Supersymmetric and Superconformal Indices

A supersymmetric index is a graded trace that counts the cohomology of one chosen supercharge. States outside that cohomology cancel in boson–fermion pairs, making the result invariant under continuous deformations that preserve the trace and its asymptotics. The definition must include the Hilbert space, supercharge, commuting charges, spin structure, gauge projection, fugacity domain, and any supersymmetric Casimir factor.

Required background. Use Q-cohomology in the path integral and the representation-theoretic BPS shortening bounds.

Helpful background. Characters and multiplet counting organize the protected states after the trace is defined.

The cohomological trace

Choose a supercharge $Q$ and define the nonnegative operator

$$\delta=\{Q,Q^\dagger\}.$$

Let $F_i$ be mutually commuting conserved charges that also commute with $Q$ and $Q^\dagger$. The index is

$$\mathcal I(\mu_i) =\operatorname{Tr}_{\mathcal H} (-1)^F e^{-\beta\delta} \prod_i\mu_i^{F_i}.$$

For every state with $\delta>0$, the action of $Q$ or $Q^\dagger$ supplies a state of opposite fermion parity and identical $F_i$. Their contributions cancel. Equivalently,

$$\frac{d\mathcal I}{d\beta} =-\operatorname{Tr}(-1)^F \{Q,Q^\dagger\}e^{-\beta\delta} \prod_i\mu_i^{F_i}=0.$$

Only $\delta=0$ states, identified with $Q$-cohomology, remain. This argument requires a trace-class regulated operator or a discrete spectrum with controlled asymptotics. In a noncompact theory, scattering states can produce a boundary contribution and spoil naive $\beta$ independence Witten 1982, §§2–3.

The trace is not thermal: $(-1)^F$ and the charge insertions are essential, and periodic fermions are normally required around the Euclidean time circle.

A four-dimensional N=1 superconformal index

On $S^3\times\mathbb R$, choose $\bar Q_{\dot -}$ so that

$$\delta=E-2j_2-\frac32R.$$

Two independent combinations commuting with this supercharge lead to

$$\mathcal I(p,q;\mathbf y) =\operatorname{Tr}_{\mathcal H(S^3)} (-1)^F p^{j_1+j_2+R/2} q^{-j_1+j_2+R/2} \prod_a y_a^{F_a},$$

with only $\delta=0$ states contributing. The conventional convergence region is $|p|<1$ and $|q|<1$, followed by analytic continuation when justified Romelsberger 2006; Kinney et al. 2007, §3.

For a chiral multiplet of superconformal R-charge $r$ in representation $\mathcal R$, the single-letter contribution is

$$f_\chi(p,q,\mathbf z) =\frac{(pq)^{r/2}\chi_{\mathcal R}(\mathbf z) -(pq)^{1-r/2}\chi_{\overline{\mathcal R}}(\mathbf z)} {(1-p)(1-q)}.$$

The first numerator term is the scalar letter, the second is the equation-of-motion-subtracted fermion, and the denominator adds the two protected derivatives. A vector multiplet contributes

$$f_V(p,q,\mathbf z) =\frac{2pq-p-q}{(1-p)(1-q)} \chi_{\mathrm{adj}}(\mathbf z).$$

Multi-particle states are generated by the plethystic exponential

$$\operatorname{PE}[f] =\exp\!\left[ \sum_{n=1}^\infty\frac1n f(p^n,q^n,\mathbf z^n) \right].$$

Finally, gauge invariance is imposed by the Haar integral:

$$\mathcal I(p,q,\mathbf y) =\int_G d\mu(\mathbf z)\, \operatorname{PE}\!\left[ f_V+\sum_I f_{\chi_I} \right].$$

The R-symmetry entering this expression must be nonanomalous, every superpotential term must have R-charge two, and the global gauge group determines the integration domain and possible discrete sectors Dolan and Osborn 2009, §§2–3.

Free chiral and elliptic-gamma checks

For one free chiral with flavor fugacity $y$, the plethystic exponential resums to an elliptic Gamma function:

$$\mathcal I_\chi(p,q;y) =\Gamma_e\!\left((pq)^{r/2}y;p,q\right),$$

where

$$\Gamma_e(z;p,q) =\prod_{m,n\ge0} \frac{1-p^{m+1}q^{n+1}z^{-1}} {1-p^m q^n z}.$$

The identity

$$\Gamma_e(z;p,q)\, \Gamma_e(pq/z;p,q)=1$$

is the one-letter image of a bosonic and fermionic short contribution recombining and cancelling. It is a useful test of fugacity inversions and R-charge complements.

Path-integral normalization

The same index can be represented by a supersymmetric path integral on $S^3\times S^1$. Depending on the Hamiltonian and local-counterterm convention, the path integral may evaluate

$$Z_{S^3\times S^1} =e^{-\beta E_{\mathrm{susy}}}\mathcal I,$$

where $E_{\mathrm{susy}}$ is the supersymmetric Casimir energy. The prefactor is anomaly controlled in many schemes but is not part of the bare Hilbert-space index. Comparisons must either retain it on both sides or remove it consistently.

Gauge holonomies around $S^1$ implement the Haar projection. Nontrivial bundles, discrete theta angles, and one-form backgrounds can require additional sectors; the Lie algebra alone does not determine the answer.

What the index forgets

The index is invariant when short multiplets recombine into a long multiplet. It therefore determines a class of protected spectra modulo recombination, not an unambiguous list of short multiplets. It also loses generic OPE coefficients, long-multiplet dimensions, and most unprotected states.

Further limitations include:

• accidental R-symmetry mixing can invalidate ultraviolet fugacity assignments;
• a continuum can leave an asymptotic contribution not captured by discrete cohomology;
• poles and series expansions depend on the fugacity chamber;
• a vanishing index may mean exact cancellation, not absence of supersymmetric states;
• equality of indices is a strong protected-sector test, not by itself a full duality proof.

Exercises

Prove the elliptic-Gamma reflection identity from the infinite product.

Solution

Substituting $pq/z$ for $z$ interchanges the numerator and denominator factors: the numerator at $(m,n)$ becomes $1-p^m q^n z$, while the denominator becomes $1-p^{m+1}q^{n+1}z^{-1}$. Multiplication cancels every factor, giving one wherever both products converge; analytic continuation extends the identity.

References

• Dolan, F. A., and H. Osborn. “Applications of the Superconformal Index for Protected Operators and $q$-Hypergeometric Identities to $N=1$ Dual Theories.” Nuclear Physics B 818 (2009): 137–178. 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.
• Romelsberger, C. “Counting Chiral Primaries in $N=1$, $d=4$ Superconformal Field Theories.” Nuclear Physics B 747 (2006): 329–353. DOI; Open PDF.
• Witten, E. “Constraints on Supersymmetry Breaking.” Nuclear Physics B 202 (1982): 253–316. DOI.