GLSMs, Phases, and Quantum Kähler Moduli

A gauged linear sigma model (GLSM) is an 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 a Fayet–Iliopoulos (FI) parameter alone. It is a low-energy description derived from the global gauge group, charge matrix, superpotential, D- and F-term equations, excluded locus, residual gauge symmetry, anomaly data, and a hierarchy that makes every omitted field heavy.

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 a two-dimensional $(2,2)$ theory with 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

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

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

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

For an honest $U(1)$ factor with minimal electric charge one and flux $\frac1{2\pi}\int F^a\in\mathbb Z$, $\theta_a\sim\theta_a+2\pi$. More generally, the cocharacter lattice of the global gauge group fixes the allowed fluxes and hence the periodic lattice of the theta angles. If all displayed charges share a common divisor, one must say whether the ineffective subgroup is divided out: a theory retaining a trivially acting $\mathbb Z_d$ and one with a faithful quotient have different bundle sums and can differ by decomposition or discrete theta data.

With canonical matter kinetic terms and no twisted masses, the Abelian bosonic potential is

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

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 $U(1)^k$, a set of nonzero fields can solve the D equations only when $r=(r_a)$ lies in the cone spanned by their charge vectors. The resulting cones divide FI space into chambers, each with 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 descriptions are developed in Hori et al. 2003, §15.4, pp. 357–383. 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 $x_i=0$	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 are derived in Witten 1993, §§3.1–3.2, pp. 16–24.

FI running and the axial anomaly

A charged chiral multiplet shifts the FI coupling at one loop. With a reference scale $\mu$ and the flow convention fixed by the displayed equation,

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

so for $\mu_2>\mu_1$ and positive charge sum, $r_a$ grows toward the ultraviolet. 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 the perturbative condition for a nonrunning FI coupling and an unbroken continuous axial R symmetry in this $(2,2)$ matter system. In the standard toric and hypersurface constructions it is also the charge form of the Calabi–Yau condition, but that geometric interpretation is not automatic for an arbitrary GLSM. 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. These one-loop statements and their theta-angle companion are derived in Hori et al. 2003, §15.3.4, pp. 353–356.

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. When $\sum_iQ_i=0$, the FI coupling does not run and the natural quantum coordinate is complex,

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

If the charge sum is nonzero, $q(\mu)$ runs and an RG-invariant dynamical scale replaces a Kähler modulus. In the nonrunning case 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 where every $Q_i\sigma\ne0$ 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 definition of $q$ and on finite linear counterterms, while the existence of the singular locus is invariant. The refinement from the real wall to one point in the complexified parameter cylinder is derived in Hori et al. 2003, §15.5.3, pp. 391–393. The effective twisted-superpotential derivation fixes the sign, branch, and renormalization conventions used here.

The chapter’s phase–mirror–tt* map keeps this quintic chamber certificate separate from the exact $\mathbb{CP}^1$ protected dictionary, so the semiclassical and protected arrows are not mistaken for one RG flow.

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.

The phase diagram is therefore a controlled map from microscopic data to effective theories, not a partition of the real $r$-axis by labels. For the quintic, the two semiclassical ends are the hypersurface sigma model and the $\mathbb Z_5$ LG orbifold; the finite Coulomb discriminant is the stop point at which neither Higgs-only description is valid.

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, ch. 15. Clay Mathematics Institute PDF.
• 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.