Rigid Superconformal Properties

• Rigid superconformal properties are geometric and algebraic features that impose strict constraints on supersymmetric field theories in fixed background geometries, characterized by conformal Killing spinors.
• They are encoded in finite-dimensional Lie superalgebras combining conformal isometries, R-symmetries, and twistor spinors, leading to powerful classification frameworks and non-renormalization theorems.
• The rigid limit in conformal supergravity fixes key background fields like the metric and auxiliary tensors, enabling analytic control over partition functions, operator anomalies, and supersymmetric localization.

Rigid superconformal properties refer to geometric and algebraic features of supersymmetric field theories in fixed (non-dynamical) background geometries that admit nontrivial superconformal symmetry algebras. These properties arise in the context of superconformal field theories (SCFTs) formulated on various curved manifolds, and they tightly constrain the background metric, auxiliary fields, and allowed operator content, often leading to robust classification and powerful non-renormalization theorems. Rigid superconformal symmetry is characterized by the existence of conformal Killing spinors and the related structures induced on the underlying geometric background. This article surveys the notions, equations, and consequences of rigid superconformal properties across dimensions, with emphasis on modern developments and classification frameworks.

1. Geometric Foundations: Conformal Killing Spinors and Vectors

Rigid superconformal symmetry is realized when the background admits nontrivial solutions of conformal Killing spinor equations. In a general $m$-dimensional (pseudo-)Riemannian manifold $(M, g)$, a twistor (conformal Killing) spinor $\epsilon$ satisfies

$\nabla_\mu \epsilon = \tfrac{1}{m} \gamma_\mu (\slashed{\nabla} \epsilon)$

(see (Medeiros, 2012, Medeiros et al., 2013)). The Dirac current $v^\mu = \bar{\epsilon} \gamma^\mu \epsilon$ is a conformal Killing vector: $$\nabla_{(\mu} v_{\nu)} = \frac{1}{m} g_{\mu\nu} \nabla\cdot v$$ For minimal 5D rigid superconformal backgrounds, vanishing of the gravitino variation in conformal supergravity imposes the existence of such a conformal Killing vector, giving a necessary and sufficient condition for unbroken rigid supersymmetry: a nonzero conformal Killing vector on $(M^5,g)$ (Pini et al., 2015). In backgrounds where the $SU(2)_R$ curvature abelianizes, a transversally holomorphic foliation emerges in the four-plane orthogonal to $v$.

2. Algebraic Structures: Superconformal Symmetry Superalgebras

Rigid superconformal properties are encoded in finite-dimensional Lie superalgebras whose even part contains conformal isometries and constant R-symmetries, and whose odd part consists of twistor spinors valued in R-symmetry representations (Medeiros et al., 2013, Pini et al., 2015). The general structure is

$\mathfrak{g} = X^c(M) \oplus R \oplus F$

where $X^c(M)$ is the Lie algebra of conformal Killing vector fields, $R$ is the Lie algebra of R-symmetries, and $F$ the odd part (twistor spinors). The superbracket has the form:

• $[X_1, X_2] = \mathcal{L}_{X_1} X_2$
• $[r_1, r_2] = [r_1, r_2]_R$
• $[X, \epsilon] = \hat{L}_X \epsilon$, $[r, \epsilon] = r \cdot \epsilon$
• $$[\epsilon, \epsilon] = \xi_\epsilon + \rho_\epsilon$$ (with $\xi_\epsilon$ a conformal Killing vector and $\rho_\epsilon$ an R-symmetry generator) Jacobi identities generally require $m < 7$ for nontrivial generic solutions. In conformally flat cases, these superalgebras match those classified by Nahm and Kac (Medeiros et al., 2013).

3. Rigid Limits and Conformal Supergravity

The rigid limit refers to coupling matter multiplets to off-shell conformal supergravity backgrounds where dynamical gravity is decoupled and all geometric fields are fixed (metric, R-symmetry gauge fields, auxiliary tensors) (Pini et al., 2015). Precisely, one gauge-fixes dilatation and treats the Weyl multiplet fields as non-dynamical background data:

• $g_{\mu\nu}$, $V_\mu^{ij}$, $T_{\mu\nu}$, $D$ Under Weyl rescalings, the weights are $[g_{\mu\nu}] = 0$, $[V_\mu]=0$, $[T_{\mu\nu}] = 1$, $[D] = 2$, demonstrating rigidity. Supersymmetry variations close only when the conformal Killing spinor and vector conditions described above are met.

4. Classification of Rigid Superconformal Backgrounds

Rigid superconformal backgrounds are systematically classified by their conformal Killing spinor structure and associated geometric constraints. Key examples (Pini et al., 2015) include:

• $\mathbb{R}^5$: flat, maximal symmetry, multiple superconformal spinors
• $S^5$: admits Killing vectors, standard SYM localization backgrounds
• $\mathbb{R} \times S^4$: only conformal Killing, forbids constant Maxwell coupling
• Sasaki–Einstein 5-manifolds: Reeb Killing vector, transversally holomorphic foliation, standard YM allowed
• Topological twists: e.g., $\mathbb{R}\times M_4$ with $v=\partial_\tau$ Killing In all cases, the chain $$\epsilon^i \leftrightarrow v^\mu \leftrightarrow P_{(\mu\nu)}$$ determines whether the background admits a rigid superconformal (Chern–Simons-type, $g_{YM}$ position-dependent) theory or a rigid Poincaré theory (with standard Maxwell term) based on whether $v$ is conformal Killing (but not Killing) or Killing.

5. Rigidity, Abelianization, and Transversally Holomorphic Foliation

Whenever the background R-symmetry curvature becomes abelian ($R^Q_{\mu\nu}$ in $U(1) \subset SU(2)_R$) and the auxiliary connection $Q_\mu^{ij}$ admits a covariantly constant projector $H^i_j$, the structure induced on the horizontal 4-plane orthogonal to $v$ yields a transversally holomorphic foliation (THF) (Pini et al., 2015). This is characterized by the vanishing of certain commutators and cocycle conditions: $$D^Q_\mu H^i_j = \partial_\mu H^i_j + [Q_\mu, H]^i_j = 0, \quad [R^Q_{\mu\nu}, H] = 0$$ Such backgrounds support supersymmetric localization schemes that exploit holomorphic data on the leaf space.

6. Supersymmetry Constraints, Operator Content, and Non-Renormalization

Rigid superconformal symmetry drastically restricts the possible operator content, moduli, and anomalies in SCFTs. In $4d$ $\mathcal N=2$ theories, the bundle of Higgs-branch superconformal primaries over the conformal manifold admits a flat connection (vanishing Berry phase), indicating rigidity (Niarchos, 2018). Non-renormalization theorems for two- and three-point functions in protected sectors follow from shortening conditions. In $4d$ $\mathcal N=1$ theories, scale invariance in unitary, R-symmetric fixed points generically implies superconformality unless multiple dimension-two singlets exist (Antoniadis et al., 2011). The rigid geometric structure ensures that the trace anomaly is purely topological under conformal Killing spinor backgrounds (Cassani et al., 2013), and rigid supersymmetry is tightly linked to the cancellation of certain anomaly terms via superspace Weyl invariants.

7. Mathematical Characterization and Moduli Spaces

For supermanifolds, rigid superconformal structures are described via fatgraph combinatorics and Čech cocycles encoding odd deformations (Schwarz et al., 2023). Rigidity arises as vanishing of the cohomological dimension for odd moduli: $$\dim_\mathbb{C} H^1(F, L \oplus (L \otimes T_F)) = 4g-4 + 2n + r = 0$$ Results such as the classification theorem of Schwarz–Zeitlin provide explicit descriptions and examples (e.g., rigid $N=1$ super Riemann surfaces exist only for genus/NS/R-punctures satisfying $2g-2 + n + r/2 = 0$).

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In summary, rigid superconformal properties entail stringent algebraic and geometric conditions on the background geometry, auxiliary fields, operator content, and anomaly structure of supersymmetric field theories. They enable powerful classification results, provide the foundation for analytic (and sometimes localization-based) control over partition functions and observables, and link the physical theory to deep mathematical structures in differential geometry, representation theory, and moduli spaces. These properties underpin the modern theory of supersymmetric QFTs in curved backgrounds and their connections to string/M-theory and holography.

References: (Pini et al., 2015, Medeiros et al., 2013, Antoniadis et al., 2011, Niarchos, 2018, Cassani et al., 2013, Schwarz et al., 2023, Akhond et al., 2021, Medeiros, 2012)