Abelian Electric–Magnetic Dualization, Charge Lattices, and Global Form

Abelian electric–magnetic dualization is locally a Gaussian transformation, but quantum equivalence depends on integral fluxes, zero modes, boundary polarization, and the spectrum of genuine lines. The transformation $\tau\mapsto-1/\tau$ is therefore only the middle of the argument: the parent theory must reproduce both local dynamics and global sectors.

Required background. Supersymmetric Yang–Mills conventions supplies the nonabelian trace convention that must be translated to the unit-flux $U(1)$ normalization below, while global form and matter representations distinguishes a Lie algebra from a gauge theory. Helpful background. See electric and magnetic one-form symmetries and BPS charge lattices.

Local first-order dualization

Start on an oriented Lorentzian four-manifold with the (+---) metric, so $*^2=-1$ on two-forms, and normalize $A$ so a unit Wilson line is $e^{i\oint A}$ and $F/2\pi$ has integral periods. Choose

$$S[A]=-\frac{1}{2e^2}\int F\wedge *F +\frac{\theta}{8\pi^2}\int F\wedge F, \qquad \tau=x+iy, \qquad x=\frac{\theta}{2\pi}, \qquad y=\frac{2\pi}{e^2}.$$

The factor $2\pi/e^2$ is tied to the displayed kinetic term, unit electric charge, and unit $2\pi$ flux. The common nonabelian convention $\theta/2\pi+4\pi i/g^2$ uses a different generator normalization; for the same quadratic form, Witten’s coupling satisfies $g^2=2e^2$. Mixing these packages changes the dual coupling by a factor of two.

For the local derivation, temporarily regard $F$ as an unconstrained two-form and impose its Bianchi identity with a dual one-form $A_D$:

$$S_{\mathrm P}[F,A_D] =S[F]+\frac{1}{2\pi}\int A_D\wedge dF.$$

Varying $A_D$ gives $dF=0$, so on a contractible patch $F=dA$ and the original theory returns. After integration by parts, varying $F$ instead gives

$$F_D=dA_D=y*F-xF.$$

This relation can be inverted without guessing. Since $*^2=-1$,

$$F=-\frac{xF_D+y*F_D}{x^2+y^2}.$$

Writing the parent bulk action as

$$S_{\mathrm P} =\frac{1}{4\pi}\int \left(-yF\wedge*F+xF\wedge F+2F_D\wedge F\right),$$

and substituting the solution gives

$$S_D[A_D] =\frac{1}{4\pi}\int \left(-y_DF_D\wedge*F_D+x_DF_D\wedge F_D\right),$$

where

$$x_D=-\frac{x}{x^2+y^2}, \qquad y_D=\frac{y}{x^2+y^2}.$$

Thus the same Maxwell form returns with

$$\tau_D=-\frac{1}{\tau}.$$

Equivalently, the equations of motion and Bianchi identity form a doublet. In the same convention,

$$G=\frac{2\pi}{e^2}*F-\frac{\theta}{2\pi}F, \qquad dF=0,\quad dG=0,$$

and $F_D=G$. The $S$ transformation exchanges $(F,G)$ with the orientation sign required by $S^2=-1$. This derivation fixes relative factors only after the action, Hodge star, unit charge, and definition of $G$ are displayed; translating conventions entry by entry is safer than copying a matrix from another normalization.

Explicitly, if $(F_D,G_D)$ denotes the field-strength doublet of the dual description, then

$$\begin{pmatrix}F_D\\ G_D\end{pmatrix} = \begin{pmatrix}0&1\\ -1&0\end{pmatrix} \begin{pmatrix}F\\ G\end{pmatrix} = \begin{pmatrix}G\\ -F\end{pmatrix}.$$

The first component reproduces the parent-action result $F_D=G$; the second is the dual constitutive relation. This is the same ordered field-strength map imposed by the $S$ wall later in the chapter.

The Gaussian step in Euclidean signature

For a convergent path integral, retain the chosen orientation while Wick rotating and use

$$S_E[F]=\frac{1}{2e^2}\int F\wedge *F -\frac{i\theta}{8\pi^2}\int F\wedge F.$$

On an oriented Riemannian four-manifold, $*^2=+1$ on two-forms, so $F=F_++F_-$ with $*F_\pm=\pm F_\pm$. With the Euclidean parent term $-i(2\pi)^{-1}\int A_D\wedge dF$, its bulk part is

$$S_{\mathrm P,E} =-\frac{i}{4\pi}\int \left(\tau F_+\wedge F_+ +\bar\tau F_-\wedge F_- +2F_D\wedge F\right).$$

The two independent completions are

$$\begin{aligned} \tau F_+^2+2F_{D,+}F_+ &=\tau\left(F_++\frac{F_{D,+}}{\tau}\right)^2 -\frac{F_{D,+}^2}{\tau},\\ \bar\tau F_-^2+2F_{D,-}F_- &=\bar\tau\left(F_-+\frac{F_{D,-}}{\bar\tau}\right)^2 -\frac{F_{D,-}^2}{\bar\tau}. \end{aligned}$$

The remaining quadratic form is the dual Euclidean action with $\tau_D=-1/\tau$. This calculation also identifies the Gaussian determinants that the purely classical substitution omits.

The determinant from nonzero modes and the harmonic-mode Gaussian do not generally cancel to one. Witten’s Euclidean convention assigns the powers $(\chi-\sigma)/4$ to $\tau$ and $(\chi+\sigma)/4$ to $\bar\tau$. The theta term in the Euclidean action above has the opposite sign relative to that convention at the fixed displayed orientation, so translating to the present $\tau$ exchanges the holomorphic and antiholomorphic weights and gives

$$Z(-1/\tau) =\tau^{(\chi+\sigma)/4} \bar\tau^{(\chi-\sigma)/4}Z(\tau),$$

Thus “the actions have the same form” is weaker than “the partition functions are identical scalars.” Local gravitational counterterms and the precise normalization of the measure belong in the latter statement Witten 1995, §2, especially eq. (2.4), arXiv PDF.

Flux sectors and compactness

For a compact $U(1)$ connection,

$$\frac{1}{2\pi}\int_{\Sigma_2}F\in\mathbb Z$$

for every closed two-cycle $\Sigma_2$, subject to possible shifts in spin-$c$ or quotient constructions. An ordinary integral over a globally defined real two-form does not impose this condition. The global parent path integral must sum over line bundles, include harmonic fluxes, and treat torsion sectors. Compactness of $A_D$ implements the integral constraint rather than only $dF=0$.

This distinction already affects theta periodicity. On a spin four-manifold the relevant intersection form is even, and the usual $\theta\sim\theta+2\pi$ can hold for a purely bosonic Maxwell sector with the stated charge lattice. On a general oriented nonspin manifold, odd self-intersections can reduce the manifest modular subgroup or require additional structure. The duality group therefore depends on the class of manifolds and on whether the theory is bosonic, spin, or spin-$c$.

Zero modes require separate gauge-volume and determinant factors. Torsion fluxes can pair through linking forms even though differential-form representatives vanish. A local first-order action sees neither feature unless the fields are formulated globally, for example in differential cohomology.

Charges, lines, and the symplectic lattice

Let a Wilson–’t Hooft line have magnetic and electric charges

$$\gamma=\binom{p}{q}\in\Gamma,$$

with Dirac pairing

$$\langle\gamma,\gamma'\rangle=pq'-qp'.$$

In an active charge convention, let an integral matrix

$$M_\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix} \in SL(2,\mathbb Z)$$

act by $\gamma\mapsto M_\gamma\gamma$ and preserve this pairing. For

$$S_\gamma=\begin{pmatrix}0&1\\-1&0\end{pmatrix},$$

one has $(p,q)\mapsto(q,-p)$ in the displayed column convention. If a period column transforms as $\Pi\mapsto M_\Pi\Pi$, central-charge invariance requires $M_\gamma=M_\Pi^{-T}$. For $S$, these matrices coincide; for $T$, they do not. Stating which representation is being used prevents a passive basis change from being mistaken for an active motion of a physical line.

The allowed set of genuine lines need not be the full lattice. Matter screens some electric charges; a nontrivial global form restricts Wilson representations and changes magnetic sectors; a discrete theta angle can correlate $p$ and $q$. A matrix preserving the ambient Dirac pairing is a duality of the chosen theory only if it maps its allowed line set to the target’s allowed line set.

For nonabelian theories with the same Lie algebra, this is decisive. $SU(N)$ and $PSU(N)=SU(N)/\mathbb Z_N$ have different genuine lines and background bundles. An $S$ operation can exchange a theory with one global form for a theory with another, sometimes also shifting discrete theta data. It should not be described as a self-duality of the Lie algebra alone; Aharony, Seiberg, and Tachikawa 2013, §§1–2, arXiv v5 PDF classify precisely these global-form and line-operator choices.

The global-form, line, anomaly, and wall map works out the three $\mathfrak{su}(2)$ center-charge choices as a finite $S$/$T$ orbit and keeps that orbit explicitly separate from the continuous Maxwell background system.

Boundaries and polarization

The integration by parts

$$\int_M A_D\wedge dF =\int_M F_D\wedge F -\int_{\partial M}A_D\wedge F$$

produces a boundary term. Dropping it silently changes the variational problem. Electric boundary conditions fix a tangential gauge potential or electric polarization; magnetic boundary conditions fix the dual data. Dualization exchanges these choices and can generate a three-dimensional boundary theory or contact term Kapustin and Tikhonov 2009, §§2–4, arXiv PDF.

At an interface across which $\tau$ is transformed, the parent coupling can be localized on the wall. Transporting a Wilson line through the wall turns it into the corresponding ’t Hooft line. Whether the wall is invertible depends on the global charge lattice and boundary degrees of freedom, not only on the bulk equations.

Supersymmetric periods and BPS masses

In a four-dimensional $\mathcal N=2$ abelian theory of rank $r$, assemble periods and charges as symplectic vectors. For rank one,

$$Z_\gamma=q\,a+p\,a_D.$$

If the period column obeys

$$\binom{a_D}{a}\longmapsto M_\Pi\binom{a_D}{a},$$

then $\gamma\mapsto M_\Pi^{-T}\gamma$ in the compatible ordering. Consequently $Z_\gamma=\gamma^T\Pi$ and the BPS mass $|Z_\gamma|$ are unchanged. A transformation of $\tau=da_D/da$ without the accompanying charge map would instead relabel physical states incorrectly.

This covariance is local on moduli space. Monodromy can prevent a single electric polarization from covering all vacua. The lattice local system, not one preferred basis, is the global object.

A complete dualization check

Before accepting the result, verify:

1. the Lorentzian and Euclidean sign conventions and $*^2$;
2. the Gaussian transformation of $\tau$;
3. integral free fluxes, torsion, zero modes, and the measure;
4. boundary terms and the chosen polarization;
5. preservation of the integral Dirac pairing;
6. the map of genuine lines, screening, global form, and discrete theta data;
7. covariance of periods, charges, and central charges;
8. any modular weight or local gravitational counterterm in the partition function.

Passing only the first two items establishes a local classical dualization, not a complete quantum equivalence.

Common pitfalls

Imposing only $dF=0$. Closedness does not impose integral periods or sum over bundles. Compact global fields are required.

Transforming $\tau$ but not charges. The coupling, periods, sources, and charge basis form one symplectic package.

Ignoring the boundary term. Electric–magnetic duality changes boundary polarization and can require interface degrees of freedom.

Exercises

Let $\Gamma=\mathbb Z^2$ with pairing $\langle(p,q),(p',q')\rangle=pq'-qp'$. For

$$M_\gamma=\begin{pmatrix}1&1\\0&1\end{pmatrix},$$

1. show that the pairing is preserved;
2. find the image of $(p,q)$;
3. explain why this is not a symmetry of a theory whose genuine lines obey $p\equiv0\pmod 2$ with unrestricted $q$.

Solution

Since $M_\gamma^TJM_\gamma=J$ for $J=\bigl(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\bigr)$, the pairing is preserved. In the displayed active charge convention, $(p,q)\mapsto(p+q,q)$. The allowed line $(0,1)$ maps to $(1,1)$, which violates the condition that $p$ be even. Thus $M_\gamma$ preserves the ambient lattice and pairing but not the theory’s genuine-line set; it maps to a different global-form or discrete-theta theory rather than acting as a symmetry of the original one.

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.
• Kapustin, Anton, and Mikhail Tikhonov. “Abelian Duality, Walls and Boundary Conditions in Diverse Dimensions.” Journal of High Energy Physics 11 (2009): 006. DOI. Open PDF.
• Witten, Edward. “On S-Duality in Abelian Gauge Theory.” Selecta Mathematica 1 (1995): 383–410. DOI. Open PDF.

Further reading

• Freed, Daniel S. “Dirac Charge Quantization and Generalized Differential Cohomology.” Surveys in Differential Geometry 7 (2002): 129–194. DOI. Open PDF.