Yang–Mills and Chern–Simons–Matter Actions

A three-dimensional $\mathcal N=2$ gauge theory is specified not only by a local Lagrangian but also by a compact global gauge group, a lattice of allowed Chern–Simons and BF couplings, matter representations, a superpotential, and the required tangential structure—usually spin, or a specified spin$_c$ formulation when the charges permit it. This page assembles those ingredients in one convention and derives the scalar potential and representative BPS equations that will be used in phase and duality checks.

Required background. We use the three-dimensional multiplets, real masses, FI terms, and topological current and the global quantization conditions for Chern–Simons actions. Helpful background. The relation between superspace terms and components is developed in action principles and component reduction.

Local terms and normalization

Work on an oriented Lorentzian spin three-manifold with signature $(+--)$ and $\epsilon^{012}=+1$. Gauge fields and Lie-algebra generators are Hermitian; for $SU(N)$, $\operatorname{Tr}_{\boldsymbol N}(T^aT^b)=\frac12\delta^{ab}$. Locally we use the connection normalization

$$D_\mu=\partial_\mu-i\rho(A_\mu), \qquad F=\mathrm dA-iA\wedge A,$$

so for charge-$q$ Abelian matter $D_\mu=\partial_\mu-iq a_\mu$. The coupling $e$ multiplies the vector-multiplet kinetic term instead of appearing in $D_\mu$.

Before writing it, fix the following theory card. Two cards with the same Lie algebra and polynomial Lagrangian but different entries can define inequivalent quantum theories.

datum	what must be specified
gauge sector	compact global group, allowed bundles, trace or Abelian charge lattice, and genuine line operators
matter	representations, flavor charges, multiplicities, and reality properties
supersymmetric couplings	Yang–Mills couplings, the complete Chern–Simons/BF bilinear form, FI terms, real masses, and $W$
background response	flavor, $R$, mixed, and gravitational counterterms in one regulator convention
global definition	spin or spin$_c$ prescription, orientation, and any transparent invertible sector
boundary data	closed manifold, or explicit boundary conditions and anomaly-canceling boundary degrees of freedom

The first three rows determine the classical equations below. The remaining rows become indispensable when fermions are regulated, manifolds or bundles are topologically nontrivial, or two theories are compared by duality.

A Yang–Mills term for an $\mathcal N=2$ vector multiplet has bosonic part

$$\mathcal L_{\mathrm{YM,bos}} =\frac{1}{e^2}\operatorname{Tr}\!\left( -\frac14F_{\mu\nu}F^{\mu\nu} +\frac12D_\mu\sigma D^\mu\sigma +\frac12D^2\right).$$

Here $[e^2]=1$ in mass units. The omitted gaugino terms and Yukawa couplings are fixed by supersymmetry. Yang–Mills interactions are therefore superrenormalizable and usually serve as an ultraviolet regulator for an infrared Chern–Simons–matter fixed point.

Because the gauge coupling is outside the kinetic term and not inside $D_\mu$, this convention has

$$[A_\mu]=[\sigma]=1, \qquad [\lambda]=\frac32, \qquad [D]=2, \qquad [e^2]=1.$$

Consequently $k$ and $k_{BF}$ are dimensionless, while $m_i$ and $\zeta$ have dimension one. These assignments check every term displayed below. A canonical rescaling of the vector multiplet changes the field dimensions and moves powers of $e$ into covariant derivatives and interactions, but cannot change a quantized level.

For a compact charge-one $U(1)$ connection,

$$\mathcal L_{\mathrm{CS}} =\frac{k}{4\pi}\epsilon^{\mu\nu\rho}a_\mu\partial_\nu a_\rho +\frac{k}{2\pi}D\sigma +\mathcal L_{\lambda\lambda}.$$

The precise sign of the gaugino mass follows the fermion convention; the displayed bosonic $D\sigma$ term fixes the sign used below. For a simple non-Abelian factor the first term becomes

$$\frac{k}{4\pi}\epsilon^{\mu\nu\rho} \operatorname{Tr}\!\left( A_\mu\partial_\nu A_\rho -\frac{2i}{3}A_\mu A_\nu A_\rho\right).$$

The integer called “$k$” depends on this trace and on the global group, not just its Lie algebra. For quotient groups and product groups, allowed invariant bilinear forms obey additional integrality conditions. A half-integral expression can define a spin Chern–Simons theory only when the spin dependence and fermion regulator are included; it is not an ordinary bosonic level Closset et al. 2012, §§2–3.

For two Abelian vector multiplets $V_a,V_b$, the mixed coupling is normalized by

$$\mathcal L_{\mathrm{BF,bos}} =\frac{k_{BF}}{2\pi}\left( \epsilon^{\mu\nu\rho}a_\mu\partial_\nu b_\rho +D_a\sigma_b+D_b\sigma_a\right).$$

With both gauge fields compact and minimally normalized, $k_{BF}\in\mathbb Z$. Taking $b$ to be a background field couples it to the topological current of $a$.

Matter, F-terms, and D-terms

Let chirals $\Phi_i$ have $U(1)$ charges $q_i$, background real masses $m_i$, and scalar components $\phi_i$. The relevant bosonic terms are

$$\begin{aligned} \mathcal L_{\mathrm{matter,bos}} =\sum_i\bigl[&D_\mu\phi_i^\dagger D^\mu\phi_i +F_i^\dagger F_i+q_iD|\phi_i|^2\\ &-(q_i\sigma+m_i)^2|\phi_i|^2\bigr]. \end{aligned}$$

The physical scalar potential is minus the nondynamical part of the Lorentzian Lagrangian. A holomorphic, gauge-invariant superpotential contributes

$$\mathcal L_W=\sum_i F_i\,\partial_iW -\frac12\sum_{i,j}\psi_i\psi_j\,\partial_i\partial_jW +\text{h.c.}$$

up to the convention used to order Grassmann fields. Supersymmetry requires $R(W)=2$. At a classically scale-invariant fixed-point presentation, $[W]=2$ because $[d^2\theta]=1$ in three dimensions.

For one $U(1)$ factor, add an FI coupling $-\zeta D$ and the Chern–Simons term above. Eliminating $F_i$ and $D$ gives

$$V=\sum_i|\partial_iW|^2 +\sum_i(q_i\sigma+m_i)^2|\phi_i|^2 +\frac{e^2}{2}\left( \sum_iq_i|\phi_i|^2-\zeta+\frac{k}{2\pi}\sigma \right)^2.$$

This component structure and the resulting quantum Coulomb-branch problem are standard features of three-dimensional $\mathcal N=2$ gauge dynamics Aharony et al. 1997, §§2–3, pp. 3–18. The formula fixes several later signs. In particular, a supersymmetric vacuum must satisfy

$$\partial_iW=0, \qquad (q_i\sigma+m_i)\phi_i=0, \qquad \sum_iq_i|\phi_i|^2-\zeta+\frac{k}{2\pi}\sigma=0.$$

For a non-Abelian group these are Lie-algebra-valued moment-map equations, with $\sigma$ acting in each representation and with one equation per generator. One must still quotient the solutions by the compact gauge group. Solving only the polynomial equations overcounts gauge-equivalent vacua.

Worked phase analysis

Consider $U(1)$ with two charge-$+1$ chirals, $W=0$, vanishing real masses, and $k=0$. This matter content satisfies the parity-anomaly condition because $\frac12\sum_iq_i^2=1$. The potential is

$$V=\sigma^2\bigl(|\phi_1|^2+|\phi_2|^2\bigr) +\frac{e^2}{2}\bigl(|\phi_1|^2+|\phi_2|^2-\zeta\bigr)^2.$$

If $\zeta>0$, the classical Higgs locus has $|\phi_1|^2+|\phi_2|^2=\zeta$ and $\sigma=0$; quotienting by $U(1)$ gives $\mathbb{CP}^1$, and the gauge group is Higgsed. If $\zeta<0$, the D-term equation has no solution, so the model has no supersymmetric vacuum at finite fields. At $\zeta=0$, $\phi_1=\phi_2=0$ leaves $\sigma$ classically unconstrained. Quantum effects and monopole operators determine whether that apparent Coulomb direction survives. This elementary example shows why both the parity-anomaly condition and the sign convention for $-\zeta D$ must be carried into every deformation map.

With $k\neq0$ and $\phi_1=\phi_2=0$, the same equation instead fixes $\sigma=2\pi\zeta/k$. The Chern–Simons term lifts the continuous Coulomb direction and makes a bare flux insertion electrically charged.

Vortex equations and the FI central charge

In the Higgs phase of the preceding $k=0$, $\zeta>0$ model, choose the consistent one-component vortex embedding $\phi_2=0$, write $\phi=\phi_1$, and take static configurations on $\mathbb R^2$. Eliminating $D$ and completing squares gives, for a consistent correlated choice of signs,

$$E=\int d^2x\left[ |(D_1\pm iD_2)\phi|^2 +\frac{1}{2e^2}\bigl(B\mp e^2(\zeta-|\phi|^2)\bigr)^2 \right] \pm\zeta\int d^2x\,B.$$

Thus

$$E\ge 2\pi|\zeta n|, \qquad n=\frac{1}{2\pi}\int_{\mathbb R^2}B\,d^2x\in\mathbb Z,$$

and equality requires

$$(D_1\pm iD_2)\phi=0, \qquad B=\pm e^2(\zeta-|\phi|^2).$$

The same integer $n$ is the topological charge. Mirror symmetry can therefore exchange a flavor-charged particle with a vortex, provided it also exchanges the associated real mass with the correctly normalized topological source $m_J=-2\pi\zeta$, rather than with an unlabeled FI coefficient Aharony et al. 1997, § 5, pp. 18–21.

N=4 constraints

An $\mathcal N=4$ vector written as an $\mathcal N=2$ vector $V$ plus an adjoint chiral $\Phi$ couples to a hypermultiplet $(Q,\widetilde Q)$ through

$$W=\widetilde Q\,\Phi Q$$

with normalization tied to the gauge coupling. Together with the D-term, its F-term equations form the triplet of hyperkähler moment maps. FI parameters and real masses likewise form triplets, exchanged together with the two $SU(2)$ R-symmetry factors under mirror symmetry Intriligator and Seiberg 1996, § 1, pp. 1–3. Keeping only the displayed $\mathcal N=2$ pieces is useful for calculations, but assigning arbitrary independent coefficients generally breaks $\mathcal N=4$.

Ordinary $\mathcal N=2$ Chern–Simons couplings give the vector multiplet a topological mass. Preserving more supersymmetry requires extra multiplets and correlated interactions; one cannot obtain a generic $\mathcal N=4$ Chern–Simons theory merely by appending the $\mathcal N=2$ term to an $\mathcal N=4$ Yang–Mills Lagrangian.

Euclidean action and boundary qualifications

Under Wick rotation, the positive Yang–Mills and scalar kinetic terms enter $e^{-S_E}$ with positive real part, whereas a real Lorentzian Chern–Simons coupling becomes a phase,

$$S_{E,\mathrm{CS}}=i\frac{k}{4\pi}\int a\wedge da.$$

Barred and unbarred fermions are independent complex variables, and localization often uses a complex contour for $D$. A Euclidean saddle equation is therefore meaningful only together with its contour; it should not be read as a new Lorentzian reality condition.

On a manifold with boundary, the variation of the Chern–Simons action produces a boundary term. Gauge invariance then requires a boundary condition and possibly anomalous boundary degrees of freedom. The closed-manifold level conditions stated here are necessary but not sufficient boundary data.

Common pitfalls

Quoting a level without a trace. $SU(N)_k$, $U(N)_k$, and quotient groups do not share one universal integer convention. State the global group, trace, Abelian factors, and spin dependence.

Eliminating $D$ with an inconsistent FI sign. Derive its algebraic equation from the displayed Lagrangian. A sign error reverses the Higgs chamber and corrupts all subsequent real-mass flows.

Treating Yang–Mills and Chern–Simons terms as equally important in the IR. $e^2$ is dimensionful, whereas $k$ is quantized and dimensionless. Yang–Mills commonly regulates a fixed point whose topological data retain the Chern–Simons level.

Exercises

1. Starting from the $D$-dependent terms

$$\frac{1}{2e^2}D^2+D\left(\sum_iq_i|\phi_i|^2-\zeta+\frac{k}{2\pi}\sigma\right),$$

eliminate $D$ and verify the last term in the potential.

Solution

The equation is $D=-e^2X$, where $X$ is the expression in parentheses. Substitution gives $\mathcal L_D=-e^2X^2/2$. Since nondynamical potential terms enter the Lorentzian Lagrangian as $-V$, this yields $V_D=e^2X^2/2$.

2. Let $a$ and $b$ be compact $U(1)$ connections and take $S_{BF}=\frac{k_{BF}}{2\pi}\int a\wedge db$. Under a large gauge transformation of $a$, explain why $k_{BF}$ must be integral.

Solution

Represent the large transformation by a closed one-form $d\lambda$ whose periods are $2\pi\mathbb Z$. Because $db/(2\pi)$ is an integral cohomology class, $\Delta S_{BF}=2\pi k_{BF}n$ for some $n\in\mathbb Z$. The phase $e^{iS}$ is invariant for all bundles precisely when $k_{BF}\in\mathbb Z$.

3. Return to the two-charge-one-chiral model, but take an allowed nonzero integer level $k$. Solve the classical vacuum equations for vanishing real masses and $W=0$. Which vacua exist for positive and negative $\zeta$?

Solution

Writing $R^2=|\phi_1|^2+|\phi_2|^2$, the equations are

$$\sigma\phi_i=0, \qquad R^2-\zeta+\frac{k}{2\pi}\sigma=0.$$

There is always an isolated classical solution

$$\phi_1=\phi_2=0, \qquad \sigma=\frac{2\pi\zeta}{k}.$$

For $\zeta>0$ there is also a Higgs branch with $\sigma=0$, $R^2=\zeta$, whose gauge quotient is $\mathbb{CP}^1$. For $\zeta<0$ that Higgs branch is absent. At $\zeta=0$ the two loci meet at the origin. This is only the classical answer: on the isolated locus the signs of the chiral masses are set by $\sigma$, so the next page must be used to replace the bare level by the correctly regulated effective level.

References

• Aharony, O., Hanany, A., Intriligator, K., Seiberg, N., and Strassler, M. J. (1997), “Aspects of $N=2$ Supersymmetric Gauge Theories in Three Dimensions,” Nuclear Physics B 499, 67–99. doi:10.1016/S0550-3213(97)00323-4. Open PDF
• Closset, C., Dumitrescu, T. T., Festuccia, G., Komargodski, Z., and Seiberg, N. (2012), “Comments on Chern–Simons Contact Terms in Three Dimensions,” Journal of High Energy Physics 2012(09), 091. doi:10.1007/JHEP09(2012)091. Open PDF
• Intriligator, K., and Seiberg, N. (1996), “Mirror Symmetry in Three Dimensional Gauge Theories,” Physics Letters B 387, 513–519. doi:10.1016/0370-2693(96)01088-X. Open PDF

Next steps

The local action is consistent only after the fermion regulator satisfies the parity-anomaly and contact-term constraints. Once those are fixed, flux boundary conditions define monopole operators and quantum Coulomb coordinates.