From 4d N=2 to 3d N=4: Wilson lines and dual photons

A four-dimensional Abelian vector multiplet contains one complex scalar. Why does its three-dimensional Coulomb branch have four real scalar coordinates? One new scalar is a gauge-field component around the compact circle. The other replaces the remaining photon. Their different origins determine their periods and kinetic terms.

Start from four-dimensional N=2 multiplets and the Seiberg–Witten charge dictionary. We use pure compact $SU(2)$ theory in a regular Coulomb vacuum, where the light gauge group has rank one. Compactify a spatial direction $y\sim y+L$, $L=2\pi R$, with periodic bosons and fermions and no R-symmetry twist. All eight real Poincaré supercharges survive: they are called N=2 in four dimensions and N=4 in three. A three-dimensional N=2 superfield description would make only four manifest. This circle is not a thermal boundary condition.

Reduce the fields before dualizing anything

Expand a neutral field as $\Phi(x,y)=\sum_n\Phi_n(x)e^{iny/R}$. Nonzero modes cost at least $|n|/R$. At energies $E\ll1/R$, the light vector fields are

Four-dimensional field	Three-dimensional zero modes	Real bosonic propagating modes
Complex Coulomb scalar $a$	$\operatorname{Re}a,\operatorname{Im}a$	2
Connection $v$	$v_y$ and a three-dimensional photon	$1+1$

The photon has one transverse polarization in three dimensions. Locally replacing it by a scalar leaves the total at four, not five. For a regular rank-$r$ Coulomb patch there are $4r$ such coordinates. This is the count in Seiberg–Witten, section 2.1; it does not apply unchanged at a locus where extra charged fields become massless.

Here $[a]=[v_y]=1$ and $[R]=-1$ in mass units. The zero modes are still written in four-dimensional field normalization. Rescaling by $\sqrt L$ would give canonically normalized three-dimensional fields, but would obscure the holonomy periods.

The global group fixes the first angle

In the inherited period convention,

$$Z_\gamma=n_ma_D+n_ea,\qquad M_{\rm BPS}=\sqrt2|Z_\gamma|,\qquad \tau_{\rm eff}=\frac{da_D}{da}. \tag{X05.1}$$

The W boson has $n_e=1$; a fundamental Wilson probe has $n_e=1/2$. The absorbed connection is $v\,t^3$, with $t^3=\operatorname{diag}(1,-1)/2$. A single-valued $SU(2)$ gauge transformation shifts $\oint v$ by $4\pi$, not $2\pi$. The latter shift is single-valued only after quotienting by the center, which would change the specified global theory.

It is convenient to introduce a unit-charge compact connection and periods

$$\begin{gathered} A=\frac v2,\quad \theta_e=\oint_{S^1}A=L A_y\sim\theta_e+2\pi,\\ a_e=\frac a{\sqrt2},\quad a_m=\sqrt2a_D,\quad \tau_c=\frac{da_m}{da_e}=2\tau_{\rm eff}=s+it,\quad t>0. \end{gathered} \tag{X05.2}$$

These new subscripts label electric and magnetic units, not additional fields. A charge $(p,q)=(2n_e,n_m)$ has physical central charge $\mathcal Z=p a_e+q a_m=\sqrt2 Z_\gamma$, and mass $|\mathcal Z|$ if it is an existing BPS particle. The W has $p=2$. A probe with $p=1$ is allowed even though no fundamental particle is present. Magnetic flux is $\int F/(2\pi)\in\mathbb Z$ for $F=dA$, equivalently $\int dv\in4\pi\mathbb Z$.

The same global-group choice and doubling of the torus coupling are explicit in SW3 (3.2)–(3.3). Thus the compactification torus must not be assigned $\tau_{\rm eff}$ merely because that parameter described the original normalized curve.

A locally constant $A_y$ has zero field strength but can have observable holonomy. A field of electric charge $p$ obeys

$$D_y=\partial_y-ip A_y,\qquad m_n^2=M^2+\frac1{R^2}\left(n-\frac{p\theta_e}{2\pi}\right)^2. \tag{X05.3}$$

$M$ is its four-dimensional mass. Under $\theta_e\mapsto\theta_e+2\pi$, the whole tower is unchanged by relabeling $n\mapsto n+p$. A single selected mode need not be unchanged. This is why a periodic Wilson line is not globally an unrestricted real scalar.

Keep the theta term during reduction

For the calculation use Euclidean $\mathbb R^3\times S^1$ with orientation $dx^1dx^2dx^3dy$. The site’s continued theta term has negative imaginary sign. In the units (X05.2), the local two-derivative action is

$$S_4=\frac1{4\pi}\int \left[t\,da_e\wedge *_4d\bar a_e +t F\wedge *_4F-is F\wedge F\right]. \tag{X05.4}$$

For example, $F=dv/2$ and $t=2\operatorname{Im}\tau_{\rm eff}$ recover the photon coefficient $\operatorname{Im}\tau_{\rm eff}(dv)_{\mu\nu}^2/(16\pi)$. There is no extra matrix trace in this Abelian formula. Functions $s,t$ may depend on $a_e$; we do not differentiate them away or replace $\tau_c$ by $a_m/a_e$.

With $F=F_3+L^{-1}d\theta_e\wedge dy$, the mixed term is $F\wedge F=2L^{-1}F_3\wedge d\theta_e\wedge dy$. Integrating $y$ gives

$$\begin{aligned} S_3=\int\Bigl[& \frac{Lt}{4\pi}da_e\wedge *_3d\bar a_e +\frac{Lt}{4\pi}F_3\wedge *_3F_3\\ & +\frac{t}{4\pi L}d\theta_e\wedge *_3d\theta_e -\frac{is}{2\pi}F_3\wedge d\theta_e\Bigr]. \end{aligned} \tag{X05.5}$$

The Maxwell coefficient defines $g_{3,A}^2=2\pi/(Lt)=g_{4,A}^2/L$, where $g_{4,A}^2=2\pi/t$. Consequently $[g_{3,A}^2]=1$. These are couplings for $A=v/2$, not for a silently rescaled unit-W connection.

Replace the remaining photon by its Fourier angle

Treat $F_3$ as independent and add $i\int F_3\wedge d\theta_m/(2\pi)$. Variation of $\theta_m$ imposes $dF_3=0$ locally. Integral magnetic flux must also be retained in the global construction. Its Fourier conjugate is an angle $\theta_m\sim\theta_m+2\pi$, with integer Fourier characters. For signs, integration by parts gives the kernel $\exp[i\int\theta_m dF_3/(2\pi)]$ in $e^{-S}$. To enforce positive oriented three-dimensional flux $dF_3=2\pi n\,\delta^{(3)}(x)\mathrm{vol}_3$, insert $e^{-in\theta_m}$ in that kernel. A smooth local equation alone would not fix the period.

Write $b=*_3F_3$ and $w=d\theta_m-s\,d\theta_e$. With vanishing boundary terms, the photon part is

$$\frac{Lt}{4\pi}b^2+\frac{i}{2\pi}b\cdot w =\frac{Lt}{4\pi}\left(b+\frac{i w}{Lt}\right)^2 +\frac{w^2}{4\pi Lt}. \tag{X05.6}$$

The Gaussian over real $b$ gives a positive last term. Its saddle is complex, $b=-iw/(Lt)$; this is not an equality of two real Euclidean classical fields. The local kinetic calculation does not by itself identify global path-integral normalizations or sum over every bundle.

In a local magnetic frame, $\theta_m$ can also be represented as the holonomy of the dual connection. This is the magnetic Wilson-line interpretation in GMN (2.32).

Define the target metric by $\mathcal L_3=\tfrac12G_{AB}\partial X^A\partial X^B$. The four scalar kinetic terms become

$$\boxed{ds_{\rm sf}^2 =R t\,|da_e|^2 +\frac{|d\theta_m-\tau_c d\theta_e|^2}{4\pi^2Rt}.} \tag{X05.7}$$

This is the semiflat metric: at each fixed Coulomb scalar, the two angular directions form a flat torus. It agrees with GMN (2.33)–(2.37) after the stated period and Euclidean-sign dictionary. The torus metric in the order $(\theta_e,\theta_m)$ has determinant $(4\pi^2R)^{-2}$, so its area for two $2\pi$ periods is $1/R$ in our metric convention. The radius dependence is physical; an unannounced factor of two in the sigma-model definition would change the reported area.

The angles are only local coordinates over the Coulomb branch. Electric–magnetic transition matrices also act on their period lattice. In general, constant affine shifts can accompany them; GMN section 2.3 explains why there need not be a globally preferred angular origin. Equation (X05.7) is unaffected by a constant shift. It does not assert a global product of a plane and a torus.

What eight supercharges require of the metric

A complex structure is a rule $I$ on tangent vectors with $I^2=-1$, generalizing multiplication by $i$ in a complex plane. A hyperkähler metric admits three metric-compatible, covariantly constant structures $I,J,K$ with $IJ=K=-JI$. This is a restriction on the kinetic geometry, not merely the statement that the dimension is divisible by four. Eight-supercharge sigma models of the present ungauged kind have this geometry; see GMN (3.1).

There is a concrete local check. Write $a_e=x+iy$ and use the orthonormal one-forms $e^1=\sqrt{Rt}\,dx$, $e^2=\sqrt{Rt}\,dy$, $e^3=\sqrt t\,d\theta_e/(2\pi\sqrt R)$, $e^4=(d\theta_m-sd\theta_e)/(2\pi\sqrt{Rt})$. The corresponding two-forms can be organized as

$$\begin{aligned} \omega_I&=Rt\,dx\wedge dy+\frac{d\theta_e\wedge d\theta_m}{4\pi^2R},\\ \omega_J+i\omega_K&=\frac{i}{2\pi}da_e\wedge(d\theta_m-\tau_c d\theta_e). \end{aligned} \tag{X05.8}$$

In the orthonormal frame these are the usual quaternionic forms. They are closed: $dt$ has only $dx,dy$ components, and holomorphy gives $da_e\wedge d\tau_c=0$. Notice that $d\theta_m-\tau_c d\theta_e$ alone is generally not closed. This checks the special geometry used in the semiclassical metric without assuming that a varying coupling gives a globally flat four-manifold.

Worked example: circle modes reveal the omitted scale

Consider one charged massive mode with the spectrum (X05.3). Its zero-three-momentum propagator summed over circle modes is

$$\mathcal G(M,\theta_e)=\sum_{n\in\mathbb Z} \frac1{M^2+(n-p\theta_e/(2\pi))^2/R^2} =\frac{\pi R}{M} \frac{\sinh(LM)}{\cosh(LM)-\cos(p\theta_e)}. \tag{X05.9}$$

One way to derive the equality is to Fourier transform the real-line propagator: the decaying Green function of $-\partial_y^2+M^2$ is $e^{-M|y|}/(2M)$. Summing its images separated by $L$, with phase $e^{iwp\theta_e}$, gives $L\sum_w e^{-M|w|L}e^{iwp\theta_e}/(2M)$. The positive and negative images pair, and their geometric series yields (X05.9). Therefore

$$\frac{\mathcal G}{\pi R/M} =1+2\sum_{w\geq1}e^{-wLM}\cos(wp\theta_e). \tag{X05.10}$$

This is an actual finite-radius effect and a check of the angular period. It is a propagator diagnostic, not the full interacting moduli-space metric. For $RM=1$ and $p\theta_e=\pi/2$, the first image cancels and the ratio is $\tanh(2\pi)=0.9999930253$. The remaining effect has not vanished. After keeping images through $w=W$, the absolute error in this ratio is at most $2e^{-(W+1)LM}/(1-e^{-LM})$.

In the interacting theory an existing four-dimensional BPS particle can wrap the circle. Its Euclidean worldline action has real part $LM_\gamma$ and angular phase $p\theta_e+q\theta_m$. This explains the suppression $e^{-2\pi R M_\gamma}$ discussed in GMN’s introduction. The positive magnetic character here describes a particle wrapping the positive circle. It is consistent with the flux-insertion sign above: with $\mathrm{vol}_4=\mathrm{vol}_3\wedge dy$, its oriented current is $J_\ell=-\delta^{(3)}(x)\mathrm{vol}_3$, defined by $\int\alpha\wedge J_\ell=\int_\ell\alpha$. Thus $dF=2\pi qJ_\ell$ has transverse flux $n=-q$, and $e^{-in\theta_m}=e^{iq\theta_m}$. Charge and orientation must travel together when comparing a wrapped particle with a three-dimensional monopole insertion. The exact prefactors and combined corrections require the BPS spectrum; the charge formula alone does not manufacture particles.

For a useful diagnostic, the prerequisite’s regular weak-chamber point $u=3\Lambda^2$ has $M_m/\Lambda=1.28197599566$ and $M_W/\Lambda=3.43938640041$. A radius that suppresses the W contribution strongly need not suppress the lighter monopole equally strongly. At $R\Lambda=0.8$, the corresponding one-wrapping exponential factors are approximately $0.00159017$ and $3.10322\times10^{-8}$. These are weights, not bounds on the complete metric correction including its prefactors and other sectors. The reduction of the four-dimensional low-energy action is controlled when $E\ll1/R$, $E\ll M_{\rm charged}$ and $RM_{\rm charged}\gg1$ for the omitted lightest charged sectors. Near a massless monopole the last condition fails, even at a large fixed $R$. The semiflat singularity then signals missing physics, not a license to discard that state.

Taking $R\to0$ is a different limit: at fixed four-dimensional coupling, $g_{3,A}^2$ diverges, and wrapped corrections cease to be small. A genuine three-dimensional limit must specify the coupling and scale matching. It is not obtained by claiming that (X05.7) remains the exact finite-radius metric. The detailed corrected geometry is a further problem.

Optional comparison with the elementary three-dimensional Coulomb patch

This panel uses the SQED dual photon. At $s=0$, (X05.6) gives $g_{3,A}^2(d\theta_m)^2/(8\pi^2)$, exactly its Maxwell dual coefficient. That lesson used $+i\int\gamma\,dF/(2\pi)$, while here $+i\int F\wedge d\theta_m/(2\pi)=-i\int\theta_m\,dF/(2\pi)$; thus $\theta_m=-\gamma$ for identical flux orientation. Both angles have period $2\pi$.

A generic three-dimensional N=2 vector has only one real vector scalar before dualization. Four real supercharges do not impose the same hyperkähler geometry or protect the same Coulomb directions. For example, AHISS (6.2) gives a monopole-generated $W\propto1/Y$ in pure N=2 $SU(2)$, with no finite supersymmetric vacuum. Here $Y$ is the complex Coulomb coordinate of that different theory. This does not describe the eight-supercharge, undeformed compactification studied above.

Optional orientation: the physical solution and the topological theory

The physical Seiberg–Witten solution gives low-energy couplings and BPS central charges. A topological twist instead combines spacetime rotation and R-symmetry assignments so that a supercharge becomes a scalar on an oriented four-manifold. Selected correlation functions can then define four-manifold invariants. Witten’s original introduction explains how physical infrared information enters this different construction. Neither that twist nor the curved reduction used to obtain three-dimensional rigid backgrounds is the ordinary periodic spatial compactification performed here.

Exercises

1. Can the Wilson-line period be halved?

At $\theta_e=\pi/3$, compare $\theta_e$ and $\theta_e+\pi$ using the full W tower and a fundamental Wilson probe. Is their equality in one test enough to make this a gauge identification in the stated $SU(2)$ theory?

Hint 1

The W has $p=2$, while the fundamental holonomy has eigenvalues $e^{i\theta_e}$ and $e^{-i\theta_e}$.

Hint 2

A gauge transformation must preserve every allowed probe, not only the dynamical adjoint spectrum.

Solution

The W shift changes $p\theta_e/(2\pi)$ by one, absorbed by $n\mapsto n+1$ in (X05.3). Its complete tower is unchanged. The ordinary fundamental trace is $2\cos\theta_e$: it is $1$ at $\pi/3$ and $-1$ at $4\pi/3$. These holonomies therefore cannot be gauge equivalent in $SU(2)$. Declaring the center-valued transformation to be a redundancy changes the global theory and its allowed probes. No rescaling of a local kinetic term settles that choice. The Weyl transformation $\theta_e\mapsto-\theta_e$ preserves the trace but does not turn these two particular conjugacy classes into one.

2. Patch the two angles with the periods

The inherited monopole monodromy acts by $a_D'=a_D$, $a'=a-2a_D$. Find its action in (X05.2)‘s units and show that (X05.7) agrees in the two frames. Track a charge $(p,q)$ and its angular phase. Work with local angular origins; constant affine offsets can be added separately.

Hint 1

Both columns $(a_m,a_e)^T$ and $(\theta_m,\theta_e)^T$ transform by the same integral matrix. The charge row in this order is $(q,p)$.

Hint 2

Compute $da_e'$ and $d\theta_m'-\tau_c'd\theta_e'$ before taking norms.

Solution

The transformation is $a_m'=a_m$, $a_e'=a_e-a_m$, and $\theta_m'=\theta_m$, $\theta_e'=\theta_e-\theta_m$. Hence

$$\begin{gathered} \tau_c'=\frac{\tau_c}{1-\tau_c},\qquad t'=\frac{t}{|1-\tau_c|^2},\qquad da_e'=(1-\tau_c)da_e,\\ d\theta_m'-\tau_c'd\theta_e' =\frac{d\theta_m-\tau_c d\theta_e}{1-\tau_c}. \end{gathered}$$

The factors cancel in both metric terms. The angle matrix has determinant one and integral entries, so it preserves the $2\pi$ lattice. Inverse transformation of the row gives $p'=p$, $q'=q+p$. Then $p'a_e'+q'a_m'=pa_e+qa_m$ and $p'\theta_e'+q'\theta_m'=p\theta_e+q\theta_m$. The W row $(2,0)$ becomes $(2,2)$ in these coordinates; no new particle has been inferred. An arbitrary real linear transformation could preserve a quadratic form locally while failing this global lattice check.

3. Change the boundary condition and the limit

Suppose the fermions are instead antiperiodic around the same spatial circle, with no compensating R-symmetry source. Does the eight-supercharge conclusion survive? Separately, in the periodic theory, how must $g_{4,A}^2$ scale with $R$ to keep a chosen $g_{3,A}^2$ fixed? Explain why neither operation is justified by a large-$R$ metric formula alone.

Hint 1

The flat supersymmetry parameters are neutral constant spinors. Compare their boundary condition with that of the gauginos.

Hint 2

Use $g_{3,A}^2=g_{4,A}^2/(2\pi R)$, then inspect $e^{-2\pi RM}$ at fixed nonzero $M$ as $R\to0$.

Solution

Antiperiodic fermions have momenta $(n+1/2)/R$. A nonzero constant supersymmetry parameter cannot be antiperiodic, so the stated flat background preserves no such supercharge. Bosonic zero modes by themselves do not establish an N=4 effective theory; their quantum potential need no longer vanish. This is a boundary-condition change, not a relabeling of N.

In the periodic case, fixed $g_{3,A}^2$ requires $g_{4,A}^2=2\pi Rg_{3,A}^2$, equivalently $t=1/(R g_{3,A}^2)$. For an interacting four-dimensional theory this also requires specifying the matching scale and its running parameters; it is not a statement that a fixed microscopic theory has constant coupling at every scale. Meanwhile $e^{-2\pi RM}\to1$ for fixed $M$. Thus the large-radius suppression that supported the semiflat approximation is lost. Periodic reduction still supplies the multiplet and supersymmetry count, but not the full small-radius quantum metric.

Compactified Coulomb branches and their geometry follows the light-particle correction beyond the semiflat approximation. Sum its momentum tower, recover the periodic fiber connection, and check the local smooth center before interpreting the result inside the full theory.