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 already identifies distinct nonnegative spectra, and discarded fugacities can create further identifications. A defensible inversion therefore reports recombination-invariant combinations, the convergence domain, and the remaining kernel—not merely one convenient list of multiplets.

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

Helpful background. Conformal characters supply ordinary character decompositions; the protected problem adds a recombination quotient.

The index is a map, not a spectrum

Fix a superconformal algebra and let $K_{\mathrm{short}}$ be the free Abelian group generated by its irreducible short multiplets. Sending each generator to its index character defines a homomorphism

$$\Phi:K_{\mathrm{short}}\longrightarrow\mathscr R, \qquad [\mathcal M]\longmapsto\mathcal I_{\mathcal M},$$

where $\mathscr R$ is the declared ring of formal series or meromorphic functions in the fugacities. When a long multiplet reaches a unitarity threshold,

$$\mathcal L\big|_{\Delta=\Delta_*} =\mathcal M_1\oplus\cdots\oplus\mathcal M_s,$$

its total index vanishes. If $R_{\mathrm{rec}}$ is generated by all vectors $[\mathcal M_1]+\cdots+[\mathcal M_s]$, then

$$R_{\mathrm{rec}}\subseteq\ker\Phi, \qquad \Phi:\frac{K_{\mathrm{short}}}{R_{\mathrm{rec}}} \longrightarrow\mathscr R.$$

The inclusion need not be an equality. Setting flavor fugacities to one, identifying two fugacities, or taking a specialized limit can make previously distinct characters coincide and enlarge $\ker\Phi$. Complete shortening and recombination rules are therefore necessary, but not sufficient, input for inversion Córdova, Dumitrescu, and Intriligator 2019, §§2–4.

A physical spectrum is a vector of nonnegative multiplicities in $K_{\mathrm{short}}$. Coordinates in the quotient can be signed: positivity of microscopic multiplicities does not imply positivity of every index coefficient.

A four-dimensional N=1 recombination channel

In this representation-theoretic section, $j_1,j_2\in\tfrac12\mathbb Z_{\geq0}$ denote the highest weights, or spins, of the two Lorentz $SU(2)$ factors. They are not the state-by-state Cartan eigenvalues denoted by the same letters in the index trace on the preceding page.

In the notation of Rastelli and Razamat, one left-shortening threshold with $j_1>0$ is

$$\mathcal A^{\,2+2j_1-\frac32r}_{r(j_1,j_2)} \longrightarrow \mathcal C_{r(j_1,j_2)} \oplus \mathcal C_{r-1(j_1-\frac12,j_2)}.$$

The conjugate right-shortening rule, for $j_2>0$, is

$$\mathcal A^{\,2+2j_2+\frac32r}_{r(j_1,j_2)} \longrightarrow \overline{\mathcal C}_{r(j_1,j_2)} \oplus \overline{\mathcal C}_{r+1(j_1,j_2-\frac12)}.$$

The endpoint $j_1=j_2=r=0$ is especially important:

$$\mathcal A^{2}_{0(0,0)} \longrightarrow \widehat{\mathcal C}_{(0,0)} \oplus\mathcal B_{-2(0,0)} \oplus\overline{\mathcal B}_{2(0,0)}.$$

$\widehat{\mathcal C}_{(0,0)}$ contains a conserved current. $\mathcal B_{-2(0,0)}$ contains a dimension-three chiral primary and its marginal F-term; $\overline{\mathcal B}_{2(0,0)}$ is the conjugate multiplet. The right-handed index used on the preceding page sees $\overline{\mathcal B}_{2(0,0)}$ and $\widehat{\mathcal C}_{(0,0)}$, whose characters cancel; the left-handed index sees the conjugate pair. Thus either handedness determines only

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

after descendants and nontrivial spin characters at the same order have been removed.

More generally, for the left-handed index set $\widetilde r=2j_1-r$. The short multiplets with fixed $(\widetilde r,j_2)$ split into two classes according to whether $j_1$ is integral or half-integral, and their characters obey

$$\mathcal I^L_{[\widetilde r,j_2]_+} =-\mathcal I^L_{[\widetilde r,j_2]_-} =(-1)^{2j_2+1} \frac{(pq)^{(\widetilde r+2)/2}\chi_{j_2}(p/q)} {(1-p)(1-q)}.$$

The unitarity bounds leave finitely many representatives in each such class. The coefficient that can be extracted is the net degeneracy $\#(+)-\#(-)$; the right-handed statement is its conjugate Rastelli and Razamat 2017, §5, pp. 276–280.

This difference can equal the local complex dimension of the conformal manifold when the marginal operators have no further obstructions and the relevant continuous global symmetries are broken at a generic point. In general, the conformal manifold is locally a quotient of marginal couplings by the complexified continuous symmetry group, so the separate current content still matters Green et al. 2010, §§2–3.

Why nonnegative spectra remain nonunique

Suppose two short characters obey $\mathcal I_B=-\mathcal I_A$. A spectrum with multiplicities $(n_A,n_B)$ has

$$\mathcal I=(n_A-n_B)\mathcal I_A.$$

If the observed coefficient is two, every spectrum

$$(n_A,n_B)=(2+k,k), \qquad k\in\mathbb Z_{\ge0},$$

is allowed. Each increment adds a complete recombination package. Positivity restricts the affine kernel to a cone, but does not select one point in that cone.

This also explains why a negative coefficient in a plethystic logarithm is not automatically “one relation.” It may arise from a fermionic generator, an equation of motion, a recombination subtraction, or a genuine relation. A Hilbert-series interpretation requires an independently established finitely generated cohomological ring.

Formal sieves and integral kernels

Choose quotient characters $\chi_a(p,q,\mathbf y)$ that are linearly independent in one common series domain. Then

$$\mathcal I=\sum_a N_a\chi_a.$$

There are two distinct operations often called inversion:

• A formal sieve orders characters by their leading monomial, reads the first coefficient, subtracts that character, and iterates. For a four-dimensional index it is often useful first to form $(1-p)(1-q)(\mathcal I-1)$ and decompose each order into $SU(2)$ and flavor characters.
• An integral inversion assumes a dual family $K_a$ and contour $\Gamma$ such that

$$N_a=\oint_\Gamma d\mu(x)\,K_a(x)\mathcal I(x).$$

The second formula is not universal. It requires a specified measure, linear independence, a common convergence annulus, and control of every pole crossed during analytic continuation. A meromorphic function can have different Laurent expansions in different annuli; mixing their coefficients is not an inversion.

Even a successful operation returns quotient coefficients $N_a$. It yields an individual multiplet multiplicity only when representation theory proves that the relevant equivalence class has a single unitary representative.

Finite-order reconstruction as an integer problem

Suppose the index is known through filtration degree $N$. A reproducible analysis is:

1. fix the exact infrared R-symmetry, all flavor refinements, and one convergence chamber;
2. list every short character whose leading term can contribute through degree $N$;
3. impose all recombination relations that affect those terms, including relations whose other members begin later;
4. reduce the character matrix to an independent quotient basis;
5. solve the coefficient equations over the integers, not floating point;
6. intersect the solution set with nonnegative microscopic multiplicities;
7. report fixed combinations and a basis for the surviving kernel.

At finite order, two characters with identical initial terms may separate beyond the truncation. Such a degeneracy is “unresolved to order $N$,” not an exact recombination relation. Conversely, omitting a character that first contributes at order $N$ can manufacture false uniqueness.

R-symmetry mixing, specializations, and continua

If $R_{\mathrm{IR}}=R_0+\sum_As_AF_A$, the right-handed four-dimensional index is regraded by

$$y_A\longmapsto(pq)^{s_A/2}y_A.$$

An accidental current or a decoupled free operator can therefore move terms between apparent R-charge orders. One must regrade with the exact infrared symmetry and factor known free sectors before identifying multiplets.

Unrefining also loses information. For example, an $SU(2)$ doublet has character $z+z^{-1}$, which becomes two at $z=1$ and is then indistinguishable from two singlets. Keeping every available flavor fugacity makes the kernel smaller, though never smaller than recombination requires.

For an ordinary unitary SCFT quantized on compact $S^3$, radial quantization gives a discrete spectrum, so no continuum correction is expected. Continuum and regulator caveats instead apply to noncompact target spaces, supersymmetric quantum mechanics, defect sectors, and generalized indices with asymptotic scattering states. In those settings a holomorphic piece may omit nonholomorphic spectral-asymmetry data, and inversion must state which regulated observable is being expanded.

What can be concluded

In decreasing order of strength, an index can provide:

• an exact multiplicity when a refined quotient class has one allowed unitary representative;
• a recombination-invariant net degeneracy such as marginal operators minus currents;
• bounds after nonnegativity, symmetry, anomaly, or independent OPE data are imposed;
• a consistency check between proposed dual descriptions.

It does not, by itself, determine long-multiplet dimensions, OPE coefficients, a unique short spectrum, or a full duality.

Exercises

1. Solve a recombination ambiguity

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

Solution

The coefficient equation is $n_A-n_B=2$ with $n_A,n_B\ge0$. Hence

$$(n_A,n_B)=(2+k,k), \qquad k\in\mathbb Z_{\ge0}.$$

Each increment adds the index-zero package $A\oplus B$.

2. Marginals and currents

A right-handed index gives net degeneracy $N_{\mathrm{marginal}}-N_{\mathrm{current}}=3$. List all nonnegative pairs and state what extra hypothesis makes three the conformal-manifold dimension.

Solution

All pairs are

$$(N_{\mathrm{marginal}},N_{\mathrm{current}})=(3+k,k), \qquad k\in\mathbb Z_{\ge0}.$$

The conformal-manifold dimension is three if the counted marginal operators have no additional obstructions and the relevant continuous symmetries are broken at a generic point, so the quotient by the complexified symmetry group removes precisely the current directions.

3. Information lost by unrefining

Show that setting an $SU(2)$ fugacity to one makes one doublet indistinguishable from two singlets.

Solution

The characters are $\chi_{1/2}(z)=z+z^{-1}$ and $\chi_0(z)=1$. At $z=1$,

$$\chi_{1/2}(1)=2=2\chi_0(1).$$

The refined characters are linearly independent, but their unrefined values are not. This is an additional kernel caused by specialization, unrelated to recombination.

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.
• Green, D., Z. Komargodski, N. Seiberg, Y. Tachikawa, and B. Wecht. “Exactly Marginal Deformations and Global Symmetries.” Journal of High Energy Physics 2010, no. 6 (2010): 106. 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.