Exact Partition Functions, Indices, and Instanton Counting
Exact supersymmetric observables compress different parts of a quantum field theory into partition functions, graded traces, or instanton integrals. This chapter explains how each object is defined, which ambiguities survive, and what information can be recovered without mistaking exact computation for a complete solution.
Enter this chapter
Section titled “Enter this chapter”Begin with Supersymmetric Partition Functions, Counterterms, and Universal Data to separate a curved-background partition function from its scheme-independent content.
Helpful background. Renormalization Conditions, Schemes, and Finite Parts; Instanton Measures, Zero Modes, and Determinants; Characters and Conformal Multiplet Counting provide the finite counterterms, instanton measures, and character decompositions used here.
Topics
Section titled “Topics”- Supersymmetric Partition Functions, Counterterms, and Universal Data — Tracks allowed counterterms and universal combinations.
- Sphere Partition Functions and Matrix Models — Reduces specified sphere observables to matrix integrals.
- F-Maximization and Accidental Symmetries — Determines an R-current with accidental-symmetry qualifications.
- Supersymmetric and Superconformal Indices — Defines the protected graded trace.
- Twisted Indices and Elliptic Genera — Retains flux and anomaly data in twisted observables.
- Index Inversion, Recombination, and Protected-Spectrum Limits — Explains why cancellation obstructs unique spectrum reconstruction.
- Instanton Counting and the Omega Background — Localizes instanton moduli in an equivariant background.
- Factorization, Holomorphic Blocks, and Gluing — States the boundary and vacuum assumptions needed for factorization.
- Exact-Observable Identities as Duality Tests — Compares exact answers after matching all background data.
How the ideas fit together
Section titled “How the ideas fit together”An index pairs states of opposite grading and records only the contributions that survive the cancellation. Distinct short multiplets can therefore combine into an invisible long-multiplet contribution, making inversion from an index to a spectrum nonunique. A partition function can retain different information, including dependence on finite counterterms. Agreement of either object is a sharp test once definitions match, but it cannot recover the information that the chosen reduction discards. Kinney et al. 2007, §§ 2–3.
Where to go next
Section titled “Where to go next”Continue to Protected Operators, BPS Defects, and Exact Correspondences to identify the mechanism protecting the particular operator or observable.