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 can 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 conclusions concern a UV Lagrangian, a protected sector, a low-energy regime, or the complete 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 target geometry visible. A Landau–Ginzburg model makes a 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 torus trace, while tt* geometry restores metric and transport information about 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 increasingly strong assertions:

1. equality inside a cohomological or index-like sector;
2. a common set of massive vacua or protected ring relations;
3. agreement of specified 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.

Check your preparation

Try the following short diagnostic before choosing a route. It is not a test to pass; each item points to the background that repairs a specific gap.

1. Can you explain why $P_+$ and $P_-$ permit independent supersymmetry generators of opposite spin in two dimensions? If not, review dimensions and spinor reality conditions before the algebra page.
2. Can you derive a component constraint from an equation such as $\bar D_\pm\Phi=0$? If not, review superspace covariant derivatives and chirality.
3. Can you find the critical locus of a holomorphic function and form its quotient by the gradient ideal? If not, begin with Wess–Zumino models and then read the Landau–Ginzburg page.
4. Can you solve F- and D-flatness equations and divide by a compact gauge group, including residual stabilizers? If not, review F- and D-flatness as a gauge quotient before GLSM phases.
5. Can you say which fields may be integrated out in a Wilsonian action and what fails at a massless threshold? If not, review Wilsonian and 1PI effective actions before effective twisted superpotentials.
6. Can you distinguish an ordinary Hilbert-space trace from a cohomological index when a continuum is present? If not, review ground-state bundles and continuum boundaries before tt* geometry.
7. Can you compute a two-dimensional chirality-weighted anomaly trace? If not, review ’t Hooft anomaly matching before elliptic genera and c-extremization.

You are ready for the main sequence if you can translate the first two items into equations and know where to repair any later gap. The pages restate their own required background, so a specialized reader can safely enter midway.

Conventions used in the chapter

We use the site-wide Lorentzian metric convention $(+,-)$ 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 a $(p,q)$ algebra, $p$ counts right-moving and $q$ counts left-moving supersymmetries. For $(2,2)$ superspace we choose

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

with all omitted anticommutators zero. A chiral field satisfies $\bar D_\pm\Phi=0$; a twisted-chiral field satisfies $\bar D_+Y=D_-Y=0$. The Abelian field strength $\Sigma$ is twisted chiral. Complex conjugation, Euclidean continuation, and superspace measures are stated locally when they matter.

The vector R symmetry defines the A-twist and the axial R symmetry defines the B-twist in this chapter. Whether either current survives quantization is checked before twisting; in an ordinary Kähler sigma model the axial symmetry can be obstructed by $c_1(TX)$.

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

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

We reserve $\mathfrak q=e^{2\pi i\tau}$ for the modular parameter in an elliptic-genus trace. For anomaly formulas, $\gamma^3=+1$ on right-moving Weyl fermions and $\gamma^3=-1$ on left-moving ones, so

$$k^{IJ}=\operatorname{Tr}\!\left(\gamma^3 Q^I Q^J\right).$$

Changing a charge lattice, global gauge group, FI normalization, R-current convention, or logarithm branch changes intermediate formulas. Every phase, mirror, index, and tt* comparison therefore declares those inputs before using a dictionary.

Choose a route

Goal	Required route	Capability gained
Decode two-dimensional chirality and multiplets	Algebras, multiplets, and superspace	Translate a $(p,q)$ algebra into chiral, twisted-chiral, vector, and bounded $(0,2)$ multiplets
Compute vacua, rings, and soliton central data	Algebra $\rightarrow$ Landau–Ginzburg models and chiral rings	Compute critical loci, Jacobi rings, massive vacua, and protected IR tests
Relate supersymmetry to target geometry	Algebra $\rightarrow$ sigma models and Kähler geometry	Separate Kähler geometry, $B$-field data, beta functions, and the axial anomaly
Construct and test GLSM phases	Sigma models $\rightarrow$ GLSM phases and quantum Kähler moduli	Reconstruct phases from charges, F/D equations, excluded loci, residual groups, anomalies, and scale hierarchies
Find Coulomb vacua	GLSM phases $\rightarrow$ effective twisted superpotentials	Derive exponentiated vacuum equations while tracking branches and excluded massless loci
Compute cohomological observables	Algebra $\rightarrow$ A- and B-twists	Identify scalar supercharges, anomaly conditions, descent observables, and A/B rings
Test a mirror proposal	GLSM phases $\rightarrow$ mirror-symmetry dictionaries	Match parameters, operators, rings, vacua, branes, defects, anomalies, and stated evidence levels
Use torus traces or exact R symmetry	Algebra and anomaly matching $\rightarrow$ elliptic genera and c-extremization	State a well-defined trace, read anomaly-controlled transformations, and extremize a valid $(0,2)$ trial R current
Follow vacua through parameter space	Landau–Ginzburg models and ground-state bundles $\rightarrow$ tt* geometry	Construct the Berry connection, ring action, tt* equations, and their collision or continuum boundary

For a first complete reading, follow algebra $\rightarrow$ Landau–Ginzburg and sigma models $\rightarrow$ GLSMs $\rightarrow$ effective twisted superpotentials. Then read twists, mirror symmetry, elliptic genera, and tt* geometry as four different protected views of the same families of theories.

What each page establishes

1. Algebras, multiplets, and superspace. Start here to translate chirality into algebra rather than importing four-dimensional notation. You should leave able to read every later superspace constraint and R-charge assignment.
2. Landau–Ginzburg models and chiral rings. Use a holomorphic superpotential to compute vacua, the Jacobi ring, BPS soliton data, and candidate fixed-point quantities. The page also marks where degeneracy, noncompactness, or orbifolding invalidates a naive quotient calculation.
3. Sigma models and Kähler geometry. Derive why chiral multiplets produce a Kähler target, then distinguish classical supersymmetry from quantum conformal invariance. This is the geometric input for GLSMs and the anomaly input for the B-twist.
4. GLSM phases and quantum Kähler moduli. Begin from the global gauge group, charge matrix, superpotential, and FI–theta parameters. A phase label is accepted only after the vacuum equations, excluded locus, residual gauge symmetry, anomaly data, and effective-field-theory regime agree.
5. Effective twisted superpotentials. Integrate out massive charged fields on a valid Coulomb patch, exponentiate the branch-dependent derivative, and solve for quantum vacua. This page supplies the algebraic bridge from a GLSM to many mirror and Bethe-type equations.
6. A- and B-twists. Construct the scalar supercharges and ask which observables survive in cohomology. The result explains why A-model quantities depend on Kähler data while B-model quantities depend on complex and superpotential data.
7. Mirror-symmetry dictionaries. Replace the slogan “Kähler becomes complex structure” with an object-by-object map. The worked Abelian examples separate a protected ring or vacuum match from a stronger infrared-QFT claim.
8. Elliptic genera and c-extremization. Define the spin structure and protected trace before invoking modularity, and distinguish a genuine weak Jacobi form from a regulator-sensitive noncompact answer. The c-extremization section applies to the proper unitary $(0,2)$ fixed-point class and explicitly checks accidental-current branches.
9. tt* geometry. Restore the Hermitian metric and Berry transport that a ring multiplication table forgets. The construction is valid on a parameter region with a finite, normalizable, gapped ground-state bundle and stops at collisions or continuum thresholds.

Recurring calculations

Aₖ Landau–Ginzburg family

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

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

Relevant deformations split the degenerate critical point into massive vacua. This family tests the LG ring, B-twist, elliptic-genus contribution, soliton central data, and tt* transport in a setting where each protected step can be computed explicitly.

Abelian GLSMs: projective space and the quintic

The $U(1)$ model with $N$ charge-$+1$ chirals is the smallest exact bridge from a geometric phase to a Coulomb-ring equation and its Toda-type mirror. Its $N=2$ case gives the transparent dictionary

$$\left(\frac{\sigma}{\mu}\right)^2=q \quad\longleftrightarrow\quad \frac{\widetilde W(x)}{\mu}=x+\frac{q}{x},$$

including the two vacua and their monodromy around $q=0$.

The quintic adds a genuine geometric/Landau–Ginzburg phase pair. Take five chirals $X_i$ of charge $+1$, one chiral $P$ of charge $-5$, and

$$W=P\,G_5(X_1,\ldots,X_5), \qquad \sum_{i=1}^{5}|x_i|^2-5|p|^2=r,$$

with $G_5$ transverse. 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 controlled phase analysis originates in Witten 1993, §§3–4.

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

Synthesis: several protected descriptions

For a compact massive $(2,2)$ theory with isolated vacua, the following structures fit together without carrying identical information:

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 objects and, when established, full-QFT data	A partial dictionary is not automatically an equivalence

Use the table diagnostically. Matching rings with different anomaly matrices rules out the proposed mirror. Matching formal elliptic genera when one side has an unaccounted continuum calls for a regulator-sensitive refinement. A singular tt* connection at a vacuum collision marks the boundary of the finite-rank ground-state description rather than a harmless coordinate artifact.

Review the chapter

Each prompt names an observable success criterion. If your calculation stalls, follow the repair route before comparing with the solution on the relevant page.

1. Translate the algebra. Starting from the $(2,2)$ algebra, identify the vector and axial R symmetries used by the A- and B-twists. Success means deriving which supercharges become scalars and explaining which quantum anomaly can obstruct the B-twist in a generic Kähler sigma model. Repair: algebra, then A/B twists.
2. Compute a deformed ring. For $W=X^4/4-uX$, compute the Jacobi ring at $u=0$, the massive vacua at $u\ne0$, and the differences of critical values. Success means separating the degenerate conformal point from the massive regime and identifying the BPS central data. Repair: Landau–Ginzburg models.
3. Reconstruct both GLSM chambers. For charges $(1,1,-2)$, write the D-term equation, find the excluded set and residual group in each sign of $r$, and determine whether $r$ runs. Success means giving every phase label together with its quotient and validity regime. Repair: sigma models, then GLSM phases.
4. Derive a quantum Coulomb relation. Obtain $\sigma^N=q\mu^N$ for the massless $\mathbb{CP}^{N-1}$ model. Success means showing how logarithm branches disappear after exponentiation and listing the massless point excluded from the derivation. Repair: GLSM phases, then effective twisted superpotentials.
5. Test a mirror at several levels. For $\mathbb{CP}^1$ and $\widetilde W/\mu=x+q/x$ on $\mathbb C^*$, match rings, vacua, and critical values. Success means stating which additional anomaly, partition-function, defect, or brane checks would support a stronger equivalence claim. Repair: effective twisted superpotentials, then mirror symmetry.
6. Qualify an elliptic genus. State the spin, compactness, discreteness, anomaly, and R-symmetry assumptions needed before calling the trace a holomorphic weak Jacobi form. Success means predicting the modular and elliptic multipliers from $k^{IJ}$ and identifying what a continuum can change. Repair: anomaly matching, then elliptic genera.
7. Diagnose a tt boundary.* Explain what the Hermitian metric and Berry connection add to a chiral ring and what happens when two vacua collide. Success means distinguishing singular gauge choices from failure of a finite-rank gapped bundle. Repair: ground-state bundles, then tt* geometry.

Where to go next

• Continue to three-dimensional supersymmetric gauge theory and duality webs to see how parity anomalies, contact terms, monopole operators, and real-mass flows replace the special two-dimensional toolkit.
• Use rigid backgrounds, twists, and localization to turn the cohomological constructions into controlled path-integral calculations.
• Return to duality dictionaries and global data when a proposed mirror requires a systematic claim and evidence hierarchy.
• Study two-dimensional CFT for the full conformal representation theory that this chapter uses only at supersymmetric fixed points.
• Follow geometric mirror symmetry in the string-theory volume and theorem-first mirror constructions in the mathematical-structures volume when those routes are materialized; this chapter retains the QFT dictionary and its physical evidence.

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.