GLSMs, Phases, and Quantum Kähler Moduli

A gauged linear sigma model is a ultraviolet gauge theory whose vacuum equations can produce nonlinear sigma models, Landau–Ginzburg orbifolds, hybrid theories, and singular Coulomb regimes. A phase is not determined by the sign of an FI parameter alone. It requires the global gauge group, charge matrix, superpotential, D- and F-term equations, excluded locus, residual gauge symmetry, anomaly data, and a scale hierarchy that makes the proposed low-energy description reliable.

Required background. We use Kähler sigma models and their quantum limitations and F- and D-flatness as a gauge quotient. Helpful background. Landau–Ginzburg vacua and orbifolds supply the non-geometric phase language.

Abelian GLSM data

Consider gauge group $G=U(1)^k$ and chiral multiplets $\Phi_i$, $i=1,\ldots,n$, with integral charge matrix $Q_i^{\ a}$. The defining data are

$$\left(G,\;Q_i^{\ a},\;W(\Phi),\;t_a=2\pi r_a-i\theta_a\right),$$

together with R charges and the global action of $G$. We normalize the FI and theta terms by

$$\mathcal L_{\mathrm{FI},\theta} =-\sum_a r_aD_a+\sum_a\frac{\theta_a}{2\pi}F^a_{01}, \qquad q_a=e^{-t_a}=e^{-2\pi r_a+i\theta_a}.$$

For an honest $U(1)$ with minimal electric charge one, $\theta_a\sim\theta_a+2\pi$. If all displayed charges share a common divisor, one must say whether the ineffective subgroup is divided out. $U(1)$, $U(1)/\mathbb Z_d$, and a theory with a trivially acting $\mathbb Z_d$ have different bundle sums and can differ by decomposition or discrete theta data.

Ignoring twisted masses for the moment, the bosonic potential is

$$\begin{aligned} V={}&\sum_i\left|\frac{\partial W}{\partial\phi_i}\right|^2 +\sum_a\frac{e_a^2}{2} \left(\sum_iQ_i^{\ a}|\phi_i|^2-r_a\right)^2\\ &+\sum_i\left|\sum_aQ_i^{\ a}\sigma_a\right|^2|\phi_i|^2 +\text{non-Abelian commutator terms}. \end{aligned}$$

Supersymmetric Higgs vacua satisfy

$$\partial_iW=0, \qquad \sum_iQ_i^{\ a}|\phi_i|^2=r_a, \qquad \left(\sum_aQ_i^{\ a}\sigma_a\right)\phi_i=0,$$

modulo $G$. A semiclassical Higgs branch usually has $\sigma_a=0$ and is a Kähler quotient further cut by the F equations.

Excluded loci and the phase fan

The D-term equation says which sets of fields cannot vanish simultaneously. In one $U(1)$ theory:

• if $r\gg0$, at least one positively charged field must be nonzero;
• if $r\ll0$, at least one negatively charged field must be nonzero.

The forbidden coordinate set is the excluded locus. Removing it before quotienting is essential: it distinguishes a projective quotient from a singular affine quotient. For $U(1)^k$, cones spanned by subsets of the charge vectors divide FI space into chambers. Each chamber has a fixed collection of allowed nonzero coordinate sets; their combinatorics is the secondary, or phase, fan.

The holomorphic quotient, and its relation to the symplectic and toric phase descriptions, is reviewed in Hori et al. 2003, chs. 12–13 and may be written schematically

$$X_r=\bigl(\mathbb C^n\setminus Z_r\bigr)/(\mathbb C^*)^k,$$

where $Z_r$ is the chamber-dependent excluded set. The equality with the symplectic quotient assumes stable orbits and is a version of the Kempf–Ness correspondence. F-term equations must still be imposed, and finite stabilizers produce orbifolds rather than smooth manifolds.

Worked example: the quintic

Take one $U(1)$, five fields $X_i$ of charge $+1$, a field $P$ of charge $-5$, and

$$W=P\,G_5(X_1,\ldots,X_5),$$

with $G_5$ a transverse homogeneous polynomial of degree five. The complete charge and R data relevant to the phase analysis are

Field	$U(1)$ charge	vector R charge	F equation
$X_1,\ldots,X_5$	$+1$	$0$	$p\,\partial_iG_5=0$
$P$	$-5$	$2$	$G_5(x)=0$

The D equation and anomaly sum are

$$\sum_{i=1}^5|x_i|^2-5|p|^2=r, \qquad \sum_iQ_i=5-5=0.$$

Geometric chamber: $r\gg0$

The excluded set is $x_1=\cdots=x_5=0$. Transversality means the equations $\partial_iG_5=0$ have no common nonzero projective solution. Therefore the F equations force $p=0$ and $G_5(x)=0$. Quotienting by $U(1)$ gives

$$X_+=\{G_5=0\}\subset\mathbb{CP}^4.$$

The gauge multiplet and radial mode have masses of order $e\sqrt r$. Far below that scale, and when the target curvature is small in cutoff units, the EFT is a nonlinear sigma model on the quintic.

Landau–Ginzburg chamber: $r\ll0$

Now $p\ne0$. Gauge fixing its phase leaves the subgroup satisfying $e^{-5i\alpha}=1$, namely $\mathbb Z_5$. The $X_i$ remain light near the origin and interact through $G_5(X)$. Thus the low-energy description is

$$\text{LG}[G_5]/\mathbb Z_5,$$

not the ungauged LG model. The orbifold acts diagonally on all $X_i$. The massive radial and vector modes again have masses of order $e\sqrt{|r|}$.

The phase information is summarized without suppressing the conditions:

Chamber	Excluded set	Residual gauge group	Light theory	Semiclassical requirement
$r\gg0$	all $x_i=0$	generically trivial	quintic sigma model	$e\sqrt r$ above the sigma-model scale
$r\ll0$	$p=0$	$\mathbb Z_5$	$G_5$ LG orbifold	$e\sqrt{\lvert r\rvert}$ above LG scales
near a quantum discriminant	Coulomb fields become light	unbroken $U(1)$ component	no pure Higgs EFT	retain $\Sigma$ and other light fields

This construction and its geometric/LG interpretation originate in Witten 1993, §§3–4.

FI running and the axial anomaly

A charged chiral multiplet shifts the FI coupling at one loop. With a reference scale $\mu$,

$$r_a(\mu_2)=r_a(\mu_1) +\frac1{2\pi}\left(\sum_iQ_i^{\ a}\right) \log\frac{\mu_2}{\mu_1},$$

up to the stated direction convention for RG flow. The same sum controls the gauge contribution to the axial R anomaly:

$$\partial_\mu j_A^\mu \propto\sum_{a,i}Q_i^{\ a}F^a_{01}.$$

Thus $\sum_iQ_i^{\ a}=0$ for every gauge factor is both the Calabi–Yau charge condition and the perturbative condition for an unbroken continuous axial R symmetry. The quintic satisfies it. The $\mathbb{CP}^{N-1}$ GLSM with $N$ fields of charge $+1$ does not: $r$ runs, a dynamical scale is generated, and the axial symmetry is reduced.

Gauge anomalies are a different question. A $(2,2)$ chiral has left- and right-moving fermions whose gauge-anomaly contributions cancel. General $(0,2)$ matter instead requires the quadratic condition

$$\sum_{\text{right fermions}}Q_i^aQ_i^b -\sum_{\text{left fermions}}Q_\alpha^aQ_\alpha^b=0$$

or a specified inflow mechanism.

Quantum Kähler moduli and singular loci

Classically $r=0$ looks like a wall. Quantum mechanically the natural coordinate is complex,

$$q=e^{-2\pi r+i\theta}\in\mathbb C^*.$$

One can often move between large positive and negative $r$ by varying $\theta$ and avoiding singular points. This does not mean every path is nonsingular. On a Coulomb branch, a field of charge $Q_i$ has mass $Q_i\sigma$, and integrating it out yields the condition

$$\prod_i\left(\frac{Q_i\sigma}{\mu}\right)^{Q_i}=q.$$

If $\sum_iQ_i=0$, the powers of $\sigma$ cancel and leave a special value of $q$. For the quintic,

$$q=(-5)^{-5}.$$

At this discriminant, the Coulomb direction is not lifted and the Higgs-only description fails. The numerical location depends on the convention for $q$ and finite renormalization, while the existence of the singular divisor is invariant. The effective twisted-superpotential derivation makes the branch structure explicit.

A reusable phase-analysis procedure

For any proposed GLSM phase:

1. specify the global gauge group and all charges, including fields that become heavy;
2. write $W$, every F equation, and every D equation;
3. solve the D equation to identify the excluded locus;
4. solve the F equations on the allowed set, checking transversality;
5. divide by the gauge group and compute every residual stabilizer;
6. check gauge and R anomalies and the FI beta function;
7. list the masses of integrated-out fields and state the energy hierarchy;
8. inspect Coulomb or mixed branches and remove the quantum discriminant;
9. only then name the phase geometric, LG, orbifold, hybrid, or mixed.

Mixed phases occur when some fields form a compact base while others remain an LG fiber. A single label such as “hybrid” is incomplete without the base, fiber superpotential, orbifold action, and singular fibers.

Common pitfalls

Drawing a phase from $r$ alone. Two theories with the same charge signs can have different F equations, stabilizers, and singular loci. The superpotential and global group are part of the phase definition.

Equating a classical wall with a quantum singularity. The complexified parameter can go around the real wall. The true obstruction is the discriminant where extra degrees of freedom become massless.

Integrating out the field that diagnoses failure. A Coulomb-branch calculation assumes charged matter is massive. At $Q_i\sigma+m_i=0$, that assumption fails and the field must be restored.

Exercises

1. Analyze a $U(1)$ model with charges $(1,1,-2)$ and no superpotential in the two signs of $r$.

Solution

The D equation is $|x_1|^2+|x_2|^2-2|p|^2=r$. For $r>0$, $(x_1,x_2)\ne(0,0)$; quotienting gives the total space of $\mathcal O(-2)\to\mathbb{CP}^1$, with $p$ the fiber coordinate. For $r<0$, $p\ne0$, leaving a residual $\mathbb Z_2$ acting by sign on $(x_1,x_2)$; without $W$ these directions are noncompact. Since $1+1-2=0$, the perturbative FI coupling does not run.

2. Explain why transversality of $G_5$ forces $p=0$ in the quintic’s $r>0$ chamber.

Solution

The F equations are $G_5=0$ and $p\partial_iG_5=0$. If $p\ne0$, all $\partial_iG_5$ vanish. Transversality says their common zero in affine space is only $x=0$, but that point is excluded for $r>0$. Hence $p=0$.

3. Derive the residual group in the $r<0$ chamber for a field of charge $-d$ acquiring a nonzero expectation value.

Solution

A gauge rotation $e^{i\alpha}$ preserves the expectation value when $e^{-id\alpha}=1$. The solutions are $\alpha=2\pi n/d$, so the unbroken subgroup is $\mathbb Z_d$, subject to any prior quotient in the specified global gauge group.

References

• Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., and Zaslow, E. Mirror Symmetry. Clay Mathematics Monographs 1. Providence, RI: American Mathematical Society and Clay Mathematics Institute, 2003, chs. 12–13. Clay Mathematics Institute book page.
• Witten, E. “Phases of $N=2$ Theories in Two Dimensions.” Nuclear Physics B 403 (1993): 159–222. doi:10.1016/0550-3213(93)90033-L; arXiv:hep-th/9301042.