The Superconformal Index¶

The superconformal index is the Witten index of a [[Superconformal Field Theory|superconformal field theory]] in the radial quantization.

\(\mathcal{N} = 1\)¶

The superconformal index is schematically

\[ \mathcal{I}(\mu_i) = \text{Tr}\: (-1)^F x^\delta \mu_i^{\mathcal{M}_i} \]

where \(\delta = \frac{1}{2} \{Q, Q^\dagger\}\). Only \(\delta = 0\) states can contribute, and so the index is independent of \(x\). The Fermi-Bose cancellation can only occur if the fugacities are those for which the charges \(\mathcal{M}_i\) commute with the supercharge \(Q\). This theory has an infinite number of states with \(\delta = 0\), and in fact an infinite number of single letter states with \(\delta = 0\). The fugacities help to regulate this divergence.

The supercharges are \(\{ Q_\alpha, S^\alpha \equiv Q^{\dagger \alpha}, \tilde{Q}_{\dot{\alpha}}, \tilde{S}^{\dot{\alpha}} \equiv \tilde{Q}^{\dagger \dot{\alpha}} \}\), where \(\alpha = \pm\) and \(\dot{\alpha} = \dot{\pm}\) are \(SU(2)_{1}\) and \(SU(2)_2\) indices, respectively, with \(SU(2)_1 \times SU(2)_2 = \text{Spin}(4)\) the isometry group of \(S^3\). The superconformal algebra satisfies:

\[ \{Q_\alpha, Q^{\dagger \beta}\} = \Delta + 2 M_{\alpha}^{\beta} + \frac{3}{2} r, \quad \{\tilde{Q}_{\dot{\alpha}}, \tilde{Q}^{\dagger \dot{\beta}}\} = \Delta + 2 \tilde{M}_{\alpha}^{\beta} - \frac{3}{2} r \]

where \(\Delta\) is the conformal dimension, \(M\) and \(\tilde{M}\) are the \(SU(2)_1\) and \(SU(2)_2\) generators, and \(r\) is the generator of the \(U(1)_r\) R-symmetry. Here \(Q\) has \(r = -1\) and \(\tilde{Q}\) has \(r = 1\), with daggers flipping the sign of \(r\).

Choosing \(Q \equiv Q_-\) to define the index, we have \(\delta = \Delta - 2j_1 + \frac{3}{2} r\). \(Q\) commutes with \(SU(2, 1)\) within the superconformal algebra \(SU(2, 2|1)\), which has rank \(2\). We choose the charges \(\mathcal{M}_i = \frac{1}{3} (\Delta + j_1) \pm j_2\) by convention. The index is then:

\[ \mathcal{I}(p, q) \equiv \text{Tr}\: (-1)^F p^{\frac{1}{3} (\Delta + j_1) + j_2} q^{\frac{1}{3} (\Delta + j_1) - j_2} = \text{Tr}\: (-1)^F p^{j_1 - \frac{1}{2} r + j_2} q^{j_1 - \frac{1}{2} r - j_2} \]

because \(\delta = \Delta - 2j_1 + \frac{3}{2} r\), but only \(\delta = 0\) states contribute.

The Chiral Multiplet¶

Consider the free chiral multiplet \(\Phi\) obeying \(\bar{Q} \Phi = 0\), which has superspace expansion

\[ \Phi = \phi + \sqrt{2} \theta \phi + i \theta^\dagger \bar{\sigma}^\mu \theta \partial_\mu \phi \]

The r-charge of \(\phi\) is \(\frac{2}{3}\). The spectrum of local operators are constructed from our chiral multiplet and its derivatives, and so our creation operators are \(\phi, \partial_\mu \phi, \partial_\mu \partial_\nu \phi, \dots, \psi, \partial_\mu \psi, \partial_\mu \partial_\nu \psi, \dots\) and their conjugates, though note \(\partial_\mu \partial^\mu \phi = 0\) and \(\partial_{\alpha \dot{\alpha}} \phi^{\alpha} = 0\). We can now tabulate the letters and compute the single letter index:

Letter	\(\Delta\)	\(j_1\)	\(j_2\)	\(r\)	\(\delta\)	\(\mathcal{I}\)
\(\phi\)	1	0	0	2/3	2	-
\(\psi_+\)	3/2	1/2	0	-1/3	0	\(-(pq)^{2/3}\)
\(\psi_-\)	3/2	-1/2	0	-1/3	2	-
\(\partial_{\dot{+}}^{\alpha} \psi_{\alpha}\)	5/2	0	1/2	-1/3	2	-
\(\partial_{\dot{-}}^{\alpha} \psi_{\alpha}\)	5/2	0	-1/2	-1/3	2	-
\(\Box \phi\)	3	0	0	2/3	4	-
\(\bar{\phi}\)	1	0	0	-2/3	0	\((pq)^{1/3}\)
\(\psi_{\dot{+}}\)	3/2	0	1/2	1/3	2	-
\(\psi_{\dot{-}}\)	3/2	0	-1/2	1/3	2	-
\(\partial_{+}^{\dot{\alpha}} \psi_{\dot{\alpha}}\)	5/2	1/2	0	1/3	2	-
\(\partial_{-}^{\dot{\alpha}} \psi_{\dot{\alpha}}\)	5/2	-1/2	0	1/3	4	-
\(\Box \bar{\phi}\)	3	0	0	-2/3	2	-
\(\partial_{+\dot{+}}\)	1	1/2	1/2	0	0	p
\(\partial_{-\dot{+}}\)	1	-1/2	1/2	0	0	q
\(\partial_{+\dot{-}}\)	1	1/2	-1/2	0	2	-
\(\partial_{-\dot{-}}\)	1	-1/2	-1/2	0	2	-

and so

\[ i_\phi(p, q) = \frac{(pq)^{1/3} - (pq)^{2/3}}{(1-p)(1-q)} \]

If we refine this with the fugacities of the Cartan generators, then we have:

\[ i_\phi(a_i; p, q)= \frac{(pq)^{1/3} \chi_{\bar{R}}(a_i) - (pq)^{2/3} \chi_R (a_i)}{(1-p)(1-q)} \]

The full index, obtained via plethystic exponentiation, is

\[ \mathcal{I}_\phi(a; p, q) = \Gamma\left( (pq)^{1/2} a^{-1}; p, q \right) \]

where \(\Gamma(z; p, q)\) is the elliptic gamma function.

The Superpotential¶

A superpotential interaction will change the \(R\)-charge of the chiral multiplet, which is determined by requiring that the superpotential itself have \(R\)-charge \(2\). If the chiral multiplet has \(R\)-charge \(r\), then the right-handed fermion \(\psi_+\) and the conjugate boson \(\bar{\phi}\) (the only two particles contributing to the index) will have \(R\)-charges \(r-1\) and \(-r\), respectively. Therefore, the index will be:

\[ i_{\phi r}(p, q) = \frac{(pq)^{r/2} - (pq)^{1-r/2}}{(1-p)(1-q)} \implies \mathcal{I}_{\phi r}(a; p, q) = \Gamma\left( (pq)^{r/2} a ; p, q\right) \]

For instance, a massive chiral multiplet has \(W(\Phi) = m \Phi^2\), which fixes the \(\phi\) \(R\)-charge at \(1\), giving \(\mathcal{I}_{\phi 1}(a; p, q) = \Gamma\left((pq)^{1/2}; p, q\right) = 1\), which is consistent with the expectation that the massive theory have a unique supersymmetric ground state. In fact, a theory with two chiral multiplets and \(W(\Phi_1, \Phi_2) = m \Phi_1 \Phi_2\) will also have an index of identically \(1\).

Consider another example with linear superpotential \(W(\Phi) = \eta \Phi\), which spontaneously breaks supersymmetry. The \(R\)-charge of \(\Phi\) is necessarily \(2\), and so we have \(\mathcal{I}_{\phi 2}(p, q) = \Gamma(pq; p, q) = 0\). This is sensible, as the \(\phi_+ \vert 0 \rangle\) state will have the same energy as \(\vert 0 \rangle\), and so the ground state drops out of the index. This also means that \(\phi_+\) acts as a Goldstino mode for spontaneous supersymmetry breaking. Any such theory has a neutral chiral multiplet with \(R\)-charge \(2\), and therefore vanishing superconformal index.

The Vector Multiplet¶

We are also generically interested in gauge theory and therefore the vector multiplet. The letters with \(\delta = 0\) are:

Letter	\(\Delta\)	\(j_1\)	\(j_2\)	\(r\)	\(\mathcal{I}\)
\(\bar{\lambda}_{\dot{+}}\)	3/2	0	1/2	-1	-p
\(\bar{\lambda}_{\dot{-}}\)	3/2	0	-1/2	-1	-q
\(F_{++}\)	2	1	0	0	pq
\(\partial_{+}^{\dot{\alpha}} \bar{\lambda}_{\dot{\alpha}}\)	5/2	1/2	0	-1	pq
\(\partial_{+\dot{+}}\)	1	1/2	1/2	0	p
\(\partial_{-\dot{+}}\)	1	-1/2	1/2	0	q

The single letter index is therefore:

\[ i_V(a_i; p, q) = \frac{-p -q + 2pq}{(1-p)(1-q)} \chi_{\text{adj}}(a_i) = \left( -\frac{p}{1-p} - \frac{q}{1-q} \right) \left( \sum_{\alpha} a^{\alpha} + N \right) \]

Gauge Theory¶

In gauge theories, we are only interested in operators which transform trivially under the gauge group, and so the index is:

\[ \mathcal{I}(b_k; p, q) = \frac{1}{\vert W \vert} \oint \left(\prod_{i=1}^{N} \frac{da_i}{2\pi i a_i}\right) \Delta(a_i) \mathcal{I}_V(a_i; p, q) \prod_{\phi_i} \mathcal{I}_{\phi_i}(a_i, b_k; p, q) \]

where \(a_i\) are the gauge symmetry fugacities and \(b_k\) are the fugacities of the other symmetries under which the chiral multiplets transform. However:

\[ \Delta(a_i) \mathcal{I}_V(a_i; p, q) = \text{PE}\left[ \left( -\frac{p}{1-p} - \frac{q}{1-q} \right) \chi_{\text{adj}}(a_i) - \sum_{\alpha} a^{\alpha} \right] = \kappa^N \prod_{\alpha} \Gamma(pqa^\alpha; p, q) \]

for

\[ \kappa \equiv (p, p)(q, q), \quad (a, q) \equiv \prod_{i=0}^{\infty} (1-aq^i) \]

We then have:

\[ \mathcal{I}(b_k; p, q) = \frac{\kappa^N}{\vert W \vert} \oint \frac{da_i}{(2\pi i)^N} \Gamma(pqa^\alpha; p, q) \prod_{\phi_i} \prod_{\rho \in R_i^G, \rho' \in {R'}_{i}^{G}} \Gamma\left( (pq)^{r_i/2} a^\rho b^{\rho'} ; p, q\right) \]

For instance, consider \(\mathcal{N} = 1\) \(SU(2)\) gauge theory with three flavors of (anti-)quarks in the (anti-)fundamental representation. The right \(R\) charges are \(1/3\), and so the index is

\[ \mathcal{I} = \kappa \oint \frac{dz}{4\pi i z} \frac{1}{\Gamma(z^{\pm 2}; p, q)} \prod_{i=1}^{3} \Gamma\left( (pq)^{1/6} b u_i z^{\pm 1} \right) \Gamma\left( (pq)^{1/6} b^{-1} v_i z^{\pm 1}l p, q \right) \]

where \(\prod_{i=1}^{3} u_i = \prod_{i=1}^{3} v_i = 1\), where these fugacities \(u_i\) and \(v_i\) parameterize the \(SU(3)_u \times SU(3)_v\) flavor symmetry, while \(b\) parameterizes the baryonic \(U(1)_b\). This integral happens to have closed form solution

\[ \mathcal{I} = \prod_{i < j} \Gamma\left( (pq)^{1/3} t_i t_j \right), \quad \left\{ t_i \right\} = \left\{ b u_i, b^{-1} v_i \right\} \]

This is equivalent to the index of a theory with fifteen chiral multiplets and a superpotential yielding and \(R\)-charge of \(2/3\). Indeed, there is a duality between gauge theory and a theory of only chiral multiplets.