Index Inversion, Recombination, and Protected-Spectrum Limits

A supersymmetric index can be expanded in short-multiplet characters, but it cannot generally be inverted into a unique protected spectrum. Recombination makes the index a linear functional on equivalence classes of short multiplets: distinct nonnegative spectra can have exactly the same index. Inversion is meaningful only after a superconformal algebra, fugacity domain, character basis, and recombination quotient have been fixed.

Required background. Use the definition and single-letter construction of supersymmetric indices together with the superconformal shortening data.

Helpful background. Conformal characters supply the ordinary character inner products that the protected inversion modifies.

The index lives in a recombination quotient

Let $K_{\mathrm{short}}$ be the free Abelian group generated by isomorphism classes of short multiplets of a fixed superconformal algebra. When a long multiplet reaches a unitarity threshold, it decomposes schematically as

$$L\big|_{\Delta=\Delta_*} =A\oplus B\oplus\cdots.$$

Because a long multiplet has zero index,

$$\mathcal I_A+\mathcal I_B+\cdots=0.$$

Let $R_{\mathrm{rec}}$ be the subgroup generated by all such relations. The index factors through

$$K_{\mathrm{short}}/R_{\mathrm{rec}}.$$

It can distinguish only classes in this quotient. Complete recombination rules therefore belong to the definition of any claimed inversion Córdova, Dumitrescu, and Intriligator 2019, §§2–4.

A minimal nonuniqueness example

Suppose two short characters obey one threshold relation

$$\mathcal I_A+\mathcal I_B=0.$$

Then a spectrum with multiplicities $(n_A,n_B)=(3,1)$ has

$$\mathcal I=3\mathcal I_A+\mathcal I_B=2\mathcal I_A.$$

The spectra $(4,2)$, $(5,3)$, and every $(3+k,1+k)$ with $k\ge0$ give the same answer. Positivity of the microscopic multiplicities does not remove the ambiguity because a complete recombination package adds only nonnegative multiplicities.

Real superconformal algebras contain chains of such relations with shifted spin and R-charges. The example captures the essential obstruction without tying it to one notation for multiplet labels.

Character decomposition and inversion kernels

After quotienting, choose protected characters $\chi_{[\mathcal M]}(x)$ that form a basis in a declared series domain. Then

$$\mathcal I(x)=\sum_{[\mathcal M]}N_{[\mathcal M]}\, \chi_{[\mathcal M]}(x).$$

If a dual kernel $K_{[\mathcal M]}(x)$ exists, an inversion formula takes the form

$$N_{[\mathcal M]} =\oint_{\Gamma}d\mu(x)\, K_{[\mathcal M]}(x)\mathcal I(x).$$

This equation requires more than formal coefficient matching:

• the characters must be linearly independent in the quotient;
• $\Gamma$ must lie in a common convergence annulus;
• poles crossed during analytic continuation must be included;
• the measure and kernel must be compatible with Weyl and flavor identifications;
• the index must use the exact infrared R-symmetry.

The extracted $N_{[\mathcal M]}$ counts an equivalence class or a protected combination, not necessarily one irreducible multiplet.

What low-order coefficients can say

In a four-dimensional $N=1$ index, the coefficient in the marginal-operator channel often takes the form

$$N_{\mathrm{marginal}}-N_{\mathrm{current}},$$

after descendants and known short contributions are removed. This difference is stable under recombination: a marginal chiral operator can pair with a conserved-current multiplet to become long. The index can therefore constrain the local dimension of a conformal manifold under additional assumptions, but it does not separately determine the two nonnegative integers Dolan and Osborn 2003, §§5–6.

Similarly, a negative coefficient in a plethystic logarithm need not mean “one relation.” It can represent a fermionic generator, an equation of motion, a recombination subtraction, or a genuine algebraic relation. A Hilbert-series interpretation requires an independently established cohomological ring with appropriate finiteness and positivity properties.

A finite-order reconstruction procedure

For a practical expansion through order $x^N$:

1. fix the exact superconformal R-symmetry and all flavor-fugacity conventions;
2. list every short character that can contribute through $x^N$;
3. impose all recombination relations at the same order;
4. reduce to a linearly independent quotient basis;
5. solve the coefficient system and propagate exact or numerical uncertainty;
6. test whether at least one nonnegative microscopic spectrum realizes the result;
7. display the kernel of the map, which parameterizes indistinguishable spectra.

If the kernel is nonzero, reporting one convenient representative as “the spectrum” is an overstatement. Report the fixed combinations and the remaining ambiguity.

Accidental symmetries and continuum effects

An ultraviolet R-charge assignment can place the expansion in the wrong fugacity grading. If an operator decouples or an accidental current appears, first rewrite the index in terms of the exact infrared charges and factor any free sector. Otherwise a formally correct inversion returns physically misidentified multiplets.

For noncompact targets or continuous spectra, the trace can depend on regulators and boundary conditions. Nonholomorphic completions and wall crossing may carry information that is absent from a formal power series. An inversion of the holomorphic piece alone must state that restriction Rastelli and Razamat 2017, §§2–3.

Exercises

Find every nonnegative spectrum compatible with $\mathcal I=2\mathcal I_A$ when $\mathcal I_B=-\mathcal I_A$.

Solution

The condition is $n_A-n_B=2$ with $n_A,n_B\ge0$. Hence $(n_A,n_B)=(2+k,k)$ for every integer $k\ge0$. Each increment adds the complete recombination pair $A\oplus B$ and leaves the index unchanged.

References

• Córdova, C., T. T. Dumitrescu, and K. Intriligator. “Multiplets of Superconformal Symmetry in Diverse Dimensions.” Journal of High Energy Physics 2019, no. 3 (2019): 163. DOI; Open PDF.
• Dolan, F. A., and H. Osborn. “On Short and Semi-Short Representations for Four-Dimensional Superconformal Symmetry.” Annals of Physics 307 (2003): 41–89. DOI; Open PDF.
• Rastelli, L., and S. S. Razamat. “The Supersymmetric Index in Four Dimensions.” In Localization Techniques in Quantum Field Theories, 261–305. Cham: Springer, 2017. DOI; Open PDF.