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A nonzero 'basepoint anomaly' — a phase acquired under a rigid symmetry transformation that leaves the background fixed — forces the partition function to vanish, and this single rule reproduces and extends the Freed-Witten, M5-brane, and S

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2026-08-04 08:33 UTC pith:IVK7GQDA

load-bearing objection Careful reformulation of anomaly-induced vanishing that cleanly re-derives known brane constraints and offers a new M5 quantization condition; the new S-fold claims are honest postulates that still need proof. the 3 major comments →

arxiv 2510.19935 v4 pith:IVK7GQDA submitted 2025-10-22 hep-th math-phmath.MP

Anomaly-induced vanishing of brane partition functions

Felix B. Christensen , I\~naki Garc\'ia Etxebarria , Enoch Leung This is my paper
classification hep-th math-phmath.MP MSC 81T3081T50
keywords anomalypartition functionFreed-Witten anomalyM5-braneD3-braneS-folddifferential cohomologyflux quantization
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.

This paper establishes a general vanishing criterion: if a quantum field theory has a 'basepoint anomaly,' a nontrivial phase acquired under a rigid symmetry transformation that leaves the background field unchanged, then its partition function on a closed spacetime is exactly zero. The authors argue this purely field-theoretic fact explains a range of known brane consistency conditions and generates new ones. In particular, they derive the Freed-Witten cancellation condition for D-branes, the shifted flux-quantization condition for the M5-brane, and a new twisted-cohomology condition for D3-branes in S-fold backgrounds. A sympathetic reader should care because the argument works off the anomaly theory alone, without needing a Lagrangian or worldsheet description, and the paper is explicit that the conditions are sufficient: failure means the partition function vanishes, not that the configuration is forbidden.

Core claim

The central claim is Section 2.3's vanishing theorem: for a background differential cocycle  and any flat rigid automorphism â satisfying  − dâ = Â, the partition function obeys Z[Â] = exp(2πi A[Â,0,â]) Z[Â], with A[Â,0,â] ∈ R/Z the basepoint anomaly. Hence a nontrivial phase leaves only Z[Â] = 0. Applied to brane worldvolumes, the theorem gives non-vanishing conditions: [H3]+[W3]=0 (Freed-Witten for D3-branes), [G4]=½[Λ4] for the M5-brane (reducing to [G4]=¼ p1 on spin manifolds), and [H3]_ρ + [W3]_ρ = 0 for D3-branes in S-folds, with stacks satisfying [H3]+β([ξ2])=0. The M5 result reproduces the well-known shifted quantization of M-theory flux, and the S-fold conditions are new.

What carries the argument

The named object is the basepoint anomaly A[Â,0,â]: the phase the partition function acquires when acted on by a flat rigid gauge transformation â that fixes the background Â. The identity Z[Â] = e^{2πi A[Â,0,â]} Z[Â] is the load-bearing equation, and it is evaluated on the mapping torus X^d × S^1 in differential cohomology, with flat differential cocycles representing the automorphisms. For the brane applications, the key derived identity is the M5 condition [G4]=½[Λ4], where Λ4 is an integral lift of the fourth Wu class; dimensional reduction over T^2 turns this into the D3-brane conditions, and for S-folds the reduction uses a proposed twisted fiber-integration map whose output lives in c

Load-bearing premise

The new S-fold constraints rest on the unproven existence of twisted Stiefel-Whitney classes w_i and a twisted fiber-integration map with the properties asserted in Section 6.3 and Appendix B; if those constructions fail, conditions (6.39) and (6.55) collapse, although the Maxwell/M5 results would survive.

What would settle it

Compute H^2(S^3/Z_4;(Z2⊕Z2)_ρ4) and its ring structure from the standard cell decomposition; the paper's w_2=0 conclusion for n=1 assumes the twisted cohomology is generated by a_1 or (a_1,b_2) with ordinary relations. If the ring differs, the predicted vanishing of the D3 partition function on S^1×S^3/Z_4 with discrete torsion is not established — and a nonzero partition function there would falsify the S-fold application.

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

If this is right

  • A single mechanism reproduces the Freed-Witten condition for D3-branes in weakly coupled Type IIB, with [H3]+[W3]=[G3]+[W3]=0, without invoking open-string worldsheets.
  • For the M5-brane, the worldvolume partition function vanishes unless [G4]=½[Λ4]; on spin manifolds this is the familiar shifted quantization [G4]=¼ p1, tying anomaly cancellation to M-theory flux quantization.
  • For D3-branes in S-fold backgrounds with nontrivial SL(2,Z) monodromy, a single brane's partition function vanishes whenever the discrete torsion is nontrivial (k=2,3,4), while stacks with suitable Chan-Paton gauge group can restore non-vanishing via the condition [H3]+β([ξ2])=0.
  • The vanishing criterion also applies to theories without Lagrangian descriptions, such as the 3d minimal TQFT, where it requires the background Z_N connection to be cohomologically trivial, stronger than triviality of the characteristic class.
  • In 2d, the same argument reproduces the Arf-invariant criterion: the free-fermion partition function vanishes exactly when Arf(q)=1.

Where Pith is reading between the lines

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

  • Editorial extension: because the vanishing criterion depends only on the anomaly polynomial, it suggests anomaly coefficients themselves act as selection rules; nonzero basepoint anomalies could serve as order parameters for anomaly-enforced triviality in any QFT whose anomaly theory is known, not just brane systems.
  • Editorial extension: if the twisted Stiefel-Whitney and twisted Wu class axioms are made rigorous, the same reduction should apply to other compactifications with nontrivial fundamental group — e.g. lens spaces, nilmanifolds, or general Class S constructions — yielding analogous flux-cancellation constraints.
  • Editorial extension: the stack condition (6.55) predicts a sharp rank dependence — for k=2,3,4 S-fold torsion is cancellable only by ranks divisible by 2, 3, or 2 respectively — which could be tested in the holographic dual N=3 SCFT by computing BPS spectra in the discrete-torsion sectors.

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 / 3 minor

Summary. The paper develops a general field-theoretic criterion for the vanishing of partition functions in the presence of 't Hooft anomalies. Working in differential cohomology, it defines a 'basepoint anomaly' A[Â,0,â] associated with rigid automorphisms of the background gauge field, and proves (Sec. 2.3, eqs. 2.37–2.38) that a non-trivial such phase forces Z[Â]=0. This is applied to a range of systems: generalized Maxwell theory, Spin^c Maxwell theory reproducing the Freed–Witten condition (3.26), Dijkgraaf–Witten theory, 3d minimal TQFTs, 2d fermions with the Arf-invariant structure, the M5-brane worldvolume theory yielding the shifted flux quantization [G4]=1/2[Λ4] (5.31) and its Spin reduction [G4]=1/4[p1] (5.38), and D3-branes in F-theory backgrounds. For trivial or constant local systems the D3 analysis recovers the standard Freed–Witten conditions, while for non-trivial local systems the paper proposes new constraints, in particular for D3-branes on S-folds: [H3]_{Z^2_ρ}+[W3]_{Z^2_ρ}=0 (6.39) and, for stacks, [H3]+β([ξ2])=0 (6.55).

Significance. If the central theorem and its applications are correct, the paper provides a unifying anomaly-based explanation of known brane flux-quantization conditions and extends them to previously unstudied S-fold and class-S backgrounds. The derivations for Maxwell, Spin^c, Dijkgraaf–Witten, 2d fermions, and M5 are careful and are anchored to external benchmarks: the Maxwell vanishing reproduces Hsieh–Tachikawa–Yonekura, the M5 condition reduces to the 1/4 p1 quantization on Spin manifolds, and the 2d example reproduces Atiyah's Arf theorem. The paper is transparent in labeling its new S-fold machinery as postulates, which is a strength in terms of honesty but leaves the headline new results conditional. The treatment of the simpler, known cases gives confidence in the general framework; the S-fold claims, however, require substantial additional justification before they can be considered established.

major comments (3)
  1. [§6.3, eqs. (6.32)–(6.37)] The twisted fiber integration map π̃_! (eq. B.14) is defined only for classes x that factorize as x = f ∪ π̃^*(b), with b∈H^p(B;Z^2_ρ) and f∈coker(π̃^*)⊂H^1(X;Z). The conditions E^{p+1,0}_∞=0, non-trivial H^0(X4;Z^2_ρ), and the normalization of [F] are asserted rather than proven, and no argument is given that the classes obtained from the M5 basepoint anomaly (6.31), (6.37) admit such a factorization. Without a proof of well-definedness and independence of the choices, the basepoint anomaly (6.38) and the vanishing condition (6.39) are unsupported. This is load-bearing for the paper's new S-fold results, even though the earlier Maxwell and M5 results do not rely on it.
  2. [§6.4.2, eq. (6.55)] The derivation of w2=0 for k=4 assumes that the twisted cohomology ring H^*(S^3/Z4;(Z2⊕Z2)_ρ4) has the same ring structure as the ordinary lens-space ring: generated by a1 alone or by (a1,b2) with a1^2 = m b2. With local coefficients, cup products need not satisfy these relations, and no spectral sequence or explicit cochain computation is supplied. The conclusion w2=0 (and hence W3=0) is used critically to reduce (6.39) to the non-Abelian condition (6.55). If w2≠0 for k=4, the claimed infinite family of non-Abelian corrections in §6.4.2 is not established. This is a concrete, load-bearing gap, not merely a presentation issue.
  3. [§6.4.1] The existence of twisted Stiefel-Whitney classes w_i∈H^i(BO(n);(Z2⊕Z2)_ρ) satisfying the twisted Wu formula w=Sq(v) and Bockstein relations W_{i+1}=β(w_i) is postulated. While motivated by analogy with the ordinary case and by references [85,86] for the coefficient system Z_{w1}, the specific multi-twisted case with (Z2⊕Z2)_ρ requires a proof or a precise reference. The twisted Wu classes v_i are defined by eq. (6.35) assuming Steenrod squares with local coefficients, but the existence of such operations with the required axioms is not demonstrated. Since [W3]_{Z^2_ρ} in (6.39) is the key new object, this postulate is central to the S-fold claims.
minor comments (3)
  1. [§5.3, eqs. (5.31)–(5.38)] The diagram defining the Spin^c cocycle ˇw is somewhat terse; the chain of inclusions modulo 2 and the relation to the Bockstein homomorphism would be clearer if the connecting maps were labeled explicitly.
  2. [§6.1] The notation [s1]_Z and [s'_1]_Z for the two 1-cycles of the torus is easy to confuse with the S^1 of the mapping torus; consider using α,β or similar.
  3. [§3.1.2] There are a few typographical slips, e.g. 'e.i.' for 'i.e.' and some unmatched parentheses in eq. (3.14) and the discussion of Q. These do not affect the mathematics.

Circularity Check

0 steps flagged

No significant circularity: the vanishing theorem and its applications are derived from stated anomaly inputs, and the S-fold results rest on explicitly labeled assumptions rather than on circular reasoning.

full rationale

The paper's central derivation chain is not circular. The vanishing theorem (2.37)-(2.38) is a direct consistency consequence of the definition of the basepoint anomaly as a phase acquired under automorphisms of the background: if Z[Â] = e^{2πi A[Â,0,â]} Z[Â] with A non-integral, then Z[Â]=0. The Maxwell/Freed-Witten condition (3.26) is obtained by evaluating the BF anomaly theory (3.2) on the mapping torus, not by assuming the result. The M5 condition (5.31) follows from the quadratic refinement (5.7) together with the explicit integral lift (5.28), and the spin-manifold specialization (5.38) correctly reproduces the known 1/4 p1 quantization, which is an external benchmark rather than an input. The paper explicitly says it is "rederiving" existing results and uses them as checks (e.g., the 2d example reproduces Atiyah's Arf theorem). The only candidate for a circular-looking step is in Section 6.1, where integral lifts of Wu classes on T^2 are chosen, e.g. 2([s1]+[s1']) for v1(T2). The paper acknowledges that one "could have chosen the integral lift to simply be zero," but explains the apparently non-unique choice is without loss of generality because the relevant class W3(X4) is 2-torsion, and interprets the choice as a tangential (SO^c) structure. This is a choice of Wu structure, not a parameter fitted to the desired conclusion; the final condition is conditional on that structure. The new S-fold claims (6.39), (6.55) do rest on unproven postulates: the twisted Stiefel-Whitney/Wu classes (6.32)-(6.36), the twisted fiber integration of Appendix B, and the assumed ring structure of H*(S^3/Z4; (Z2⊕Z2)_ρ4) in §6.4.1. These are asserted as postulates and conjectures, and the paper labels them as such; they are gaps or correctness risks, not circular reductions. The few self-citations (e.g., [35], [94], [29]) are used for technical background or standard consequences and are not load-bearing for the vanishing theorem or the new conditions.

Axiom & Free-Parameter Ledger

3 free parameters · 8 axioms · 2 invented entities

The central re-derivations (Maxwell, Freed-Witten, M5 spin case) rest on standard inputs from prior literature and check against external benchmarks. The new S-fold results rest on: two ad-hoc mathematical entities (twisted SW/Wu classes and twisted fiber integration), an assumed ring structure for twisted lens-space cohomology, and three hand-chosen parameters (Wu-class lifts and a normalisation). This is the honest price of the new claims.

free parameters (3)
  • Integral lift of Wu class v1(T2) = 2([s1]Z + [s1']Z) ∈ H1(T2;Z)
    Chosen by hand in §6.1 (eq. 6.9) so the v3(X4) W3-term survives the T2 reduction; the paper notes the zero lift would remove this contribution and give 'a less general constraint', so the choice affects the derived D3 condition (6.15).
  • Integral lift of Wu class v2(T2) = 2[ω2]Z ∈ H2(T2;Z)
    Chosen in §6.1 instead of the canonical zero Euler-class lift, which 'would give rise to a constraint that might be too restrictive'; this choice feeds into the final conditions (6.15).
  • Normalisation of [F]_Z^2_ρ in twisted fiber integration = 'a sum of the independent generator(s)... up to a preferred choice of normalisation' (App. B)
    The twisted pushforward in eq. (B.14) is not unique without fixing this choice, and the S-fold anomaly (6.38) inherits the ambiguity.
axioms (8)
  • domain assumption M5-brane anomaly theory is the level-κ self-dual Chern-Simons theory with quadratic refinement (5.29) built from G4 and the integral Wu-class lift Λ4.
    Input from [5,17,67]; the M5 and all subsequent D3/F-theory derivations start from (5.30). If the anomaly theory had corrections, all downstream conditions shift.
  • domain assumption Generalized Maxwell theory anomaly is the BF-type product ˇC ⋆ ˇB in (3.2), with the Spin^c replacement ˇB→ˇB+ˇw, ˇC→ˇC+ˇw.
    Standard result of [5]; the Freed-Witten re-derivation (3.26) and the stack condition (3.49) depend on it.
  • domain assumption Partition functions are sections of a rank-1 line bundle over the space of gauge fields (mod gauge equivalence).
    Stated in §2.2; the vanishing conclusion Z[Â]=0 follows from the rigid-automorphism argument of §2.3 only in the line-bundle/rank-1 case.
  • ad hoc to paper Existence of twisted Stiefel-Whitney classes wi ∈ H^i(BO(n); (Z2⊕Z2)_ρ) satisfying the twisted Wu formula w = Sq(v) and Bockstein relations W_{i+1} = β(wi).
    Postulated in §6.3 (eqs. 6.32-6.37) 'taking inspiration from [85,86]'; the S-fold non-vanishing condition (6.39) depends on it.
  • ad hoc to paper The twisted fiber integration map π̃!: H^{p+2}(X;Z) → H^p(B;Z^2_ρ) of Appendix B is well-defined for the backgrounds used, with E^{p+1,0}_∞ = 0 and non-trivial H^0(X4;Z^2_ρ).
    'We propose in Appendix B that under suitable conditions...' — the construction is a proposal, not a proven theorem; eq. (6.38) relies on it.
  • ad hoc to paper The twisted cohomology ring of S^{2n+1}/Z4 with (Z2⊕Z2)_ρ4 coefficients is generated by a1 (or by a1,b2 with a1^2 = mb2) exactly as in the ordinary lens-space case.
    Assumed in §6.4.1 ('We assume the same for the twisted cohomology groups...'); the k=4 conclusion w2=0 and the resulting vanishing statement depend on it.
  • standard math Standard tools: differential cohomology (Hopkins-Singer), Poincaré-Pontryagin duality pairing (2.35), Leray-Serre spectral sequence with edge homomorphisms, universal coefficient theorem.
    Used throughout; these are established background results.
  • domain assumption S-fold backgrounds are characterized by SL(2,Z) representations ρk (k=2,3,4,6) acting on the (B2,C2) doublet with matrices (6.41).
    From [1,89]; the lens-space cohomology computations (6.43)-(6.45) follow from this data.
invented entities (2)
  • Twisted Stiefel-Whitney classes wi ∈ H^i(BO(n); (Z2⊕Z2)_ρ) and twisted Wu classes vi no independent evidence
    purpose: Define the twisted third integral Stiefel-Whitney class [W3]_Z^2_ρ used in the S-fold D3 vanishing condition (6.39) and the non-Abelian cancellation condition (6.55).
    Postulated in §6.3; no independent verification is supplied. The resulting conditions are checkable in principle against other methods, but the classes themselves are introduced ad hoc.
  • Twisted fiber integration map π̃!: H^{p+2}(X;Z) → H^p(B;Z^2_ρ) no independent evidence
    purpose: Reduce the M5 basepoint anomaly over the T2 fiber when π1(X4) acts non-trivially on the fiber, producing the S-fold anomaly (6.38).
    Proposed in Appendix B 'under suitable conditions'; the codomain and normalisation are not uniquely fixed, so the construction carries inherent ambiguity.

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read the original abstract

In the presence of 't Hooft anomalies, backgrounds for the symmetries of a quantum field theory can lead to non-conservation of Noether currents, or more generally, to the presence of charged insertions in the path integral. When there is a net background charge, the partition function evaluated on closed manifolds will vanish. For anomalous symmetries, this statement can also be understood as the anomaly theory giving rise to a non-trivial anomalous phase for the partition function even for "rigid" transformations which leave all background fields unchanged. We use the generalisation of this second viewpoint to the setting of anomalous higher-form symmetries in order to show vanishing of the partition function for a number of examples, both with and without a Lagrangian description. In particular, we show how to derive from these considerations the analogue of the Freed-Witten anomaly cancellation condition for the M5-brane, and also that for the D3-brane in S-fold backgrounds.

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Reference graph

Works this paper leans on

99 extracted references · 62 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Witten,Baryons and branes in anti-de Sitter space,JHEP07(1998) 006 [hep-th/9805112]

    E. Witten,Baryons and branes in anti-de Sitter space,JHEP07(1998) 006 [hep-th/9805112]

    Pith/arXiv arXiv 1998

  2. [2]

    Freed and E

    D.S. Freed and E. Witten,Anomalies in string theory with D-branes,Asian J. Math.3 (1999) 819 [hep-th/9907189]

    Pith/arXiv arXiv 1999

  3. [3]

    Hsin, H.T

    P.-S. Hsin, H.T. Lam and N. Seiberg,Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d,SciPost Phys.6(2019) 039 [1812.04716]

    Pith/arXiv arXiv 2019

  4. [4]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett,Generalized Global Symmetries,JHEP 02(2015) 172 [1412.5148]

    Pith/arXiv arXiv 2015

  5. [5]

    Hsieh, Y

    C.-T. Hsieh, Y. Tachikawa and K. Yonekura,Anomaly Inflow and p-Form Gauge Theories, Commun. Math. Phys.391(2022) 495 [2003.11550]

    arXiv 2022

  6. [6]

    Freed,Anomalies and Invertible Field Theories,Proc

    D.S. Freed,Anomalies and Invertible Field Theories,Proc. Symp. Pure Math.88(2014) 25 [1404.7224]

    Pith/arXiv arXiv 2014

  7. [7]

    Monnier,A Modern Point of View on Anomalies,Fortsch

    S. Monnier,A Modern Point of View on Anomalies,Fortsch. Phys.67(2019) 1910012 [1903.02828]

    Pith/arXiv arXiv 2019

  8. [8]

    Zumino, Y.-S

    B. Zumino, Y.-S. Wu and A. Zee,Chiral Anomalies, Higher Dimensions, and Differential Geometry,Nucl. Phys. B239(1984) 477

    1984

  9. [9]

    Maeda and Y

    J. Maeda and Y. Tanizaki,Twisted partition functions as order parameters,JHEP08(2025) 128 [2505.16546]

    Pith/arXiv arXiv 2025

  10. [10]

    C´ ordova and K

    C. C´ ordova and K. Ohmori,Anomaly Constraints on Gapped Phases with Discrete Chiral Symmetry,Phys. Rev. D102(2020) 025011 [1912.13069]

    Pith/arXiv arXiv 2020

  11. [11]

    Adler,Axial-vector vertex in spinor electrodynamics,Phys

    S.L. Adler,Axial-vector vertex in spinor electrodynamics,Phys. Rev.177(1969) 2426

    1969

  12. [12]

    Bell and R

    J.S. Bell and R. Jackiw,A pcac puzzle:π 0 →γγin theσ-model,Il Nuovo Cimento A (1965-1970)60(1969) 47

    1965

  13. [13]

    Dai and D.S

    X.-z. Dai and D.S. Freed,eta invariants and determinant lines,J. Math. Phys.35(1994) 5155 [hep-th/9405012]

    Pith/arXiv arXiv 1994

  14. [14]

    Fujikawa and H

    K. Fujikawa and H. Suzuki,Path Integrals and Quantum Anomalies, Oxford University Press (04, 2004), 10.1093/acprof:oso/9780198529132.001.0001. – 59 –

    arXiv 2004

  15. [15]

    Meynet, D

    S. Meynet, D. Migliorati, R. Savelli and M. Tortora,Duality-Symmetry Enhancement in Maxwell Theory,SciPost Phys.19(2025) 089 [2504.20141]

    arXiv 2025

  16. [16]

    Cheeger and J

    J. Cheeger and J. Simons,Differential characters and geometric invariants, inGeometry and Topology, (Berlin, Heidelberg), pp. 50–80, Springer Berlin Heidelberg, 1985

    1985

  17. [17]

    Hopkins and I.M

    M.J. Hopkins and I.M. Singer,Quadratic functions in geometry, topology, and M theory,J. Diff. Geom.70(2005) 329 [math/0211216]

    Pith/arXiv arXiv 2005

  18. [18]

    Diaconescu, G.W

    E. Diaconescu, G.W. Moore and D.S. Freed,The M theory three form and E(8) gauge theory, hep-th/0312069

    Pith/arXiv arXiv

  19. [19]

    Moore,Anomalies, Gauss laws, and Page charges in M-theory,Comptes Rendus Physique6(2005) 251 [hep-th/0409158]

    G.W. Moore,Anomalies, Gauss laws, and Page charges in M-theory,Comptes Rendus Physique6(2005) 251 [hep-th/0409158]

    Pith/arXiv arXiv 2005

  20. [20]

    Belov and G.W

    D. Belov and G.W. Moore,Conformal blocks for AdS(5) singletons,hep-th/0412167

    Pith/arXiv arXiv

  21. [21]

    Belov and G.W

    D. Belov and G.W. Moore,Holographic Action for the Self-Dual Field,hep-th/0605038

    Pith/arXiv arXiv

  22. [22]

    Freed, G.W

    D.S. Freed, G.W. Moore and G. Segal,Heisenberg Groups and Noncommutative Fluxes, Annals Phys.322(2007) 236 [hep-th/0605200]

    Pith/arXiv arXiv 2007

  23. [23]

    ’t Hooft,Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking,NATO Sci

    G. ’t Hooft,Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking,NATO Sci. Ser. B59(1980) 135

    1980

  24. [24]

    Simons and D

    J. Simons and D. Sullivan,Structured vector bundles define differential K-theory,arXiv e-prints(2008) arXiv:0810.4935 [0810.4935]

    Pith/arXiv arXiv 2008

  25. [25]

    Seiberg, Y

    N. Seiberg, Y. Tachikawa and K. Yonekura,Anomalies of Duality Groups and Extended Conformal Manifolds,PTEP2018(2018) 073B04 [1803.07366]

    Pith/arXiv arXiv 2018

  26. [26]

    Hosseini, Y

    S.S. Hosseini, Y. Tachikawa and H.Y. Zhang,Type I anomaly cancellation revisited, 2505.07933

    arXiv

  27. [27]

    Apruzzi, I

    F. Apruzzi, I. Bah, F. Bonetti and S. Schafer-Nameki,Noninvertible Symmetries from Holography and Branes,Phys. Rev. Lett.130(2023) 121601 [2208.07373]

    Pith/arXiv arXiv 2023

  28. [28]

    Choi, H.T

    Y. Choi, H.T. Lam and S.-H. Shao,Non-invertible Gauss law and axions,JHEP09(2023) 067 [2212.04499]

    Pith/arXiv arXiv 2023

  29. [29]

    I. Bah, E. Leung and T. Waddleton,Non-invertible symmetries, brane dynamics, and tachyon condensation,JHEP01(2024) 117 [2306.15783]

    Pith/arXiv arXiv 2024

  30. [30]

    Apruzzi, F

    F. Apruzzi, F. Bonetti, I. Garc ´ ıa Etxebarria, S.S. Hosseini and S. Schafer-Nameki,Symmetry TFTs from String Theory,Commun. Math. Phys.402(2023) 895 [2112.02092]

    Pith/arXiv arXiv 2023

  31. [31]

    Freed, G.W

    D.S. Freed, G.W. Moore and C. Teleman,Topological symmetry in quantum field theory, 2209.07471

    Pith/arXiv arXiv

  32. [32]

    Witten,Duality relations among topological effects in string theory,JHEP05(2000) 031 [hep-th/9912086]

    E. Witten,Duality relations among topological effects in string theory,JHEP05(2000) 031 [hep-th/9912086]

    Pith/arXiv arXiv 2000

  33. [33]

    Yonekura,Constraints on the topology of Type IIB string theory,JHEP07(2024) 112 [2403.04127]

    K. Yonekura,Constraints on the topology of Type IIB string theory,JHEP07(2024) 112 [2403.04127]

    Pith/arXiv arXiv 2024

  34. [34]

    Harlow and H

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

    Pith/arXiv arXiv 2021

  35. [35]

    Garc ´ ıa Etxebarria and S.S

    I. Garc ´ ıa Etxebarria and S.S. Hosseini,Some aspects of symmetry descent,JHEP12(2025) 223 [2404.16028]. – 60 –

    Pith/arXiv arXiv 2025

  36. [36]

    Kapustin,Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology,1403.1467

    A. Kapustin,Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology,1403.1467

    Pith/arXiv arXiv

  37. [37]

    Thorngren,Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d,JHEP02(2015) 152 [1404.4385]

    R. Thorngren,Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d,JHEP02(2015) 152 [1404.4385]

    Pith/arXiv arXiv 2015

  38. [38]

    Wang, X.-G

    J. Wang, X.-G. Wen and E. Witten,A New SU(2) Anomaly,J. Math. Phys.60(2019) 052301 [1810.00844]

    Pith/arXiv arXiv 2019

  39. [39]

    H. Kim, J. Manschot, G.W. Moore, R. Tao and X. Zhang,Path Integral Derivations Of K-Theoretic Donaldson Invariants,2509.23042

    arXiv

  40. [40]

    Diaconescu, G.W

    D.-E. Diaconescu, G.W. Moore and E. Witten,E(8) gauge theory, and a derivation of K theory from M theory,Adv. Theor. Math. Phys.6(2003) 1031 [hep-th/0005090]

    Pith/arXiv arXiv 2003

  41. [41]

    Teichner and E

    P. Teichner and E. Vogt,All oriented 4-manifolds have spin-c structures,

  42. [42]

    Maldacena and N

    J. Maldacena and N. Seiberg,D-brane instantons and k-theory charges,Journal of High Energy Physics2001(2001) 062–062

    2001

  43. [43]

    Dijkgraaf and E

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

    1990

  44. [44]

    Kapustin and N

    A. Kapustin and N. Seiberg,Coupling a QFT to a TQFT and Duality,JHEP04(2014) 001 [1401.0740]

    Pith/arXiv arXiv 2014

  45. [45]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg,Theta, Time Reversal, and Temperature,JHEP05(2017) 091 [1703.00501]

    Pith/arXiv arXiv 2017

  46. [46]

    Brennan and K

    T.D. Brennan and K. Intriligator,Anomalies of 4d Spin G theories,JHEP07(2024) 157 [2312.04756]

    Pith/arXiv arXiv 2024

  47. [47]

    Kapustin,D-branes in a topologically nontrivial B field,Adv

    A. Kapustin,D-branes in a topologically nontrivial B field,Adv. Theor. Math. Phys.4(2000) 127 [hep-th/9909089]

    Pith/arXiv arXiv 2000

  48. [48]

    Kaidi, K

    J. Kaidi, K. Ohmori and Y. Zheng,Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories,Phys. Rev. Lett.128(2022) 111601 [2111.01141]

    Pith/arXiv arXiv 2022

  49. [49]

    Y. Choi, C. Cordova, P.-S. Hsin, H.T. Lam and S.-H. Shao,Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions,Commun. Math. Phys.402(2023) 489 [2204.09025]

    Pith/arXiv arXiv 2023

  50. [50]

    Karasik,On anomalies and gauging of U(1) non-invertible symmetries in 4d QED, SciPost Phys.15(2023) 002 [2211.05802]

    A. Karasik,On anomalies and gauging of U(1) non-invertible symmetries in 4d QED, SciPost Phys.15(2023) 002 [2211.05802]

    Pith/arXiv arXiv 2023

  51. [51]

    Thomas,A generalization of the pontrjagin square cohomology operation,Proceedings of the National Academy of Sciences of the United States of America42(1956) 266

    E. Thomas,A generalization of the pontrjagin square cohomology operation,Proceedings of the National Academy of Sciences of the United States of America42(1956) 266

    1956

  52. [52]

    Whitehead,On simply connected, 4-dimensional polyhedra,Commentarii Mathematici Helvetici22(1949) 48

    J.H.C. Whitehead,On simply connected, 4-dimensional polyhedra,Commentarii Mathematici Helvetici22(1949) 48

    1949

  53. [53]

    Aharony, N

    O. Aharony, N. Seiberg and Y. Tachikawa,Reading between the lines of four-dimensional gauge theories,JHEP08(2013) 115 [1305.0318]

    Pith/arXiv arXiv 2013

  54. [54]

    Benini, C

    F. Benini, C. C´ ordova and P.-S. Hsin,On 2-Group Global Symmetries and their Anomalies, JHEP03(2019) 118 [1803.09336]

    Pith/arXiv arXiv 2019

  55. [55]

    Steenrod,Products of cocycles and extensions of mappings,Annals of Mathematics48 (1947) 290

    N.E. Steenrod,Products of cocycles and extensions of mappings,Annals of Mathematics48 (1947) 290. – 61 –

    1947

  56. [56]

    Karch, D

    A. Karch, D. Tong and C. Turner,A web of 2d dualities:z 2 gauge fields and arf invariants, SciPost Physics7(2019)

    2019

  57. [57]

    K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa et al.,Mirror Symmetry, vol. 1 ofClay Mathematics Monographs, American Mathematical Society, Providence, RI and Cambridge, MA (2003)

    2003

  58. [58]

    Atiyah,Riemann surfaces and spin structures,Annales scientifiques de l’ ´Ecole Normale Sup´ erieure4(1971) 47

    M.F. Atiyah,Riemann surfaces and spin structures,Annales scientifiques de l’ ´Ecole Normale Sup´ erieure4(1971) 47

    1971

  59. [59]

    Guillemin and A

    V. Guillemin and A. Pollack,Differential Topology, AMS Chelsea Publishing, AMS Chelsea Pub. (2010)

    2010

  60. [60]

    Johnson,Spin structures and quadratic forms on surfaces,Journal of the London Mathematical Societys2-22(1980) 365

    D. Johnson,Spin structures and quadratic forms on surfaces,Journal of the London Mathematical Societys2-22(1980) 365

    1980

  61. [61]

    Delmastro, D

    D. Delmastro, D. Gaiotto and J. Gomis,Global anomalies on the Hilbert space,JHEP11 (2021) 142 [2101.02218]

    Pith/arXiv arXiv 2021

  62. [62]

    Witten and K

    E. Witten and K. Yonekura,Anomaly inflow and theη-invariant, 2020

    2020

  63. [63]

    Tachikawa and K

    Y. Tachikawa and K. Yonekura,Why are fractional charges of orientifolds compatible with Dirac quantization?,SciPost Phys.7(2019) 058 [1805.02772]

    Pith/arXiv arXiv 2019

  64. [64]

    Kirby and L

    R. Kirby and L. Taylor,Pin structures on low-dimensional manifolds, inGeometry of Low-Dimensional Manifolds: Symplectic Manifolds and Jones-Witten Theory, S.K. Donaldson and C.B. Thomas, eds., London Mathematical Society Lecture Note Series, p. 177–242, Cambridge University Press (1991)

    1991

  65. [65]

    M. Guo, K. Ohmori, P. Putrov, Z. Wan and J. Wang,Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms,Commun. Math. Phys.376(2020) 1073 [1812.11959]

    Pith/arXiv arXiv 2020

  66. [66]

    Grigoletto and P

    A. Grigoletto and P. Putrov,Spin-Cobordisms, Surgeries and Fermionic Modular Bootstrap, Commun. Math. Phys.401(2023) 3169 [2106.16247]

    Pith/arXiv arXiv 2023

  67. [67]

    Witten,On flux quantization in M theory and the effective action,J

    E. Witten,On flux quantization in M theory and the effective action,J. Geom. Phys.22 (1997) 1 [hep-th/9609122]

    Pith/arXiv arXiv 1997

  68. [68]

    Steenrod and D.B.A

    N.E. Steenrod and D.B.A. Epstein,Cohomology Operations (AM-50): Lectures by N.E. Steenrod. (AM-50), Princeton University Press (1962)

    1962

  69. [69]

    Milnor and J.D

    J.W. Milnor and J.D. Stasheff,Characteristic Classes. Annals of Mathematics Studies AM-76, Princeton University Press (1974)

    1974

  70. [70]

    May,A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press (1999)

    J. May,A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press (1999)

    1999

  71. [71]

    Thomas,On the cohomology of the real grassmann complexes and the characteristic classes of n-plane bundles,Transactions of the American Mathematical Society96(1960) 67

    E. Thomas,On the cohomology of the real grassmann complexes and the characteristic classes of n-plane bundles,Transactions of the American Mathematical Society96(1960) 67

    1960

  72. [72]

    Sati,Twisted topological structures related to M-branes II: Twisted Wu andW u c structures,Int

    H. Sati,Twisted topological structures related to M-branes II: Twisted Wu andW u c structures,Int. J. Geom. Meth. Mod. Phys.09(2012) 1250056 [1109.4461]

    Pith/arXiv arXiv 2012

  73. [73]

    Witten,Five-brane effective action inM-theory.,J

    E. Witten,Five-brane effective action inM-theory.,J. Geom. Phys.22(1997) 103 [hep-th/9610234]

    Pith/arXiv arXiv 1997

  74. [74]

    Witten,Five-branes and M-theory on an orbifold,Nucl

    E. Witten,Five-branes and M-theory on an orbifold,Nucl. Phys. B463(1996) 383 [hep-th/9512219]. – 62 –

    Pith/arXiv arXiv 1996

  75. [75]

    Killingback,World Sheet Anomalies and Loop Geometry,Nucl

    T.P. Killingback,World Sheet Anomalies and Loop Geometry,Nucl. Phys. B288(1987) 578

    1987

  76. [76]

    Stolz and P

    S. Stolz and P. Teichner,What is an elliptic object?, inTopology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, U. Tillmann, ed., London Mathematical Society Lecture Note Series, p. 247–343, Cambridge University Press (2004)

    2002

  77. [77]

    H. Sati, U. Schreiber and J. Stasheff,Differential twisted String and Fivebrane structures, Commun. Math. Phys.315(2012) 169 [0910.4001]

    Pith/arXiv arXiv 2012

  78. [78]

    Sati,Geometric and topological structures related to M-branes II: Twisted String and String c structures,J

    H. Sati,Geometric and topological structures related to M-branes II: Twisted String and String c structures,J. Austral. Math. Soc.90(2011) 93 [1007.5419]

    Pith/arXiv arXiv 2011

  79. [79]

    Johnson,From M theory to F theory, with branes,Nucl

    C.V. Johnson,From M theory to F theory, with branes,Nucl. Phys. B507(1997) 227 [hep-th/9706155]

    Pith/arXiv arXiv 1997

  80. [80]

    Steenrod,Homology with local coefficients,Annals of Mathematics44(1943) 610

    N.E. Steenrod,Homology with local coefficients,Annals of Mathematics44(1943) 610

    1943

Showing first 80 references.