnside's class-algebra formula plus the localization of the Hurwitz worldsheet to covering maps — the paper provides a concrete bridge between string-worldsheet BUFT and asymptotic representation theory. The decomposition is parameter-free, the measure is manifestly positive, and the Hartle–Hawking state is explicitly the Poissonized Plancherel ensemble. The advertised CLT/pseudorandomness claim, if established from the worldsheet amplitudes themselves, would give a striking physical reinterpretation of Kerov's theorem and is falsifiable via the joint moments in Eqs. (15)–(16). However, the CLT step is not proved in the manuscript: it rests on an unquantified assertion about disconnected-worldsheet dominance. The paper's significance therefore depends on closing that gap; the exact α-state part is solid and is not affected by the gap.

major comments (3)
  1. [§4, Eqs. (15)–(16)] The Gaussian moment formulas are the linchpin of the advertised CLT/pseudorandomness result, but they are justified only by the assertion that 'the dominant contribution ... comes from disconnected worldsheets that connect the insertions μ_i pairwise when possible' and by references to [43] and [34]. Neither reference proves this suppression for the Hurwitz numbers H^•_{μ_1,...,μ_k} for arbitrary single-cycle profiles. [27] proves Kerov's theorem for characters but does not address this worldsheet expansion; [34] concerns symmetric-orbifold correlators but does not analyze the connected/disconnected competition at the required order. Because Eq. (10) is exact, one could derive (15)–(16) from known character asymptotics, but then the 'worldsheet derivation' would be importing Kerov's theorem rather than deriving it. Please provide a self-contained asymptotic estimate, or a precise externa
  2. [§4 and Appendix] The passage from convergence of all joint moments to convergence in distribution is delegated to [27] 'under minor technical assumptions satisfied here', but the relevant boundedness or moment-determinacy condition is not verified for the random variables |μ| χ_R(μ)/dim R under the Poissonized Plancherel measure. If Eqs. (15)–(16) are meant as a self-contained worldsheet derivation, this verification is part of the proof. In addition, the theorem as stated includes X_k for k ≤ n, while the limit in Eq. (20) omits X_1; since the body uses single-cycle insertions and the case μ_i = 1 has different behavior, the precise range of k and the convention for X_1 need to be stated.
  3. [§3, Eq. (14)] The equivalence between the microcanonical state |HH_N⟩ and the grand-canonical state |HH⟩ at leading order for O(1) windings is quoted from [34] without an error estimate. This equivalence is used to connect the p → ∞ CLT in the worldsheet ensemble to the standard Plancherel-measure CLT on S_N. A precise statement of the equivalence — for example, uniform in the relevant Young diagrams and character values — is needed, or the CLT should be formulated directly in the grand-canonical ensemble without invoking the microcanonical limit.
minor comments (4)
  1. [§2–§4] The symbol μ is overloaded: it denotes a conjugacy class, the length of a single cycle, and via |μ| the size of the class. In Eqs. (15)–(17) it is genuinely hard to tell whether the exponent p^{μ_i} is the cycle length or the class size. Suggest writing ℓ_i for the cycle length and c_i for the class size. Also state the proportionality constant in g_s^{-2} ∝ p if it is convention-dependent.
  2. [§3, Eq. (13)] The exact isomorphism H_BU ≅ ⊕_N H_N is asserted in Eq. (13) after the baby-universe Hilbert space was defined by a null-state quotient. It would be helpful to specify the inner product on ⊕_N H_N and to show explicitly why the quotient leaves exactly this space.
  3. [§3] The statement that α-states correspond to cluster-decomposing states in Hurwitz TQFT is cited to the authors' forthcoming work [41]. If this identification is essential to the advertised boundary interpretation, it should be stated self-containedly (or at least with a precise definition of 'cluster-decomposing'); otherwise it should be labeled as a conjecture.
  4. [Appendix] The proof of Kerov's theorem is not given, but since the theorem is a central mathematical input, the statement should be aligned with the body's normalization. In particular, the relation between the theorem's X_k(R) and the body's ⟨R|Ẑ(k)|R⟩ should be written out: for a k-cycle, X_k = k ⟨R|Ẑ(k)|R⟩. This would also make the variance μ_i^{-1} in Eq. (17) transparent.

Circularity Check

0 steps flagged

No circular reduction: α-states follow from Burnside's formula and the CLT is identified with Kerov's theorem; self-citations and the asserted disconnected-worldsheet dominance are caveats but not circularity.

full rationale

The paper's central identification is not circular. Eq. (10) is obtained by substituting Burnside's class-algebra formula (Eq. (9), cited to [35], an external 1911 result) into the Hurwitz amplitude (7); the α-state dictionary (Eqs. (11)-(12)) is a coefficient match to the BUFT moment formula (5). No parameter is fitted to force the result. The CLT section imports Kerov's theorem [24-27] as external mathematics rather than deriving it from a fitted input; the worldsheet genus-expansion input (Eqs. (15)-(16)) is asserted with details delegated to [27] and [34], which is a gap in self-containedness for the pseudorandomness claim, but not a definitional circular reduction. Specifically, the passage "The dominant contribution ... comes from disconnected worldsheets that connect the insertions μ_i pairwise when possible [43]. All other contributions are suppressed by powers of p" (Section 'Kerov's central limit theorem from the worldsheet') is an unproven asymptotic assumption; I weigh it as a correctness risk, not circularity. The only self-citations ([41], and background refs. [12],[13],[42]) are used for interpretation, future work, and motivation; they are not load-bearing for the α-state or CLT claims. Overall the paper is a translation of known representation theory into BUFT language, with no circular step that reduces a prediction to its input.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

No free parameters are introduced: p and N are physical ensemble variables, not fitted constants, and the μ_i label conjugacy classes. The axioms are mostly standard mathematics (Burnside/Kerov) plus domain assumptions about the worldsheet/BUFT dictionary and the AdS3/Hurwitz realization. No new entities are postulated; α-states and the Hartle–Hawking state are pre-existing BUFT concepts, and Young diagrams label known irreducible representations.

axioms (7)
  • domain assumption Hurwitz numbers count ramified covers and equal the class-algebra structure constants of the symmetric group (Eq. 8).
    Used to identify worldsheet amplitudes with Hurwitz numbers; standard in Hurwitz theory, cited to [20–22].
  • domain assumption The worldsheet path integral localizes to covering surfaces of the S² boundary with weight g_s^{-2N}.
    This localization is imported from [22,29–31]; it is the bridge between string amplitudes and Hurwitz numbers.
  • standard math Burnside's class-algebra formula expands Hurwitz numbers as sums over irreducible representations (Eq. 9).
    Central mathematical identity; standard result, cited to [35].
  • domain assumption The baby universe inner product is positive semidefinite, so the GNS construction yields a Hilbert space H_BU.
    Required for the BUFT interpretation; the paper notes positivity and in this model positivity follows from the Plancherel weights, but the general BUFT premise is imported from [4].
  • domain assumption The microcanonical and grand-canonical ensembles are equivalent in the thermodynamic limit p ∼ N → ∞.
    Used to exchange the Plancherel and Poissonized Plancherel measures and to connect the string genus expansion to Kerov's CLT; cited to [34,57].
  • standard math Kerov's central limit theorem for normalized characters under the Plancherel measure.
    The mathematical content being reinterpreted physically; cited to [24–27].
  • domain assumption The dual of the Hurwitz worldsheet is Hurwitz TQFT, and α-states map to cluster-decomposing states.
    Boundary interpretation; supported by [20–23] but partly deferred to the unpublished companion [41].

pith-pipeline@v1.3.0-alltime-deepseek · 9716 in / 18138 out tokens · 188527 ms · 2026-08-01T23:38:38.165453+00:00 · methodology

0 comments
read the original abstract

The worldsheet description of string amplitudes can be reinterpreted as a baby universe field theory. Under this reinterpretation, we determine the $\alpha$-states of a topological string theory: the Hurwitz worldsheet. We find that their relation to the Hartle--Hawking state encodes foundational results in the asymptotic representation theory of the symmetric group. In the holographic dual, our results characterize the pseudorandom nature of an ensemble of cluster-decomposing correlators.

Figures

Figures reproduced from arXiv: 2607.15336 by Elliott Gesteau, Suzanne Bintanja.

Figure 1
Figure 1. Figure 1: FIG. 1: The Young diagram associated with the partition [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

58 extracted references · 34 linked inside Pith

  1. [1]

    S. R. Coleman, Black holes as red herrings: Topological fluctuations and the loss of quantum coherence, Nucl. Phys. B307, 867 (1988)

    1988

  2. [2]

    S. B. Giddings and A. Strominger, Loss of incoherence and determination of coupling constants in quantum gravity, Nucl. Phys. B307, 854 (1988)

    1988

  3. [3]

    S. B. Giddings and A. Strominger, Baby Universes, Third Quantization and the Cosmological Constant, Nucl. Phys. B321, 481 (1989)

    1989

  4. [4]

    Marolf and H

    D. Marolf and H. Maxfield, Transcending the ensem- ble: baby universes, spacetime wormholes, and the or- der and disorder of black hole information, (2020), arXiv:2002.08950 [hep-th]

    Pith/arXiv arXiv 2020

  5. [5]

    J. M. Maldacena and L. Maoz, Wormholes in AdS, JHEP 02, 053, arXiv:hep-th/0401024

    Pith/arXiv arXiv

  6. [6]

    Penington, Entanglement Wedge Reconstruction and the Information Paradox, (2019), arXiv:1905.08255 [hep- th]

    G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, (2019), arXiv:1905.08255 [hep- th]

    Pith/arXiv arXiv 2019

  7. [7]

    McNamara and C

    J. McNamara and C. Vafa, Baby Universes, Holography, and the Swampland, (2020), arXiv:2004.06738 [hep-th]

    Pith/arXiv arXiv 2020

  8. [8]

    Usatyuk and Y

    M. Usatyuk and Y. Zhao, Closed universes, fac- torization, and ensemble averaging, JHEP02, 052, arXiv:2403.13047 [hep-th]

    Pith/arXiv arXiv

  9. [9]

    Usatyuk, Z.-Y

    M. Usatyuk, Z.-Y. Wang, and Y. Zhao, Closed universes in two dimensional gravity, SciPost Phys.17, 051 (2024), arXiv:2402.00098 [hep-th]

    Pith/arXiv arXiv 2024

  10. [10]

    Harlow, M

    D. Harlow, M. Usatyuk, and Y. Zhao, Quantum mechan- ics and observers for gravity in a closed universe, (2025), arXiv:2501.02359 [hep-th]

    Pith/arXiv arXiv 2025

  11. [11]

    A. I. Abdalla, S. Antonini, L. V. Iliesiu, and A. Levine, The gravitational path integral from an observer’s point of view, JHEP05, 059, arXiv:2501.02632 [hep-th]

    Pith/arXiv arXiv

  12. [12]

    Engelhardt and E

    N. Engelhardt and E. Gesteau, Further Evidence Against a Semiclassical Baby Universe in AdS/CFT, (2025), arXiv:2504.14586 [hep-th]

    Pith/arXiv arXiv 2025

  13. [13]

    Gesteau, A no-go theorem for largeNclosed universes, (2025), arXiv:2509.14338 [hep-th]

    E. Gesteau, A no-go theorem for largeNclosed universes, (2025), arXiv:2509.14338 [hep-th]

    arXiv 2025

  14. [14]

    P. Saad, S. H. Shenker, and D. Stanford, JT gravity as a matrix integral, (2019), arXiv:1903.11115 [hep-th]

    Pith/arXiv arXiv 2019

  15. [15]

    Casali, D

    E. Casali, D. Marolf, H. Maxfield, and M. Rangamani, Baby universes and worldline field theories, Class. Quant. Grav.39, 134004 (2022), arXiv:2101.12221 [hep-th]

    Pith/arXiv arXiv 2022

  16. [16]

    B. Post, J. van der Heijden, and E. Verlinde, A universe field theory for JT gravity, JHEP05, 118, arXiv:2201.08859 [hep-th]

    Pith/arXiv arXiv

  17. [17]

    Harlow, Observers,α-parameters, and the Hartle- Hawking state, (2026), arXiv:2602.03835 [hep-th]

    D. Harlow, Observers,α-parameters, and the Hartle- Hawking state, (2026), arXiv:2602.03835 [hep-th]

    arXiv 2026

  18. [18]

    Zhao, ”It from Bit”: The Hartle-Hawking state and quantum mechanics for de Sitter observers, (2026), arXiv:2602.05939 [hep-th]

    Y. Zhao, ”It from Bit”: The Hartle-Hawking state and quantum mechanics for de Sitter observers, (2026), arXiv:2602.05939 [hep-th]

    arXiv 2026

  19. [19]

    In fact, the dual TQFT relevant to this letter does factorize

    This does not mean that a dual gauge theory does not factorize. In fact, the dual TQFT relevant to this letter does factorize. In this case, theα-states should be identi- fied with different states of the dual TQFT, see e.g. [58]

  20. [20]

    Okounkov and R

    A. Okounkov and R. Pandharipande, Gromov-witten theory, hurwitz theory, and completed cycles (2002), arXiv:math/0204305 [math.AG]

    Pith/arXiv arXiv 2002

  21. [21]

    Benizri and J

    L. Benizri and J. Troost, Symmetric group gauge theories and simple gauge/string dualities, J. Phys. A57, 505401 (2024), arXiv:2404.12543 [hep-th]

    Pith/arXiv arXiv 2024

  22. [22]

    Komatsu and P

    S. Komatsu and P. Maity, String Duals of Two- Dimensional Yang-Mills Theory and Symmetric Prod- uct Orbifolds, Phys. Rev. Lett.135, 231603 (2025), arXiv:2506.21663 [hep-th]

    arXiv 2025

  23. [23]

    Gaberdiel, B

    M. Gaberdiel, B. Knighton, and K. Naderi, To Appear, (2026)

    2026

  24. [24]

    V. N. Ivanov and S. V. Kerov, The algebra of conju- gacy classes in symmetric groups, and partial permuta- tions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)256, 95 (1999), teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 3

    1999

  25. [25]

    V. N. Ivanov and S. V. Kerov, The algebra of conjugacy classes in symmetric groups and partial permutations, Journal of Mathematical Sciences107, 4212 (2001)

    2001

  26. [26]

    Kerov, Gaussian limit for the plancherel measure of the symmetric group, Comptes Rendus de l Acad´ emie des Sciences - Series I - Mathematics316(2000)

    S. Kerov, Gaussian limit for the plancherel measure of the symmetric group, Comptes Rendus de l Acad´ emie des Sciences - Series I - Mathematics316(2000)

    2000

  27. [27]

    Ivanov and G

    V. Ivanov and G. Olshanski, Kerov’s central limit theo- rem for the Plancherel measure on Young diagrams, in Symmetric functions 2001: surveys of developments and perspectives(Springer, 2002) pp. 93–151

    2001

  28. [28]

    Knighton, (private communication), (2026)

    B. Knighton, (private communication), (2026)

    2026

  29. [29]

    Eberhardt, M

    L. Eberhardt, M. R. Gaberdiel, and R. Gopakumar, De- riving the AdS 3/CFT2 correspondence, JHEP02, 136, arXiv:1911.00378 [hep-th]

    Pith/arXiv arXiv 1911

  30. [30]

    A. Dei, M. R. Gaberdiel, R. Gopakumar, and B. Knighton, Free field world-sheet correlators for AdS 3, JHEP02, 081, arXiv:2009.11306 [hep-th]

    Pith/arXiv arXiv 2009

  31. [31]

    Knighton, Higher genus correlators for tensionless AdS3 strings, JHEP04, 211, arXiv:2012.01445 [hep-th]

    B. Knighton, Higher genus correlators for tensionless AdS3 strings, JHEP04, 211, arXiv:2012.01445 [hep-th]

    Pith/arXiv arXiv 2012

  32. [32]

    The symmetry factor is equal to the inverse of the order of the automorphism group of the covering map

  33. [33]

    See [34] for a discussion on the choices of normalization for the vertex operators

  34. [34]

    Aharony and E

    O. Aharony and E. Y. Urbach, Type II string theory on AdS3×S3×T4 and symmetric orbifolds, Phys. Rev. D 7 110, 046028 (2024), arXiv:2406.14605 [hep-th]

    Pith/arXiv arXiv 2024

  35. [35]

    Burnside,Theory of Groups of Finite Order, 2nd ed

    W. Burnside,Theory of Groups of Finite Order, 2nd ed. (Cambridge University Press, Cambridge, 1911) reprinted by Dover, New York, 1955. The class algebra constants formula appears at§221, p. 316

    1911

  36. [36]

    L. V. Bogachev and Z. G. Su, Central limit theorem for random partitions under the plancherel measure, arXiv preprint math/0607635 (2006), arXiv:math/0607635 [math.PR]

    Pith/arXiv arXiv 2006

  37. [37]

    Knighton, Deriving the long-string CFT in AdS 3, JHEP07, 260, arXiv:2410.16904 [hep-th]

    B. Knighton, Deriving the long-string CFT in AdS 3, JHEP07, 260, arXiv:2410.16904 [hep-th]

    Pith/arXiv arXiv

  38. [38]

    Dijkgraaf and E

    R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys.129, 393 (1990)

    1990

  39. [39]

    Dijkgraaf, Mirror symmetry and elliptic curves, inThe Moduli Space of Curves, Progress in Mathematics, Vol

    R. Dijkgraaf, Mirror symmetry and elliptic curves, inThe Moduli Space of Curves, Progress in Mathematics, Vol. 129, edited by R. H. Dijkgraaf, C. F. Faber, and G. B. M. van der Geer (Birkh¨ auser Boston, 1995) pp. 149–163

    1995

  40. [40]

    Li and J

    S. Li and J. Troost, The Topological Symmetric Orbifold, JHEP10, 201, arXiv:2006.09346 [hep-th]

    Pith/arXiv arXiv 2006

  41. [41]

    Bintanja and E

    S. Bintanja and E. Gesteau, To Appear, (2026)

    2026

  42. [42]

    Gesteau, M

    E. Gesteau, M. Marcolli, and J. McNamara, Wormhole Renormalization: The gravitational path integral, holog- raphy, and a gauge group for topology change, (2024), arXiv:2407.20324 [hep-th]

    Pith/arXiv arXiv 2024

  43. [43]

    Related to this fact is the string genus expansion [34]

  44. [44]

    Y. Chen, M. Junge, and N. Lashkari, Operator Algebras and Third Quantization, (2025), arXiv:2509.02293 [hep- th]

    Pith/arXiv arXiv 2025

  45. [45]

    Belin and J

    A. Belin and J. de Boer, Random Statistics of OPE Coefficients and Euclidean Wormholes, (2020), arXiv:2006.05499 [hep-th]

    Pith/arXiv arXiv 2020

  46. [46]

    Belin, J

    A. Belin, J. de Boer, and D. Liska, Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes, JHEP06, 116, arXiv:2110.14649 [hep-th]

    Pith/arXiv arXiv

  47. [47]

    Chandra, S

    J. Chandra, S. Collier, T. Hartman, and A. Maloney, Semiclassical 3D gravity as an average of large-c CFTs, JHEP12, 069, arXiv:2203.06511 [hep-th]

    Pith/arXiv arXiv

  48. [48]

    de Boer, D

    J. de Boer, D. Liska, B. Post, and M. Sasieta, A principle of maximum ignorance for semiclassical gravity, JHEP 2024, 003, arXiv:2311.08132 [hep-th]

    Pith/arXiv arXiv 2024

  49. [49]

    Belin, J

    A. Belin, J. de Boer, D. L. Jafferis, P. Nayak, and J. Sonner, Approximate CFTs and random tensor mod- els, JHEP09, 163, arXiv:2308.03829 [hep-th]

    Pith/arXiv arXiv

  50. [50]

    de Boer, D

    J. de Boer, D. Liska, and B. Post, Multibound- ary wormholes and OPE statistics, JHEP10, 207, arXiv:2405.13111 [hep-th]

    Pith/arXiv arXiv

  51. [51]

    D. L. Jafferis, L. Rozenberg, and G. Wong, 3d gravity as a random ensemble, JHEP02, 208, arXiv:2407.02649 [hep-th]

    Pith/arXiv arXiv

  52. [52]

    Belin, S

    A. Belin, S. Collier, L. Eberhardt, D. Liska, and B. Post, A universal sum over topologies in 3d gravity, (2026), arXiv:2601.07906 [hep-th]

    arXiv 2026

  53. [53]

    Gutperle, Y.-Y

    M. Gutperle, Y.-Y. Li, D. Rathore, and K. Roumpedakis, Non-invertible symmetries in S N orbifold CFTs and holography, JHEP09, 110, arXiv:2405.15693 [hep-th]

    Pith/arXiv arXiv

  54. [54]

    Knighton, V

    B. Knighton, V. Sriprachyakul, and J. Voˇ smera, Topo- logical defects and tensionless holography, JHEP07, 083, arXiv:2406.03467 [hep-th]

    Pith/arXiv arXiv

  55. [55]

    Benjamin, S

    N. Benjamin, S. Bintanja, Y.-J. Chen, M. Gutperle, C. Luo, and D. Rathore, Generalized symmetries and deformations of symmetric product orbifolds, JHEP04, 008, arXiv:2509.12180 [hep-th]

    arXiv

  56. [56]

    Harris, Y

    S. Harris, Y. Hikida, V. Schomerus, and T. Tsuda, Holo- graphic interfaces in symmetric product orbifolds, JHEP 12, 114, arXiv:2504.00078 [hep-th]

    Pith/arXiv arXiv

  57. [57]

    This is true up to subleading corrections in the weak- coupling limit

  58. [58]

    Sharpe, Dilaton shifts, probability measures, and decomposition, J

    E. Sharpe, Dilaton shifts, probability measures, and decomposition, J. Phys. A57, 445401 (2024), arXiv:2312.08438 [hep-th]

    Pith/arXiv arXiv 2024