Boundary Global Symmetry, Bulk Gauge Symmetry, and Global Form

An ordinary boundary global symmetry supplies a bulk gauge field and fixes its local gauge algebra through the current multiplet. It does not by itself fix the bulk gauge group’s global form, charge lattice, allowed bundles, discrete theta data, or complete spectrum of line and brane operators. Those data are detected by extended observables and higher-form symmetries, not by local current correlators alone.

Required background. Currents, stress tensor, and bulk gauge fields supplies the local current/gauge map. Helpful background. Generalized symmetries, global forms, and anomalies, electric–magnetic charge lattices, and the symmetry TFT supply the global data that local Lie algebras omit.

What a conserved current determines

Let the boundary theory have a continuous global symmetry with Lie algebra $\mathfrak g$ and current $J_i^a$. Coupling a background connection $A_{(0)}$ through

$$\delta S_{\partial}=\int\mathrm d^d x\,A_{(0)i}^aJ^{ia}$$

maps holographically to a bulk gauge field with the same local algebra. Current two- and three-point functions determine its kinetic normalization and local cubic couplings. Boundary anomalies determine bulk Chern–Simons, inflow, or topological terms. Under the usual assumptions of AdS/CFT, this realizes the principle that exact boundary global symmetries are gauged in the bulk, developed in general form by Harlow and Ooguri 2021, §§2–4.

This information is local. The Lie algebras of $SU(n)$ and $PSU(n)=SU(n)/\mathbb Z_n$ are identical. Their small gauge transformations and perturbative gauge bosons are therefore indistinguishable, but their allowed representations, bundles, and line operators differ.

The missing global data

A complete gauge-sector entry records at least:

Datum	Question answered	Local currents sufficient?
global form $G$	which representations exponentiate to genuine charges?	no
electric and magnetic lattice	which Wilson and ‘t Hooft lines are genuine?	no
bundles and discrete theta angles	which topological sectors enter the path integral?	no
higher-form symmetry	which extended charges are conserved?	no
anomaly and inflow	which background transformations carry phases?	partly, with global anomaly data added
charged spectrum	which allowed charges are actually populated?	no

Two theories can share all perturbative correlators of $J_i^a$ and still differ on a lens space, in the presence of a background bundle, or when probed by linked line operators. The generalized-symmetry framework makes these extended charges precise Gaiotto et al. 2015, §§2–3. This is why a global-form claim must name more than $\mathfrak g$.

First application: SU(N) and PSU(N)

In four-dimensional $\mathcal N=4$ SYM, choosing gauge group $SU(N)$ or $PSU(N)$ does not change the local adjoint Lagrangian or stress-tensor multiplet. It changes genuine Wilson–‘t Hooft line operators and the associated one-form symmetries. The bulk dual must therefore change its boundary conditions for discrete two-form gauge fields and its sum over topological sectors, even though the local type-IIB supergravity equations are unchanged. The classification of such choices is explained by Aharony, Seiberg, and Tachikawa 2013, §§2–4.

This example requires careful language: $SU(N)$ versus $PSU(N)$ here is the global form of the boundary theory’s gauge group, not the ordinary boundary global symmetry that sources one bulk vector. It is nevertheless holographic global data, encoded through higher-form symmetries and discrete bulk gauge sectors. The example demonstrates the broader lesson that the boundary local operator algebra does not complete the bulk global gauge theory.

Adversarial check: identical current correlators

Imagine two candidate dictionaries with the same $\mathfrak g$, $C_J$, and perturbative three-current coefficient but different charge lattices. Every local current test passes. A Wilson line in a representation that exists for one global form and not the other distinguishes them; so can a partition function in a nontrivial background bundle.

The strongest claim licensed by current correlators is a match of local gauge algebra and perturbative couplings. A complete gauge dictionary additionally needs the global form, charge spectrum, extended operators, anomalies, and boundary conditions for topological fields. Even specifying the allowed lattice does not prove that every allowed charge is dynamically realized; that is a separate completeness claim.

Controlled limits and handoff

This page distinguishes local Lie-algebra data from global sectors; it does not assert a charge-completeness theorem or derive the spectrum of a compactification. Extended Operators, Defects, and Brane Charges identifies probes of those sectors, and Dictionary Normalization and Global-Data Audit keeps charge units synchronized with gauge-field rescalings.

Exercise

Give a simple abelian reason why the Lie algebra $\mathfrak u(1)$ does not determine charge quantization.

Solution

The local transformation parameter and gauge field are the same for any normalization of a compact $U(1)$ Lie algebra, and they also resemble a noncompact $\mathbb R$ gauge field locally. The global identification of the gauge parameter fixes whether charges lie on an integer lattice and what the minimal charge is. A Wilson line detects this periodicity; a local field-strength correlator does not.

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 2013, 115 (2013). arXiv. DOI.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, 172 (2015). arXiv. DOI.
• Harlow, Daniel, and Hirosi Ooguri. “Symmetries in Quantum Field Theory and Quantum Gravity.” Communications in Mathematical Physics 383 (2021): 1669–1804. arXiv. DOI.