Demystifying integrable QFTs in AdS: No-go theorems for higher-spin charges

TL;DR

This work investigates whether integrable quantum field theories can exist on AdS$_2$ by enforcing local higher-spin (HS) currents and their associated charges. It shows that in AdS$_2$ the isometry algebra is simple, which forces full conservation of HS currents and drastically constrains dynamics, unlike flat space where partial conservation is possible. Through detailed analysis of generalized free boson/fermion theories and Virasoro bulk CFTs, the authors derive Ward identities and sum rules, culminating in No-Go theorems: continuous deformations that preserve HS charges are generically impossible, both for boundary theories and bulk CFTs in AdS; long-range models likewise cannot host HS-preserving fixed points. The results together imply the nonexistence of HS-integrable QFTs in AdS$_2$ with local HS charges, guiding future explorations toward non-local charges or non-AdS geometries. The paper further develops sum rules linking bulk and boundary data and clarifies how HS symmetry organizes spectra into integer-spaced towers, with implications for bootstrap and holography.

Abstract

Higher-spin conserved currents and charges feature prominently in integrable 2d QFTs in flat space. Motivated by the question of integrable field theories in AdS space, we consider the consequences of higher-spin currents for QFTs in AdS$_2$, and find that their effect is much more constraining than in flat space. Specifically, it is impossible to preserve: (a) any higher-spin charges when deforming a free field of generic mass by interactions (even boundary-localized), or (b) any spin-4 charges when deforming a CFT by a Virasoro primary. Therefore, in these settings, there are no integrable theories in AdS with higher-spin conserved charges. Along the way, we explain how higher-spin charges lead to integer spacing in the spectrum of primaries, sum rules on the OPE data, and constraints on correlation functions. We also explain a key difference between AdS and flat space: in AdS one cannot `partially' conserve a higher-spin current along particular directions, since the AdS isometries imply full conservation. Finally, we describe the consequences of higher-spin symmetry breaking on the spectrum of long-range models.

Figures (8)

• Figure 1: Deforming contours both parallel and perpendicular to the boundary of AdS to obtain the action on a local boundary operator. The dashed line emphasizes that the integrand in \ref{['eq:Chargesspin2']} must vanish, up to a total derivative at most, when pushed to the boundary away from operator insertions. Then, a countour ending on the boundary can be deformed as in the right panel.
• Figure 2: In blue, a compact surface $\Sigma$ leading to a higher-spin charge with potentially divergent segments that approach the boundary. Bulk and boundary operators are denoted by black dots.
• Figure 3: On the left, the same contour as Figure \ref{['fig:befordrop']}, with the horizontal segments replaced by their endpoint contributions. On the right, the countours are further pushed towards the boundary, leading to a finite result once the appropriate counterterms are included.
• Figure 4: Poincaré AdS configuration to compute the matrix element of a charge between two boundary operators $\mathcal{O}_1$ and $\mathcal{O}_2$. The bulk current $T$ is integrated over its $y$ coordinate.
• Figure 5: Witten diagram describing the action of the momentum generator $P$ on the two boundary operators. The black lines denote bulk-boundary propagators and the blue line denotes the region of the co-dimension 1 integral over the position of $T_{xx}$.