Boundary Global Symmetry, Bulk Gauge Symmetry, and Global Form

A current remembers a Lie algebra; a complete holographic gauge sector remembers a group, its bundles, its genuine extended operators, its boundary conditions, and the charges that actually occur. Two theories can therefore agree on all tested local correlators in the trivial background sector and still be different theories. The key is to separate two related dictionaries: a connected compact boundary global symmetry sources a bulk one-form gauge field, whereas the global form of a boundary theory’s dynamical gauge group is encoded by bundle sums, line operators, higher-form symmetries, and bulk form-field boundary data.

Required background. Currents, stress tensor, and bulk gauge fields supplies the local current/gauge map and its normalization conventions. Helpful background. Generalized symmetries, global forms, and anomalies, electric–magnetic charge lattices, and the symmetry TFT develop the general machinery used here.

Two symmetry dictionaries that must not be conflated

Suppose first that a boundary CFT has a connected compact internal symmetry $G$. Its gauge-invariant conserved current $J_i^a$ couples to a nondynamical background connection $A_{(0)}$:

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

In a semiclassical AdS dual, the source $A_{(0)}$ is the boundary value of a bulk one-form gauge field. Harlow and Ooguri argue that the boundary symmetry is realized as a long-range bulk gauge symmetry when the symmetry is splittable—its action can be localized to separated boundary regions, up to operators near their boundaries—and when a semiclassical dual with a low-energy bulk Lagrangian is already assumed Harlow and Ooguri 2021, § 4.4, pp. 93–95, PDF. This is not an assumption-free converse theorem. A discrete zero-form symmetry has no current and instead maps to a discrete or topological bulk gauge sector. More generally, a current sees only $\operatorname{Lie}(G)=\operatorname{Lie}(G_0)$; disconnected components of a compact group require additional discrete background and topological data.

Now suppose instead that the boundary CFT is itself presented as a gauge theory. Its gauge transformations are redundancies, not physical global transformations, so they have no gauge-invariant Noether current. Choosing $SU(N)$ rather than $PSU(N)=SU(N)/\mathbb Z_N$ specifies the global form of that boundary gauge redundancy. The choice is visible in admissible bundles and genuine Wilson–‘t Hooft lines. In holography it is encoded by a topological sector and boundary conditions for bulk two-form gauge fields, not by changing the Lie algebra of one ordinary bulk vector.

Keeping these lanes separate prevents the most common category error on this page: the $SU(N)$/$PSU(N)$ example does not say that the boundary’s $SU(N)$ gauge redundancy becomes an $SU(N)$ vector field in AdS.

What local current data determine

The current OPE and Ward identities determine the local Lie algebra $\mathfrak g$ and its action on local operators. Once the generator basis and source normalization are fixed, $C_J$ fixes the quadratic on-shell normalization of the leading bulk gauge field. Three-current structures constrain on-shell cubic combinations. Perturbative ‘t Hooft-anomaly coefficients constrain bulk Chern–Simons or inflow classes.

Those conclusions still carry an effective-field-theory qualifier. Field redefinitions can move terms between vertices, higher-derivative operators can contribute the same tensor structures at a fixed order, and local counterterms change contact terms. Global anomalies require large transformations or nontrivial background bundles and cannot be reconstructed from ordinary separated-point current correlators alone. A useful claim ladder is

$$\begin{aligned} \text{current data} &\Longrightarrow \text{local algebra and normalized on-shell data},\\ \text{fixed semiclassical truncation} &\Longrightarrow \text{perturbative bulk-EFT couplings at that order}. \end{aligned}$$

Neither arrow determines a global group or a dynamical charge spectrum.

The table organizes the additional evidence in increasingly global layers. Each layer answers a new question rather than merely improving the precision of the preceding one. On a narrow screen, scroll horizontally so the entries remain at a readable size.

What different boundary data establish in the bulk gauge dictionary.

Boundary datum	Diagnostic observable	Bulk conclusion	Still not determined
Current algebra	Ward identities and separated-point current correlators in a fixed source basis	Local gauge algebra and normalized on-shell couplings within a stated truncation	Faithful global group, nontrivial bundles, and charge lattice
Faithful symmetry action and backgrounds	Representations of genuine operators and partition functions in background G-bundles	Which central quotient acts faithfully and which bulk bundles are admissible	Which allowed charges are dynamically populated
Genuine lines and mutual braiding	Wilson–'t Hooft defects, screening, linking phases, and attached-surface dependence	A mutually local line lattice and boundary conditions for bulk form fields	Relative weights of all topological sectors and the massive spectrum
Sector sums and anomaly phases	Partition functions on nontrivial manifolds, discrete backgrounds, and large transformations	Discrete theta terms, topological couplings, and global anomaly or inflow data	A proof that the two full theories are equivalent
Charged operator and state spectrum	Actual local operators, lines, branes, and states in each charge sector	Dynamically realized bulk charges	Completeness outside the tested domain or an arbitrary compactification

From a Lie algebra to a faithful group

Return first to the ordinary-global-symmetry lane. For a semisimple Lie algebra, let $\widetilde G$ be the simply connected compact group with that algebra. Connected compact groups with the same local algebra take the form

$$G=\widetilde G/\Gamma, \qquad \Gamma\subseteq Z(\widetilde G).$$

A representation of $\mathfrak g$ defines a representation of $G$ only when every element of $\Gamma$ acts trivially. Conversely, if a central subgroup acts trivially on every operator or state included in the full physical symmetry action, retaining it would make the purported symmetry action nonfaithful. “Full” here includes genuine local and extended observables, not gauge-variant fields. Schematically,

$$\begin{aligned} \Gamma_{\mathrm{triv}} &=\{z\in Z(\widetilde G):z\text{ acts trivially on every physical operator and state}\},\\ G_{\mathrm{faithful}} &=\widetilde G/\Gamma_{\mathrm{triv}}. \end{aligned}$$

Current correlators see $\mathfrak g$, not the periodic identifications in $G$. Charged-operator representations and admissible background $G$-bundles supply that missing information. In an exact holographic framework, the boundary and long-range bulk groups can then be matched as global groups; the current alone does not perform this step. Harlow and Ooguri 2021, §§ 4.4–5, pp. 93–98, PDF gives the conditional holographic argument.

Return now to the dynamical-gauge-group lane. For a four-dimensional gauge theory, even the quotient group is not always the end of the story. One must also choose a mutually local set of genuine dyonic lines and any discrete theta data. Wilson charges begin as weights and ‘t Hooft charges as coweights. Adjoint matter screens roots and coroots; for gauge algebra $\mathfrak{su}(N)$, the remaining classes reduce to

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

Here $e$ is the electric $N$-ality and $m$ is the magnetic center class. For $\gamma=(e,m)$ and $\gamma'=(e',m')$, the Dirac pairing is

$$\omega\bigl((e,m),(e',m')\bigr) =em'-me'\pmod N.$$

The corresponding mutual-braiding phase is

$$\exp\!\left(\frac{2\pi i}{N}\,\omega(\gamma,\gamma')\right).$$

Two genuine lines must have vanishing pairing. A standalone (absolute) theory with a complete mutually local set of genuine lines chooses a maximal isotropic subgroup $L$. In the standard spin-theory convention,

$$\begin{aligned} L_{SU(N)}&=\{(r,0):r\in\mathbb Z_N\},\\ L_{PSU(N)_p}&=\{(pr,r):r\in\mathbb Z_N\}. \end{aligned}$$

with $p\in\mathbb Z_N$. Thus “the $PSU(N)$ theory” is incomplete notation: there are distinct $PSU(N)_p$ choices. This label assumes a fixed $2\pi$ branch for the continuous theta angle; in the convention used here, $\theta\mapsto\theta+2\pi$ sends $p\mapsto p+1$, or equivalently one may use an extended $2\pi N$ theta range. These reduced center charges suppress the full weight and coweight labels. In particular, $L_{SU(N)}$ does not mean that no ‘t Hooft defect can be written; it says that a nontrivial magnetic center-charge line is not a genuine line without additional surface data in this absolute theory. The classification, theta shift, and pairing appear in Aharony, Seiberg, and Tachikawa 2013, § 1.1, pp. 2–4, and §§ 2.1–2.4, pp. 12–20, PDF.

For $N=2$, the three choices are especially transparent:

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

Each subgroup has vanishing internal pairing and contains half of the four charge classes, so each is maximal isotropic. The global-theory orbit figure displays these three lattices and their duality transport; the equations here isolate the data used in the holographic dictionary. Nonspin manifolds require extra line-spin and quadratic-refinement data; the formulas above declare the spin convention rather than concealing that refinement.

Worked holographic application: SU(N) and PSU(N)ₚ

Consider four-dimensional $\mathcal N=4$ super-Yang–Mills with adjoint dynamical fields, fixed $N$, a fixed continuous-theta branch, and fixed coupling, on an oriented spin four-manifold $M_4$. The $SU(N)$ and $PSU(N)_p$ theories share the local algebra $\mathfrak{su}(N)$ and the same local adjoint Lagrangian. Local gauge-invariant correlators in the trivial bundle sector—including stress-tensor and $SU(4)_R$-current correlators—do not determine which global theory was chosen.

The allowed bundle sectors and their weights do. A $PSU(N)$ bundle has a $\mathbb Z_N$-valued obstruction to lifting to an $SU(N)$ bundle, often called a generalized Stiefel–Whitney class or ‘t Hooft flux,

$$w_2(P)\in H^2(M_4,\mathbb Z_N).$$

For $N>2$, this is not the ordinary tangent-bundle Stiefel–Whitney class. Every $SU(N)$ bundle induces a liftable $PSU(N)$ bundle with $w_2(P)=0$, so the $SU(N)$ path integral includes only that part of the quotient bundle sum. A $PSU(N)_p$ theory also includes nonliftable sectors and, on the chosen continuous-theta branch, weights them with its discrete theta datum. On $S^2\times S^2$ or $T^4$, where $H^2(M_4,\mathbb Z_N)$ is nonzero, this gives a dimension-matched experiment that local flat-space correlators cannot imitate. Gauging the electric one-form symmetry, summing over the corresponding two-form backgrounds, and including the allowed discrete counterterm produces the quotient theories; Gaiotto et al. 2015, Introduction, pp. 2–4, PDF explains this relation among genuine lines, one-form symmetry, bundle sums, and discrete theta weights.

In the type-IIB construction on asymptotically $AdS_5\times S^5$, reduce on $S^5$ and let $X_5$ denote the five-dimensional effective AdS spacetime. Normalize the compact NSNS and RR two-form gauge potentials $B_2$ and $C_2$ so that unit F1 and D1 worldsheets have holonomies $\exp(i\int B_2)$ and $\exp(i\int C_2)$; their periods are then $2\pi$. Five-form flux $N$ induces the Euclidean topological term

$$S_{\mathrm{top}}^{E} =-\frac{iN}{2\pi}\int_{X_5} C_2\wedge \mathrm d B_2,$$

in Witten’s orientation and field-order convention. Reversing orientation changes the displayed sign. Integrating by parts also produces a boundary term proportional to $\int_{\partial X_5}C_2\wedge B_2$; changing the field ordering describes the same bulk topological sector only when the corresponding boundary term and variational convention are changed with it. In the long-distance topological sector, the flat boundary holonomies of $B_2$ and $C_2$ are conjugate variables, so one chooses a polarization rather than fixing both independently.

• Fixing the flat $B_2$ boundary holonomy and summing over its conjugate $C_2$ datum yields the $SU(N)$ theory and its electric $\mathbb Z_N$ one-form symmetry.
• Exchanging $B_2$ and $C_2$ yields $PSU(N)_0$ and its magnetic $\mathbb Z_N$ one-form symmetry.
• Mixed polarizations and boundary topological terms yield the other $PSU(N)_p$ choices and their dyonic one-form symmetries.

The fundamental string couples electrically to $B_2$, and its boundary endpoint is a Wilson line. The D1-brane couples to $C_2$, and its endpoint is an ‘t Hooft line. A D5-brane wrapped on $S^5$ absorbs $N$ F1 strings; its S-dual wrapped NS5-brane absorbs $N$ D1 strings. These baryon vertices leave the finite $\mathbb Z_N$ charge classes. The chosen polarization decides which endpoint defines a genuine boundary line and which requires an attached surface. Witten derives the level-$N$ sector and its string probes Witten 1998, Introduction, pp. 1–3; § 2, pp. 5–10; and § 5, pp. 28–31, PDF. General mixed polarizations and their line spectra are derived directly by Bergman and Hirano 2022, § 2.3, pp. 7–9, and § 2.5, pp. 10–11, PDF and reviewed in modern background-field language by Bhardwaj et al. 2024, § 6.2, arXiv v2 pp. 134–136, PDF.

The table summarizes the result. On a narrow screen, scroll horizontally so the entries remain at a readable size.

Global data in the top-down AdS5/CFT4 example.

Datum	SU(N)	PSU(N)0	PSU(N)p ≠ 0
Local adjoint sector	Same local algebra and local adjoint Lagrangian; trivial-sector local correlators do not select the global form.
Genuine center-charge generator	Electric (1,0)	Magnetic (0,1)	Dyonic (p,1)
Fundamental Wilson line	Genuine	Not genuine without extra surface data	Not genuine without extra surface data
Bundle sectors	Only sectors lifting to SU(N)	Nonliftable sectors included	Nonliftable sectors included with a p-dependent weight
One-form symmetry	Electric ZN	Magnetic ZN	Dyonic ZN
Bulk polarization	Flat B2 holonomy fixed; conjugate C2 datum summed	Flat C2 holonomy fixed; conjugate B2 datum summed	Mixed flat-holonomy polarization plus boundary topological data
Resolving probe	Genuine line spectrum, a nontrivial bundle or one-form background, and the corresponding partition function phase

The line and bundle classification is exact QFT data in the stated spin convention. The five-dimensional topological sector is a top-down part of the IIB reduction. A weakly curved local supergravity description additionally requires the familiar large-$N$, strong-‘t Hooft-coupling regime; the finite $\mathbb Z_N$ sector must not be discarded merely because it is invisible in the local classical equations.

Adversarial check: identical local correlators

Take two proposed dictionaries, one for $SU(N)$ and one for $PSU(N)_p$, and compare only local gauge-invariant correlators in the trivial bundle sector. The stress tensor, $SU(4)_R$ current, and all local gauge-invariant operators built from adjoint fields can pass the comparison. This does not test the boundary gauge group’s global form because that gauge redundancy has no physical current of its own.

Three resolving probes expose the omitted hypothesis:

1. Ask whether the fundamental Wilson line is genuine. It is for $SU(N)$ but needs extra surface data for $PSU(N)_p$.
2. Put the theory on a spin four-manifold with $H^2(M_4,\mathbb Z_N)\ne0$ and turn on a nonzero lifting obstruction. The $SU(N)$ bundle sum rejects that sector; the quotient theory includes it.
3. Couple the one-form symmetry to a background two-form field. Its partition function and discrete-theta phase distinguish the choices.

The failed hypothesis is that local, trivial-sector agreement exhausts the observable algebra and all background sectors. The strongest surviving claim is narrower: the theories share the tested local algebra and local correlators, and hence the associated perturbative bulk fields and on-shell couplings within the chosen truncation. They need not have the same global bulk gauge sector or define equivalent full theories.

Allowed charges are not yet a populated spectrum

Three uses of “complete” should remain separate.

• Representation descent is kinematic: it says which Lie-algebra representations are allowed by the quotient group.
• A maximal line lattice is also kinematic: it selects a maximal mutually local set of genuine line charges.
• Dynamical charge completeness is spectral: it asks whether actual states, operators, lines, or branes populate every allowed charge class.

Neither of the first two proves the third. Matter screening can also change the conserved line classes without changing the local gauge algebra. Harlow and Ooguri derive a bulk representation-completeness result within their exact holographic assumptions Harlow and Ooguri 2021, § 5, pp. 96–98, PDF; it is not an unconditional theorem about an arbitrary low-energy EFT or compactification. The charge-lattice and gauge-completeness tests develop that quantum-gravity question, while the later dictionary-completeness audit tests how global data can be lost in a proposed duality.

Common pitfalls

“Every boundary symmetry current names the complete bulk gauge group.” The current names the local algebra. The faithful quotient requires representation and background-bundle data, and a discrete symmetry has no current at all.

“SU(N) versus PSU(N) is an example of the ordinary current/vector map.” It is instead a choice of global form for a dynamical boundary gauge group. Its holographic image lies in the two-form topological sector and its boundary polarization.

“The quotient group fixes every line operator.” A quotient constrains the possibilities, but a maximal mutually local dyonic lattice and discrete theta datum still have to be chosen. The label $PSU(N)_p$ records this extra choice in the spin convention.

“Allowed means dynamically realized.” A representation can descend to the group without any state carrying it. Conversely, screening can identify infrared line charges even when microscopic charged fields exist.

“Identical flat-space correlators prove the same theory.” They prove agreement only for the tested operators, backgrounds, and contact-term scheme. Bundles, extended operators, global anomalies, and sector weights remain independent tests.

Scope and handoffs

This discussion focuses on the holographic separation between local current data and global gauge-sector data. General higher-form symmetry, gauging, and symmetry-TFT constructions are developed on the linked Volume III pages; the complete $PSU(N)_p$ duality orbit is analyzed in the Volume X line-operator treatment. General conformal representations, currents, defects, OPE data, and deformations are developed in Conformal Field Theory and Bootstrap; here they are used to construct the holographic bulk dictionary. General self-adjoint extensions and QFT on timelike boundaries are developed in QFT in Curved Spacetime; here asymptotically AdS boundary choices enter through their dictionary role.

Extended operators, defects, and brane charges continues with the dynamics and probe regimes of strings and branes. Dictionary normalization and global-data audit keeps charge units synchronized with gauge-field rescalings. No-global-symmetry and compactness results conditional on AdS/CFT develops the quantum-gravity consequences, while rigorous theorem status is treated in Mathematical QFT.

Exercises

1. Compact U(1) versus the real line

Fix a generator convention in which the compact gauge parameter obeys $\alpha\sim\alpha+2\pi$. Show that a one-dimensional representation $e^{iq\alpha}$ requires $q\in\mathbb Z$. Explain why this does not identify the smallest dynamically realized charge.

Solution

Single-valuedness requires

$$e^{iq(\alpha+2\pi)}=e^{iq\alpha},$$

so $e^{2\pi iq}=1$ and therefore $q\in\mathbb Z$. A noncompact $\mathbb R$ group has no such periodic identification, so the local Lie algebra $\mathfrak u(1)$ does not distinguish the two global choices. The integer lattice is the allowed character lattice after a normalization is fixed. The actual spectrum may occupy only a sublattice, such as $k\mathbb Z$, or no nonzero charge sector at all. Local field-strength correlators see neither the periodicity nor which allowed charges are populated.

2. Descent from SU(N) to PSU(N)

An $SU(N)$ representation has $N$-ality $k$ when the center generator $z=e^{2\pi i/N}\mathbf 1$ acts by $e^{2\pi i k/N}$. Show that the representation descends to $PSU(N)$ exactly when $k=0\pmod N$. Apply the test to the fundamental and adjoint representations.

Solution

A representation descends to the quotient only if every identified central element acts trivially. For the generator this requires

$$e^{2\pi i k/N}=1,$$

or $k=0\pmod N$. The fundamental has $k=1$ and therefore does not define a representation of $PSU(N)$. The adjoint has $k=0$ because the center acts trivially by conjugation, so it descends. This is why the same local adjoint Lagrangian is compatible with both global forms even though their Wilson lines differ.

3. The three SU(2)/SO(3) line lattices

Use the mod-two Dirac pairing to verify that the subgroups generated by $(1,0)$, $(0,1)$, and $(1,1)$ are maximal isotropic in $\mathbb Z_2^2$. Identify the corresponding theories.

Solution

The pairing is alternating, so every generator has zero pairing with itself. Each nonzero generator spans a two-element subgroup,

$$\begin{aligned} \langle(1,0)\rangle&=\{(0,0),(1,0)\},\\ \langle(0,1)\rangle&=\{(0,0),(0,1)\},\\ \langle(1,1)\rangle&=\{(0,0),(1,1)\}. \end{aligned}$$

Every pair of distinct nonzero vectors has nonvanishing pairing. For example,

$$\begin{aligned} \omega\bigl((1,0),(0,1)\bigr)&=1\pmod 2,\\ \omega\bigl((1,0),(1,1)\bigr)&=1\pmod 2. \end{aligned}$$

and the remaining distinct pair gives the same result. No second independent nonzero vector can therefore be added while preserving mutual locality, so each subgroup is maximal isotropic. In the spin convention they define $SU(2)$, $SO(3)_+$, and $SO(3)_-$, respectively.

4. Boundary variation and polarization

Vary the topological action

$$S_{\mathrm{top}}^{E} =-\frac{iN}{2\pi}\int_{X_5}C_2\wedge \mathrm dB_2.$$

Identify the boundary term in $\delta S_{\mathrm{top}}^{E}$. Explain why the flat $B_2$ and $C_2$ boundary holonomies are conjugate data and why a boundary condition should select a polarization rather than fix both independently.

Solution

Using $C_2\wedge\mathrm d(\delta B_2)=\mathrm d(C_2\wedge\delta B_2)-\mathrm dC_2\wedge\delta B_2$ gives

$$\begin{aligned} \delta S_{\mathrm{top}}^{E} =-\frac{iN}{2\pi}\biggl[ &\int_{X_5}\delta C_2\wedge\mathrm dB_2 -\int_{X_5}\mathrm dC_2\wedge\delta B_2\\ &+\int_{\partial X_5}C_2\wedge\delta B_2 \biggr]. \end{aligned}$$

Within the topological truncation, the bulk equations are $\mathrm dB_2=0$ and $\mathrm dC_2=0$. The remaining boundary variation vanishes when the $B_2$ holonomy is fixed, so $C_2$ is its conjugate response variable. Adding the appropriate boundary term exchanges the roles and makes fixed $C_2$ the natural variational problem. Because this is a first-order topological system, choosing both independent holonomies would overconstrain the conjugate pair; a consistent boundary condition instead selects a maximal commuting, or Lagrangian, subgroup. That is the bulk origin of the genuine-line lattice.

5. An adversarial dictionary

Two holographic candidates have the same $\mathfrak g$, $C_J$, and separated-point three-current structures. State the strongest conclusion this comparison supports and list four kinds of global information it leaves open.

Solution

After fixing the generator/source normalization and a semiclassical truncation, the comparison supports the same local gauge algebra, quadratic on-shell normalization, and the tested cubic on-shell structures. It does not by itself fix:

1. the faithful global group $\widetilde G/\Gamma$;
2. admissible nontrivial bundles and discrete theta weights;
3. the mutually local spectrum of genuine Wilson–‘t Hooft lines and form-field boundary conditions;
4. the dynamically populated charge spectrum or a completeness theorem.

Global anomalies and contact-term choices may require additional tests as well. A charged-operator representation, a nontrivial background bundle, or an extended-operator braiding experiment can distinguish candidates that pass the current-correlator comparison.

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). DOI. Open PDF, arXiv:1305.0318.
• Bergman, Oren, and Shinji Hirano. “The Holography of Duality in $\mathcal N=4$ SYM.” Journal of High Energy Physics 11 (2022): 069. 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.” Physics Reports 1051 (2024): 1–87. DOI. Open PDF, arXiv v2.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, 172 (2015). 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.