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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)(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.

Enter this chapter

Section titled “Enter this chapter”

Begin with Two-Dimensional Supersymmetry Algebras, Multiplets, and Superspace to organize supercharges and multiplets by two-dimensional chirality.

Helpful background. Complex Coordinates and Local Conformal Symmetry; Regulated Jacobians and Measure Variation; de Rham Cohomology, Periods, Duality, and Intersection provide the two-dimensional conformal kinematics, anomalies, and cohomology used here.

Topics

Section titled “Topics”
  1. Two-Dimensional Supersymmetry Algebras, Multiplets, and Superspace — Defines the chiral supersymmetry algebra and superspace.
  2. Landau–Ginzburg Models and Chiral Rings — Extracts chiral rings from a superpotential.
  3. Supersymmetric Sigma Models and Kähler Geometry — Relates supersymmetry to target-space geometry.
  4. GLSMs, Phases, and Quantum Kähler Moduli — Connects gauge-theory parameters to distinct low-energy phases.
  5. Effective Twisted Superpotentials and Quantum Vacua — Finds quantum vacua through twisted chiral data.
  6. A- and B-Twists and Cohomological Observables — States which R-symmetries permit cohomological twists.
  7. Two-Dimensional Mirror Symmetry and Object-Level Dictionaries — Maps specified chiral, twisted-chiral, and geometric objects.
  8. Elliptic Genera, Anomaly Data, and c-Extremization — Separates protected traces from exact R-current inference.
  9. tt* Geometry and Ground-State Bundles — Studies metrics and transport on the ground-state bundle.

How the ideas fit together

Section titled “How the ideas fit together”

A GLSM can have geometric and Landau–Ginzburg regimes within one ultraviolet gauge theory. Changing its parameters therefore changes which low-energy variables are useful, while protected quantities can sometimes be continued between regimes. Such continuation does not automatically transport an unprotected target-space metric or every correlation function. Mirror symmetry exchanges specified structures, and singular loci or noncompact directions can obstruct a naive extrapolation. Witten 1993, §§ 3–4.

Where to go next

Section titled “Where to go next”

Continue to Three-Dimensional Supersymmetric Gauge Theory and Duality Webs to identify the real supercharges and scalar content of three-dimensional multiplets.

References

Section titled “References”