Exact Partition Functions, Indices, and Instanton Counting

Supersymmetric partition functions and indices can turn an infinite-dimensional path integral, a Hilbert-space trace, or an instanton-moduli integral into a finite-dimensional expression. “Exact” does not mean that the entire quantum field theory has been solved. It means that a particular, fully specified observable is protected strongly enough to be computed without an uncontrolled approximation. This chapter builds those specifications, carries their ambiguities through the calculation, and explains how much physical information each answer retains.

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

Three distinct meanings of exact

The three central constructions look similar only after much of their defining information has been suppressed.

1. A localized Euclidean path integral starts from a named supersymmetric background and an integration cycle. It reduces to a measure on fixed loci, possibly summed over flux or instanton sectors. A sphere matrix model belongs here.
2. A protected trace starts from a Hilbert space, a chosen supercharge, and commuting fugacities. Bosonic and fermionic states with positive $\{Q,Q^\dagger\}$ pair and cancel. A superconformal index belongs here; it is not an ordinary thermal trace.
3. An equivariant instanton sum starts from a framed moduli problem in an Omega background. Torus-fixed instantons replace a moduli integral by a weighted sum over partitions. The result can recover specified protected limits, but it is not the complete nonperturbative theory.

Schematically, the outputs are

$$\begin{aligned} Z_{\mathcal B} &=\sum_{\mathfrak m}\int_{\Gamma_{\mathfrak m}} \mathrm d\mu_{\mathfrak m}\, e^{-S_{\mathrm{cl},\mathfrak m}} Z_{\text{1-loop},\mathfrak m}Z_{\mathrm{np},\mathfrak m},\\ \mathcal I(\boldsymbol x) &=\operatorname{Tr}_{\mathcal H}(-1)^F e^{-\beta\Delta_Q}\prod_i x_i^{F_i}, \qquad \Delta_Q=\frac12\{Q,Q^\dagger\},\\ Z_{\mathrm{inst}}(q) &=\sum_{\vec Y}q^{|\vec Y|} Z_{\vec Y}(a,m;\epsilon_1,\epsilon_2). \end{aligned}$$

The symbols in one row cannot simply be reinterpreted as those in another. Pestun’s $S^4$ construction, for example, has a four-dimensional $\mathcal N=2$ locus and north/south-pole instanton factors Pestun 2012, §§3–5, whereas the four-dimensional superconformal index is a protected trace on the cylinder Kinney et al. 2007, §§2–3. Nekrasov’s partition function instead depends on the equivariant rotations $\epsilon_1,\epsilon_2$ of the Omega background Nekrasov 2003, §§2–3.

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. Nor is there a single sphere formula valid in all dimensions: $S^2$, $S^3$, and $S^4$ theories have different multiplets, fixed loci, measures, and nonperturbative sectors.

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

This table is a decision tree, not a menu of interchangeable techniques. For example, to determine an infrared R-current in a three-dimensional $\mathcal N=2$ theory, first define and validate the $S^3$ matrix model, then extremize the appropriate real free energy while checking accidental currents and decoupled operators. Computing an index first may reveal protected operators, but it does not replace that extremization problem.

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,

$$\frac{d\log Z_t}{dt}=-\langle QV\rangle_t=0.$$

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. Two additional steps must remain separate:

• finite-$t$ deformation independence requires a valid Ward identity, an invariant cycle and measure, and no boundary term in field space;
• evaluating the $t\to\infty$ limit requires control of all fixed-locus branches, zero modes, noncompact directions, sectors, and convergence.

The preceding localization chapter owns those validity conditions. This chapter starts only after they have been met and asks what exact observable they produce.

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.

For reproducibility, append one more field to every numerical result: the evaluation method and its error control. A residue sum needs the chosen poles and truncation; a quadrature needs the contour, working precision, and convergence test; a fugacity or instanton series needs its truncation order and a remainder estimate or a statement that it is being used only as a formal series.

Factorization requires additional hypotheses

Some three-dimensional sphere partition functions and indices can be written as bilinear pairings of vacuum-labeled holomorphic blocks,

$$Z_{\mathcal M}(x,\widetilde x;q,\widetilde q) =e^{P_g}\sum_{\alpha,\beta} B_<^\alpha(x;q)\, (K_g)_{\alpha\beta}\, B_>^\beta(\widetilde x;\widetilde q).$$

The labels $\alpha$ and $\beta$ name massive vacua or, equivalently in the appropriate construction, cycles; $K_g$ is the gluing kernel and $e^{P_g}$ retains the anomaly and contact-term prefactor. A diagonal sum is a special basis-dependent case, not the general definition. The pairing, conjugate parameters, and flux sum depend on the observable being reconstructed. Across a Stokes wall the basis of blocks may jump while the correctly transported glued answer remains unchanged Beem, Dimofte, and Pasquetti 2014, §§2–4. Consequently, neither a displayed product form nor one convenient block basis is meaningful without its chamber and gluing data.

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.

Record two logically separate labels for every comparison. The validation mode says how the equality was established:

• an analytic function identity is proved on a stated domain, with any continuation specified;
• a formal-series identity holds coefficient by coefficient without an additional convergence claim;
• a precision-bounded numerical match states the truncation, working precision, and error bound;
• a conditional calculation follows from declared localization, contour, regulator, and convergence assumptions.

The evidentiary reach says what the result supports: a protected-sector match establishes agreement of the named cohomological data after all parameter and global-sector maps, while broader duality evidence combines that match with logically independent observables or structural checks. A theorem can establish an analytic identity, but “theorem” describes proof status rather than evidentiary reach. Even an analytic identity can test only what its two sides encode. The scientific gain comes from a precise match plus an honest ceiling, not from calling every equality a proof of duality.

References

• Beem, C., T. Dimofte, and S. Pasquetti. “Holomorphic Blocks in Three Dimensions.” Journal of High Energy Physics 2014, no. 12 (2014): 177. DOI; Open PDF.
• Kinney, J., J. Maldacena, S. Minwalla, and S. Raju. “An Index for 4 Dimensional Super Conformal Theories.” Communications in Mathematical Physics 275 (2007): 209–254. DOI; Open PDF.
• Nekrasov, N. A. “Seiberg–Witten Prepotential from Instanton Counting.” Advances in Theoretical and Mathematical Physics 7 (2003): 831–864. DOI; Open PDF.
• Pestun, V. “Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops.” Communications in Mathematical Physics 313 (2012): 71–129. DOI; Open PDF.
• 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.