Dictionary Completeness and Global Data

A local operator dictionary is only one layer of a holographic correspondence. Two theories may have the same local Lie algebra, perturbative fields, and correlators of neutral local operators, yet differ in their genuine line operators, allowed bundles, topological sectors, or boundary conditions. A completeness claim must therefore say which observables and sectors are mapped, how they are normalized, and which global choices define each theory.

Required background. Holographic Duality: Claims, Dictionaries, and Regimes defines the theory-pair contract, and Relational, Boundary, and Asymptotic Observables identifies the observables whose completeness is at issue.

Helpful background. Generalized Symmetries, Global Forms, and Anomalies under Duality develops the underlying QFT distinctions and includes a charge-lattice figure for the $\mathfrak{su}(2)$ case. Additivity, Haag Duality, and Information Completeness gives a complementary operator-algebra perspective.

What a complete dictionary must specify

It helps to separate a local core from its global completion. The local core includes the operator map, couplings, correlation functions, and their domain of validity. The completion says which theory those local data belong to.

The terms in the checklist have operational meanings. The Lie algebra controls infinitesimal gauge transformations; the global form specifies which finite transformations are identified and therefore which representations and bundles exist. A line is genuine when it is defined without an attached surface; a surface-attached line may still be a useful probe, but it is not a standalone operator. A one-form symmetry acts on genuine lines, and a two-form background probes that action. An anomaly is an obstruction to gauging a symmetry while preserving all required invariances, not merely a failure of a classical current equation.

A superselection sector is always relative to a chosen observable algebra: by definition, no operator in that algebra connects distinct sectors. An interface, a defect, or an enlarged algebra may relate them, but then that enlargement is additional dictionary data. This page uses “complete” to mean complete for the declared holographic dictionary. It does not mean maximality of a local operator algebra or the quantum-gravity conjecture that every allowed gauge charge is dynamically realized.

Minimum information carried by a global holographic dictionary

Layer	What must be recorded	A comparison that can expose a mismatch
Local observables	Operator map, sources, contact-term scheme, couplings, and normalization	Ward identities and normalized correlation functions
Global groups	Faithful global and gauge groups, including quotients by centers	Representations and bundles allowed by the quotient
Extended operators	Genuine lines, surfaces, defects, junctions, and their fusion rules	Linking, braiding, fusion, and endpoint tests
Charge data	Charge groups, screening relations, pairings, and the genuine subset	Dirac or mutual-locality phases
Generalized symmetries	Higher-form and higher-group symmetries and their charged objects	Background-field transformations and symmetry defects
Anomalies	Perturbative, global, torsion, and mixed anomaly classes	Inflow or partition functions on nontrivial backgrounds
Topological sectors	Allowed bundles, fluxes, discrete theta terms, and sector weights	Partition functions resolved by topology or flux
Boundary data	Boundary conditions, boundary degrees of freedom, and allowed endpoints	Boundary operator and defect spectra
State space	Hilbert space or observable algebra in each sector and its superselection structure	State counting and sector-changing operations
Prescription	Fixed sector, sum over sectors, gauging, state, ensemble, and measure	Factorization and Fourier-transform tests
Claim domain	Spacetime, topology, regulator, parameter regime, approximation, and uncertainty	A probe just outside the declared domain

This is not a demand to compute every entry before making any useful statement. It is a demand to keep the conclusion no broader than the entries actually fixed. Matching anomalies, for example, is a strong necessary test but is not sufficient for equivalence: distinct theories can share the same anomaly. The background-field and extended-operator framework is developed systematically in Gaiotto, Kapustin, Seiberg, and Willett 2015, §§2–4 and 6, Open PDF.

Electric and magnetic line charges for su(N)

The cleanest counterexample uses four-dimensional gauge theories with Lie algebra $\mathfrak{su}(N)$ and only adjoint dynamical fields. We work on spin four-manifolds and first consider flat $\mathbb R^4$ with no line insertions. This scope includes $\mathcal N=4$ super-Yang–Mills theory. Nonspin manifolds require additional information about line statistics and discrete topological terms.

Retain only the finite center-charge data by quotienting the electric weight lattice by the root lattice and the magnetic coweight lattice by the coroot lattice. A Wilson–’t Hooft line then has a class

$$\gamma=(e,m)\in \Gamma=\mathbb Z_N\times\mathbb Z_N.$$

Here $e$ is electric $N$-ality and $m$ is its magnetic counterpart. Taking a line $\gamma'$ once around $\gamma$ produces the phase

$$\exp\!\left[\frac{2\pi i}{N}\, \omega(\gamma,\gamma')\right], \qquad \omega\bigl((e,m),(e',m')\bigr) =em'-me'\pmod N.$$

Two lines can both be genuine line operators in the same theory only if they are mutually local, so this phase must be one. Thus their charges must obey $\omega=0$ modulo $N$. A complete choice of genuine center-charge classes is a maximal mutually local, or maximal isotropic, subgroup $L\subset\Gamma$: its elements pair trivially with one another, and no further charge class can be added without violating mutual locality Aharony, Seiberg, and Tachikawa 2013, Introduction and §§1.1–1.3, Open PDF.

This finite quotient suppresses the root- and coroot-lattice labels carried by individual lines. In particular, the formula below does not say that an $SU(N)$ theory has no ’t Hooft lines; it says that their magnetic center-charge class is trivial.

For the simply connected group, the genuine classes are purely electric,

$$L_{SU(N)} =\bigl\{(r,0):r\in\mathbb Z_N\bigr\}.$$

In particular, the fundamental Wilson line $(1,0)$ is genuine. For the quotient group $PSU(N)=SU(N)/\mathbb Z_N$, there are $N$ spin-theory choices customarily denoted $PSU(N)_n$, with

$$L_n =\bigl\{(nr,r)\pmod N:r\in\mathbb Z_N\bigr\}, \qquad n=0,1,\ldots,N-1.$$

The generator is a magnetic line for $n=0$ and a dyonic line for $n\ne0$. The label $n$ is discrete theta-like data: a $2\pi$ shift of the ordinary theta angle permutes these theories rather than acting trivially on a fixed line spectrum. These statements, including the composite-$N$ completeness condition, are derived in Aharony, Seiberg, and Tachikawa 2013, §2.1 and §6.2, Open PDF.

The $N=2$ case is a useful picture in words. There are three maximal choices:

$$\begin{aligned} L_{SU(2)}&=\{(0,0),(1,0)\},\\ L_{SO(3)_+}&=\{(0,0),(0,1)\},\\ L_{SO(3)_-}&=\{(0,0),(1,1)\}. \end{aligned}$$

They select an electric, magnetic, or dyonic nontrivial line, respectively. The $+$ and $-$ labels here use the spin-manifold convention; on a nonspin manifold the line spins and refined topological terms must also be declared.

The distinction is invisible to correlators of local operators on $\mathbb R^4$ without extended insertions. It becomes visible immediately to a line probe, on a compactification where a wrapped line becomes local, or on a manifold carrying a nontrivial bundle. “Same local correlators” therefore means same tested local sector, not “the same theory on every spacetime.”

The bulk remembers the choice

The standard $\mathcal N=4$ example makes the holographic consequence concrete. Type-IIB string theory on $\mathrm{AdS}_5\times S^5$ with $N$ units of five-form flux contains the Neveu–Schwarz and Ramond–Ramond two-form gauge fields, $B_2$ and $C_2$. Normalize them so that a unit fundamental- or D-string worldsheet contributes $\exp(i\int B_2)$ or $\exp(i\int C_2)$; their periods are then $2\pi$. In Witten’s Euclidean convention, the five-dimensional low-energy action includes

$$S_{\mathrm{top}}^{E} =-\frac{iN}{2\pi}\int_X C_2\wedge dB_2.$$

This is only the topological part of the effective action, with its global definition understood in differential cohomology, but it is precisely the part that remembers finite flux data. The first-order term makes the two boundary fields a conjugate pair, so a boundary condition chooses a polarization: which member is held fixed and which is allowed to fluctuate. Dirichlet boundary conditions fix a boundary value as a background source; Neumann boundary conditions instead sum over that value while fixing its conjugate response, thereby changing which complementary one-form symmetry is global or gauged. Because $X$ has a boundary, replacing $C_2\wedge dB_2$ by $B_2\wedge dC_2$ changes the action by a boundary term; that term belongs to the variational problem and must be fixed before “Dirichlet” and “Neumann” are assigned Witten 1998, Introduction and §§2–3, Open PDF.

A fundamental string couples to $B_2$ and can end on a Wilson line; a D-string couples to $C_2$ and can end on an ’t Hooft line. More generally, a $(p,q)$ string endpoint carries the corresponding dyonic charge. The allowed boundary endpoints must reproduce the chosen subgroup $L$. In modern language, changing the boundary polarization of the two-form topological sector changes the global completion of the boundary theory even when the propagating supergravity spectrum is unchanged Witten 1998, §5, Open PDF.

The two simplest polarizations make the result explicit. Use the boundary action and notation of Bhardwaj et al., with $b_2$ descending from $B_2$ and $c_2$ from $C_2$. Dirichlet $b_2$ with Neumann $c_2$ gives the $SU(N)$ theory and its electric $\mathbb Z_N$ one-form symmetry. A wrapped D5-brane is a baryon vertex on which $N$ fundamental strings can end. Exchanging the boundary conditions—Dirichlet $c_2$ with Neumann $b_2$—gives $PSU(N)_0$ and its magnetic $\mathbb Z_N$ one-form symmetry; a wrapped NS5-brane screens $N$ D-strings. Mixed polarizations encode the other discrete-theta variants Bhardwaj et al. 2023, §6.2, Open PDF.

What the first holographic application must distinguish

Datum	SU(N)	PSU(N) with label n
Local gauge algebra	su(N)	su(N)
Local adjoint correlators on flat space	Can agree at the same local coupling and theta value
Genuine center-charge generator	Electric class (1, 0)	Dyonic class (n, 1)
Fundamental Wilson line	Genuine	Not a standalone genuine line
Bundle sectors	Bundles lift to SU(N)	Non-liftable bundles with discrete magnetic flux are allowed
One-form symmetry realization	Electric center symmetry	Magnetic or dyonic realization determined by n
Bulk datum that must change	Boundary polarization and allowed endpoints in the two-form topological sector

The table does not assert that the two boundary theories require different local supergravity Lagrangians. Its point is the opposite: the same propagating bulk fields do not determine the boundary global form. The topological sector, its boundary condition, and its allowed string endpoints are indispensable dictionary entries. Within asymptotically AdS quantum gravity with a CFT dual, Harlow and Ooguri analyze compact internal, splittable boundary symmetries and compact long-range bulk gauge symmetries; the reverse construction also needs assumptions about a semiclassical bulk. Their result constrains faithful groups and charged objects, but it does not prove completeness of an arbitrary holographic dictionary Harlow and Ooguri 2021, §§4.4–5 and 8, Open PDF.

Background fields and sector sums are part of the equality

Let $B$ be a flat discrete two-form background, or cocycle, for the electric $\mathbb Z_N$ one-form symmetry of the $SU(N)$ theory. Its cohomology class records the obstruction to lifting a $PSU(N)$ bundle to an $SU(N)$ bundle. The source-dependent object

$$Z_{SU(N)}[M;B]$$

is a family of background-resolved, or twisted, partition functions. Gauging the one-form symmetry constructs a quotient theory by a finite sum of the schematic form

$$Z_{PSU(N)_n}[M] =\frac{1}{\mathcal N(M)} \sum_{[B]\in H^2(M,\mathbb Z_N)} Z_{SU(N)}[M;B]\,e^{iS_n[B]}.$$

Here $S_n[B]$ is the allowed discrete topological phase and $\mathcal N(M)$ includes the finite-gauge-theory normalization appropriate to the chosen convention. Its detailed cochain formula depends on the topology and on spin versus nonspin refinements; suppressing those details is harmless only if the equality itself is explicitly labeled schematic. Fixing $B=0$, summing over $B$, and Fourier transforming among flux sectors are different operations Aharony, Seiberg, and Tachikawa 2013, §§6.1–6.4, Open PDF.

This distinction is not a statistical ensemble over unrelated Hamiltonians. It is a gauging or sector prescription within a specified theory construction. The next page, Fixed-Theory, Ensemble, and Superselection Claims, separates these operations and gives their factorization tests.

A reproducible completeness test

A researcher can test a proposed global dictionary without assuming the desired conclusion.

1. Declare the domain. Record the spacetime topology and tangential structure—for example, oriented, spin, or nonspin—together with boundary conditions, couplings, regulator, approximation, and whether defects are inserted.
2. Fix each theory. State the faithful groups, genuine extended operators, charge group, pairing, allowed bundles, topological actions, and sector-sum prescription on both sides.
3. Map the data. Give an integral pairing-preserving map of charge classes, a map of backgrounds and anomalies, and a map of boundary conditions and allowed endpoints.
4. Choose a resolving probe. Select a charge that belongs to exactly one of the two proposed genuine subsets, or a bundle sector included on only one side.
5. Compute the same object. Compare either the linked-line phase, a background- or flux-resolved partition function, or the amplitude for the corresponding bulk object to end at the boundary. Use the same normalization and counterterm convention.
6. State the surviving claim. If the resolving probe fails, retain only the common local or sectoral statement; do not call the dictionary complete.

For $SU(N)$ versus $PSU(N)_n$, the fundamental electric class $(1,0)$ is already a resolving probe: it belongs to $L_{SU(N)}$ but not to $L_n$. Conversely, $(n,1)$ belongs to $L_n$ but not to $L_{SU(N)}$. Agreement of every tested local adjoint correlator cannot repair either mismatch.

Common pitfalls

Treating the Lie algebra as the gauge group. The algebra determines local gauge bosons, but the global group determines representations, bundles, and genuine Wilson lines. Write both.

Calling every line with a label “genuine.” A non-genuine line requires an attached surface or other auxiliary data. Such an object is useful, but it is not interchangeable with a standalone line operator.

Comparing a fixed sector with a sector sum. Equality at $B=0$ is not equality after summing over $[B]$. State the measure, phases, and normalization before comparing partition functions.

Using anomaly matching as a proof of duality. Anomalies are robust necessary data. They do not determine all operator products, spectra, sector weights, or dynamics.

Forgetting the manifold class. The simple $PSU(N)_n$ presentation above assumes spin four-manifolds. Nonspin manifolds can refine theta periodicities and line statistics, so the dictionary must record those choices rather than extrapolate silently.

Exercises

Show directly that both $L_{SU(N)}$ and $L_n$ are mutually local. Why are they maximal isotropic subgroups of $\Gamma=\mathbb Z_N^2$?

Solution

For two elements $(r,0)$ and $(r',0)$ of $L_{SU(N)}$,

$$\omega\bigl((r,0),(r',0)\bigr)=0.$$

For two elements $(nr,r)$ and $(nr',r')$ of $L_n$,

$$\omega\bigl((nr,r),(nr',r')\bigr) =nrr'-rnr'=0\pmod N.$$

For the nondegenerate finite pairing,

$$|L|\,|L^\perp|=|\Gamma|=N^2.$$

Isotropy gives $L\subseteq L^\perp$, hence $|L|\le N$. Each displayed subgroup has exactly $N$ elements, so $L=L^\perp$ and neither can be enlarged while remaining isotropic. This argument also works when $N$ is composite.

For $N=4$, list the genuine center-charge classes of $SU(4)$ and $PSU(4)_1$. Give one line probe that distinguishes the two theories in each direction.

Solution

The two subsets are

$$L_{SU(4)}=\{(0,0),(1,0),(2,0),(3,0)\},$$

and

$$L_1=\{(0,0),(1,1),(2,2),(3,3)\}.$$

The fundamental Wilson class $(1,0)$ is genuine in $SU(4)$ and not in $PSU(4)_1$. The dyonic class $(1,1)$ is genuine in $PSU(4)_1$ and not in $SU(4)$. Either probe distinguishes the global theories although local adjoint correlators on $\mathbb R^4$ can agree.

Suppose a proposed dictionary equates $Z_A[M;B=0]$ with

$$Z_B[M]=\frac{1}{\mathcal N(M)} \sum_{[B]}Z_A[M;B]e^{iS[B]}.$$

What additional statement would be needed for this to be an equality of the displayed objects rather than an accidental equality in one sector?

Solution

The proposal must specify a gauging or sector transform that maps the full family $Z_A[M;B]$ to $Z_B[M]$, including the set of summed classes, the phase $S[B]$, and the normalization $\mathcal N(M)$. Equality of the $B=0$ term alone cannot establish equality with the sum. It would suffice only after proving, in the declared domain, that all other weighted contributions vanish or combine to the required value—a much stronger and explicitly testable statement.

Scope and handoff

This page supplies a refutation-oriented completeness test; passing the listed checks does not prove that no relevant observable remains unexamined. Symmetry and Gauge Theory, Conformal Field Theory and the Bootstrap, and Supersymmetry and Duality own the detailed symmetry, CFT, and protected-sector constructions. Boundary Global Symmetry, Bulk Gauge Symmetry, and Global Form develops the Chapter 3 dictionary. Charge-Lattice Completeness and Spectrum Tests owns the distinct Chapter 25 charge-completeness question, and Mathematical QFT owns theorem-level formulations. Nonperturbative Definition and Completion Criteria asks the separate question of whether a proposed construction defines all of its claimed observables at finite coupling and finite $N$.

Evidence cutoff. Examples and literature-status statements are fixed to 25 July 2026. The gauge-theory and topological-field-theory formulas above are established results in the stated domain; the cutoff does not turn the checklist into a proof of any complete holographic equivalence.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 08 (2013): 115. DOI. Open PDF.
• Bhardwaj, Lakshya, Lea E. Bottini, Ludovic Fraser-Taliente, Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. “Lectures on Generalized Symmetries.” arXiv:2307.07547 [hep-th] (2023). Open PDF.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF.
• Harlow, Daniel, and Hirosi Ooguri. “Symmetries in Quantum Field Theory and Quantum Gravity.” Communications in Mathematical Physics 383 (2021): 1669–1804. DOI. Open PDF.
• Witten, Edward. “AdS/CFT Correspondence and Topological Field Theory.” Journal of High Energy Physics 12 (1998): 012. DOI. Open PDF.