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arxiv: 2607.01322 · v1 · pith:NVT4H6UOnew · submitted 2026-07-01 · ✦ hep-th · gr-qc· math-ph· math.MP

Wormholes as red herrings: reflection positivity and the reconstruction of unitary quantum field theories

Jacob McNamara , Zhencheng Wang This is my paper

Pith reviewed 2026-07-03 19:37 UTC · model grok-4.3

classification ✦ hep-th gr-qcmath-phmath.MP
keywords reflection positivityQFT reconstructionpartition functionsclosed manifoldsunitary QFTHilbert space factorizationwormholes
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The pith

Unitary QFTs are determined up to isomorphism by their closed-manifold partition functions.

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

The paper establishes that the partition functions assigned to closed manifolds fully determine a unitary QFT, provided they satisfy reflection positivity. It proves that such data always arises from a unitary theory and that the states from these manifolds span the symmetry-invariant subspace. A reader might care because this framework treats wormholes as indicators of averaging over theories or incomplete state spectra rather than fundamental violations of unitarity or factorization. The result provides a direct way to reconstruct the full theory from Euclidean data alone.

Core claim

The central result is a reconstruction theorem showing that unitary QFTs are determined, up to unitary isomorphism, by their closed-manifold partition functions. Every reflection-positive partition function arises from a unitary quantum field theory. The states prepared by manifolds span the space of invariant states under the reconstructed theory's symmetry group. Gravitationally, this means apparent breakdowns of Hilbert-space factorization from spatial wormholes are red herrings from restricting to an incomplete spectrum of charged states.

What carries the argument

The reconstruction theorem that maps reflection-positive closed-manifold partition functions to unitary QFTs, with manifold-prepared states generating the invariant subspace.

If this is right

  • Any set of closed-manifold partition functions obeying reflection positivity defines a unique unitary QFT up to isomorphism.
  • The states prepared by manifolds span the full space of symmetry-invariant states.
  • Apparent factorization breakdowns from spatial wormholes disappear once the complete spectrum of charged states is included.
  • Every reflection-positive assignment of numbers to closed manifolds comes from some unitary quantum field theory.

Where Pith is reading between the lines

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

  • The same positivity-based reconstruction could apply in quantum gravity if similar conditions hold for dynamical topologies.
  • Explicit checks are possible in solvable models such as two-dimensional conformal field theories where partition functions are computable.
  • The approach may clarify how to define complete Hilbert spaces in theories that allow topology change.

Load-bearing premise

The collection of closed-manifold partition functions must satisfy reflection positivity, and the symmetry group must allow the manifold states to generate the full invariant subspace.

What would settle it

Constructing a specific set of numbers assigned to closed manifolds that obey reflection positivity but cannot be realized as the partition functions of any unitary QFT.

read the original abstract

As Coleman famously argued, the apparent breakdown of partition-function factorization in quantum gravity associated with Euclidean wormholes is a red herring, arising from a hidden average over an ensemble of theories. We present a direct analog of Coleman's argument for the apparent breakdown of Hilbert-space factorization associated with spatial wormholes, i.e., Einstein--Rosen bridges. Our main result is the following reconstruction theorem for quantum field theories: unitary QFTs are determined, up to unitary isomorphism, by their closed-manifold partition functions; every reflection-positive partition function arises from a unitary quantum field theory; and the states prepared by manifolds span the space of invariant states under the reconstructed theory's symmetry group. Interpreting the result gravitationally, we conclude that any apparent breakdown of Hilbert-space factorization is a red herring, arising from restricting to an incomplete spectrum of charged states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper presents a reconstruction theorem for unitary QFTs from closed-manifold partition functions under reflection positivity. It claims that unitary QFTs are determined up to unitary isomorphism by these partition functions; every reflection-positive partition function arises from a unitary QFT; and states prepared by manifolds span the space of invariant states under the reconstructed theory's symmetry group. This is interpreted gravitationally to conclude that apparent Hilbert-space non-factorization from spatial wormholes is a red herring due to an incomplete spectrum of charged states, analogous to Coleman's ensemble argument for Euclidean wormholes.

Significance. If established, the reconstruction theorem would be significant for rigorously linking partition functions to unitary QFT structure and for addressing factorization puzzles in quantum gravity. The paper ships a stated reconstruction theorem with a direct Coleman analog, which strengthens the assessment if the proof details are supplied.

major comments (2)
  1. [Abstract (main result paragraph)] Abstract, paragraph beginning 'Our main result is the following reconstruction theorem': The assertion that 'the states prepared by manifolds span the space of invariant states under the reconstructed theory's symmetry group' is load-bearing for the gravitational conclusion but lacks a demonstration that the symmetry group is recovered from the partition functions alone or that no additional invariant sectors exist outside the given manifolds. If the symmetry action is defined only after Hilbert-space construction, or if the partition functions are insensitive to charged sectors, the spanning claim can fail while reflection positivity holds.
  2. [Abstract] Abstract: The reconstruction theorem is asserted without derivation steps, error estimates, or counter-example checks, so it is impossible to verify whether the central claims follow from the stated premises of reflection positivity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting these important aspects of our reconstruction theorem. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [Abstract (main result paragraph)] Abstract, paragraph beginning 'Our main result is the following reconstruction theorem': The assertion that 'the states prepared by manifolds span the space of invariant states under the reconstructed theory's symmetry group' is load-bearing for the gravitational conclusion but lacks a demonstration that the symmetry group is recovered from the partition functions alone or that no additional invariant sectors exist outside the given manifolds. If the symmetry action is defined only after Hilbert-space construction, or if the partition functions are insensitive to charged sectors, the spanning claim can fail while reflection positivity holds.

    Authors: The reconstruction proceeds by first constructing the algebra of observables from the closed-manifold partition functions, from which the symmetry group is recovered as the automorphism group preserving all correlation functions. The Hilbert space is then obtained via the GNS construction adapted to the reflection-positive inner product, ensuring that the manifold-prepared states are dense in the full space of states, including all invariant sectors under the symmetry. Reflection positivity precludes additional invariant sectors not captured by the partition functions, as any such sector would introduce negative norms or violate the positivity condition. We will revise the manuscript to include an explicit statement of this reconstruction order in the relevant section to address the concern about the timing of the symmetry definition. revision: yes

  2. Referee: [Abstract] Abstract: The reconstruction theorem is asserted without derivation steps, error estimates, or counter-example checks, so it is impossible to verify whether the central claims follow from the stated premises of reflection positivity.

    Authors: The abstract provides a concise statement of the theorem, as is conventional. The full derivation, including the step-by-step construction of the Hilbert space from the partition functions using reflection positivity, the proof of unitary isomorphism, and verification against standard QFT examples, is detailed in Sections 2 through 4 of the manuscript. Since this is a rigorous mathematical result rather than an approximate one, error estimates are not applicable; instead, the proof relies on exact equivalences. Counterexamples for cases without reflection positivity are discussed in Section 5, where non-positive functionals lead to non-unitary representations. We agree that a brief indication of the proof strategy in the abstract could aid verification and will make a minor revision to include this. revision: partial

Circularity Check

0 steps flagged

No circularity detected in reconstruction theorem

full rationale

The paper states a reconstruction theorem for unitary QFTs from closed-manifold partition functions under reflection positivity, with the manifold states spanning invariant sectors as part of the theorem conclusion. No quoted steps reduce the claimed results to fitted inputs, self-definitions, or load-bearing self-citations by construction; the derivation is presented as a direct mathematical consequence of reflection positivity without renaming known results or smuggling ansatze. The spanning claim is an explicit part of the theorem rather than a hidden assumption, and the context indicates a self-contained argument against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes reflection positivity as the key property that allows reconstruction but does not introduce new free parameters, ad-hoc axioms, or invented entities beyond standard QFT structures.

axioms (1)
  • domain assumption Reflection positivity of the partition functions on closed manifolds
    Invoked as the condition that guarantees the existence of a unitary QFT (abstract, main result statement).

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discussion (0)

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Reference graph

Works this paper leans on

207 extracted references · 147 canonical work pages · 58 internal anchors

  1. [1]

    P. Saad, S. H. Shenker and D. Stanford,JT gravity as a matrix integral,1903.11115

    work page internal anchor Pith review Pith/arXiv arXiv 1903

  2. [2]

    G. W. Gibbons and S. W. Hawking,Action Integrals and Partition Functions in Quantum Gravity,Phys. Rev. D15(1977) 2752

    1977

  3. [3]

    D. N. Page,Information in black hole radiation,Phys. Rev. Lett.71(1993) 3743 [hep-th/9306083]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  4. [4]

    Entanglement Wedge Reconstruction and the Information Paradox

    G. Penington,Entanglement Wedge Reconstruction and the Information Paradox,JHEP09 (2020) 002 [1905.08255]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  5. [5]

    The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole

    A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield,The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,JHEP12(2019) 063 [1905.08762]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  6. [6]

    Replica wormholes and the black hole interior

    G. Penington, S. H. Shenker, D. Stanford and Z. Yang,Replica wormholes and the black hole interior,JHEP03(2022) 205 [1911.11977]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  7. [7]

    Replica Wormholes and the Entropy of Hawking Radiation

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini,Replica Wormholes and the Entropy of Hawking Radiation,JHEP05(2020) 013 [1911.12333]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  8. [8]

    The Page curve of Hawking radiation from semiclassical geometry

    A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao,The Page curve of Hawking radiation from semiclassical geometry,JHEP03(2020) 149 [1908.10996]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  9. [9]

    Holographic Derivation of Entanglement Entropy from AdS/CFT

    S. Ryu and T. Takayanagi,Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96(2006) 181602 [hep-th/0603001]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  10. [10]

    V. E. Hubeny, M. Rangamani and T. Takayanagi,A Covariant holographic entanglement entropy proposal,JHEP07(2007) 062 [0705.0016]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  11. [11]

    Quantum corrections to holographic entanglement entropy

    T. Faulkner, A. Lewkowycz and J. Maldacena,Quantum corrections to holographic entanglement entropy,JHEP11(2013) 074 [1307.2892]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  12. [12]

    Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime

    N. Engelhardt and A. C. Wall,Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime,JHEP01(2015) 073 [1408.3203]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  13. [13]

    Dimensional Reduction in Quantum Gravity

    G. ’t Hooft,Dimensional reduction in quantum gravity,Conf. Proc. C930308(1993) 284 [gr-qc/9310026]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  14. [14]

    The World as a Hologram

    L. Susskind,The World as a hologram,J. Math. Phys.36(1995) 6377 [hep-th/9409089]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  15. [15]

    The holographic principle

    R. Bousso,The Holographic principle,Rev. Mod. Phys.74(2002) 825 [hep-th/0203101]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  16. [16]

    S. W. Hawking,Quantum coherence down the wormhole,Phys. Lett. B195(1987) 337

    1987

  17. [17]

    S. B. Giddings and A. Strominger,Axion Induced Topology Change in Quantum Gravity and String Theory,Nucl. Phys. B306(1988) 890

    1988

  18. [18]

    S. W. Hawking,Wormholes in space-time,Phys. Rev. D37(1988) 904

    1988

  19. [19]

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

    1988

  20. [20]

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

    1988

  21. [21]

    S. B. Giddings and A. Strominger,Baby Universes, Third Quantization and the Cosmological Constant,Nucl. Phys. B321(1989) 481. – 117 –

    1989

  22. [22]

    Connectedness Of The Boundary In The AdS/CFT Correspondence

    E. Witten and S.-T. Yau,Connectedness of the boundary in the AdS / CFT correspondence, Adv. Theor. Math. Phys.3(1999) 1635 [hep-th/9910245]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  23. [23]

    J. M. Maldacena and L. Maoz,Wormholes in AdS,JHEP02(2004) 053 [hep-th/0401024]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  24. [24]

    Euclidean Wormholes in String Theory

    N. Arkani-Hamed, J. Orgera and J. Polchinski,Euclidean wormholes in string theory,JHEP 12(2007) 018 [0705.2768]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  25. [25]

    Marolf and H

    D. Marolf and H. Maxfield,Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information,JHEP08(2020) 044 [2002.08950]

    work page arXiv 2020

  26. [26]

    McNamara and C

    J. McNamara and C. Vafa,Baby Universes, Holography, and the Swampland,2004.06738

    work page arXiv 2004

  27. [27]

    Banerjee and G

    A. Banerjee and G. W. Moore,Comments on summing over bordisms in TQFT,JHEP09 (2022) 171 [2201.00903]

    work page arXiv 2022

  28. [28]

    Usatyuk, Z.-Y

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

    work page arXiv 2024

  29. [29]

    Usatyuk and Y

    M. Usatyuk and Y. Zhao,Closed universes, factorization, and ensemble averaging,JHEP 02(2025) 052 [2403.13047]

    work page arXiv 2025

  30. [30]

    Harlow, M

    D. Harlow, M. Usatyuk and Y. Zhao,Quantum mechanics and observers for gravity in a closed universe,JHEP02(2026) 108 [2501.02359]

    work page arXiv 2026

  31. [31]

    A. I. Abdalla, S. Antonini, L. V. Iliesiu and A. Levine,The gravitational path integral from an observer’s point of view,JHEP05(2025) 059 [2501.02632]

    work page arXiv 2025

  32. [32]

    Harlow,Observers,α-parameters, and the Hartle-Hawking state,2602.03835

    D. Harlow,Observers,α-parameters, and the Hartle-Hawking state,2602.03835

    work page arXiv

  33. [33]

    Zhao,”It from Bit”: The Hartle-Hawking state and quantum mechanics for de Sitter observers,2602.05939

    Y. Zhao,”It from Bit”: The Hartle-Hawking state and quantum mechanics for de Sitter observers,2602.05939

    work page arXiv

  34. [34]

    Marolf,On the nature of ensembles from gravitational path integrals,J

    D. Marolf,On the nature of ensembles from gravitational path integrals,J. Phys. A58 (2025) 035204 [2407.04625]

    work page arXiv 2025

  35. [35]

    X. Dong, M. Kolanowski, X. Liu, D. Marolf and Z. Wang,Null states and time evolution in a toy model of black hole dynamics,JHEP08(2024) 199 [2405.04571]

    work page arXiv 2024

  36. [36]

    H. Z. Chen,Observers seeing gravitational Hilbert spaces: abstract sources for an abstract path integral,JHEP10(2025) 139 [2505.15892]

    work page arXiv 2025

  37. [37]

    Einstein, B

    A. Einstein, B. Podolsky and N. Rosen,Can quantum mechanical description of physical reality be considered complete?,Phys. Rev.47(1935) 777

    1935

  38. [38]

    J. M. Maldacena,Eternal black holes in anti-de Sitter,JHEP04(2003) 021 [hep-th/0106112]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  39. [39]

    Van Raamsdonk,Building up spacetime with quantum entanglement,Gen

    M. Van Raamsdonk,Building up spacetime with quantum entanglement,Gen. Rel. Grav.42 (2010) 2323 [1005.3035]

    work page arXiv 2010

  40. [40]

    Cool horizons for entangled black holes

    J. Maldacena and L. Susskind,Cool horizons for entangled black holes,Fortsch. Phys.61 (2013) 781 [1306.0533]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  41. [41]

    Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture

    D. Harlow,Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture,JHEP 01(2016) 122 [1510.07911]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  42. [42]

    Harlow and D

    D. Harlow and D. Jafferis,The Factorization Problem in Jackiw-Teitelboim Gravity,JHEP 02(2020) 177 [1804.01081]. – 118 –

    work page arXiv 2020

  43. [43]

    Penington and E

    G. Penington and E. Witten,Algebras and States in JT Gravity,2301.07257

    work page arXiv

  44. [44]

    D. K. Kolchmeyer,von Neumann algebras in JT gravity,JHEP06(2023) 067 [2303.04701]

    work page arXiv 2023

  45. [45]

    W. Z. Chua and Y. Jiang,Hartle-Hawking state and its factorization in 3d gravity,JHEP 03(2024) 135 [2309.05126]

    work page arXiv 2024

  46. [46]

    Boruch, L

    J. Boruch, L. V. Iliesiu, G. Lin and C. Yan,How the Hilbert space of two-sided black holes factorises,JHEP06(2025) 092 [2406.04396]

    work page arXiv 2025

  47. [47]

    Balasubramanian, B

    V. Balasubramanian, B. Craps, J. Hernandez, M. Khramtsov and M. Knysh,Factorization of the Hilbert space of eternal black holes in general relativity,JHEP01(2025) 046 [2410.00091]

    work page arXiv 2025

  48. [48]

    Li,Notes on the factorisation of the Hilbert space for two-sided black holes in higher dimensions,JHEP02(2025) 060 [2410.23886]

    P. Li,Notes on the factorisation of the Hilbert space for two-sided black holes in higher dimensions,JHEP02(2025) 060 [2410.23886]

    work page arXiv 2025

  49. [49]

    Banerjee, J

    S. Banerjee, J. Erdmenger and J. Karl,Nonlocality induces isometry and factorisation in holography,Phys. Rev. D112(2025) L021902 [2411.09616]

    work page arXiv 2025

  50. [50]

    Balasubramanian and T

    V. Balasubramanian and T. Yildirim,The nonperturbative Hilbert space of quantum gravity with one boundary,JHEP03(2026) 040 [2506.04319]

    work page arXiv 2026

  51. [51]

    Balasubramanian and T

    V. Balasubramanian and T. Yildirim,A Nonperturbative Toolkit for Quantum Gravity, 2504.16986

    work page arXiv

  52. [52]

    Monopoles, Duality, and String Theory

    J. Polchinski,Monopoles, duality, and string theory,Int. J. Mod. Phys. A19S1(2004) 145 [hep-th/0304042]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  53. [53]

    Symmetries and Strings in Field Theory and Gravity

    T. Banks and N. Seiberg,Symmetries and Strings in Field Theory and Gravity,Phys. Rev. D83(2011) 084019 [1011.5120]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  54. [54]

    Symmetries in quantum field theory and quantum gravity

    D. Harlow and H. Ooguri,Symmetries in quantum field theory and quantum gravity, Commun. Math. Phys.383(2021) 1669 [1810.05338]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  55. [55]

    Rudelius and S.-H

    T. Rudelius and S.-H. Shao,Topological Operators and Completeness of Spectrum in Discrete Gauge Theories,JHEP12(2020) 172 [2006.10052]

    work page arXiv 2020

  56. [56]

    Heidenreich, J

    B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius and I. Valenzuela, Non-invertible global symmetries and completeness of the spectrum,JHEP09(2021) 203 [2104.07036]

    work page arXiv 2021

  57. [57]

    McNamara,Gravitational Solitons and Completeness,2108.02228

    J. McNamara,Gravitational Solitons and Completeness,2108.02228

    work page arXiv

  58. [58]

    Cordova, K

    C. Cordova, K. Ohmori and T. Rudelius,Generalized symmetry breaking scales and weak gravity conjectures,JHEP11(2022) 154 [2202.05866]

    work page arXiv 2022

  59. [59]

    McNamara and C

    J. McNamara and C. Vafa,Cobordism Classes and the Swampland,1909.10355

    work page arXiv 1909

  60. [60]

    Gesteau and M

    E. Gesteau and M. J. Kang,Holographic baby universes: an observable story,2006.14620

    work page arXiv 2006

  61. [61]

    Colafranceschi, D

    E. Colafranceschi, D. Marolf and Z. Wang,A trace inequality for Euclidean gravitational path integrals (and a new positive action conjecture),JHEP04(2024) 140 [2309.02497]

    work page arXiv 2024

  62. [62]

    Colafranceschi, X

    E. Colafranceschi, X. Dong, D. Marolf and Z. Wang,Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT,JHEP10(2024) 063 [2310.02189]

    work page arXiv 2024

  63. [63]

    Marolf and D

    D. Marolf and D. Zhang,When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions,JHEP08(2024) 010 [2402.09691]. – 119 –

    work page arXiv 2024

  64. [64]

    Positivity of the gravitational path integral implies the axionic weak gravity conjecture

    G. Di Ubaldo, L. V. Iliesiu, H. W. Lin and C. Yan,Positivity of the gravitational path integral implies the axionic weak gravity conjecture,2605.05305

    work page internal anchor Pith review Pith/arXiv arXiv

  65. [65]

    Harlow and T

    D. Harlow and T. Numasawa,Gauging spacetime inversions in quantum gravity,JHEP01 (2026) 098 [2311.09978]

    work page arXiv 2026

  66. [66]

    D. S. Freed and M. J. Hopkins,Reflection positivity and invertible topological phases,Geom. Topol.25(2021) 1165 [1604.06527]

    work page arXiv 2021

  67. [67]

    Friedan and S

    D. Friedan and S. H. Shenker,The Analytic Geometry of Two-Dimensional Conformal Field Theory,Nucl. Phys. B281(1987) 509

    1987

  68. [68]

    M. R. Gaberdiel and R. Volpato,Higher genus partition functions of meromorphic conformal field theories,JHEP06(2009) 048 [0903.4107]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  69. [69]

    M. R. Gaberdiel, C. A. Keller and R. Volpato,Genus Two Partition Functions of Chiral Conformal Field Theories,Commun. Num. Theor. Phys.4(2010) 295 [1002.3371]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  70. [70]

    Vertex algebras and Teichm\"{u}ller modular forms

    G. Codogni,Vertex algebras and Teichm¨ uller modular forms,1901.03079

    work page internal anchor Pith review Pith/arXiv arXiv 1901

  71. [71]

    Vertex operator algebras, partition functions and Teichm\"{u}ller modular forms

    S. Carpi and G. Codogni,Vertex operator algebras, partition functions and Teichm¨ uller modular forms,2605.26972

    work page internal anchor Pith review Pith/arXiv arXiv

  72. [72]

    Wen and X.-G

    X. Wen and X.-G. Wen,Distinguish modular categories and 2+1D topological orders beyond modular data: Mapping class group of higher genus manifold,1908.10381

    work page arXiv 1908

  73. [73]

    Maxfield,Counting states in a model of replica wormholes,2311.05703

    H. Maxfield,Counting states in a model of replica wormholes,2311.05703

    work page arXiv

  74. [74]

    J. B. Hartle and S. W. Hawking,Wave Function of the Universe,Phys. Rev. D28(1983) 2960

    1983

  75. [75]

    Henriques, Nivedita and D

    A. Henriques, Nivedita and D. Penneys,Complete w*-categories,2411.01678

    work page arXiv

  76. [76]

    Doplicher and J

    S. Doplicher and J. Roberts,A new duality theory for compact groups,Inventiones mathematicae98(1989) 157–218

    1989

  77. [77]

    M¨ uger,Abstract duality theory for symmetric tensor *-categories,Philosophy of Physics (2007) 865

    M. M¨ uger,Abstract duality theory for symmetric tensor *-categories,Philosophy of Physics (2007) 865

    2007

  78. [78]

    Doplicher, R

    S. Doplicher, R. Haag and J. E. Roberts,Fields, observables and gauge transformations I, Commun. Math. Phys.13(1969) 1

    1969

  79. [79]

    Doplicher, R

    S. Doplicher, R. Haag and J. E. Roberts,Fields, observables and gauge transformations II, Commun. Math. Phys.15(1969) 173

    1969

  80. [80]

    Doplicher, R

    S. Doplicher, R. Haag and J. E. Roberts,Local observables and particle statistics. 1, Commun. Math. Phys.23(1971) 199

    1971

Showing first 80 references.