Two-Dimensional Supersymmetric QFT, GLSMs, and Mirror Symmetry

Two-dimensional supersymmetry is unusually concrete: chirality separates left- and right-moving supercharges, holomorphy turns many vacuum questions into algebra, and gauge theories interpolate between geometric and Landau–Ginzburg descriptions. This chapter develops one connected toolkit—from $(2,2)$ superspace to GLSM phases, mirror symmetry, elliptic genera, and tt* transport—while keeping clear which statements concern an ultraviolet Lagrangian, a protected sector, or the full infrared quantum field theory.

Helpful background. The chapter uses complex coordinates and left–right factorization in two-dimensional CFT, regulated anomalies and measure variation, and de Rham cohomology, periods, and intersection pairings.

Enter this chapter

The same theory can admit several useful descriptions. A nonlinear sigma model makes the target geometry visible. A Landau–Ginzburg model makes the superpotential and its critical points visible. A gauged linear sigma model (GLSM) can contain both as low-energy regimes. A topological twist discards most local dynamics but retains a protected ring of observables. A mirror description exchanges chiral and twisted-chiral data. An elliptic genus compresses the spectrum into a protected trace, while tt* geometry restores metric information about the bundle of supersymmetric ground states. Their common framework is developed systematically in Hori et al. 2003, chs. 11–15.

These descriptions are related, but they are not interchangeable without hypotheses. Throughout the chapter we distinguish four levels of assertion:

1. an equality inside a cohomological or index-like sector;
2. a common set of massive vacua or chiral-ring relations;
3. agreement of partition functions, defects, or brane categories;
4. equivalence of the complete infrared QFT.

Evidence at one level need not prove the next. That distinction is especially important for noncompact targets, singular phase boundaries, accidental symmetries, and theories with a continuum of states.

Conventions used in the chapter

We use the site-wide Lorentzian metric convention $(+,-)$ in two dimensions and light-cone coordinates

$$x^\pm=x^0\pm x^1, \qquad \partial_\pm=\frac12(\partial_0\pm\partial_1).$$

The subscripts $+$ and $-$ label right- and left-moving spin. In $(2,2)$ superspace our covariant derivatives obey

$$\{D_\pm,\bar D_\pm\}=2i\partial_\pm, \qquad \{D_+,D_-\}=\{D_+,\bar D_-\}=0,$$

with all omitted anticommutators zero. A chiral superfield satisfies $\bar D_\pm\Phi=0$; a twisted-chiral field satisfies $\bar D_+Y=D_-Y=0$. The field strength $\Sigma$ of an Abelian vector multiplet is twisted chiral.

For a $U(1)$ GLSM we normalize integer matter charges so the smallest allowed charge is one and write

$$\mathcal L_{\mathrm{FI},\theta}=-rD+\frac{\theta}{2\pi}F_{01}, \qquad q=e^{-2\pi r+i\theta}, \qquad \theta\sim\theta+2\pi.$$

Changing any of these normalizations changes intermediate formulas. Each comparison in the chapter therefore records the charge lattice, global gauge group, and definition of $q$ before using a phase or mirror dictionary.

A route through the material

Question	Start here	Main output
Which supercharges and multiplets exist?	Algebras, multiplets, and superspace	$(p,q)$ algebra, R symmetries, chiral and twisted-chiral constraints
What does a holomorphic superpotential determine?	Landau–Ginzburg models and chiral rings	Critical points, Jacobi ring, soliton central charges, IR tests
How does supersymmetry constrain target geometry?	Sigma models and Kähler geometry	Kähler metric, $B$-field, beta function, R-symmetry anomaly
How can one UV theory produce several IR regimes?	GLSM phases and quantum Kähler moduli	Charge matrix, D/F equations, excluded loci, phase fan, discriminant
How are Coulomb vacua computed?	Effective twisted superpotentials	One-loop $\widetilde W_{\mathrm{eff}}$, exponentiated vacuum equations, Hessian
Which observables survive a topological twist?	A- and B-twists	Scalar supercharge, descent, quantum and Jacobi rings
What exactly does mirror symmetry exchange?	Mirror symmetry dictionaries	Parameters, rings, vacua, branes, defects, and evidence levels
What can anomalies and a torus trace determine?	Elliptic genera and c-extremization	Anomaly matrix, Jacobi behavior, gauge residues, exact trial R symmetry
How do ground states vary with couplings?	tt* geometry	Berry connection, ring action, tt* equations, Stokes limits

The most efficient first pass is algebra $\rightarrow$ Landau–Ginzburg and sigma models $\rightarrow$ GLSMs $\rightarrow$ twisted superpotentials. The twists, mirror dictionary, elliptic genus, and tt* pages can then be read as four different protected views of the same families of theories.

Two recurring test models

The $A_k$ Landau–Ginzburg model

For one chiral field and

$$W(X)=\frac{X^{k+2}}{k+2},$$

the Jacobi ring is $\mathbb C[X]/(X^{k+1})$. Assigning $X$ weight $1/(k+2)$ makes $W$ quasi-homogeneous, and the candidate infrared central charge is

$$c=3\left(1-\frac{2}{k+2}\right)=\frac{3k}{k+2}.$$

The model tests the LG ring, B-twist, elliptic genus, mirror, and tt* constructions in a setting where every step can be done explicitly.

The quintic GLSM

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

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

where $G_5$ is transverse. The D-term equation is

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

For $r\gg0$ the low-energy target is the quintic hypersurface $G_5=0$ in $\mathbb{CP}^4$; for $r\ll0$ it is a $\mathbb Z_5$ Landau–Ginzburg orbifold. Because $\sum_aQ_a=0$, the perturbative FI beta function and axial gauge anomaly vanish. This is the standard controlled example of a geometric/LG phase interpolation introduced in Witten 1993, §§3–4.

The statement is deliberately local in quantum Kähler moduli space: semiclassical regions can be connected through complexified $q$, but singular Coulomb loci must be removed. A real-axis cartoon alone does not establish a nonsingular interpolation.

Synthesis: one theory, several protected shadows

For a compact massive $(2,2)$ theory with isolated vacua, the following data fit together:

Structure	What it remembers	What it can forget
Chiral or twisted-chiral ring	Multiplication in supercharge cohomology	Norms of states and unprotected excitations
Effective twisted superpotential	Coulomb vacua and BPS central data	Light fields omitted from its domain of validity
Topological twist	Metric-independent correlators	Ordinary unitary time evolution
Elliptic genus	Protected signed spectrum and anomalies	Paired states; continuum subtleties can spoil holomorphy
tt* geometry	Hermitian metric and Berry transport on vacua	Requires a well-separated finite ground-state bundle
Mirror dictionary	Corresponding protected and, when established, full-QFT objects	A partial dictionary is not automatically an equivalence

This table is also a diagnostic. If two proposed mirrors have matching rings but different anomaly matrices, the claim fails. If their elliptic genera agree but one side has an unaccounted continuum, the equality requires a regulator-sensitive refinement. If a tt* connection becomes singular where vacua collide, the finite-rank ground-state description has reached its boundary.

Review the chapter

1. Starting from the $(2,2)$ algebra, identify which R symmetry is needed for the A-twist and which for the B-twist. Explain why a quantum anomaly can obstruct only one of them in a generic Kähler sigma model.
2. For $W=X^4/4-uX$, compute the Jacobi ring at $u=0$, the massive vacua at $u\ne0$, and the soliton central charges between them.
3. Given a $U(1)$ charge vector $(1,1,-2)$, write the D-term equation, determine the semiclassical excluded set in each sign of $r$, and check whether $r$ runs.
4. Derive the massless $\mathbb{CP}^{N-1}$ Coulomb-vacuum equation $\sigma^N=q\mu^N$ and state which points must be excluded from the derivation.
5. For the mirror pair $\mathbb{CP}^1$ and $\widetilde W=x+q/x$ on $\mathbb C^*$, match rings, vacua, and critical values. Which additional checks would support a full-QFT equivalence?
6. State the compactness, discreteness, anomaly, and R-symmetry assumptions needed before calling an elliptic genus a holomorphic weak Jacobi form.
7. Explain why tt* equations contain more information than the multiplication table of a chiral ring, and why they cease to define a smooth finite-rank bundle at a vacuum collision.

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. 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.