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Symmetry theta angles and topological Witten effects

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper defines theta angles purely from symmetry and derives every topological Witten effect from them.

desk verdict A genuinely new way to define theta angles from symmetry data, with the main universality claim (Prop. 3.2) asserted rather than proved; otherwise a thoughtful, useful paper that deserves refereeing. read the letter →

arxiv 2507.00220 v1 pith:FC4N6QUF submitted 2025-06-30 hep-th hep-ph

Shi Chen , Aleksey Cherman , Maria Neuzil This is my paper
classification hep-thhep-ph
keywords symmetrythetaangletopologicalWitteneffectchargehigher-formAharonov-BohminvertiblefieldtheoriesKennedy-Tasakitransformation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces a class of $\theta$ angles, called symmetry $\theta$ angles, that are defined entirely by the symmetries of a quantum field theory rather than by a chosen Lagrangian or path integral. For any non-anomalous finite Abelian invertible symmetry, and for U(1)-type symmetries, a symmetry $\theta$ angle is produced by a $\Theta = S^{\dagger} T S$ operation: gauge the symmetry, stack an invertible theory, then un-gauge. The paper argues that every topological Witten effect, meaning the reshuffling of which operators sit in which twisted sectors of a symmetry, arises from such a $\theta$ angle, and that the familiar charge-fractionalization Witten effect is only a special case requiring an extra symmetry. If correct, discrete symmetry $\theta$ angles are new RG-invariant labels of quantum field theories, and the Maxwell $\theta$ term remains observable even in the Higgs phase through string Aharonov-Bohm phases.

What carries the argument

The load-bearing object is the $\Theta$ transformation $\Theta = S^{\dagger} T S$, built from three operations on a $G$-symmetric QFT: $S$ gauges the non-anomalous symmetry $G$ to pass to the dual symmetry $\hat{G}$, $T$ stacks a $\hat{G}$-symmetric invertible theory, and $S^{\dagger}$ undoes the gauging. A Dijkgraaf-Witten theory, meaning a TQFT that becomes invertible after gauging a finite symmetry, is what remains of the invertible theory after conjugation, and its partition function on the background gauge field encodes the $\theta$ angle. The core identity is Eq. (3.42): a $\Theta$ transformation sends the subsector Hamiltonian $H^\alpha_a$ to $H^{\alpha+\theta(a^\vee)}_a$, so it shuffles twists at fixed charge. This spectral-table shuffle is precisely the topological Witten effect, and the possible $\theta$ angles are classified by the bordism classification of invertible theories, $\mathrm{Hom}(\widetilde{\Omega}^{SO}_d(B\hat{G}), \mathbb{R}/2\pi\mathbb{Z})$ or the spin analogue.

What would settle it

In the unit-charge Higgs phase of $U(1)$ gauge theory at $\theta = \pi$, two magnetic vortex strings whose worldsheets intersect once should acquire a relative phase $e^{i\pi}$; a lattice or analogue experiment showing no such phase would falsify the claim that the topological Witten effect persists without an electric symmetry.

Watch

Extended reading notes

Core claim

The central discovery is that $\theta$ angles can be promoted from auxiliary input in a path integral to intrinsic data of a QFT: a symmetry $\theta$ angle for a non-anomalous Abelian invertible symmetry $G$ is defined by a $\Theta = S^{\dagger} T S$ transformation, where $S$ gauges $G$, $S^{\dagger}$ ungauges it, and $T$ stacks a $\hat{G}$-symmetric invertible theory. Concretely, $\Theta$ acts on the spectral table of the theory by a vertical shuffle: it permutes the $G$-twisted sectors within each fixed $G$-charge sector while leaving charges untouched. This shuffle is the topological Witten effect, and Proposition 3.2 states that all topological Witten effects, and therefore all charge Witten effects, come from symmetry $\theta$ angles. The paper shows that familiar examples, including the Maxwell $\theta$ term, are special cases, and constructs new ones such as electric $\theta$ angles, Cheshire $\theta$ angles, and cubic $\theta$ angles.

Load-bearing premise

The definition of symmetry $\theta$ angles assumes that gauging a non-anomalous Abelian symmetry and then ungauging it are exact inverse operations on every spacetime manifold, so that $\Theta = S^{\dagger} T S$ is a well-defined map from QFTs to QFTs.

Editorial extensions

If this is right

  • The Maxwell theta term is a symmetry theta angle for the magnetic 1-form symmetry $U(1)^{(1)}_m$, so its definition does not require the Maxwell Lagrangian.
  • The topological Witten effect survives the breaking of the electric symmetry: in the unit-charge Higgs phase of $U(1)$ gauge theory, the theta angle produces a generalized Aharonov-Bohm phase for magnetic vortex strings even though the ordinary charge Witten effect is gone.
  • Discrete symmetry theta angles are preserved along renormalization-group flows and can label universality classes of otherwise identical QFTs.
  • Rational electric theta angles can act as Kennedy-Tasaki transformations, connecting symmetry-broken phases to SPT phases, as illustrated by charge-$p$ Abelian Higgs models and by su(2) Yang-Mills, where the six topological manipulations fall into three pairs related by $\Theta$.
  • Some symmetry theta angles are not Lagrangian theta angles, so they supply new discrete parameters in effective field theories, such as a Cheshire theta angle in the large-$N_c$ chiral Lagrangian that would give Skyrmions a $U(1)_A$ charge of $\theta/2\pi$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If discrete symmetry theta angles are as universal as the construction suggests, existing effective field theories may be missing discrete labels, and matching UV theories to IR EFTs will require computing these angles along the flow rather than setting them to zero.
  • The string Aharonov-Bohm prediction in the Higgs phase is a concrete target: lattice or cold-atom analogues of $U(1)$ gauge theory at $\theta = \pi$ could look for an $e^{i\pi}$ intersection phase between magnetic strings.
  • The su(2) and Standard Model global-structure analyses suggest that counting $\Theta$ transformations may overcount physical theories whenever degeneracies signal additional 0-form, possibly non-invertible, symmetries, so counting degeneracies is also a way to detect new symmetries.
  • The construction is restricted to Abelian invertible symmetries; the natural next step, left implicit by the paper, is to ask which non-invertible or non-Abelian symmetries admit an analogous intrinsic theta angle.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces a class of parameters called symmetry θ angles, defined intrinsically from a QFT's non-anomalous Abelian invertible symmetry rather than from a particular Lagrangian. The construction is based on Θ = S†T S transformations, where S gauges the symmetry, T stacks an invertible (dual-symmetry) theory, and S† ungauges. The authors show that such transformations act on spectral tables by shuffling twisted sectors at fixed charge, a phenomenon they call the topological Witten effect, and that this generalizes the standard Witten effect in Maxwell theory. They develop the finite Abelian case in detail, extend the construction to U(1)-type symmetries using differential cohomology, discuss the relation between symmetry θ angles and conventional Lagrangian θ angles, and give examples in 4d gauge theories, Cheshire θ angles, large-N QCD, and cubic θ angles in 3d and 4d.

Significance. If the central universality claim holds, the paper provides a useful conceptual unification: the Maxwell θ angle is a special case of a symmetry θ angle for the magnetic 1-form symmetry, and the topological Witten effect is a robust phenomenon that persists even when the electric symmetry is broken. The finite-Abelian spectral-table derivation in §3.2.2 is explicit and coherent, the Maxwell/Higgs-phase AB effect in §2.4.1 is worked out in detail, and the use of differential cohomology in §5.1.2 is a technically careful treatment of a delicate point. The examples in §7, including the electric θ angle, Cheshire θ angles, and cubic θ angles, are concrete and should stimulate further work. The paper does not ship machine-checked proofs, but the analytic derivations are detailed and largely reproducible.

major comments (3)
  1. [§3.2.2 and Proposition 3.2] The central universality claim is not established. Equation (3.42) shows that every Θ transformation induces the vertical shuffle H^α_a → H^{α+θ(a∨)}_a on the spectral table. The converse—that every topological Witten effect, defined as a locality-preserving reorganization of twisted sectors at fixed G-charge, is of this form—requires a classification of the group S(G) of symmetry-preserving topological manipulations. The sentence in §3.2.2 stating that stacking with invertible theories produces the most general possible horizontal shuffles is an assertion, not a derivation; §4.1 later asserts without proof that for non-self-dual G the group S(G) is generated by SΘ and ST. Because the SPT set Hom(Ω_d(BbG), U(1)) is much smaller than the set of table permutations in general, surjectivity is not automatic. Until this generation statement is proved or the proposition is weakened, Proposition 3.2 remains conditional.
  2. [§4.1] The generation statement for S(G) is load-bearing and unproved. The paper states that for non-self-dual G, SΘ(G) and ST(G) generate all of S(G), and that for self-dual G one must add S; it also states that these generators satisfy relations such as (ST)^3 ≃ 1. No theorem or reference is supplied for either the generation statement or the claimed relations beyond the specific G × bG and Z_N^[1] examples in Appendix B. The spectral-table arguments in §3.2.2 only establish inclusions for the operations considered. The order-72 counting for the Standard-Model Z_6^[1] symmetry in §8 also relies on this unproved generation statement, so the gap has consequences beyond the abstract formulation.
  3. [§3.1.3 / Definition 3.1] The definition of a symmetry θ angle assumes that S and S† are inverse operations on the space of QFTs, but this is only verified at the level of closed-manifold partition functions with the normalization Eq. (3.13). The paper does not discuss whether the Θ transformation remains well-defined as an operation on a QFT with boundaries, defects, or on non-closed manifolds, nor how the normalization behaves under gluing. Since Definition 3.1 elevates Θ to an intrinsic parameter of a QFT, the class of manifolds and geometric structures on which S and S† are inverse should be stated explicitly.
minor comments (5)
  1. [§7.2.3] The text contains the typo 'Chershire θ angle' where 'Cheshire θ angle' is meant.
  2. [Eq. (3.20)] The transformed theory is written as Z ∗ I, but the new theory is not given its own symbol; introducing Z_θ = Z ∗ I at the point of Definition 3.1 would make the subsequent spectral-table discussion easier to follow.
  3. [§2.2.2] The passage from Eq. (2.25) to Eq. (2.27) assumes that the n-sum can be organized as a trace over charge-m subsectors; a brief statement of the relevant Hilbert-space decomposition would improve readability.
  4. [Footnote 18] The assertion that H^•(M × S^1, Z) is torsion-free for any oriented closed 2-manifold M is correct, but it relies on the classification of closed surfaces and could be stated with that context.
  5. [§5.1.2] Equation (5.6) is written as a path integral, but the discussion makes clear that the BF coupling is only a shorthand for a differential-cohomology expression; moving that caveat from the later paragraph into the main text would prevent a casual misreading.

Circularity Check

2 steps flagged · score 4.0 of 10

Partial circularity: Proposition 3.2's universality claim is largely a definitional consequence of the spectral-table shuffle, with the 'all' resting on an asserted equivalence rather than an independent proof.

  1. self definitional [Definition 3.1; Section 3.2.2, Eq. (3.42); Proposition 3.2]
    "Definition 3.1: For a QFT with a non-anomalous finite Abelian invertible symmetry G, a G-symmetry θ angle is defined by a Θ transformation with respect to G. ... Hα a (Z ∗ I, M) Θ= Hα+θ(a∨) a (Z, M). ... This means that the effect of the symmetry θ angle is to shuffle the twisted sectors of each given charge under the symmetry G. This is a topological Witten effect. ..."

    The topological Witten effect was defined in Definition 2.2/Eq. (2.28) as exactly the twist-shuffle H^α_m(θ) ≃ H^{α+θm}_m(0) at fixed charge. Eq. (3.42) is the same shuffle pattern for a general finite Abelian symmetry, and the paper explicitly identifies it as a topological Witten effect. The converse half of Proposition 3.2—that all topological Witten effects arise this way—is then justified only by the statement that T-stacking gives the most general horizontal shuffles. Since Θ is defined as S†TS, that statement is the dualized version of the desired surjectivity; the 'all' is therefore asserted by construction rather than derived from an independent classification or theorem.

  2. self definitional [Section 5.1.1, last paragraph; Section 6.2, footnote 27]
    "If we apply the construction in the section below to such a U (1)[−1] symmetry, we will simply rediscover the parameter φ itself. This means that a 'U (1)[−1]-symmetry θ angle' is tautologically equivalent to having a 'U (1)[−1] symmetry' in the first place."

    This is a self-admitted tautology in the proposed framework. The paper explicitly excludes U(1)[−1] symmetries from the allowed U in the definition, and the remark is not used as a load-bearing premise for any later result, so it makes only a minor contribution to the circularity score.

full rationale

The paper contains substantial independent derivations: the Maxwell θ-angle analysis in Section 2 is carried out by direct path-integral evaluation; the S, S†, and T transformation identities in Section 3 are explicit algebraic statements; the U(1) construction in Section 5 is worked out with differential-cohomology input; and the examples in Section 7 are self-contained QFT computations. There is no fitted parameter renamed as a prediction, and the self-citation to the authors' earlier Cheshire θ-angle paper is illustrative rather than load-bearing. The main circularity concern is confined to the strong universality claim in Proposition 3.2. The paper shows that every symmetry θ angle produces a spectral-table vertical shuffle, which matches the previously defined topological Witten effect; the reverse direction—that every possible topological Witten effect is a symmetry θ angle—is supported only by the sentence asserting that stacking with invertible theories yields the most general horizontal shuffles. That assertion is not proven, nor is it cited to an external or machine-checked result, and it is equivalent to the surjectivity half of the proposition. The unproved generation statement about S(G) in Section 4.1 is a further correctness risk, but it is an unsupported assertion rather than a circular reduction. Overall, the central 'all' claim carries a definitional component, but the paper's concrete results retain independent content; hence score 4 rather than a higher circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central construction does not fit any numerical constant to data; the theta parameters are introduced as intrinsic couplings, not fitted values. The paper instead imports three load-bearing pieces: the bordism classification of invertible theories, the invertibility of gauging operations, and the spectral-table decomposition of Hilbert spaces on M x S^1. No new particles, forces, or dimensions are postulated.

assumptions (5)
  • domain assumption G-symmetric invertible field theories are classified by Hom(Omega_d^Spin(B bG), R/2pi Z) or Hom(Omega_d^SO(B bG), R/2pi Z).
    Used to count symmetry theta angles in Eqs. (3.23) and (5.24); the classification is imported from the bordism and SPT literature and not proved in this paper.
  • domain assumption Gauging a non-anomalous finite Abelian symmetry and ungauging it are inverse operations with the normalization N(X,G) given in Eq. (3.13).
    The S and S-dagger transformations are the backbone of Theta = S-dagger T S; if S is not invertible on arbitrary manifolds, the definition of symmetry theta angles fails. Section 3.1.1.
  • standard math The BF coupling between U(1) gauge fields is globally well-defined via the Beilinson-Deligne cup product on differential cohomology.
    Needed to define U(1) S transformations when deformation classes are nontrivial; Section 5.1.2.
  • domain assumption On M x S^1 the symmetry sector decomposition into spatial and temporal gauge fields gives the Hamiltonian interpretation of Eq. (3.30) and the spectral table.
    The topological Witten effect is derived through spectral tables; this decomposition is standard but is assumed throughout Section 3.2.
  • domain assumption The unit-charge Higgs phase of U(1) gauge theory has unbroken U(1)_m^{[1]} symmetry and finite-tension magnetic string excitations.
    The claimed physical consequence of the string Aharonov-Bohm effect in Section 2.4.1 depends on this assumption.

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Pith. "Pith review of Symmetry theta angles and topological Witten effects." pith.science (2026). https://pith.science/paper/FC4N6QUF

@misc{pith2026250700220,
  author       = {Pith},
  title        = {Pith review of: Symmetry theta angles and topological Witten effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FC4N6QUF}},
  note         = {Machine review of arXiv:2507.00220}
}
abstract

We introduce a large class of $\theta$ angles in quantum field theory that we call symmetry $\theta$ angles. Unlike conventional $\theta$ angles whose definition depends on a choice of a path integral, symmetry $\theta$ angles are intrinsic parameters of a quantum field theory that depend only on its symmetries. A frequent consequence of symmetry $\theta$ angles is a phenomenon we call the topological Witten effect, which is a generalization of the standard Witten effect. Topological Witten effects modify which charged operators are attached to topological operators as a function of $\theta$. Physically, topological Witten effects induce generalized Aharonov-Bohm effects. We show that these new $\theta$ angles and Witten effects can appear in many familiar field theories.

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Reference graph

Works this paper leans on

107 extracted references · 8 canonical work pages · cited by 2 Pith papers

  1. [23]

    Garcia Garcia, M

    I. Garcia Garcia, M. Kongsore and K. Van Tilburg,Dyon Loops and Abelian Instantons, 2506.14867

    arXiv

  2. [1]

    Witten,Dyons of Chargeeθ/2π, Phys

    E. Witten,Dyons of Chargeeθ/2π, Phys. Lett. B86 (1979) 283

    1979

  3. [2]

    Y. Choi, H. T. Lam and S.-H. Shao,Non-invertible Global Symmetries in the Standard Model, 2205.05086

    arXiv

  4. [3]

    D. S. Freed,Pions and Generalized Cohomology, J. Diff. Geom.80 (2008) 45 [hep-th/0607134]

    arXiv 2008

  5. [4]

    Seiberg,Modifying the Sum Over Topological Sectors and Constraints on Supergravity, JHEP 07 (2010) 070 [1005.0002]

    N. Seiberg,Modifying the Sum Over Topological Sectors and Constraints on Supergravity, JHEP 07 (2010) 070 [1005.0002]

    arXiv 2010

  6. [5]

    Kapustin and N

    A. Kapustin and N. Seiberg,Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [1401.0740]

    arXiv 2014

  7. [6]

    D. S. Freed and M. J. Hopkins,Reflection positivity and invertible topological phases, 1604.06527

    arXiv

  8. [7]

    D. S. Freed, Z. Komargodski and N. Seiberg,The Sum Over Topological Sectors andθ in the 2+1-Dimensional CP1 σ-Model, Commun. Math. Phys.362 (2018) 167 [1707.05448]

    arXiv 2018

Show all 107 references
  1. [8]

    Cordova, P.-S

    C. Cordova, P.-S. Hsin and N. Seiberg,Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups, SciPost Phys. 4 (2018) 021 [1711.10008]

    2018 arXiv

  2. [9]

    Yonekura,On the cobordism classification of symmetry protected topological phases, Commun

    K. Yonekura,On the cobordism classification of symmetry protected topological phases, Commun. Math. Phys.368 (2019) 1121 [1803.10796]

    2019 arXiv

  3. [10]

    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]

    2019 arXiv

  4. [11]

    Hsin and A

    P.-S. Hsin and A. Turzillo,Symmetry-enriched quantum spin liquids in (3 + 1)d, JHEP 09 (2020) 022 [1904.11550]

    2020 arXiv

  5. [12]

    Hsin and H

    P.-S. Hsin and H. T. Lam,Discrete theta angles, symmetries and anomalies, SciPost Phys. 10 (2021) 032 [2007.05915]

    2021 arXiv

  6. [13]

    Chen and Y

    S. Chen and Y. Tanizaki,Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients, 2307.00939

    arXiv

  7. [14]

    Chen and Y

    S. Chen and Y. Tanizaki,Solitonic Symmetry beyond Homotopy: Invertibility from Bordism and Noninvertibility from Topological Quantum Field Theory, Phys. Rev. Lett.131 (2023) 011602 [2210.13780]

    2023 arXiv

  8. [15]

    Gaiotto, A

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

    2015 arXiv

  9. [16]

    S. Chen, A. Cherman, G. Choi and M. Neuzil,Cheshire θ terms, Aharonov-Bohm effects, and axions, 2410.23355

    arXiv

  10. [17]

    Thorngren,Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d, JHEP 02 (2015) 152 [1404.4385]

    R. Thorngren,Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d, JHEP 02 (2015) 152 [1404.4385]

    2015 arXiv

  11. [18]

    S. M. Kravec, J. McGreevy and B. Swingle,All-fermion electrodynamics and fermion number anomaly inflow, Phys. Rev. D92 (2015) 085024 [1409.8339]

    2015 arXiv

  12. [19]

    Wang, X.-G

    J. Wang, X.-G. Wen and E. Witten,A New SU(2) Anomaly, J. Math. Phys.60 (2019) 052301 [1810.00844]. – 75 –

    2019 arXiv

  13. [20]

    J. P. Ang, K. Roumpedakis and S. Seifnashri,Line Operators of Gauge Theories on Non-Spin Manifolds, 1911.00589

    1911 arXiv

  14. [21]

    N. Kan, K. Kawabata and H. Wada,Symmetry fractionalization and duality defects in Maxwell theory, JHEP 10 (2024) 238 [2404.14481]

    2024 arXiv

  15. [22]

    Kovner and B

    A. Kovner and B. Rosenstein,New look at QED in four-dimensions: The Photon as a Goldstone boson and the topological interpretation of electric charge, Phys. Rev. D49 (1994) 5571 [hep-th/9210154]

    1994 arXiv

  16. [24]

    Seiberg and E

    N. Seiberg and E. Witten,Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B426 (1994) 19 [hep-th/9407087]

    1994 arXiv

  17. [25]

    Seiberg and E

    N. Seiberg and E. Witten,Monopoles, duality and chiral symmetry breaking inN = 2 supersymmetric QCD, Nucl. Phys. B431 (1994) 484 [hep-th/9408099]

    1994 arXiv

  18. [26]

    Cherman and T

    A. Cherman and T. Jacobson,Emergent 1-form symmetries, Phys. Rev. D109 (2024) 125013 [2304.13751]

    2024 arXiv

  19. [27]

    H. B. Nielsen and P. Olesen,Vortex Line Models for Dual Strings, Nucl. Phys. B 61 (1973) 45

    1973

  20. [28]

    Chang, Y.-H

    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin,Topological Defect Lines and Renormalization Group Flows in Two Dimensions, JHEP 01 (2019) 026 [1802.04445]

    2019 arXiv

  21. [29]

    Cordova and K

    C. Cordova and K. Ohmori,Noninvertible Chiral Symmetry and Exponential Hierarchies, Phys. Rev. X 13 (2023) 011034 [2205.06243]

    2023 arXiv

  22. [30]

    Vafa,Quantum Symmetries of String Vacua, Mod

    C. Vafa,Quantum Symmetries of String Vacua, Mod. Phys. Lett. A4 (1989) 1615

    1989

  23. [31]

    Bhardwaj and Y

    L. Bhardwaj and Y. Tachikawa,On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [1704.02330]

    2018 arXiv

  24. [32]

    Aharony, N

    O. Aharony, N. Seiberg and Y. Tachikawa,Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [1305.0318]

    2013 arXiv

  25. [33]

    Tachikawa,On gauging finite subgroups, SciPost Phys

    Y. Tachikawa,On gauging finite subgroups, SciPost Phys. 8 (2020) 015 [1712.09542]

    2020 arXiv

  26. [34]

    E. Witten,SL(2,Z) action on three-dimensional conformal field theories with Abelian symmetry, inFrom Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan, pp. 1173–1200, 7, 2003,hep-th/0307041

    2003 arXiv

  27. [35]

    Hatcher,Algebraic topology

    A. Hatcher,Algebraic topology. Cambridge Univ. Press, 2001

    2001

  28. [36]

    Chen and A

    Y.-A. Chen and A. Kapustin,Bosonization in three spatial dimensions and a 2-form gauge theory, Phys. Rev. B100 (2019) 245127 [1807.07081]

    2019 arXiv

  29. [37]

    Chen and S

    Y.-A. Chen and S. Tata,Higher cup products on hypercubic lattices: Application to lattice models of topological phases, J. Math. Phys.64 (2023) 091902 [2106.05274]

    2023 arXiv

  30. [38]

    Jacobson and T

    T. Jacobson and T. Sulejmanpasic,Modified Villain formulation of Abelian Chern-Simons theory, Phys. Rev. D107 (2023) 125017 [2303.06160]

    2023 arXiv

  31. [39]

    Toward topological classification of phases with short-range entanglement

    A. Kitaev, “Toward topological classification of phases with short-range entanglement.” http://online.kitp.ucsb.edu/online/topomat11/kitaev/, 2011. – 76 –

    2011

  32. [40]

    On the classification of short-range entangled states

    A. Kitaev, “On the classification of short-range entangled states.” http://scgp.stonybrook.edu/archives/7874, 2013

    2013

  33. [41]

    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

    arXiv

  34. [42]

    Kapustin, R

    A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang,Fermionic Symmetry Protected Topological Phases and Cobordisms, JHEP 12 (2015) 052 [1406.7329]

    2015 arXiv

  35. [43]

    Witten,Fermion Path Integrals And Topological Phases, Rev

    E. Witten,Fermion Path Integrals And Topological Phases, Rev. Mod. Phys.88 (2016) 035001 [1508.04715]

    2016 arXiv

  36. [44]

    Gu and X.-G

    Z.-C. Gu and X.-G. Wen,Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order, Phys. Rev. B80 (2009) 155131 [0903.1069]

    2009 arXiv

  37. [45]

    Pollmann, E

    F. Pollmann, E. Berg, A. M. Turner and M. Oshikawa,Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B85 (2012) 075125 [0909.4059]

    2012 arXiv

  38. [46]

    Chen, Z.-X

    X. Chen, Z.-X. Liu and X.-G. Wen,Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations, Phys. Rev. B84 (2011) 235141 [1106.4752]

    2011 arXiv

  39. [47]

    Dijkgraaf and E

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

    1990

  40. [48]

    J. C. Baez and J. Dolan,Higher dimensional algebra and topological quantum field theory, J. Math. Phys.36 (1995) 6073 [q-alg/9503002]

    1995 arXiv

  41. [49]

    Lurie,On the Classification of Topological Field Theories, 0905.0465

    J. Lurie,On the Classification of Topological Field Theories, 0905.0465

    arXiv

  42. [50]

    Yamashita and K

    M. Yamashita and K. Yonekura,Differential models for the Anderson dual to bordism theories and invertible QFT’s. I., J. Gökova Geom. Topol. GGT16 (2023) 1 [2106.09270]

    2023 arXiv

  43. [51]

    Koide, Y

    M. Koide, Y. Nagoya and S. Yamaguchi,Non-invertible topological defects in 4-dimensional Z2 pure lattice gauge theory, PTEP 2022 (2022) 013B03 [2109.05992]

    2022 arXiv

  44. [52]

    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]

    2022 arXiv

  45. [53]

    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, 2204.09025

    arXiv

  46. [54]

    Sun and Y

    Z. Sun and Y. Zheng,When are Duality Defects Group-Theoretical?, 2307.14428

    arXiv

  47. [55]

    Shao,What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, 2308.00747

    S.-H. Shao,What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries, 2308.00747

    arXiv

  48. [56]

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

    arXiv

  49. [57]

    Kennedy and H

    T. Kennedy and H. Tasaki,Hidden symmetry breaking and the Haldane phase in S=1 quantum spin chains., Commun. Math. Phys.147 (1992) 431

    1992

  50. [58]

    Kennedy and H

    T. Kennedy and H. Tasaki,Hidden Z2×Z2 symmetry breaking in Haldane-gap antiferromagnets, Phys. Rev. B45 (1992) 304

    1992

  51. [59]

    Oshikawa,Hidden Z2*Z2 symmetry in quantum spin chains with arbitrary integer spin, J

    M. Oshikawa,Hidden Z2*Z2 symmetry in quantum spin chains with arbitrary integer spin, J. Phys.: Condens. Matter4 (1992) 7469. – 77 –

    1992

  52. [60]

    L. Li, M. Oshikawa and Y. Zheng,Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases, Phys. Rev. B 108 (2023) 214429 [2301.07899]

    2023 arXiv

  53. [61]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, D. Pajer and S. Schafer-Nameki,The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries, SciPost Phys. 18 (2025) 156 [2312.17322]

    2025 arXiv

  54. [62]

    L. Li, M. Oshikawa and Y. Zheng,Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations, SciPost Phys. 18 (2025) 153 [2307.04788]

    2025 arXiv

  55. [63]

    Parayil Mana, Y

    A. Parayil Mana, Y. Li, H. Sukeno and T.-C. Wei,Kennedy-Tasaki transformation and noninvertible symmetry in lattice models beyond one dimension, Phys. Rev. B109 (2024) 245129 [2402.09520]

    2024 arXiv

  56. [64]

    Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam and S.-H. Shao,Noninvertible duality defects in 3+1 dimensions, Phys. Rev. D105 (2022) 125016 [2111.01139]

    2022 arXiv

  57. [65]

    Córdova, D

    C. Córdova, D. S. Freed, H. T. Lam and N. Seiberg,Anomalies in the Space of Coupling Constants and Their Dynamical Applications I, SciPost Phys. 8 (2020) 001 [1905.09315]

    2020 arXiv

  58. [66]

    Córdova, D

    C. Córdova, D. S. Freed, H. T. Lam and N. Seiberg,Anomalies in the Space of Coupling Constants and Their Dynamical Applications II, SciPost Phys. 8 (2020) 002 [1905.13361]

    2020 arXiv

  59. [67]

    Tanizaki and M

    Y. Tanizaki and M. Unsal,Modified instanton sum in QCD and higher-groups, JHEP 03 (2020) 123 [1912.01033]

    2020 arXiv

  60. [68]

    Heidenreich, J

    B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius and I. Valenzuela, Chern-Weil global symmetries and how quantum gravity avoids them, JHEP 11 (2021) 053 [2012.00009]

    2021 arXiv

  61. [69]

    Aloni, E

    D. Aloni, E. García-Valdecasas, M. Reece and M. Suzuki,Spontaneously broken (-1)-form U(1) symmetries, SciPost Phys. 17 (2024) 031 [2402.00117]

    2024 arXiv

  62. [70]

    Karch and D

    A. Karch and D. Tong,Particle-Vortex Duality from 3d Bosonization, Phys. Rev. X 6 (2016) 031043 [1606.01893]

    2016 arXiv

  63. [71]

    Seiberg, T

    N. Seiberg, T. Senthil, C. Wang and E. Witten,A Duality Web in 2+1 Dimensions and Condensed Matter Physics, Annals Phys. 374 (2016) 395 [1606.01989]

    2016 arXiv

  64. [72]

    Senthil, D

    T. Senthil, D. T. Son, C. Wang and C. Xu,Duality between (2 + 1)d Quantum Critical Points, Phys. Rept. 827 (2019) 1 [1810.05174]

    2019 arXiv

  65. [73]

    Deligne,Théorie de Hodge : II, Publications Mathématiques de l’IHÉS40 (1971) 5

    P. Deligne,Théorie de Hodge : II, Publications Mathématiques de l’IHÉS40 (1971) 5

    1971

  66. [74]

    A. A. Beilinson,Higher regulators and values of l-functions, Journal of Soviet Mathematics 30 (1985) 2036

    1985

  67. [75]

    Gajer,Geometry of deligne cohomology, Inventiones mathematicae 127 (1997) 155

    P. Gajer,Geometry of deligne cohomology, Inventiones mathematicae 127 (1997) 155

    1997

  68. [76]

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

    2005 arXiv

  69. [77]

    Armoni,S-Dual of Maxwell–Chern-Simons Theory, Phys

    A. Armoni,S-Dual of Maxwell–Chern-Simons Theory, Phys. Rev. Lett.130 (2023) 141601 [2212.00513]

    2023 arXiv

  70. [78]

    Heidenreich, J

    B. Heidenreich, J. McNamara and M. Reece,Non-standard axion electrodynamics and the dual Witten effect, JHEP 01 (2024) 120 [2309.07951]. – 78 –

    2024 arXiv

  71. [79]

    Z. Duan, Q. Jia and S. Lee,Web of 4D dualities, supersymmetric partition functions and SymTFT, JHEP 01 (2025) 161 [2410.10036]

    2025 arXiv

  72. [80]

    Witten,Large N chiral dynamics, Annals Phys

    E. Witten,Large N chiral dynamics, Annals Phys. 128 (1980) 363

    1980

  73. [81]

    Witten,Global aspects of current algebra, Nucl

    E. Witten,Global aspects of current algebra, Nucl. Phys. B223 (1983) 422

    1983

  74. [82]

    Witten,Current Algebra, Baryons, and Quark Confinement, Nucl

    E. Witten,Current Algebra, Baryons, and Quark Confinement, Nucl. Phys. B 223 (1983) 433

    1983

  75. [83]

    Gaiotto, Z

    D. Gaiotto, Z. Komargodski and N. Seiberg,Time-reversal breaking in QCD4, walls, and dualities in 2 + 1 dimensions, JHEP 01 (2018) 110 [1708.06806]

    2018 arXiv

  76. [84]

    Komargodski,Baryons as Quantum Hall Droplets, 1812.09253

    Z. Komargodski,Baryons as Quantum Hall Droplets, 1812.09253

    arXiv

  77. [85]

    Y.-L. Ma, M. A. Nowak, M. Rho and I. Zahed,Baryon as a Quantum Hall Droplet and the Cheshire Cat Principle, Phys. Rev. Lett.123 (2019) 172301 [1907.00958]

    2019 arXiv

  78. [86]

    Karasik,Skyrmions, Quantum Hall Droplets, and one current to rule them all, SciPost Phys

    A. Karasik,Skyrmions, Quantum Hall Droplets, and one current to rule them all, SciPost Phys. 9 (2020) 008 [2003.07893]

    2020 arXiv

  79. [87]

    Karasik,Vector dominance, one flavored baryons, and QCD domain walls from the ”hidden” Wess-Zumino term, SciPost Phys

    A. Karasik,Vector dominance, one flavored baryons, and QCD domain walls from the ”hidden” Wess-Zumino term, SciPost Phys. 10 (2021) 138 [2010.10544]

    2021 arXiv

  80. [88]

    Bigazzi, A

    F. Bigazzi, A. L. Cotrone and A. Olzi,Hall Droplet Sheets in Holographic QCD, JHEP 02 (2023) 194 [2211.05147]

    2023 arXiv

  81. [89]

    Tong,Line Operators in the Standard Model, JHEP 07 (2017) 104 [1705.01853]

    D. Tong,Line Operators in the Standard Model, JHEP 07 (2017) 104 [1705.01853]

    2017 arXiv

  82. [90]

    M. M. Anber and E. Poppitz,Nonperturbative effects in the Standard Model with gauged 1-form symmetry, JHEP 12 (2021) 055 [2110.02981]

    2021 arXiv

  83. [91]

    Cordova, S

    C. Cordova, S. Hong, S. Koren and K. Ohmori,Neutrino Masses from Generalized Symmetry Breaking, Phys. Rev. X 14 (2024) 031033 [2211.07639]

    2024 arXiv

  84. [92]

    Reece,Axion-gauge coupling quantization with a twist, JHEP 10 (2023) 116 [2309.03939]

    M. Reece,Axion-gauge coupling quantization with a twist, JHEP 10 (2023) 116 [2309.03939]

    2023 arXiv

  85. [93]

    Y. Choi, M. Forslund, H. T. Lam and S.-H. Shao,Quantization of Axion-Gauge Couplings and Noninvertible Higher Symmetries, Phys. Rev. Lett.132 (2024) 121601 [2309.03937]

    2024 arXiv

  86. [94]

    Cordova, S

    C. Cordova, S. Hong and L.-T. Wang,Axion domain walls, small instantons, and non-invertible symmetry breaking, JHEP 05 (2024) 325 [2309.05636]

    2024 arXiv

  87. [95]

    Cordova, S

    C. Cordova, S. Hong and S. Koren,Non-Invertible Peccei-Quinn Symmetry and the Massless Quark Solution to the Strong CP Problem, 2402.12453

    arXiv

  88. [96]

    Alonso, D

    R. Alonso, D. Dimakou and M. West,Fractional-charge hadrons and leptons to tell the Standard Model group apart, Phys. Lett. B863 (2025) 139354 [2404.03438]

    2025 arXiv

  89. [97]

    Li and L.-X

    H.-L. Li and L.-X. Xu,Understanding the SM gauge group from SMEFT, JHEP 07 (2024) 199 [2404.04229]

    2024 arXiv

  90. [98]

    Koren and A

    S. Koren and A. Martin,Fractionally charged particles at the energy frontier: The SM gauge group and one-form global symmetry, SciPost Phys. 18 (2025) 004 [2406.17850]

    2025 arXiv

  91. [99]

    Dierigl and D

    M. Dierigl and D. Novičić,The axion is going dark, JHEP 12 (2024) 104 [2409.02180]

    2024 arXiv

  92. [100]

    Q.-H. Cao, S. Ge, Y. Liu and J.-C. Wang,Berry phase in axion physics, SM global structure, and generalized symmetries, 2411.04749. – 79 –

  93. [101]

    Hsin and J

    P.-S. Hsin and J. Gomis,Detecting Standard Model Gauge Group from Generalized Fractional Quantum Hall Effect, 2411.18160

    arXiv

  94. [102]

    Z. Wan, J. Wang and Y.-Z. You,Topological Responses of the Standard Model Gauge Group, 2412.21196

    arXiv

  95. [103]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg,Theta, time reversal, and temperature, JHEP 05 (2017) 091 [1703.00501]

    2017 arXiv

  96. [104]

    R. J. Milgram,Surgery with coefficients, Annals of Mathematics100 (1974) 194

    1974

  97. [105]

    Belov and G

    D. Belov and G. W. Moore,Classification of Abelian spin Chern-Simons theories, hep-th/0505235

    arXiv

  98. [106]

    van der Blij,An invariant of quadratic forms modulo 8, Indagationes Mathematicae 21 (1959) 291

    F. van der Blij,An invariant of quadratic forms modulo 8, Indagationes Mathematicae 21 (1959) 291

    1959

  99. [107]

    Turaev,Reciprocity for gauss sums on finite abelian groups, Mathematical Proceedings of the Cambridge Philosophical Society124 (1998) 205–214

    V. Turaev,Reciprocity for gauss sums on finite abelian groups, Mathematical Proceedings of the Cambridge Philosophical Society124 (1998) 205–214. – 80 –

    1998

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