Exact Partition Functions, Indices, and Instanton Counting

Supersymmetric partition functions and indices turn infinite-dimensional path integrals or Hilbert-space traces into exact finite-dimensional expressions. Their exactness is conditional: one must fix the background, preserved supercharge, global theory, flux sectors, integration cycle, regulator, local counterterms, and normalization before the answer is a well-defined object. This chapter develops that specification and shows how much physical information each exact observable retains.

Helpful background. Review renormalization schemes and finite parts, instanton measures and zero modes, and character decompositions as needed.

Choose the observable before computing

The phrase “the exact partition function” is incomplete. Even for one Lagrangian, different supersymmetric backgrounds and boundary conditions define different quantities:

$$Z[\mathcal B,\mathfrak s,\Gamma,\mathcal C] =\int_{\Gamma}[D\Phi]_{\mathfrak s} \exp\!\left[-S_{\mathcal B}[\Phi]-S_{\mathcal C}[\mathcal B]\right].$$

Here $\mathcal B$ includes the metric and background multiplets, $\mathfrak s$ includes the spin structure and topological sector, $\Gamma$ is the integration cycle, and $\mathcal C$ is the finite local-counterterm convention. Localization simplifies this already-defined object; it does not choose these data for the reader.

Use the nine leaves according to the question being asked.

Question	Start here	Output
Which part of $Z$ is scheme independent?	Counterterms and universal data	Allowed local shifts and invariant combinations
What finite-dimensional integral represents a sphere path integral?	Sphere matrix models	Measure, determinants, sectors, and contour
Which three-dimensional R-current is superconformal?	F-maximization	Mixing parameters, Hessian, and decoupling conditions
Which states survive a chosen supercharge?	Supersymmetric indices	A gauge-projected protected trace
How are fluxes and holonomies counted?	Twisted indices and elliptic genera	Flux sum and residue formula
Can an index be inverted into a spectrum?	Index inversion	Recombination classes and nonuniqueness
How are instantons counted equivariantly?	Omega-background counting	Young-diagram fixed-point sum
When does a result split into blocks?	Holomorphic blocks and gluing	Vacuum-labeled blocks, cycles, and Stokes data
How strongly does an identity test a duality?	Exact-observable duality tests	Parameter map, contact terms, and evidence ceiling

The common exactness mechanism

In the path-integral examples, a fermionic symmetry $Q$ permits the deformation

$$S\longmapsto S+t\,QV,$$

with $Q^2$ generating a compact bosonic symmetry and the measure invariant. Under the convergence and boundary conditions developed in the preceding localization chapter Pestun et al. 2017, §2,

$$\frac{dZ}{dt}=-\langle QV\rangle_t=0.$$

The $t\to\infty$ limit reduces the integral to the $QV=0$ locus and a one-loop determinant, possibly with nonperturbative sectors. In a Hilbert-space index, the same pairing appears as cancellation between states with positive $\{Q,Q^\dagger\}$ Witten 1982, §2.

These are cohomological statements. They protect the chosen observable against certain deformations; they do not make every quantity in the theory protected.

A reproducible exact-data specification

Before comparing or exporting an answer, record:

1. spacetime manifold, metric parameters, spin structure, and background bundles;
2. preserved supercharge and the bosonic generator $Q^2$;
3. gauge group including global form, matter representations, and discrete terms;
4. continuous masses, FI parameters, couplings, fugacities, and their normalization;
5. topological sectors and the lattice over which they are summed;
6. integration variables, Weyl quotient, cycle or residue prescription, and poles at infinity;
7. zero-mode treatment, determinant phase, regularization, and local counterterms;
8. convergence domain and analytic continuation, if any;
9. a benchmark limit such as a free multiplet, weak coupling, or first few series coefficients.

Without these entries, two formulas that look different may represent the same observable in different conventions—or two formulas that look identical may represent different global theories.

What exact observables do and do not determine

An index is constant under recombination of short multiplets into a long one. A localized sphere free energy may determine an exact R-symmetry, while its phase can shift under quantized contact terms. An instanton sum controls an equivariant sector and can recover a prepotential in a limit, but it does not by itself specify the full nonperturbative theory. A block decomposition depends on cycles and can jump by a Stokes transformation even when the glued observable stays fixed.

The safe hierarchy is:

$$\text{defined background observable} \Longrightarrow \text{exact protected value} \Longrightarrow \text{constraint on the theory}.$$

The reverse arrows generally fail. Equality of one protected observable is nontrivial evidence for a proposed duality only after the maps and ambiguities are controlled; it is not automatically equality of full operator algebras, OPE coefficients, or unprotected spectra.

References

• Pestun, V., et al. “Localization Techniques in Quantum Field Theories.” Journal of Physics A 50 (2017): 440301. DOI; Open PDF.
• Witten, E. “Constraints on Supersymmetry Breaking.” Nuclear Physics B 202 (1982): 253–316. DOI.