Gauge-Invariant Coordinates and Classical Moduli Varieties

Gauge-invariant operators turn a quotient of fields into an ordinary algebraic variety. Choose invariant generators $u_A(\phi)$ , impose both the F-term equations and every polynomial relation among the $u_A$ , and form the coordinate ring

\mathbb C[\mathcal M_{\mathrm{cl}}] =\left(\mathbb C[\phi]/I_F\right)^{G_{\mathbb C}} \simeq \mathbb C[u_1,\ldots,u_s]/I_{\mathrm{rel}}.

This description makes branches, intersections, and singular loci computable without choosing a gauge; its use for classical supersymmetric vacua is developed in Luty and Taylor 1996, pp. 3399–3405, arXiv:hep-th/9506098. It does not, by itself, retain stabilizers, distinguish all stability chambers, or determine the metric.

Required background. F- and D-flat quotients supplies the physical quotient, while constraints and reduction supplies the reduction logic. Helpful background. Branches and analytic continuation is useful for separating local coordinate patches from genuine components.

From fields to an invariant coordinate ring

For a reductive complexified gauge group, polynomial invariants are finitely generated. A practical construction has four layers:

Solve or quotient by the F-term ideal $I_F=(\partial_iW)$ .
Find invariant generators $u_A$ under $G_{\mathbb C}$ .
Determine the kernel of the map $\mathbb C[u_A]\to(\mathbb C[\phi]/I_F)^{G_{\mathbb C}}$ ; this kernel is the relation ideal $I_{\mathrm{rel}}$ .
Decompose the resulting ideal when the vacuum space has distinct irreducible branches, and record their intersections.

Completeness matters at both steps. A list of familiar mesons is not a coordinate system unless it generates the full invariant ring in the sector under discussion. Conversely, listing too many generators is harmless only if all syzygies—relations among relations as well as the primary relations—are handled consistently.

The Zariski tangent space at a point $p$ represented by coordinates $u_A(p)$ is found by linearizing generators $f_\alpha$ of $I_{\mathrm{rel}}$ :

T_p\mathcal M =\left\{\delta u\in\mathbb C^s: \sum_A\frac{\partial f_\alpha}{\partial u_A}(p)\,\delta u_A=0 \ \text{for all }\alpha\right\}.

If the tangent dimension exceeds the local dimension, $p$ is singular as an algebraic variety. This test is intrinsic to the coordinate ring; whether the singularity signals extra massless particles is a separate physical question.

A determinantal cone from a U(1) theory

Consider chiral multiplets $x_1,x_2$ of charge $+1$ and $y_1,y_2$ of charge $-1$ , with $W=0$ and vanishing FI parameter. The basic gauge invariants are the four mesons

M_{ij}=x_i y_j, \qquad i,j=1,2.

Because $M$ is an outer product, its rank is at most one. Therefore

\det M=M_{11}M_{22}-M_{12}M_{21}=0,

and

\mathcal M_{\mathrm{cl}} =\operatorname{Spec}\frac{\mathbb C[M_{11},M_{12},M_{21},M_{22}]} {(M_{11}M_{22}-M_{12}M_{21})}.

Away from the origin, one equation in $\mathbb C^4$ gives complex dimension three. This agrees with the field count: four complex fields minus one complexified $U(1)$ orbit. The Jacobian of the determinant is

(M_{22},-M_{21},-M_{12},M_{11}),

which vanishes only at $M=0$ . At that point the tangent space has dimension four while the cone has dimension three, so the origin is singular.

The field representatives explain why. For $M\neq0$ , at least one $x_i$ and one $y_j$ are nonzero and the $U(1)$ stabilizer is trivial. At $M=0$ , the closed orbit is represented by $x=y=0$ , whose stabilizer is the entire gauge group. The gauge boson becomes massless there. The algebraic singularity and the physical massless sector agree in this example, but that agreement was checked rather than assumed.

For $M_{11}\neq0$ , a smooth patch is parametrized by $M_{11},M_{12},M_{21}$ with

M_{22}=\frac{M_{12}M_{21}}{M_{11}}.

This formula is not valid at $M_{11}=0$ ; the apparent denominator is a patch boundary, not an additional component. Covering the variety by analogous patches prevents a coordinate artifact from being mistaken for a branch.

Mesons, baryons, and rank conditions in SQCD

In four-dimensional $\mathcal N=1$ $SU(N_c)$ gauge theory with $N_f$ flavors $Q^i_a$ and $\widetilde Q^a_j$ , classical invariants include

M^i{}_j=Q^i_a\widetilde Q^a_j,

and, when $N_f\ge N_c$ , baryons and antibaryons built from $N_c$ quarks using the epsilon tensor. The matrix inequality

\operatorname{rank}M\le N_c

imposes vanishing $(N_c+1)\times(N_c+1)$ minors. Baryons satisfy further relations with mesons. For $N_f=N_c$ , for example,

\det M-B\widetilde B=0

classically. Later quantum dynamics can deform this relation Seiberg 1994, pp. 6857–6863, arXiv:hep-th/9402044; the word classically is therefore part of the statement, not decorative qualification.

Rank strata organize the physics. A generic point of one stratum has a fixed stabilizer and a predictable number of massive vector multiplets. Lower-rank loci may have enhanced stabilizers and extra light fields. The invariant equations show where strata meet, while a representative field configuration and its mass matrix identify the actual low-energy degrees of freedom.

Branches and scheme-theoretic cautions

Suppose an F-term equation is $XY=0$ . Its zero set is the union of the branches $X=0$ and $Y=0$ , meeting at the origin. Replacing the equation by the two simultaneous equations $X=Y=0$ would destroy both branches. More subtly, an ideal such as $(X^2)$ has the same set of complex points as $(X)$ but a different nonreduced scheme: nilpotent directions retain information about infinitesimal deformations and can affect deformation theory.

For most first-pass vacuum analyses, the reduced variety is adequate. A claim about tangent complexes, obstructions, chiral-ring multiplicities, or derived intersections is not. State explicitly whether the object is the set of vacua, its reduced coordinate ring, or the full scheme defined by the F-term ideal.

Cross-checks that should agree

A robust classical answer passes three independent comparisons:

Dimension. The invariant-ring dimension agrees with the quotient count on each smooth stratum, including stabilizer corrections.
Patches. Gauge fixing and invariant coordinates give mutually invertible descriptions where both are valid.
Singular strata. Jacobian rank loss is compared with stabilizer enhancement and the quadratic mass matrix.

The comparisons can fail for an instructive reason: an affine quotient may collapse nonclosed orbits, a chosen set of invariants may be incomplete, or a particular FI chamber may use a different stable locus. Such a failure diagnoses missing input rather than an inconsistency of the physical theory.

Common pitfalls

Treating generators as independent coordinates. Invariants nearly always obey rank, determinant, or Plücker-type relations. Ignoring them gives the wrong dimension and erases singular strata.

Inferring a residual gauge group from invariants alone. Distinct stabilizer types can map to the same invariant point. Choose a closed-orbit representative and compute its stabilizer directly.

Confusing a classical relation with the exact chiral ring. Strong dynamics may deform a constraint or generate a superpotential. The later four-dimensional $\mathcal N=1$ gauge-dynamics chapter explains when that happens.

Exercises

For the hypersurface $xy-z^2=0$ in $\mathbb C^3$ :

Find its singular locus using the Jacobian criterion.
Show that the patch $x\neq0$ is smooth by solving for $y$ .
Parametrize the surface by $(u,v)\mapsto(u^2,v^2,uv)$ and identify the discrete identification.

Solution

The gradient is $(y,x,-2z)$ , so it vanishes on the hypersurface only at the origin. On $x\neq0$ , $y=z^2/x$ , leaving the smooth coordinates $(x,z)$ . The parametrization is invariant under $(u,v)\mapsto(-u,-v)$ and is otherwise generically two-to-one before quotienting, so the surface is $\mathbb C^2/\mathbb Z_2$ . The fixed point of the $\mathbb Z_2$ action maps to the singular origin.

References

Luty, Markus A., and Washington Taylor IV. “Varieties of Vacua in Classical Supersymmetric Gauge Theories.” Physical Review D 53 (1996): 3399–3405. arXiv:hep-th/9506098.
Seiberg, Nathan. “Exact Results on the Space of Vacua of Four-Dimensional SUSY Gauge Theories.” Physical Review D 49 (1994): 6857–6863. arXiv:hep-th/9402044.