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Rigid Backgrounds, Topological Twists, and Localization

Localization computes selected supersymmetric observables by deforming their path integral without changing its value. The argument requires a globally defined supercharge, compatible gauge fixing, a controlled integration cycle, and all relevant saddles and zero modes. This chapter builds those conditions before using a reduced integral.

Enter this chapter

Section titled “Enter this chapter”

Begin with Supercurrent Multiplets and Rigid-Background Compatibility to check that the theory has a current multiplet compatible with the proposed background.

Helpful background. Fredholm and Dirac Index Theorems and Zero-Mode Counting; Heat Kernels, Zeta Functions, and Spectral Determinants; BRST Cohomology and Physical Observables provide the index theory, spectral determinants, and BRST cohomology used here.

Topics

Section titled “Topics”
  1. Supercurrent Multiplets and Rigid-Background Compatibility — Tests whether background sources exist globally.
  2. Rigid Supersymmetry from Nondynamical Supergravity Backgrounds — Derives rigid supersymmetry from nondynamical supergravity fields.
  3. Generalized Killing Spinors and Global Spin-R Bundles — Distinguishes local spinor equations from global supercharges.
  4. Topological and Holomorphic Twists — Constructs scalar symmetries through appropriate twists.
  5. Finite-Dimensional and Equivariant Localization — States finite-dimensional localization theorems and hypotheses.
  6. Q-Cohomology and Path-Integral Deformation — Separates deformation invariance from the large-deformation limit.
  7. Gauge Fixing and the Localization Deformation Complex — Combines supersymmetry with the gauge-fixing complex.
  8. Localization Loci, Zero Modes, and One-Loop Determinants — Determines saddle sectors and their fluctuation measures.
  9. Complex Contours, Stokes Chambers, and Regularization — Retains contours, poles, and Stokes choices.
  10. Boundaries, Gluing, and Jeffrey–Kirwan Residues — Matches boundary data before gluing or taking residues.

How the ideas fit together

Section titled “How the ideas fit together”

A Q-exact deformation leaves a Q-closed observable unchanged only if the symmetry of the measure and integration cycle allows the variation to integrate to zero. Boundaries or noncompact directions can spoil that step. Sending the deformation large adds a second requirement: the limiting saddle expansion must account for every relevant sector and zero mode. The resulting finite-dimensional expression still depends on its contour and counterterms; the integrand alone is not the observable. Pestun 2012, §§ 3–4.

Where to go next

Section titled “Where to go next”

Continue to Exact Partition Functions, Indices, and Instanton Counting to separate a curved-background partition function from its scheme-independent content.

References

Section titled “References”