The conformal group of \(\mathbb{R}^{1,3}\) is \(SO(2,4)\).

\[ \begin{aligned}\\ [D,P_\mu] &= P_\mu\\ [D,K_\mu] &= -K_\mu\\ [M_{\mu\nu},P_\rho] &= \eta_{\nu\rho}P_\mu-\eta_{\mu\rho}P_\nu\\ [M_{\mu\nu},K_\rho] &= \eta_{\nu\rho}K_\mu-\eta_{\mu\rho}K_\nu\\ [K_\mu,P_\nu] &= 2\eta_{\mu\nu}D-2M_{\mu\nu}\\ \end{aligned} \]

Adding fermionic generators to \(\mathfrak{so}(2,4)\) and an abelian \(\mathfrak{u}(1)_R\) gives \(\mathfrak{su}(2,2|1)\), of superdimension \((16|8)\):

\( Q^\dagger{}^\alpha=S^\alpha\) and \(\bar Q^\dagger{}^{\dot\alpha} =\bar S^{\dot\alpha}\): four real supercharges of each type.

Note: \(\alpha\in\{+,-\}\) and \(\dot\alpha\in\{\dot{+},\dot{-}\}\)

Introduce new (anti)commutators to close the conformal algebra:

\[ \begin{aligned} \{Q_\alpha,\bar Q_{\dot\alpha}\} &= 2\sigma^\mu_{\alpha\dot\alpha}P_\mu\,, &\qquad \{\bar S^{\dot\alpha},S^{\alpha}\} &= 2\bar\sigma_\mu^{\dot\alpha\alpha}K^\mu\,,\\ [D,Q_\alpha] &= \tfrac12 Q_\alpha\,, & [D,S^\alpha] &= -\tfrac12 S^\alpha\,,\\ [R,Q_\alpha] &= -Q_\alpha\,, & [R,\bar Q_{\dot\alpha}] &= \bar Q_{\dot\alpha}\,, \end{aligned} \]

\[ \{Q_\alpha,S^{\beta}\}\;=\;4M_\alpha{}^{\beta} +2\delta_\alpha^\beta\big(D+\tfrac32 R\big)\,. \]

Pick a Poincaré supercharge \(\mathcal{Q}\equiv\bar{Q}_{\dot -}\) and set \(\bar M_{\dot -}{}^{\dot -}=-j_2\),

\[ \delta \;:=\; \tfrac12 \{\mathcal{Q},\mathcal{Q}^\dagger\}\;=\;E-2j_2-\tfrac32 R\;\ge\;0\,. \]

So-called short multiplets are annihilated by \(\mathcal{Q}\).

where \(v_a\) are fugacities for the global (flavour) charges \(Q_a\) commuting with \(\mathcal{Q}\).

\(p,q\) are conjugate to combinations that commute with \(\mathcal{Q}\) since \(\mathcal{Q}\) carries \((j_1,j_2,R)=(0,-\tfrac12,1)\)

Every \(\delta>0\) state pairs with a partner of opposite \((-1)^F\) and identical \((p,q,\vec v)\) charges, obtained by acting with \(\mathcal{Q}\) or \(\mathcal{Q}^\dagger\). Those pairs cancel identically.

\(\mathcal{I}\) is protected: independent of exactly marginal couplings, computable at weak coupling, invariant along the RG flow.

Chiral multiplet

Bifundamental \(\square\bar\square\) of \(SU(N)^2\) with R-charge \(r\) — a product of elliptic gamma functions:

\[ \prod_{i,j=1}^{N}\Gamma\big(t^{3r}z_i/w_j;p,q\big) \]

\(t=(pq)^{1/6}\) and \(\{z_i\},\{w_j\}\) unimodular with \(\prod_k z_k=\prod_k w_k=1\) parametrize the two Cartan subalgebras.

Vector multiplet

Adjoint of \(SU(N)\) — a Haar measure times elliptic gamma functions:

\[ \begin{aligned} \oint\!\mu(z;p,q) &:= \frac{\kappa^{N-1}}{N!} \oint_{\mathbb{T}_{N-1}}\prod_{i=1}^{N-1}\frac{dz_i}{2\pi i z_i}\\ &\quad\times\prod_{1\le k<l\le N}\frac{1}{\Gamma(z_k/z_l,z_l/z_k;p,q)} \end{aligned} \]

\(\mathbb{T}_{N-1}\) is \(N-1\) unit circles taken counterclockwise, and \(\kappa=(p;p)_\infty(q;q)_\infty\) with \((a;b)_\infty=\prod_{k\ge0}(1-ab^k)\).

The full \(4d\ \mathcal{N}=1\) Lagrangian is fixed by the quiver plus one holomorphic function:

\[ \begin{aligned} \mathcal{L} \;=\;& \int\! d^4\theta\sum_{X_{ij}}\bar X_{ij}\,e^{V_i}X_{ij}e^{-V_j}\\ &+\Big(\int\! d^2\theta\Big[\tfrac{1}{4g^2}\sum_k\mathrm{Tr}(\,\mathcal{W}^{\alpha}_k\mathcal{W}_{\alpha k}) +W(X)\Big]+\mathrm{c.c.}\Big) \end{aligned} \]

Each monomial of \(W\) is an oriented closed loop of the quiver, its gauge indices contracted around the loop,

\[ X_{i_1i_2}X_{i_2i_3}\cdots X_{i_ki_1}\,. \]

Toric condition. Every chiral multiplet appears in exactly two monomials — once with \(+\), once with \(-\).

The vacuum moduli space solves the \(F\)-terms \(\partial W/\partial X_{ij}=0\) together with the \(D\)-terms; for special abelian theories this mesonic moduli space \(\mathcal{M}_{\rm mes}\) is a toric Calabi–Yau 3-fold.

At the superconformal fixed point the R-charges obey, for every node \(k\) and every superpotential term \(W_a\),

\[ \sum_{X\,\in\,\partial k}\big(1-r_X\big)=2\,, \qquad \sum_{X\in W_a}r_X=2\,, \]

the first being the vanishing NSVZ beta function at node \(k\), the second exact marginality of \(W\). The residual family of solutions is fixed by a-maximization,

\[ a(R)=\tfrac{3}{32}\big(3\,\mathrm{Tr}R^3-\mathrm{Tr}R\big)\,, \]

after which chiral primaries have \(\Delta=\tfrac32 R\).

For the doublet \(2a\)/\(2b\) of \(H_{1,1,2,1}\), the R-charges of ten fields are exchanged in pairs and the other nine are untouched:

\(R_1,\dots,R_6\) are the R-charges of the six extremal GLSM fields (the corners of the toric diagram), with \(\sum_a R_a = 2\).

The set of R-charges is the same on both sides. This is not true in general. It is not a local move: it is tied to the global structure of the quiver, so it cannot be written as a local elliptic gamma identity.

What we have

• Toric phases of a brane tiling that share a common quiver but differ in \(W\) — five doublets and one triplet for \(H_{1,1,2,1}\).
• Their superconformal indices coincide as elliptic hypergeometric integrals, giving an exact involution on the R-charges.
• Unlike Rains's theorem, these identities are not local: they are tied to the global structure of the quiver.
• A recipe for genereting infinitely many global constraints. Look at larger brane-tiling for further identities (eg. \(PdP_{6c}\), \(H^{1,3,1,2}\), \(H^{1,1,2,2}\), etc).

Where it points

• A new way of constraining the functional form of the index: any candidate closed form, written as a combination of special functions, must reproduce these identities.
• Compute the residue sum explicitly — following van Leuven–Mathieson–Roy and Pan–Peelaers — and extend the identities to that representation.
• The triplet points to quiver-invariant dualities beyond the doublet pattern.