Supersymmetric Parents of Mirror, Particle–Vortex, and Bosonization Webs

Supersymmetric mirror and Chern–Simons dualities organize much of the modern three-dimensional particle–vortex and bosonization web. They supply exact charge maps, quantized contact terms, and controllable massive phases. They do not by themselves prove that a supersymmetry-breaking critical point survives with no intervening phase. A sound descendant claim therefore keeps the protected parent data and labels the additional nonsupersymmetric infrared assumption.

Required background. We use the full dictionaries of Aharony and Giveon–Kutasov dualities and their real-mass, FI, and compactification flows. Helpful background. The evidentiary status of each step should be assessed using duality checks and failure modes.

From a supersymmetric parent to a descendant

A protected parent duality consists of two theories $A$ and $B$ , a symmetry and operator dictionary, background contact terms, and a map of supersymmetric deformations. To obtain a nonsupersymmetric descendant one adds corresponding relevant operators,

\delta\mathcal L_A=\sum_a g_a\mathcal O_a, \qquad \delta\mathcal L_B=\sum_a g'_a\mathcal O'_a,

and separately tunes the scalar and fermion masses that supersymmetry formerly related. The infrared claim requires more than the parent equality:

both flows must land on continuous transitions rather than first-order surfaces;
no extra relevant singlet or accidental symmetry may appear on only one side;
all massive chambers must have matching topological order and response;
the operator identified as the order parameter or monopole must retain the required statistics and global charges.

Supersymmetry controls the starting multicritical point and the deformation map. Once it is broken, equality of critical exponents and correlators is a new dynamical conjecture. This distinction is explicit in derivations of nonsupersymmetric dualities from simple $\mathcal N=2$ mirror pairs Kachru et al. 2017, pp. 1–4.

S and T operations on a U(1) current

Many Abelian webs can be generated without rederiving every Lagrangian. Let a theory have a compact background field $A$ for a $U(1)$ current. Define:

$T$ : add $\operatorname{CS}[A]=\frac{1}{4\pi}\int A\wedge dA$ ;
$S$ : promote $A$ to a dynamical field $a$ and couple the new background $B$ through $\operatorname{BF}[a;B]=\frac{1}{2\pi}\int a\wedge dB$ .

Applying the same operation to both sides of a valid duality produces another candidate duality. The operations act on the integral charge lattice and exchange ordinary with topological currents. Their formal $SL(2,\mathbb Z)$ relations hold only up to charge conjugation and possible invertible spin theories. If $T$ adds an odd level, spin dependence must be recorded; if $S$ gauges a quotient rather than the full $U(1)$ , the monopole and line spectra change.

This calculus explains why a single bosonization seed generates particle–vortex, fermion–fermion, and boson–boson relations Karch and Tong 2016, §§2–3. It is a transformation of a complete generating functional, not merely a manipulation of local equations of motion.

A regulated Abelian bosonization seed

Write $\operatorname{CS}[A]=\frac{1}{4\pi}\int A\wedge dA$ and $\operatorname{BF}[b;A]=\frac{1}{2\pi}\int b\wedge dA$ . Define $\mathcal L_f[A]$ to be a single charge-one Dirac fermion together with the parity-anomaly counterterm $-\frac12\operatorname{CS}[A]$ . A representative seed, in this regulator convention and orientation, is

\mathcal L_f[A] \quad\longleftrightarrow\quad |D_b\phi|^2-r|\phi|^2-u|\phi|^4 +\operatorname{CS}[b]+\operatorname{BF}[b;A],

with $u$ tuned to the Wilson–Fisher critical surface. The signs of all Chern–Simons terms reverse under orientation reversal; changing the fermion regulator adds the corresponding integral background counterterm. The formula is shorthand for equality in the infrared after the mass parameters are mapped with the convention-dependent sign Seiberg et al. 2016, §§2.1–2.2.

The current map follows by varying $A$ :

j_f^\mu \longleftrightarrow \frac{1}{2\pi}\epsilon^{\mu\nu\rho}\partial_\nu b_\rho.

The fermion is therefore represented by a monopole operator of the bosonic gauge theory, with its spin and statistics supplied by the level-one Chern–Simons coupling. Saying only “fermion equals vortex” omits this statistical transmutation and the background contact term.

The two massive phases

The seed passes a decisive response check. With the chosen regulator, integrating out the fermion gives background levels

k_A^{\rm eff}=0\quad(M>0), \qquad k_A^{\rm eff}=-1\quad(M<0).

On the bosonic side, one sign of $r$ condenses $\phi$ and Higgses $b$ , leaving zero background response. The other leaves the scalar massive and the $U(1)_1$ Chern–Simons field unHiggsed. Integrating out $b$ in

\operatorname{CS}[b]+\operatorname{BF}[b;A]

gives $-\operatorname{CS}[A]$ . Thus the two gapped chambers match after choosing the mass-sign map. This phase agreement is necessary and highly nontrivial, but it does not prove that the separating transitions are the same interacting CFT.

Applying $S$ gauges $A$ and produces a particle–vortex form in which an ordinary particle current on one side becomes flux on the other. Applying $T$ before or after $S$ yields fermionic particle–vortex and QED $_3$ variants. At every step, Gaussian integration of the Abelian Chern–Simons matrix checks the new background response.

Supersymmetric routes into the web

There are two especially useful parents.

Mirror parents. $\mathcal N=4$ mirror symmetry already exchanges flavor particles with vortices and masses with FI parameters. Breaking to $\mathcal N=2$ and then adding soft scalar/fermion masses yields multicritical phase diagrams. In the simplest $\mathcal N=2$ mirror pairs, a free chiral description is dual to an Abelian gauge description with a charge-one chiral and the regulator-required half-level contact term. Separating the superpartners’ masses produces scalar QED, a free Dirac fermion, and Wilson–Fisher sectors in different chambers Kachru et al. 2017, pp. 1–4.

Chern–Simons parents. Giveon–Kutasov and related level/rank dualities contain both gauge rank exchange and fundamental matter. Massing superpartners in correlated ways motivates non-Abelian bosonization: roughly, a critical scalar coupled to one Chern–Simons theory is paired with a critical fermion coupled to the level/rank-dual theory. The precise finite-rank formula depends on whether the gauge group is $SU(N)$ or $U(N)$ , on its two possible $U(N)$ levels, and on the fermionic half-level convention. Large- $N$ correlators give strong evidence for this family Aharony, Gur-Ari, and Yacoby 2012, §§1 and 5.2, but should not be used to erase finite-rank global-form qualifications.

In both routes, the supersymmetric parent determines which background terms accompany the light nonsupersymmetric fields. It does not determine the sign of every radiative scalar quartic after supersymmetry breaking; that coupling must lie in the basin of the proposed critical point.

Spin, spin-c, transparent sectors, and boundaries

A single Dirac fermion is a spin theory. In electronic applications, the electromagnetic background is often a spin $_c$ connection, and the combination of fermion determinant and half-level counterterm can be defined without choosing an independent spin structure. These are different global formulations. A symbol such as $U(1)_{1/2}$ is meaningful only after the relevant spin/spin $_c$ prescription is given.

Integrating out matter can also leave an invertible spin topological field theory or a transparent fermion. Such a sector has no nontrivial bosonic local operator, yet changes the partition-function phase, chiral central charge, and line category. It must be included when comparing time reversal or gravitational response.

With a boundary, bulk Chern–Simons terms produce anomaly inflow. A boundary version of the duality requires matched boundary conditions and edge degrees of freedom; the closed-manifold relation cannot simply be restricted to a half-space. On unorientable spacetimes, a further Pin refinement is needed.

Evidence levels and failure modes

A responsible descendant claim separates the following:

exactly inherited: charge lattices, quantization laws, and algebraic $S/T$ operations once the seed is assumed;
protected in the parent: supersymmetric indices, chiral rings, localized partition functions, and BPS deformation maps;
strong consistency checks: matching massive phases, Hall/gravitational response, anomalies, and operator quantum numbers;
dynamical evidence: large- $N$ correlators, expansions, lattice results, and numerical bootstrap;
conjectural core: equality of the nonsupersymmetric infrared fixed points at finite rank.

Likely failure modes include a first-order transition, an intervening symmetry-breaking phase, a dangerously irrelevant coupling, an accidental symmetry, or a missed topological sector. The relativistic web also does not by itself fix a microscopic condensed-matter realization; lattice symmetries, electromagnetic normalization, filling, and disorder can change the application.

Common pitfalls

Manipulating half-level terms as standalone bosonic actions. They are shorthand for a regulated fermion system or a spin/spin $_c$ refinement. Keep the determinant and counterterm together.

Calling a phase match a proof of critical duality. Matching both adjacent TQFTs and responses is necessary, not sufficient. The transition can still split or become first order.

Gauging a symmetry without changing the operator spectrum. The $S$ operation turns background flux into a new conserved charge and introduces new monopoles and lines. Recompute global form and genuine operators.

Exercises

Integrate out $b$ from $\operatorname{CS}[b]+\operatorname{BF}[b;A]$ and derive the background response $-\operatorname{CS}[A]$ .

Solution

The topological equation of motion is $db=-dA$ . Completing the square at the level of the Abelian $K$ matrix, with $K=1$ and charge vector $q=1$ , gives the induced response $-q^2K^{-1}\operatorname{CS}[A]=-\operatorname{CS}[A]$ . This statement includes the appropriate global sum over bundles; a local substitution $b=-A$ is only a mnemonic.

Why does applying $S$ twice give charge conjugation rather than the identity on a theory with a $U(1)$ current?