Coulomb Branches and the Abelian Low-Energy Theory

At a generic point of a rank-$r$ $\mathcal N=2$ Coulomb branch, the nonabelian gauge group is broken to $U(1)^r$. The light two-derivative Wilsonian theory contains $r$ abelian vector multiplets. Its couplings vary holomorphically over the branch, but a local electric description fails wherever an omitted charged state becomes massless.

Required background. $\mathcal N=2$ multiplets and branches supplies the microscopic fields, and singular loci and light degrees of freedom supplies the EFT diagnostic. Helpful background. Moduli-space metrics and corrections distinguishes the classical quotient metric from the quantum one.

Abelianization at a regular point

Let the adjoint scalar expectation value be $a\in\mathfrak t_{\mathbb C}$. A root vector $W_\alpha$ has mass set by

$$M_{W_\alpha}\propto|\alpha(a)|,$$

with the proportionality fixed by the gauge-field normalization. Away from root hyperplanes and matter mass loci, every non-Cartan field is massive. At energies

$$E\ll M_{\mathrm{charged}}(u),$$

integrate those fields out and retain $r$ abelian $\mathcal N=2$ vector multiplets.

Gauge-invariant coordinates $u_k$ on the Coulomb branch are not generally the scalar fields with canonical abelian kinetic terms. On a local patch choose special coordinates $a^I(u)$, $I=1,\ldots,r$, adapted to an electric polarization. Their magnetic duals are $a_{D,I}(u)$.

The condition “regular point” includes more than a smooth $u$ coordinate: the discriminant is nonzero, the chosen light-field set is complete, and $\operatorname{Im}\tau_{IJ}$ is positive.

The two-derivative Wilsonian action

In $\mathcal N=1$ superspace, combine each abelian vector $V^I$ with a chiral field $A^I$. A local holomorphic prepotential $\mathcal F(A)$ gives

$$a_{D,I}=\frac{\partial\mathcal F}{\partial a^I}, \qquad \tau_{IJ}=\frac{\partial^2\mathcal F} {\partial a^I\partial a^J}.$$

One conventional normalization is

$$\mathcal L_{2} =\frac{1}{4\pi}\operatorname{Im}\!\left[ \int d^4\theta\, \frac{\partial\mathcal F(A)}{\partial A^I} \overline{A^I} +\frac12\int d^2\theta\, \tau_{IJ}(A)W^{I\alpha}W^J_\alpha \right].$$

It yields scalar metric

$$g_{I\bar J}=\operatorname{Im}\tau_{IJ}$$

and Maxwell/theta couplings from the real and imaginary parts of $\tau_{IJ}$. The factor $1/(4\pi)$ and the ordering of $a_D,a$ are conventions; period, charge, and action normalizations must be translated together.

The prepotential controls the Wilsonian two-derivative vector-multiplet action. It does not encode generic four-derivative interactions, the complete BPS spectrum, hypermultiplet metrics, or the theory exactly at a singular point. The abelian action and its special-coordinate interpretation are developed in Seiberg and Witten 1994, §2.3.

Wilsonian versus 1PI descriptions

Choose a Wilsonian scale $\mu_W$ satisfying

$$E\ll\mu_W\ll M_{\mathrm{charged}}.$$

Integrating out modes above $\mu_W$ produces a local derivative expansion. The holomorphic coupling in this action can be singular as $M_{\mathrm{charged}}\to0$ because the scale window collapses.

The 1PI action also integrates over massless photons and can contain nonlocal infrared terms. Calling every coupling “the effective prepotential” hides this difference. Seiberg–Witten geometry determines the exact two-derivative Wilsonian couplings on regular patches; infrared observables may require additional massless-loop resummation.

Central charges and light particles

Electric and magnetic charges form an integral lattice. In a local basis,

$$\gamma=(p^I,q_I;s_a),$$

where $s_a$ are flavor charges. The central charge is

$$Z_\gamma(u) =q_Ia^I(u)+p^Ia_{D,I}(u)+s_am^a,$$

and a BPS particle, if present, has

$$M_\gamma=|Z_\gamma|.$$

The exact periods make this an exact mass formula conditional on state existence and stability. The lattice alone does not assert a state for every $\gamma$. A textbook treatment of the $\mathcal N=2$ effective action, central charges, and duality is given in Weinberg 2000, chs. 27–29.

If $Z_\gamma(u_*)=0$, the corresponding BPS state can become massless. When all such charges are mutually local, choose a duality frame in which they are electric and include their hypermultiplets explicitly. The resulting local $\mathcal N=2$ QED description is regular even though the photon-only coupling has a logarithm.

One-loop asymptotics

For pure $SU(2)$ at large $|a|$, the charged $W$ multiplets give

$$\mathcal F_{\mathrm{1-loop}}(a) =\frac{i}{2\pi}a^2 \log\frac{a^2}{\Lambda^2}$$

up to convention-dependent quadratic terms and the normalization of $a$. Hence

$$a_D=\frac{\partial\mathcal F}{\partial a} \sim\frac{i}{\pi}a \left(\log\frac{a^2}{\Lambda^2}+1\right),$$

again modulo an allowed linear shift corresponding to a symplectic basis change. Instanton corrections are suppressed by powers of $(\Lambda/a)^4$.

The logarithm predicts monodromy around infinity. It does not locate the strong-coupling singularities by itself; global holomorphy, discrete symmetries, and the exact periods complete the solution.

Global form and charge normalization

The Lie algebra determines the local Coulomb-branch dimension, but the global gauge group determines allowed Wilson–’t Hooft charges. Choosing $SU(2)$ versus $SO(3)$ changes the genuine-line lattice and the subgroup of electromagnetic transformations acting within a fixed theory.

The low-energy $U(1)^r$ gauge fields therefore come with an integral polarization inherited from the ultraviolet theory. A real linear change of photon basis that does not preserve the lattice is not an allowed quantum duality frame.

Patch validity

A local abelian patch is valid when:

1. every omitted charged mass is larger than the Wilsonian scale;
2. all retained light charges are mutually local in the chosen polarization;
3. $\operatorname{Im}\tau$ is positive;
4. the derivative expansion is controlled;
5. the patch does not cross a branch intersection requiring hypermultiplet moduli;
6. period and charge bases are continued consistently.

Violating the second condition can signal a strongly interacting singularity rather than a missing weakly coupled field.

Common pitfalls

Using $u$ as if it were the special coordinate. In pure $SU(2)$, $u\sim a^2/2$ only asymptotically; the exact relation is corrected.

Calling the photon-only action singular physics. A divergent coupling often means a charged state that was integrated out has become massless.

Treating the prepotential as global. Monodromy mixes $a$ and $a_D$, so different patches can require different polarizations and prepotentials.

Exercises

Assume rank one with $a_D=\tau a$ and constant $\operatorname{Im}\tau>0$. For charge $(p,q)$:

1. write the central charge;
2. find the locus where it vanishes;
3. explain why this constant-coupling toy model cannot describe an isolated finite Coulomb-branch singularity unless $a=0$.

Solution

$$Z_{(p,q)}=(q+p\tau)a.$$

Since $p,q$ are real integers and $\operatorname{Im}\tau>0$, $q+p\tau$ cannot vanish for nonzero $(p,q)$. Thus $Z=0$ only at $a=0$. An isolated singularity at some other $u_*$ requires nontrivial variation and monodromy of the periods; a globally constant coupling cannot supply it.

References

• Seiberg, Nathan, and Edward Witten. “Electric–Magnetic Duality, Monopole Condensation, and Confinement in $\mathcal N=2$ Supersymmetric Yang–Mills Theory.” Nuclear Physics B 426 (1994): 19–52; erratum 430 (1994): 485–486. arXiv:hep-th/9407087.
• Weinberg, Steven. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge University Press, 2000, chs. 27–29. doi:10.1017/CBO9781139644198.