{
  "schema_version": 1,
  "id": "PHYS-2026-08-11-01",
  "canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-11-01/",
  "source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-11-01.md",
  "metadata": {
    "schema_version": 1,
    "id": "PHYS-2026-08-11-01",
    "date": "2026-08-11",
    "updated_at": "2026-08-11",
    "title": "Gauging trades a finite Abelian symmetry for its Fourier dual",
    "summary": "Derive the dual higher-form symmetry produced by finite Abelian gauging, track double gauging and charge conjugation, and locate the limits imposed by anomalies, global form, and non-invertible fusion.",
    "language": "en",
    "entry_kind": "daily",
    "status": "published",
    "level": "research",
    "user_difficulty": "unrated",
    "domains": [
      "quantum-field-theory",
      "mathematical-physics",
      "condensed-matter",
      "quantum-information"
    ],
    "estimated_minutes": 95
  },
  "content_markdown": "\n## The claim under pressure\n\n**Gauging a finite Abelian global symmetry removes its old global action, but it also creates a dual global symmetry.**\n\nLet a $d$-dimensional QFT have a non-anomalous finite Abelian $p$-form symmetry $G^{(p)}$, with\n\n$$\n0\\leq p\\leq d-2.\n$$\n\nUntwisted gauging produces\n\n$$\nG^{(p)}\n\\quad\\longrightarrow\\quad\n\\widehat G^{(d-p-2)},\n$$\n\nwhere\n\n$$\n\\widehat G=\\operatorname{Hom}(G,U(1))\n$$\n\nis the Pontryagin dual group.\n\nThe statement follows from the finite gauge sum and its character pairing. It does not determine whether the gauged theory confines, develops topological order, remains gapless, or breaks the dual symmetry.\n\n## 1. What gauging changes\n\nA global symmetry acts on physical operators and states. A gauge transformation identifies field configurations that represent the same physical configuration.\n\nGauging $G$ performs two related operations:\n\n1. it projects onto gauge-invariant states and operators;\n2. it sums over nontrivial $G$ gauge bundles or higher gauge fields.\n\nThe original $G$ transformation becomes a redundancy. That fact alone does not exhaust the operation because the new sum has topological sectors.\n\nCharacters of $G$ can measure those sectors. Their topological action defines the dual symmetry.\n\nThis is why “gauging destroys the symmetry” is incomplete. The original action is gone as a global action, while the topology of the gauge field supplies new global symmetry data.\n\n## 2. A one-sector Fourier transform\n\nStart with $G=\\mathbb Z_N$. Let $Z(n)$ denote a partition function or amplitude in a sector $n\\in\\mathbb Z_N$.\n\nDefine\n\n$$\n\\widetilde Z(m)\n=\\frac{1}{\\sqrt N}\n\\sum_{n=0}^{N-1}\ne^{2\\pi\\ii mn/N}Z(n).\n$$\n\nThe phase\n\n$$\n\\chi_m(n)=e^{2\\pi\\ii mn/N}\n$$\n\nis a character of $\\mathbb Z_N$. The label $m$ therefore belongs to\n\n$$\n\\widehat{\\mathbb Z_N}\n=\\operatorname{Hom}(\\mathbb Z_N,U(1))\n\\cong\\mathbb Z_N.\n$$\n\nThe isomorphism $\\widehat{\\mathbb Z_N}\\cong\\mathbb Z_N$ is canonical only after the pairing convention has been chosen. For a general finite Abelian group, $G$ and $\\widehat G$ are isomorphic but need not come with a preferred isomorphism.\n\nTransform once more:\n\n$$\n\\widetilde{\\widetilde Z}(k)\n=\\frac{1}{N}\n\\sum_{m,n=0}^{N-1}\ne^{2\\pi\\ii m(k+n)/N}Z(n).\n$$\n\nCharacter orthogonality gives\n\n$$\n\\sum_{m=0}^{N-1}e^{2\\pi\\ii mr/N}\n=N\\,\\delta_{r,0\\;{\\rm mod}\\;N}.\n$$\n\nHence\n\n$$\n\\boxed{\n\\widetilde{\\widetilde Z}(k)=Z(-k)\n}.\n$$\n\nThe minus sign is the inversion map $n\\mapsto-n$, which acts as charge conjugation on the finite Abelian group. Double Fourier transform returns the original data with this involution.\n\n## 3. The spacetime degree shift\n\nA $p$-form symmetry couples to a background $(p+1)$-form gauge field. For $G=\\mathbb Z_N$, write\n\n$$\nB_{p+1}\\in Z^{p+1}(M,\\mathbb Z_N).\n$$\n\nGauging promotes $B$ to a dynamical cocycle $b$. A background for the dual symmetry is\n\n$$\n\\widetilde B_{d-p-1}\n\\in Z^{d-p-1}(M,\\widehat G).\n$$\n\nThe gauged partition function has the schematic form\n\n$$\nZ_{\\mathcal T/G^{(p)}}[\\widetilde B]\n=\\mathcal N_M\n\\sum_{[b]}\nZ_{\\mathcal T}[b]\\,\n\\exp\\!\\left(\n\\frac{2\\pi\\ii}{N}\n\\int_M\\widetilde B\\cup b\n\\right).\n$$\n\nThe degrees add to the spacetime dimension:\n\n$$\n(d-p-1)+(p+1)=d.\n$$\n\nA background of degree $d-p-1$ couples to a symmetry of form degree\n\n$$\nq=(d-p-1)-1=d-p-2.\n$$\n\nTherefore the dual symmetry is\n\n$$\n\\widehat G^{(d-p-2)}.\n$$\n\nThis dimension shift comes from the topological pairing. No mass spectrum or infrared approximation entered.\n\n## 4. What the schematic gauge sum suppresses\n\nThe previous formula contains the right pairing but hides several pieces of mathematical bookkeeping.\n\n### Gauge equivalence\n\nThe sum is not over arbitrary cochains. For the untwisted finite theory, one sums over flat fields modulo higher gauge transformations.\n\nOn a closed manifold, sectors are represented by\n\n$$\n[b]\\in H^{p+1}(M,G).\n$$\n\n### Gauge automorphisms\n\nGauge fields can have stabilizers and higher automorphisms. Their orders contribute to the gauge-volume normalization $\\mathcal N_M$.\n\nThe factor $1/\\sqrt N$ in the toy transform should not be copied blindly into a spacetime path integral.\n\n### Torsion and the pairing\n\nOn a general manifold, the pairing should be formulated using the appropriate cohomology operation and the canonical evaluation\n\n$$\nG\\times\\widehat G\\longrightarrow U(1).\n$$\n\nPoincaré duality, torsion, orientations, and boundaries determine when the pairing is nondegenerate and what extra boundary data are required.\n\n### Topological counterterms\n\nOne may weight the gauge sum by a Dijkgraaf–Witten action or another allowed local counterterm. Different choices can produce distinct gauged theories and alter anomaly data.\n\nThe clean Fourier statement describes the untwisted baseline. A complete claim must name the counterterm.\n\nThe [lectures by Bhardwaj et al.](https://arxiv.org/abs/2307.07547) develop this construction, its anomaly conditions, and the SymTFT interpretation in a common language.\n\n## 5. The dual symmetry acts by linking\n\nBefore gauging, a $p$-dimensional charged operator links with a codimension-$(p+1)$ symmetry defect.\n\nAfter gauging, the dynamical field $b_{p+1}$ has topological flux sectors. A dual symmetry operator inserts a character that evaluates this flux.\n\nThe dual symmetry has form degree\n\n$$\nq=d-p-2,\n$$\n\nso its generating topological defect has dimension\n\n$$\nd-q-1=p+1.\n$$\n\nThat $(p+1)$-dimensional defect measures $b$ through linking. The algebraic pairing in the partition function and the geometric linking operation describe the same structure.\n\nThis also reorganizes genuine operators.\n\nAn operator attached to an original symmetry defect can become genuine after gauging because the attachment is now gauge data. The resulting operator carries charge under the dual symmetry.\n\nConversely, an operator charged under the gauged group must be dressed by a gauge defect or gauge Wilson operator. It is no longer genuine by itself.\n\n## 6. Three dimensions expose the degree shift\n\nTake a $2+1$-dimensional theory with an ordinary finite Abelian symmetry $G^{(0)}$.\n\nGauging gives\n\n$$\n\\widehat G^{(1)}.\n$$\n\nThe original charged local operators cease to be gauge invariant unless dressed. The dynamical finite gauge theory contains line operators labelled by characters of $G$.\n\nThe new one-form symmetry acts on those lines.\n\nThis example prevents a common mistake: gauging an ordinary symmetry does not always produce another ordinary symmetry. The dual form degree depends on spacetime dimension.\n\nIn $1+1$ dimensions, by contrast,\n\n$$\np=0,\n\\qquad d=2\n\\quad\\Longrightarrow\\quad\nq=0.\n$$\n\nThe dual is again an ordinary symmetry. This is the familiar quantum symmetry of an Abelian orbifold.\n\n## 7. Four-dimensional Yang–Mills tests global form\n\nPure $SU(2)$ Yang–Mills has an electric center one-form symmetry\n\n$$\n\\mathbb Z_2^{(1)}.\n$$\n\nIts charged genuine operator is the Wilson line in the fundamental representation. Gauging the center changes the global form of the gauge group:\n\n$$\nSU(2)\n\\quad\\longrightarrow\\quad\nSO(3)=SU(2)/\\mathbb Z_2.\n$$\n\nBecause $d=4$ and $p=1$, the dual symmetry is also a one-form symmetry:\n\n$$\n\\widehat{\\mathbb Z_2}^{(1)}\n\\cong\\mathbb Z_2^{(1)}.\n$$\n\nIt is naturally called magnetic because it acts on the genuine magnetic line selected in the $SO(3)$ theory.\n\nThis sentence still omits a choice. Four-dimensional $SO(3)$ gauge theory admits distinct line spectra associated with a discrete theta-like parameter.\n\nDepending on the counterterm used when gauging, one obtains the conventional $SO(3)_+$ or $SO(3)_-$ choice, with magnetic or dyonic genuine lines. [Aharony, Seiberg, and Tachikawa](https://arxiv.org/abs/1305.0318) explain why the Lie algebra alone does not fix this data.\n\nThe correct map is therefore\n\n$$\n\\text{global gauge-group form}\n\\;+\n\\text{discrete theta data}\n\\;\\longleftrightarrow\\;\n\\text{spectrum of genuine lines}\n\\;\\longleftrightarrow\\;\n\\text{one-form symmetry}.\n$$\n\n## 8. Symmetry existence does not determine the phase\n\nSuppose the gauged theory has $\\widehat G^{(d-p-2)}$. This is an exact statement about topological defects and charged operators.\n\nIts realization in a state or phase is another question.\n\n| Exact structural datum | Dynamical question still open |\n| --- | --- |\n| Dual topological operators exist | Is the dual symmetry spontaneously broken? |\n| A line or surface carries dual charge | Does its expectation value show area, perimeter, or another law? |\n| Double gauging recovers the original theory | Is the once-gauged theory self-dual? |\n| The gauge sum includes nontrivial sectors | Are those sectors gapped, confined, or gapless? |\n| The SymTFT is fixed | Which physical boundary condition and IR phase are realized? |\n\nFor a one-form symmetry, confinement diagnostics can often be phrased as its realization on Wilson or ’t Hooft lines. The symmetry label alone does not calculate the line law.\n\nThe [foundational generalized-symmetry analysis](https://link.springer.com/article/10.1007/JHEP02%282015%29172) separates the existence, gauging, anomaly, and spontaneous breaking of higher-form symmetries.\n\n## 9. An anomaly obstructs the ordinary gauge sum\n\nAn ’t Hooft anomaly means that the background-field partition function cannot be made gauge invariant by a $d$-dimensional local counterterm.\n\nThen the expression\n\n$$\n\\sum_{[b]}Z_{\\mathcal T}[b]\n$$\n\ndoes not define an ordinary standalone $d$-dimensional QFT.\n\nSeveral repairs are possible, but each changes the problem:\n\n- place the theory on the boundary of an anomaly-inflow bulk;\n- gauge an anomaly-free subgroup;\n- extend the symmetry so that the obstruction trivializes;\n- combine the theory with another sector whose anomaly cancels it.\n\nOne should not apply the finite Fourier argument first and discuss the anomaly later. Gauge invariance is what makes the transform a map between consistent theories.\n\nIf other symmetries remain, gauging can convert the original anomaly into a mixed anomaly, an extension, or a higher-group structure involving the dual symmetry.\n\nFor a finite normal Abelian subgroup, [Tachikawa’s analysis](https://arxiv.org/abs/1712.09542) shows explicitly how residual symmetry, the dual higher-form symmetry, and anomaly data can combine into structures more intricate than a direct product.\n\n## 10. Double gauging is invertibility of an operation\n\nThe finite Fourier transform is invertible. With background and normalization data retained, gauging $G$ and then gauging $\\widehat G$ recovers the original theory up to charge conjugation and allowed local topological factors.\n\nThis does not imply\n\n$$\n\\mathcal T\\cong\\mathcal T/G.\n$$\n\nCompare an invertible map $F:V\\to W$ with an identity map on $V$. The existence of $F^{-1}$ does not identify $V$ and $W$ as the same object with the same preferred observables.\n\nLikewise, gauging changes:\n\n- the Hilbert-space projection;\n- the sum over bundles;\n- which operators are genuine;\n- the topological sectors;\n- possible counterterms and anomalies.\n\nA self-duality requires extra input: an equivalence between $\\mathcal T$ and its gauged theory.\n\nWhen such an equivalence exists, the gauging interface can become a duality defect inside one theory. That defect may be non-invertible even though gauging, viewed as a map between appropriately decorated theories, can be undone.\n\n## 11. Kramers–Wannier duality adds one crucial ingredient\n\nIn the two-dimensional Ising CFT, gauging the spin-flip $\\mathbb Z_2$ symmetry returns an equivalent critical theory.\n\nThe interface implementing gauging can therefore be folded into a defect $\\mathcal N$ of the same theory. Its fusion obeys\n\n$$\n\\mathcal N\\times\\mathcal N\n=\\mathbf 1+\\eta,\n$$\n\nwhere $\\eta$ is the invertible $\\mathbb Z_2$ symmetry line.\n\nThus $\\mathcal N$ has no inverse. Its quantum dimension satisfies\n\n$$\nd_{\\mathcal N}^2=2,\n\\qquad\nd_{\\mathcal N}=\\sqrt2.\n$$\n\nThe non-invertible defect comes from two ingredients together:\n\n1. a gauging interface between two theories;\n2. a self-duality that identifies those theories.\n\nFinite Abelian gauging by itself does not prove that the starting theory has an internal Kramers–Wannier defect.\n\nThe [ICTP lectures on non-invertible symmetries](https://arxiv.org/abs/2305.18296) explain how gauging and topological interfaces generate this broader defect algebra.\n\n## 12. Non-Abelian gauging is not a group Fourier transform\n\nFor a finite Abelian group, every irreducible representation is one-dimensional. Its characters form another group under multiplication.\n\nFor a finite non-Abelian group $G$, irreducible representations can have dimension greater than one. Tensor products decompose as\n\n$$\nR_i\\otimes R_j\n=\\bigoplus_k N_{ij}^{\\phantom{ij}k}R_k.\n$$\n\nThe corresponding topological operators need not be invertible. Their fusion is encoded by a category rather than by a group law.\n\nIn two-dimensional QFT, the quantum symmetry after gauging an ordinary finite non-Abelian $G$ is described by the fusion category\n\n$$\n\\operatorname{Rep}(G),\n$$\n\nsubject to the anomaly and discrete-torsion choices of the orbifold.\n\nThis statement should not be copied unchanged into arbitrary dimension. Higher-dimensional defects require higher categories, and gauging only a subgroup can mix residual and dual data.\n\nThe 2023 work on [generalized gauging in two dimensions](https://arxiv.org/abs/2311.17044) formulates sequential gaugings through topological interfaces, algebra objects, and module categories.\n\n## 13. SymTFT turns gauging into a boundary change\n\nThe symmetry topological field theory lives in $d+1$ dimensions. It stores the symmetry defects, their fusion, charges, and anomalies.\n\nFor the finite Abelian case, its simplest continuum mnemonic is a BF-type coupling\n\n$$\nS_{\\rm BF}\n=\\frac{2\\pi\\ii}{N}\n\\int_{X_{d+1}}\n\\widetilde b_{d-p-1}\\cup\\delta b_{p+1}.\n$$\n\nThe degrees add to $d+1$. The expression is schematic because a fully discrete action must track cocycles, lifts, and gauge equivalences.\n\nOne topological boundary condition treats the electric variable as fixed. The conjugate boundary condition treats the magnetic variable as fixed.\n\nChanging between these boundary conditions implements the finite Fourier transform. The physical QFT occupies the other boundary of the SymTFT sandwich.\n\nThis viewpoint explains why gauging changes global form without changing the local Lie algebra. It changes the allowed boundary condition and hence the spectrum of genuine extended operators.\n\n## 14. Where the frontier has moved\n\nThe finite Abelian result is a controlled starting point. Current research extends different parts of it, with different levels of generality.\n\n### Continuous SymTFT\n\nA [2024 construction](https://arxiv.org/abs/2401.06128) incorporates continuous $U(1)$ symmetry into SymTFT and proposes extensions to general continuous groups.\n\nContinuous gauging is not a finite Fourier sum. Measures, local gauge modes, compactness, charged matter, monopoles, and counterterms affect the answer.\n\n### Symmetry constraints on gaplessness\n\nA [2025 SymTFT analysis](https://arxiv.org/abs/2504.18501) classifies broad classes of non-invertible symmetries that can enforce gaplessness or undergo spontaneous breaking in four-dimensional examples.\n\nThis uses categorical anomaly and boundary data. It does not turn the existence of any non-invertible defect into a universal proof of gaplessness.\n\n### Lattice gauging of non-invertible symmetry\n\nA [2025 lattice prescription](https://arxiv.org/abs/2503.02925) constructs Gauss-law-like constraints for selected non-invertible symmetries in $1+1$ dimensions.\n\nIt shows that “gauging” can extend beyond groups, but the input includes explicit fusion and associator data. The simple character sum no longer suffices.\n\n### Anti-unitary categorical symmetry\n\nA [2026 preprint](https://arxiv.org/abs/2603.28720) proposes real fusion categories and a time-reversal-enriched SymTFT for anti-unitary and non-invertible time-reversal structures.\n\nThis is a categorical extension under active development. It should be cited as a proposal, not as a consequence of finite Abelian Pontryagin duality.\n\n## The deliberately false inference\n\n**If gauging $G^{(p)}$ produces $\\widehat G^{(d-p-2)}$ and double gauging returns the original theory, then the original and once-gauged theories have the same phase and spectrum.**\n\nDouble gauging proves that the transformation can be inverted when the symmetry, backgrounds, normalization, and counterterms are retained.\n\nIt does not identify the intermediate theories. Their genuine operators, bundle sectors, line spectra, anomalies, and phase realizations can differ.\n\nSelf-duality is an additional equivalence, not a corollary of Fourier invertibility.\n\n## Collision: five levels\n\n### Redundancy\n\nThe original $G$ action becomes gauge equivalence. Operators charged under $G$ require gauge dressing.\n\n### Topological sectors\n\nThe gauge sum introduces bundles or higher cocycle sectors that local perturbation theory around a trivial field can miss.\n\n### Dual symmetry\n\nCharacters of those sectors generate $\\widehat G^{(d-p-2)}$ through a spacetime linking pairing.\n\n### Dynamics\n\nConfinement, deconfinement, symmetry breaking, and gaplessness depend on the action, matter, dimension, and couplings.\n\n### Categorical extension\n\nNon-Abelian or non-invertible gauging replaces group multiplication by fusion, associators, module categories, and their higher-dimensional analogues.\n\n## Do this now\n\nLet\n\n$$\n\\widetilde Z(m)\n=\\frac{1}{\\sqrt N}\n\\sum_{n=0}^{N-1}\ne^{2\\pi\\ii mn/N}Z(n).\n$$\n\n### Target A: double the transform\n\nCompute $\\widetilde{\\widetilde Z}(k)$ and show that\n\n$$\n\\widetilde{\\widetilde Z}(k)=Z(-k).\n$$\n\nIdentify the operation represented by the minus sign.\n\n### Target B: derive the form degree\n\nTake\n\n$$\nb\\in H^{p+1}(M,G),\n\\qquad\n\\widetilde B\\in H^r(M,\\widehat G).\n$$\n\nRequire $\\int_M\\widetilde B\\cup b$ to be a $d$-dimensional pairing. Find $r$ and the form degree of the dual symmetry.\n\nEvaluate the result for:\n\n1. $(d,p)=(2,0)$;\n2. $(d,p)=(3,0)$;\n3. $(d,p)=(4,1)$.\n\n### Target C: audit the gauging formula\n\nList four pieces hidden by\n\n$$\nZ_{\\mathcal T/G}\n\\propto\\sum_{[b]}Z_{\\mathcal T}[b].\n$$\n\nYour list should include gauge automorphisms, topology, counterterms, and anomaly cancellation.\n\n### Target D: separate invertibility from self-duality\n\nExplain why\n\n$$\n(\\mathcal T/G)/\\widehat G\n\\cong\\mathcal T^C\n$$\n\ndoes not imply\n\n$$\n\\mathcal T/G\\cong\\mathcal T.\n$$\n\nWhat extra input is required to turn a gauging interface into an internal duality defect?\n\n### Target E: find non-invertibility\n\nSuppose a defect obeys\n\n$$\n\\mathcal N\\times\\mathcal N\n=\\sum_{g\\in G}U_g,\n$$\n\nwhere every $U_g$ is invertible and has quantum dimension one. Find $d_{\\mathcal N}$.\n\nWhy can no defect $\\mathcal N^{-1}$ satisfy $\\mathcal N\\times\\mathcal N^{-1}=\\mathbf1$ when $|G|>1$?\n\n### Oral check 1\n\nWhich part of $SU(2)\\to SO(3)$ is determined by gauging, and which part requires a discrete theta choice?\n\n### Oral check 2\n\nWhy does the existence of a magnetic one-form symmetry fail to prove deconfinement?\n\n<details>\n<summary>Hints</summary>\n\n- Use character orthogonality in Target A.\n- Background degree equals symmetry form degree plus one.\n- An inverse operation between decorated theories is weaker than equality at the midpoint.\n- Quantum dimensions multiply under fusion and add under direct sum.\n\n</details>\n\n<details class=\"solution\">\n<summary>Solution outline</summary>\n\nFor the second transform,\n\n$$\n\\widetilde{\\widetilde Z}(k)\n=\\frac1N\n\\sum_{m,n}\ne^{2\\pi\\ii m(k+n)/N}Z(n).\n$$\n\nThe sum over $m$ is nonzero only when\n\n$$\nk+n=0\\pmod N.\n$$\n\nTherefore\n\n$$\n\\widetilde{\\widetilde Z}(k)=Z(-k).\n$$\n\nThe map $k\\mapsto-k$ is charge conjugation.\n\nFor the spacetime pairing,\n\n$$\nr+(p+1)=d,\n$$\n\nso\n\n$$\nr=d-p-1.\n$$\n\nThe background degree is one above the symmetry form degree. Thus\n\n$$\nq=r-1=d-p-2.\n$$\n\nThe three examples give\n\n$$\n(2,0):\\quad q=0,\n$$\n\n$$\n(3,0):\\quad q=1,\n$$\n\n$$\n(4,1):\\quad q=1.\n$$\n\nThe schematic gauge sum hides the quotient by gauge transformations, weights from gauge automorphisms, the cohomology and torsion of $M$, boundary conditions, normalization, possible topological counterterms, and the requirement that the gauged symmetry be anomaly free.\n\nDouble gauging supplies an inverse transformation between theories carrying the required symmetry and background data. It does not make the first transform an identity.\n\nAn internal duality defect requires an independent equivalence\n\n$$\n\\varphi:\\mathcal T/G\\longrightarrow\\mathcal T.\n$$\n\nFinally,\n\n$$\nd_{\\mathcal N}^2\n=\\sum_{g\\in G}d_{U_g}\n=|G|,\n$$\n\nso\n\n$$\nd_{\\mathcal N}=\\sqrt{|G|}.\n$$\n\nAn invertible defect must have quantum dimension one. For $|G|>1$, $\\mathcal N$ cannot be invertible.\n\n</details>\n\n## Exit ticket\n\nA four-dimensional theory has a non-anomalous $\\mathbb Z_N^{(1)}$ symmetry. After gauging it, someone writes:\n\n> The theory has a dual $\\mathbb Z_N^{(1)}$ symmetry, so it is deconfined and physically equivalent to the original theory.\n\nSeparate the exact statement from the two unsupported inferences.\n\nA strong answer should name:\n\n1. the Pontryagin-dual symmetry and its form degree;\n2. the counterterm or discrete theta choice;\n3. the line operators on which the symmetry acts;\n4. the dynamical observable that diagnoses confinement;\n5. the extra equivalence needed for self-duality.\n\n## Research checks worth doing next\n\n1. **Normalization audit.** Derive the finite gauge-volume factor on a chosen manifold instead of importing $1/\\sqrt{|G|}$ from a one-sector transform.\n2. **Line-spectrum audit.** List all genuine Wilson, ’t Hooft, and dyonic lines before and after gauging.\n3. **Counterterm scan.** Classify allowed Dijkgraaf–Witten or discrete theta weights and compute how they change dual anomaly data.\n4. **Boundary test.** Repeat the gauge sum on a manifold with boundary and identify the extra boundary degrees of freedom.\n5. **Partial gauging.** Gauge a normal Abelian subgroup $A\\subset\\Gamma$ and determine whether the residual and dual symmetries form a product, extension, or higher group.\n6. **Defect fusion.** At a self-dual point, calculate the gauging-interface fusion and its quantum dimension.\n7. **Categorical lift.** Replace $\\mathbb Z_N$ by $S_3$ in two dimensions and derive the fusion rules of $\\operatorname{Rep}(S_3)$.\n\n## Further reading\n\n- [Generalized Global Symmetries](https://link.springer.com/article/10.1007/JHEP02%282015%29172): higher-form symmetries, their charged operators, gauging, anomalies, and spontaneous breaking.\n- [Lectures on Generalized Symmetries](https://arxiv.org/abs/2307.07547): a systematic treatment of finite gauging, dual symmetry, global forms, higher groups, and SymTFT.\n- [Reading between the lines of four-dimensional gauge theories](https://arxiv.org/abs/1305.0318): genuine line spectra, global gauge-group form, and discrete theta-like parameters.\n- [On gauging finite subgroups](https://arxiv.org/abs/1712.09542): residual symmetry, Pontryagin-dual higher-form symmetry, extensions, and anomaly mixing after partial gauging.\n- [ICTP Lectures on (Non-)Invertible Generalized Symmetries](https://arxiv.org/abs/2305.18296): gauging interfaces, condensation defects, non-invertible fusion, and SymTFT.\n- [Gauging Non-Invertible Symmetries in 2d QFT](https://arxiv.org/abs/2311.17044): algebra objects, module categories, interfaces, and sequential generalized gauging.\n- [A SymTFT for Continuous Symmetries](https://arxiv.org/abs/2401.06128): $U(1)$ SymTFT and a proposed route beyond finite symmetry.\n- [SymTFT, Protected Gaplessness, and Spontaneous Breaking](https://arxiv.org/abs/2504.18501): 2025 constraints on infrared realizations of non-invertible symmetry.\n- [Gauging non-invertible symmetries on the lattice](https://arxiv.org/abs/2503.02925): a 2025 Hamiltonian construction in selected $1+1$-dimensional systems.\n- [Categorical Time-Reversal Symmetries](https://arxiv.org/abs/2603.28720): a 2026 preprint proposing real fusion-category and SymTFT structures for anti-unitary symmetry.\n\n## What to retain\n\n- Gauging converts the original global action into redundancy and introduces topological gauge sectors.\n- Characters of a finite Abelian gauge field generate the Pontryagin-dual symmetry.\n- The cup-product degree fixes the shift $p\\mapsto d-p-2$.\n- Spacetime gauging requires gauge-volume, topology, boundary, and counterterm data beyond a toy Fourier sum.\n- An anomaly obstructs ordinary gauging unless additional structure cancels it.\n- Double gauging is invertibility of a decorated operation, not equality of the intermediate theories.\n- $SU(2)$ and $SO(3)$ share a Lie algebra but differ in genuine lines, one-form symmetry, and discrete theta data.\n- Symmetry existence does not determine confinement, gaplessness, or spontaneous breaking.\n- Non-Abelian gauging produces categorical fusion rather than another ordinary dual group.\n- A non-invertible duality defect requires gauging plus a self-duality equivalence.\n\nNext: gauge a finite non-Abelian symmetry in two dimensions and derive $\\operatorname{Rep}(G)$ from Wilson-line fusion rather than assuming it as a label.\n",
  "content_sha256": "49bb78a6562feb27aacdb6ddb0b31b57ecd65b75f0032ac1273bd593588e19ae",
  "formula_count": 149,
  "formulas": [
    {
      "index": 1,
      "display": false,
      "tex": "d",
      "line": 25,
      "sha256": "18ac3e7343f016890c510e93f935261169d9e3f565436429830faf0934f4f8e4"
    },
    {
      "index": 2,
      "display": false,
      "tex": "p",
      "line": 25,
      "sha256": "148de9c5a7a44d19e56cd9ae1a554bf67847afb0c58f6e12fa29ac7ddfca9940"
    },
    {
      "index": 3,
      "display": false,
      "tex": "G^{(p)}",
      "line": 25,
      "sha256": "9bd9eb18d1f209576d1ee7f566fe8aa74d973bd21c158f0dafc6e5d531f8c16c"
    },
    {
      "index": 4,
      "display": true,
      "tex": "0\\leq p\\leq d-2.",
      "line": 27,
      "sha256": "8dca6c8d6a29052684d4d9ddf6fe74ab5227c4a61074f0aa214cd5fd9b82623c"
    },
    {
      "index": 5,
      "display": true,
      "tex": "G^{(p)}\n\\quad\\longrightarrow\\quad\n\\widehat G^{(d-p-2)},",
      "line": 33,
      "sha256": "75ede73b0f104dd76dd943cf347be4a5ec1387e15c9ee7dbec8996e200d10b91"
    },
    {
      "index": 6,
      "display": true,
      "tex": "\\widehat G=\\operatorname{Hom}(G,U(1))",
      "line": 41,
      "sha256": "aa79a65d5f40cf8f32b7c1240dd506f1f31d86a51e4d91e388a74d4f701778fa"
    },
    {
      "index": 7,
      "display": false,
      "tex": "G",
      "line": 53,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 8,
      "display": false,
      "tex": "G",
      "line": 56,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 9,
      "display": false,
      "tex": "G",
      "line": 58,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 10,
      "display": false,
      "tex": "G",
      "line": 60,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 11,
      "display": false,
      "tex": "G=\\mathbb Z_N",
      "line": 66,
      "sha256": "aa3ebe660f4133357a4a8d2fa3a17ebdf5fec8aff56e7770c08ac91ea40c6653"
    },
    {
      "index": 12,
      "display": false,
      "tex": "Z(n)",
      "line": 66,
      "sha256": "d1b4ca68301f5d73b962dac1b6ddd151f982ddfd20f3cf293df95542f61d00c7"
    },
    {
      "index": 13,
      "display": false,
      "tex": "n\\in\\mathbb Z_N",
      "line": 66,
      "sha256": "5126e6bf3e10ca6be48bf98c379b8b185a582a8a71b468ecd11ea033a6baaa65"
    },
    {
      "index": 14,
      "display": true,
      "tex": "\\widetilde Z(m)\n=\\frac{1}{\\sqrt N}\n\\sum_{n=0}^{N-1}\ne^{2\\pi\\ii mn/N}Z(n).",
      "line": 70,
      "sha256": "be01a1b43c17b3ba59deb7702ab08b6340a966feb11118348871417089f150db"
    },
    {
      "index": 15,
      "display": true,
      "tex": "\\chi_m(n)=e^{2\\pi\\ii mn/N}",
      "line": 79,
      "sha256": "49fd9fda64093fe5d2fd625338a355e47270bd22cbfabc1331d737a7bdadcedd"
    },
    {
      "index": 16,
      "display": false,
      "tex": "\\mathbb Z_N",
      "line": 83,
      "sha256": "3b3b6283a40dd6623560ed457b05fa54fb875ba7c53703b84876bb288f6a8f3c"
    },
    {
      "index": 17,
      "display": false,
      "tex": "m",
      "line": 83,
      "sha256": "62c66a7a5dd70c3146618063c344e531e6d4b59e379808443ce962b3abd63c5a"
    },
    {
      "index": 18,
      "display": true,
      "tex": "\\widehat{\\mathbb Z_N}\n=\\operatorname{Hom}(\\mathbb Z_N,U(1))\n\\cong\\mathbb Z_N.",
      "line": 85,
      "sha256": "909f93bd1f30378b36e2409ca0b662ca10bbc5514b977f5c04b815e85002e118"
    },
    {
      "index": 19,
      "display": false,
      "tex": "\\widehat{\\mathbb Z_N}\\cong\\mathbb Z_N",
      "line": 91,
      "sha256": "96d3119e09fbb00149f7a9f0cb86b5fba69ea0acd9dd2dc002e7e8e58e87f38b"
    },
    {
      "index": 20,
      "display": false,
      "tex": "G",
      "line": 91,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 21,
      "display": false,
      "tex": "\\widehat G",
      "line": 91,
      "sha256": "4ae2c725a0324b8f84d2411b5b882618ac9dc57170eb14fff35c48afbc1e6781"
    },
    {
      "index": 22,
      "display": true,
      "tex": "\\widetilde{\\widetilde Z}(k)\n=\\frac{1}{N}\n\\sum_{m,n=0}^{N-1}\ne^{2\\pi\\ii m(k+n)/N}Z(n).",
      "line": 95,
      "sha256": "805743fe4271e97f68abed950080203294016e22e9cd1d3432e07faa11ae04ba"
    },
    {
      "index": 23,
      "display": true,
      "tex": "\\sum_{m=0}^{N-1}e^{2\\pi\\ii mr/N}\n=N\\,\\delta_{r,0\\;{\\rm mod}\\;N}.",
      "line": 104,
      "sha256": "893b13a3a18feb506e0cffb3d71a7f39a26c4680a4077be996fdddd7cd741429"
    },
    {
      "index": 24,
      "display": true,
      "tex": "\\boxed{\n\\widetilde{\\widetilde Z}(k)=Z(-k)\n}.",
      "line": 111,
      "sha256": "000f190fd41838e7b07359e0c1d7ea85750c2f521df0f674599a691e0cc342cc"
    },
    {
      "index": 25,
      "display": false,
      "tex": "n\\mapsto-n",
      "line": 117,
      "sha256": "a9ce5d05006142daad609f8dcd60429599a4bbc256dbfc4b25a4833d76caea07"
    },
    {
      "index": 26,
      "display": false,
      "tex": "p",
      "line": 121,
      "sha256": "148de9c5a7a44d19e56cd9ae1a554bf67847afb0c58f6e12fa29ac7ddfca9940"
    },
    {
      "index": 27,
      "display": false,
      "tex": "(p+1)",
      "line": 121,
      "sha256": "f602e9b63296daf73e9da653a6039d67fa5414e8b1b0f4c5e51c9f84bb80cf14"
    },
    {
      "index": 28,
      "display": false,
      "tex": "G=\\mathbb Z_N",
      "line": 121,
      "sha256": "aa3ebe660f4133357a4a8d2fa3a17ebdf5fec8aff56e7770c08ac91ea40c6653"
    },
    {
      "index": 29,
      "display": true,
      "tex": "B_{p+1}\\in Z^{p+1}(M,\\mathbb Z_N).",
      "line": 123,
      "sha256": "f89bc42084b33f56f040f3f40f642f012d5496edabcb68de0491a1e78355bbc2"
    },
    {
      "index": 30,
      "display": false,
      "tex": "B",
      "line": 127,
      "sha256": "df7e70e5021544f4834bbee64a9e3789febc4be81470df629cad6ddb03320a5c"
    },
    {
      "index": 31,
      "display": false,
      "tex": "b",
      "line": 127,
      "sha256": "3e23e8160039594a33894f6564e1b1348bbd7a0088d42c4acb73eeaed59c009d"
    },
    {
      "index": 32,
      "display": true,
      "tex": "\\widetilde B_{d-p-1}\n\\in Z^{d-p-1}(M,\\widehat G).",
      "line": 129,
      "sha256": "6fb13735ee6e250848d77a95558858a6d8281e61445745eaae0041e6c878c43f"
    },
    {
      "index": 33,
      "display": true,
      "tex": "Z_{\\mathcal T/G^{(p)}}[\\widetilde B]\n=\\mathcal N_M\n\\sum_{[b]}\nZ_{\\mathcal T}[b]\\,\n\\exp\\!\\left(\n\\frac{2\\pi\\ii}{N}\n\\int_M\\widetilde B\\cup b\n\\right).",
      "line": 136,
      "sha256": "2c442eabf774f499cf7365d6c32beebcb1b6db578d38c3c40ee75574f9d8d23e"
    },
    {
      "index": 34,
      "display": true,
      "tex": "(d-p-1)+(p+1)=d.",
      "line": 149,
      "sha256": "85e812df8552593ba1ecc62f7007848465655c48979d6f595f9a44a7c167acd2"
    },
    {
      "index": 35,
      "display": false,
      "tex": "d-p-1",
      "line": 153,
      "sha256": "4b76eb2ec2e6e975b9a0995d5a02531dcb4ed3d7d5d306249df824aebec0ee2b"
    },
    {
      "index": 36,
      "display": true,
      "tex": "q=(d-p-1)-1=d-p-2.",
      "line": 155,
      "sha256": "f4a36974c6abe4fc36e48d4327f91187e13205dd76b30ab6c5af0292b8506acc"
    },
    {
      "index": 37,
      "display": true,
      "tex": "\\widehat G^{(d-p-2)}.",
      "line": 161,
      "sha256": "ee3222ea86b21f6f254f7c6a3b3a25a4af06bb14aaeaef2c3bdce4c6f6781bab"
    },
    {
      "index": 38,
      "display": true,
      "tex": "[b]\\in H^{p+1}(M,G).",
      "line": 177,
      "sha256": "028d43a8d60f84be255ae76afcfa4c587b5b0cdadda520e7ce9991dfc05bf18d"
    },
    {
      "index": 39,
      "display": false,
      "tex": "\\mathcal N_M",
      "line": 183,
      "sha256": "e0db0a8d7f40bd3945a6f08a58487e1958ae826fb74dadb5e606d5bae228e1a1"
    },
    {
      "index": 40,
      "display": false,
      "tex": "1/\\sqrt N",
      "line": 185,
      "sha256": "389fbb89a17da65cd94e5e373b7d0de950bbe4592333b7b16adcf99a11203082"
    },
    {
      "index": 41,
      "display": true,
      "tex": "G\\times\\widehat G\\longrightarrow U(1).",
      "line": 191,
      "sha256": "579f69f9ecdf16d89aedd6a08aa23ed952a613b62ead256e4a264be7c2c63d22"
    },
    {
      "index": 42,
      "display": false,
      "tex": "p",
      "line": 207,
      "sha256": "148de9c5a7a44d19e56cd9ae1a554bf67847afb0c58f6e12fa29ac7ddfca9940"
    },
    {
      "index": 43,
      "display": false,
      "tex": "(p+1)",
      "line": 207,
      "sha256": "f602e9b63296daf73e9da653a6039d67fa5414e8b1b0f4c5e51c9f84bb80cf14"
    },
    {
      "index": 44,
      "display": false,
      "tex": "b_{p+1}",
      "line": 209,
      "sha256": "53a53fe277c23cc0d2e9a626493fbc3440c0738aa1423a9f54afb5c8c1fc1bc6"
    },
    {
      "index": 45,
      "display": true,
      "tex": "q=d-p-2,",
      "line": 213,
      "sha256": "7213c83211cb62194b71e63774f52f7a3494fb82f0b7ac99866e3c885d0d412b"
    },
    {
      "index": 46,
      "display": true,
      "tex": "d-q-1=p+1.",
      "line": 219,
      "sha256": "36a25fc5c10c6b7943708ff8ffaa1e03889ab65c78b19f92a8b9046dfa81dd73"
    },
    {
      "index": 47,
      "display": false,
      "tex": "(p+1)",
      "line": 223,
      "sha256": "f602e9b63296daf73e9da653a6039d67fa5414e8b1b0f4c5e51c9f84bb80cf14"
    },
    {
      "index": 48,
      "display": false,
      "tex": "b",
      "line": 223,
      "sha256": "3e23e8160039594a33894f6564e1b1348bbd7a0088d42c4acb73eeaed59c009d"
    },
    {
      "index": 49,
      "display": false,
      "tex": "2+1",
      "line": 233,
      "sha256": "2841016d1cd9efa25e200f44d2b1b365eac01d33b3b6875c0b658258fb3f29a7"
    },
    {
      "index": 50,
      "display": false,
      "tex": "G^{(0)}",
      "line": 233,
      "sha256": "72cf3f3bece075cb0ec3651e3cd2be54c084003d3c9c7ebfa784d7dad249d04c"
    },
    {
      "index": 51,
      "display": true,
      "tex": "\\widehat G^{(1)}.",
      "line": 237,
      "sha256": "4687633060cba7d9681e85349c7cb3cafb88e7f02de9797bf217c900ea81e569"
    },
    {
      "index": 52,
      "display": false,
      "tex": "G",
      "line": 241,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 53,
      "display": false,
      "tex": "1+1",
      "line": 247,
      "sha256": "4a1b21d876ae00c8ed5c4d1cde09c61f8a61b50fe370a801b10c339831f370ab"
    },
    {
      "index": 54,
      "display": true,
      "tex": "p=0,\n\\qquad d=2\n\\quad\\Longrightarrow\\quad\nq=0.",
      "line": 249,
      "sha256": "545ea37adde25265708e451f9259ad133530b215019972000cc3fa8da3ffd4b7"
    },
    {
      "index": 55,
      "display": false,
      "tex": "SU(2)",
      "line": 260,
      "sha256": "1cc49e66b54735e6a9cda24bf4b975829904af96b480a524deb0f292f5f6267a"
    },
    {
      "index": 56,
      "display": true,
      "tex": "\\mathbb Z_2^{(1)}.",
      "line": 262,
      "sha256": "eed312593820d508877b81b073df0d5a572313e437c14442339ad0961395ed21"
    },
    {
      "index": 57,
      "display": true,
      "tex": "SU(2)\n\\quad\\longrightarrow\\quad\nSO(3)=SU(2)/\\mathbb Z_2.",
      "line": 268,
      "sha256": "c42e59c1f0288360583efb8585e0e72d075d4d4ef4cd03603bb2d9c711119c02"
    },
    {
      "index": 58,
      "display": false,
      "tex": "d=4",
      "line": 274,
      "sha256": "edc255b85c3fd80fb686206bedd10302c69055cea8926a789a3e22f562afc8e8"
    },
    {
      "index": 59,
      "display": false,
      "tex": "p=1",
      "line": 274,
      "sha256": "791a7b3a1c84e05cbf520166fc1107eebd36ea5ed05f636222d80d2bf4183966"
    },
    {
      "index": 60,
      "display": true,
      "tex": "\\widehat{\\mathbb Z_2}^{(1)}\n\\cong\\mathbb Z_2^{(1)}.",
      "line": 276,
      "sha256": "35d37f1353a0b3ae8cfc77c7049ef8381fa37c13f8b1a443de798a078758783a"
    },
    {
      "index": 61,
      "display": false,
      "tex": "SO(3)",
      "line": 281,
      "sha256": "6cbc614c7298af78ced81f03c307f76f9928c97fddbc2be6bb8373ba8277fded"
    },
    {
      "index": 62,
      "display": false,
      "tex": "SO(3)",
      "line": 283,
      "sha256": "6cbc614c7298af78ced81f03c307f76f9928c97fddbc2be6bb8373ba8277fded"
    },
    {
      "index": 63,
      "display": false,
      "tex": "SO(3)_+",
      "line": 285,
      "sha256": "ac8ee18fb24fd5ee6040b386d67a238234d707b15ec3523a02b00c4ec1524301"
    },
    {
      "index": 64,
      "display": false,
      "tex": "SO(3)_-",
      "line": 285,
      "sha256": "5dc0f1624a620433d6b5bff72bd2702aa07ec905801d3aae7bcd64856e954056"
    },
    {
      "index": 65,
      "display": true,
      "tex": "\\text{global gauge-group form}\n\\;+\n\\text{discrete theta data}\n\\;\\longleftrightarrow\\;\n\\text{spectrum of genuine lines}\n\\;\\longleftrightarrow\\;\n\\text{one-form symmetry}.",
      "line": 289,
      "sha256": "8bd2fd94f67d116da340c0b1e1720b71136cef1231df992fa388e656fca2aeb1"
    },
    {
      "index": 66,
      "display": false,
      "tex": "\\widehat G^{(d-p-2)}",
      "line": 301,
      "sha256": "61215071acc3ccea9af0ebcedcb973014b47e89c59e4c006da17dd48a8af0026"
    },
    {
      "index": 67,
      "display": false,
      "tex": "d",
      "line": 319,
      "sha256": "18ac3e7343f016890c510e93f935261169d9e3f565436429830faf0934f4f8e4"
    },
    {
      "index": 68,
      "display": true,
      "tex": "\\sum_{[b]}Z_{\\mathcal T}[b]",
      "line": 323,
      "sha256": "8e61f03bef1b29007d8787cfc5dee6e4ef8fb6831eb1aed3eaa3768d40aed181"
    },
    {
      "index": 69,
      "display": false,
      "tex": "d",
      "line": 327,
      "sha256": "18ac3e7343f016890c510e93f935261169d9e3f565436429830faf0934f4f8e4"
    },
    {
      "index": 70,
      "display": false,
      "tex": "G",
      "line": 344,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 71,
      "display": false,
      "tex": "\\widehat G",
      "line": 344,
      "sha256": "4ae2c725a0324b8f84d2411b5b882618ac9dc57170eb14fff35c48afbc1e6781"
    },
    {
      "index": 72,
      "display": true,
      "tex": "\\mathcal T\\cong\\mathcal T/G.",
      "line": 348,
      "sha256": "748b0a719aaa149348cfdeb34fc0011775ec1ea25f35542e47ae7504197e553f"
    },
    {
      "index": 73,
      "display": false,
      "tex": "F:V\\to W",
      "line": 352,
      "sha256": "577457fb7ad117955018df5a5a981c54d5f114a0190b611577cf60a5ada0b568"
    },
    {
      "index": 74,
      "display": false,
      "tex": "V",
      "line": 352,
      "sha256": "de5a6f78116eca62d7fc5ce159d23ae6b889b365a1739ad2cf36f925a140d0cc"
    },
    {
      "index": 75,
      "display": false,
      "tex": "F^{-1}",
      "line": 352,
      "sha256": "08c5fee29bc1411dcd7133f4c58c8242777600aa6046a662e3a80b9fb30d4e30"
    },
    {
      "index": 76,
      "display": false,
      "tex": "V",
      "line": 352,
      "sha256": "de5a6f78116eca62d7fc5ce159d23ae6b889b365a1739ad2cf36f925a140d0cc"
    },
    {
      "index": 77,
      "display": false,
      "tex": "W",
      "line": 352,
      "sha256": "fcb5f40df9be6bae66c1d77a6c15968866a9e6cbd7314ca432b019d17392f6f4"
    },
    {
      "index": 78,
      "display": false,
      "tex": "\\mathcal T",
      "line": 362,
      "sha256": "c6f5e71887472d5489cbd66ceb6058e23c7b8ed1f6f008777e74379f591d4ec8"
    },
    {
      "index": 79,
      "display": false,
      "tex": "\\mathbb Z_2",
      "line": 368,
      "sha256": "370c87df70534fa468a6af6914e6ec1479f7ad8525206fc9b05102456176a787"
    },
    {
      "index": 80,
      "display": false,
      "tex": "\\mathcal N",
      "line": 370,
      "sha256": "6e9e0a3f9b6130ca0fa1e9096adc4469c23282c477e9149be4e72825a22ba1fa"
    },
    {
      "index": 81,
      "display": true,
      "tex": "\\mathcal N\\times\\mathcal N\n=\\mathbf 1+\\eta,",
      "line": 372,
      "sha256": "c20f7a10ba124b0b21ca4096bb1abe371ca0ab695c6aefdf67689d02638e913d"
    },
    {
      "index": 82,
      "display": false,
      "tex": "\\eta",
      "line": 377,
      "sha256": "59a94416a2a319caf40a348286e620ef3594226a2eac1efe842231962d1e157d"
    },
    {
      "index": 83,
      "display": false,
      "tex": "\\mathbb Z_2",
      "line": 377,
      "sha256": "370c87df70534fa468a6af6914e6ec1479f7ad8525206fc9b05102456176a787"
    },
    {
      "index": 84,
      "display": false,
      "tex": "\\mathcal N",
      "line": 379,
      "sha256": "6e9e0a3f9b6130ca0fa1e9096adc4469c23282c477e9149be4e72825a22ba1fa"
    },
    {
      "index": 85,
      "display": true,
      "tex": "d_{\\mathcal N}^2=2,\n\\qquad\nd_{\\mathcal N}=\\sqrt2.",
      "line": 381,
      "sha256": "f38380850474a9c6dcea6cfadfae5a2d663b3dd38ed68c0058f1930396f19aff"
    },
    {
      "index": 86,
      "display": false,
      "tex": "G",
      "line": 400,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 87,
      "display": true,
      "tex": "R_i\\otimes R_j\n=\\bigoplus_k N_{ij}^{\\phantom{ij}k}R_k.",
      "line": 402,
      "sha256": "c2245fbdcb27abefce6ac5f7f2a0e790dfd1bbfefa60d59e0ff559961b823b7f"
    },
    {
      "index": 88,
      "display": false,
      "tex": "G",
      "line": 409,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 89,
      "display": true,
      "tex": "\\operatorname{Rep}(G),",
      "line": 411,
      "sha256": "5647cc0a02b32d52df824098f26c3dd165c315c2b5df8d38b724d0404ce3971f"
    },
    {
      "index": 90,
      "display": false,
      "tex": "d+1",
      "line": 423,
      "sha256": "16d23e3472056d06adf099ca90308bdd6fcd2d1864102ba509a128da301622e0"
    },
    {
      "index": 91,
      "display": true,
      "tex": "S_{\\rm BF}\n=\\frac{2\\pi\\ii}{N}\n\\int_{X_{d+1}}\n\\widetilde b_{d-p-1}\\cup\\delta b_{p+1}.",
      "line": 427,
      "sha256": "3c176ff36aacaa876ae7b03094c35c13179a3e8f51c0866d81d3ce8955461e14"
    },
    {
      "index": 92,
      "display": false,
      "tex": "d+1",
      "line": 434,
      "sha256": "16d23e3472056d06adf099ca90308bdd6fcd2d1864102ba509a128da301622e0"
    },
    {
      "index": 93,
      "display": false,
      "tex": "U(1)",
      "line": 448,
      "sha256": "db5f23f690458e1741ff9e859e8d525a0b730cfb76588165163812dbbad598a3"
    },
    {
      "index": 94,
      "display": false,
      "tex": "1+1",
      "line": 460,
      "sha256": "4a1b21d876ae00c8ed5c4d1cde09c61f8a61b50fe370a801b10c339831f370ab"
    },
    {
      "index": 95,
      "display": false,
      "tex": "G^{(p)}",
      "line": 472,
      "sha256": "9bd9eb18d1f209576d1ee7f566fe8aa74d973bd21c158f0dafc6e5d531f8c16c"
    },
    {
      "index": 96,
      "display": false,
      "tex": "\\widehat G^{(d-p-2)}",
      "line": 472,
      "sha256": "61215071acc3ccea9af0ebcedcb973014b47e89c59e4c006da17dd48a8af0026"
    },
    {
      "index": 97,
      "display": false,
      "tex": "G",
      "line": 484,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 98,
      "display": false,
      "tex": "G",
      "line": 484,
      "sha256": "333e0a1e27815d0ceee55c473fe3dc93d56c63e3bee2b3b4aee8eed6d70191a3"
    },
    {
      "index": 99,
      "display": false,
      "tex": "\\widehat G^{(d-p-2)}",
      "line": 492,
      "sha256": "61215071acc3ccea9af0ebcedcb973014b47e89c59e4c006da17dd48a8af0026"
    },
    {
      "index": 100,
      "display": true,
      "tex": "\\widetilde Z(m)\n=\\frac{1}{\\sqrt N}\n\\sum_{n=0}^{N-1}\ne^{2\\pi\\ii mn/N}Z(n).",
      "line": 506,
      "sha256": "be01a1b43c17b3ba59deb7702ab08b6340a966feb11118348871417089f150db"
    },
    {
      "index": 101,
      "display": false,
      "tex": "\\widetilde{\\widetilde Z}(k)",
      "line": 515,
      "sha256": "61ad62cb0f1933808ecc27bfda78abb13299151d50633c04c0b146315bb00682"
    },
    {
      "index": 102,
      "display": true,
      "tex": "\\widetilde{\\widetilde Z}(k)=Z(-k).",
      "line": 517,
      "sha256": "dfe51aff805bd14499c9613d65664c0430c7f9f9e37b058ca2ed804017ee30fc"
    },
    {
      "index": 103,
      "display": true,
      "tex": "b\\in H^{p+1}(M,G),\n\\qquad\n\\widetilde B\\in H^r(M,\\widehat G).",
      "line": 527,
      "sha256": "aa0ca3b597c705f544d82b5d24c79750a8c8dcba4ba6e89451f3e4485350ca93"
    },
    {
      "index": 104,
      "display": false,
      "tex": "\\int_M\\widetilde B\\cup b",
      "line": 533,
      "sha256": "4c3bb23aad4c1af7d3c1d0f1b168b1847c06e3f6fa4bc994751ab624a3d1fbbb"
    },
    {
      "index": 105,
      "display": false,
      "tex": "d",
      "line": 533,
      "sha256": "18ac3e7343f016890c510e93f935261169d9e3f565436429830faf0934f4f8e4"
    },
    {
      "index": 106,
      "display": false,
      "tex": "r",
      "line": 533,
      "sha256": "454349e422f05297191ead13e21d3db520e5abef52055e4964b82fb213f593a1"
    },
    {
      "index": 107,
      "display": false,
      "tex": "(d,p)=(2,0)",
      "line": 537,
      "sha256": "3a6fe61cdb0507fd0d466e8d2ac1980c41f1a74026e73c4bdf68b7e4aafea374"
    },
    {
      "index": 108,
      "display": false,
      "tex": "(d,p)=(3,0)",
      "line": 538,
      "sha256": "1c94938a8046b131865e9ea0cf92e04f641deef3a156f832c9f62f2bf687a0ce"
    },
    {
      "index": 109,
      "display": false,
      "tex": "(d,p)=(4,1)",
      "line": 539,
      "sha256": "3106c24549cf263a10dd7b6c54fe03b0ad1d56c6eb5e3369c3c427390d792fb7"
    },
    {
      "index": 110,
      "display": true,
      "tex": "Z_{\\mathcal T/G}\n\\propto\\sum_{[b]}Z_{\\mathcal T}[b].",
      "line": 545,
      "sha256": "01c2e2c6aa6a63f23f6ae0b30bd52b81ebeb73b44324c839f4ec85880f847b3a"
    },
    {
      "index": 111,
      "display": true,
      "tex": "(\\mathcal T/G)/\\widehat G\n\\cong\\mathcal T^C",
      "line": 556,
      "sha256": "815538d3e1b301ddc7e05fe4d52de05bf5a0ecd9509b8af30911fdde906f5019"
    },
    {
      "index": 112,
      "display": true,
      "tex": "\\mathcal T/G\\cong\\mathcal T.",
      "line": 563,
      "sha256": "01763a83a2edca781247cf60eee039a7574480b70349cdc8e2568a877fdb8d06"
    },
    {
      "index": 113,
      "display": true,
      "tex": "\\mathcal N\\times\\mathcal N\n=\\sum_{g\\in G}U_g,",
      "line": 573,
      "sha256": "858d084a18caebb59d799e9d19d82f77942a8167f0dcc0b5c5475a3bc65f8b0b"
    },
    {
      "index": 114,
      "display": false,
      "tex": "U_g",
      "line": 578,
      "sha256": "9f5988962b4f6877c82e64301ad4d7fb5303e90ee260e2177fb7d90e3763239c"
    },
    {
      "index": 115,
      "display": false,
      "tex": "d_{\\mathcal N}",
      "line": 578,
      "sha256": "eb27e9bc6e531b4919d2d896f11eba59da952b4a2c499350ff0d0dfdcf492dcf"
    },
    {
      "index": 116,
      "display": false,
      "tex": "\\mathcal N^{-1}",
      "line": 580,
      "sha256": "f40bbcca7096796b470f80983d744fb3ac98c118b5d7dd6d11355005fe2d771e"
    },
    {
      "index": 117,
      "display": false,
      "tex": "\\mathcal N\\times\\mathcal N^{-1}=\\mathbf1",
      "line": 580,
      "sha256": "8e5e6fb90605d7f8eb17f38567d6601d551b1693e7e7c55652ae4104f8593143"
    },
    {
      "index": 118,
      "display": false,
      "tex": "|G|>1",
      "line": 580,
      "sha256": "f14b4a03e9e99cc3daaa8086efeb14c40c1a8e3a8ed88b4632c57cb6c8299351"
    },
    {
      "index": 119,
      "display": false,
      "tex": "SU(2)\\to SO(3)",
      "line": 584,
      "sha256": "ce55d3a051090e55f744ee846e6027df8aee255603487a236b7c2d53296dfed3"
    },
    {
      "index": 120,
      "display": true,
      "tex": "\\widetilde{\\widetilde Z}(k)\n=\\frac1N\n\\sum_{m,n}\ne^{2\\pi\\ii m(k+n)/N}Z(n).",
      "line": 605,
      "sha256": "506e973f518256b3a4c0b3648962faf0682c1b4cd34d4528f6f3e003d0f7d7af"
    },
    {
      "index": 121,
      "display": false,
      "tex": "m",
      "line": 612,
      "sha256": "62c66a7a5dd70c3146618063c344e531e6d4b59e379808443ce962b3abd63c5a"
    },
    {
      "index": 122,
      "display": true,
      "tex": "k+n=0\\pmod N.",
      "line": 614,
      "sha256": "584aea4d078e98512d5fb5c6232f6f16734df64a9b29d306dc5044f9b0a3d620"
    },
    {
      "index": 123,
      "display": true,
      "tex": "\\widetilde{\\widetilde Z}(k)=Z(-k).",
      "line": 620,
      "sha256": "dfe51aff805bd14499c9613d65664c0430c7f9f9e37b058ca2ed804017ee30fc"
    },
    {
      "index": 124,
      "display": false,
      "tex": "k\\mapsto-k",
      "line": 624,
      "sha256": "d46b11f698d56f4e7b54b5832f632e351834209b4f8398c6fc2082840b12e6cd"
    },
    {
      "index": 125,
      "display": true,
      "tex": "r+(p+1)=d,",
      "line": 628,
      "sha256": "ee5b0a9009c485d450aaf3c6f8f8a1f38b6cf4c228c75ba957045e30ec9265a8"
    },
    {
      "index": 126,
      "display": true,
      "tex": "r=d-p-1.",
      "line": 634,
      "sha256": "e52ff947b8a6b5ed50c5d5ae2fc8d7563bbf9bce016a57db0c48191ec79a3dff"
    },
    {
      "index": 127,
      "display": true,
      "tex": "q=r-1=d-p-2.",
      "line": 640,
      "sha256": "1ec9a1ffff6c457cc116ced3d8c249c05f6233741248493787b55cd220ee96bc"
    },
    {
      "index": 128,
      "display": true,
      "tex": "(2,0):\\quad q=0,",
      "line": 646,
      "sha256": "a3d98c87e4a3331f8d111190fdd7a10b7c2b8586c4262b1bb39fccaf98cc19f0"
    },
    {
      "index": 129,
      "display": true,
      "tex": "(3,0):\\quad q=1,",
      "line": 650,
      "sha256": "ac25a9111d43cce67080c12c2b1b1a37aea90cc23b8aaf6855c86e245df2af00"
    },
    {
      "index": 130,
      "display": true,
      "tex": "(4,1):\\quad q=1.",
      "line": 654,
      "sha256": "9d90927245b513bcdb1274e242a2ff5dd63e694efd889a5bbe5ae1eaab3002f6"
    },
    {
      "index": 131,
      "display": false,
      "tex": "M",
      "line": 658,
      "sha256": "08f271887ce94707da822d5263bae19d5519cb3614e0daedc4c7ce5dab7473f1"
    },
    {
      "index": 132,
      "display": true,
      "tex": "\\varphi:\\mathcal T/G\\longrightarrow\\mathcal T.",
      "line": 664,
      "sha256": "164022c7e0fbfa59f8b705155382c93ebfa9d0a6a82d564cb15711dee83c07b5"
    },
    {
      "index": 133,
      "display": true,
      "tex": "d_{\\mathcal N}^2\n=\\sum_{g\\in G}d_{U_g}\n=|G|,",
      "line": 670,
      "sha256": "a20b162ba99d9e894d2039ca0195d45fb563b5f71a1cb61945c49db218bb40fd"
    },
    {
      "index": 134,
      "display": true,
      "tex": "d_{\\mathcal N}=\\sqrt{|G|}.",
      "line": 678,
      "sha256": "52c89c31e2b869b941800c8d5b42c6dee1c4aed426243771f03b91af3eea21f1"
    },
    {
      "index": 135,
      "display": false,
      "tex": "|G|>1",
      "line": 682,
      "sha256": "f14b4a03e9e99cc3daaa8086efeb14c40c1a8e3a8ed88b4632c57cb6c8299351"
    },
    {
      "index": 136,
      "display": false,
      "tex": "\\mathcal N",
      "line": 682,
      "sha256": "6e9e0a3f9b6130ca0fa1e9096adc4469c23282c477e9149be4e72825a22ba1fa"
    },
    {
      "index": 137,
      "display": false,
      "tex": "\\mathbb Z_N^{(1)}",
      "line": 688,
      "sha256": "ff336c390036bd82d468ef793753dfd8e7786c1e5a1c2cde794890e680268f31"
    },
    {
      "index": 138,
      "display": false,
      "tex": "\\mathbb Z_N^{(1)}",
      "line": 690,
      "sha256": "ff336c390036bd82d468ef793753dfd8e7786c1e5a1c2cde794890e680268f31"
    },
    {
      "index": 139,
      "display": false,
      "tex": "1/\\sqrt{|G|}",
      "line": 704,
      "sha256": "3e2c913abbe98ad8fb62cfb5fda9bdec9276de09957e52925c9090bf8c570688"
    },
    {
      "index": 140,
      "display": false,
      "tex": "A\\subset\\Gamma",
      "line": 708,
      "sha256": "051f75d416c8c448e9f0b8eff09b6bcc76bf3300346c91973ec1a10fc7ffe8d1"
    },
    {
      "index": 141,
      "display": false,
      "tex": "\\mathbb Z_N",
      "line": 710,
      "sha256": "3b3b6283a40dd6623560ed457b05fa54fb875ba7c53703b84876bb288f6a8f3c"
    },
    {
      "index": 142,
      "display": false,
      "tex": "S_3",
      "line": 710,
      "sha256": "cdc525abf7a300b59409ed00199930cad4a28003ca1c22a390c54d7cb8d8f283"
    },
    {
      "index": 143,
      "display": false,
      "tex": "\\operatorname{Rep}(S_3)",
      "line": 710,
      "sha256": "d46722a5dce795de7e77bc1928ebb77fd9cb7bb22f1e1faf7612f58151de53a2"
    },
    {
      "index": 144,
      "display": false,
      "tex": "U(1)",
      "line": 720,
      "sha256": "db5f23f690458e1741ff9e859e8d525a0b730cfb76588165163812dbbad598a3"
    },
    {
      "index": 145,
      "display": false,
      "tex": "1+1",
      "line": 722,
      "sha256": "4a1b21d876ae00c8ed5c4d1cde09c61f8a61b50fe370a801b10c339831f370ab"
    },
    {
      "index": 146,
      "display": false,
      "tex": "p\\mapsto d-p-2",
      "line": 729,
      "sha256": "8e4fe79ce0f17fb761fd784e40adf45545fda6c7350da1a19c880fd888a92edf"
    },
    {
      "index": 147,
      "display": false,
      "tex": "SU(2)",
      "line": 733,
      "sha256": "1cc49e66b54735e6a9cda24bf4b975829904af96b480a524deb0f292f5f6267a"
    },
    {
      "index": 148,
      "display": false,
      "tex": "SO(3)",
      "line": 733,
      "sha256": "6cbc614c7298af78ced81f03c307f76f9928c97fddbc2be6bb8373ba8277fded"
    },
    {
      "index": 149,
      "display": false,
      "tex": "\\operatorname{Rep}(G)",
      "line": 738,
      "sha256": "3c4e202d2468872aba12dc6d5afc5d02dc372e02e74c232fa4846c9a07a158b0"
    }
  ]
}