Carrollian Limit of OPE Blocks

• Carrollian OPE blocks are a reformulation of operator-product expansions where traditional CFT structures are contracted to accommodate null-boundary geometries and infinite soft towers.
• They are formulated via distinct methods—including projector, shadow-integral, and collinear smearing approaches—that efficiently capture heavy-light dynamics and simplify descendant structures.
• The framework bridges flat holography and celestial amplitudes, providing a unified mechanism to map bulk AdS correlators to Carrollian amplitudes and revealing deep connections with BMS-type symmetries.

The Carrollian limit of OPE blocks is the ultra-relativistic reorganization of operator-product data under Carrollian kinematics, null-boundary geometry, or BMS-type contractions. In the literature it appears in several closely related forms rather than as a single universal construction. In two-dimensional Carrollian/Galilean conformal field theories, OPE blocks arise in the conformal-block decomposition of four-point functions and simplify sharply in large-central-charge regimes, especially in heavy-light configurations where the heavy backreaction is absorbed into a Carrollian conformal coordinate transformation (Hao et al., 29 Oct 2025). In higher-dimensional Carrollian and celestial settings, OPE blocks are realized as short-distance or collinear expansions at null infinity, as smeared integrals over null generators, as light- or shadow-transformed power-law blocks, or as towers of $\mathfrak{sl}(2,\mathbb{C})$ modes obtained after the Carrollian contraction of ordinary CFT blocks (Nguyen et al., 19 Mar 2025, Mason et al., 2023, Banerjee et al., 2024, Gioia et al., 27 Aug 2025).

1. Carrollian contraction, null boundary geometry, and the meaning of the limit

A standard Carrollian contraction is implemented by introducing coordinates $(t,z,\bar z)$, scaling

$t = c\,u,$

and then taking $c\to 0$ with $u$ fixed. In the formulation developed for CFT$_3$, this contracts $\mathfrak{so}(3,2)$ to the conformal Carroll algebra $\mathfrak{ccarr}(3)$, which is stated to be isomorphic to BMS$_4$ (Gioia et al., 27 Aug 2025). In flat-holographic constructions the same limit appears as the boundary avatar of the bulk flat limit: in AdS$_4$ Bondi coordinates, the boundary metric becomes degenerate when $(t,z,\bar z)$0, and the relation $(t,z,\bar z)$1 identifies the flat bulk limit with a Carrollian boundary limit (Alday et al., 2024).

This kinematical contraction does not merely suppress time dependence. The cited works emphasize that it reorganizes operator representations, descendant towers, and singularity structure. In particular, the null direction becomes special: retarded time or Carrollian time is either ultralocal, integrated over, or converted into mode data, depending on the realization. A plausible implication is that “Carrollian limit of OPE blocks” should be read as a family of procedures adapted to null infinity rather than a single canonical limit.

In two-dimensional Carrollian/Galilean CFTs, often described as BMS field theories, the same ultra-relativistic logic governs the conformal-block problem, but now directly in a theory whose symmetry algebra is already Carrollian/Galilean rather than obtained from a higher-dimensional parent (Hao et al., 29 Oct 2025). In that setting the limit is less a contraction of an ambient relativistic block and more a semiclassical regime of an intrinsically BMS-type block decomposition.

2. OPE blocks in Carrollian theories: projector, shadow, and smeared formulations

In the two-dimensional C/G setting, a four-point function such as

$(t,z,\bar z)$2

is decomposed by inserting a projector onto the family of a primary $(t,z,\bar z)$3,

$(t,z,\bar z)$4

with descendants labeled by $(t,z,\bar z)$5. The corresponding conformal block is

$(t,z,\bar z)$6

so the OPE block is the sum over the entire descendant family in the exchanged channel (Hao et al., 29 Oct 2025).

In Carrollian CFT on $(t,z,\bar z)$7, the OPE is formulated more generally as

$(t,z,\bar z)$8

where the “subleading” sector includes BMS-supertranslation descendants and the “massive” sector is required because the product of two generic massless states has non-null total momentum (Nguyen et al., 19 Mar 2025). At finite separation, the corresponding OPE block is written as

$(t,z,\bar z)$9

with $t = c\,u,$0 a Carrollian three-point function with shadow dimensions and $t = c\,u,$1 a symmetry-invariant integration domain (Nguyen et al., 19 Mar 2025).

A third realization appears in Carrollian amplitudes derived from collinear limits. There the OPE is explicitly smeared along a generator of null infinity and includes the full tower of $t = c\,u,$2- and $t = c\,u,$3-derivatives. The integral representation of the Carrollian OPE and its descendant expansion are presented as the operator-algebraic form of collinear factorization (Mason et al., 2023). The basic structural point is that the null generator is not an auxiliary direction: it is part of the OPE block itself.

These formulations already show that the Carrollian notion of an OPE block is broader than in ordinary Euclidean CFT. Depending on the context, the block is a projector-defined four-point building block, a shadow-integral operator, or a collinear smearing that resums Carrollian descendants.

3. Two-dimensional C/G conformal blocks and heavy-light reduction

The intrinsic two-dimensional C/G algebra used in the block analysis is

$t = c\,u,$4

with highest-weight primaries labeled by $t = c\,u,$5, the eigenvalues of $t = c\,u,$6 (Hao et al., 29 Oct 2025). In the light-light regime, when external quantum numbers remain $t = c\,u,$7 and either $t = c\,u,$8 with $t = c\,u,$9 or $c\to 0$0 with $c\to 0$1 finite, non-global descendants are suppressed and the full C/G block reduces to the global block (Hao et al., 29 Oct 2025).

The heavy-light regime is the main development. Two heavy operators with parameters scaling linearly with $c\to 0$2 are treated as backreacting on the Carrollian geometry. The heavy effect is absorbed into a C/G conformal transformation

$c\to 0$3

with

$c\to 0$4

where $c\to 0$5 and $c\to 0$6 in the replica construction (Hao et al., 29 Oct 2025). The stated purpose of this map is to cancel the leading heavy dependence in the operator insertions and transfer it into the background geometry. The transformed cylinder identification becomes

$c\to 0$7

which the paper interprets as a rescaled and tilted cylinder sourced by the heavy state (Hao et al., 29 Oct 2025).

After this transformation, the large-$c\to 0$8 heavy-light block reduces to the global C/G block in the new coordinates, multiplied by Jacobian factors fixed by the primary transformation law. For the vacuum exchange in particular, the vacuum block takes an explicit closed form in $c\to 0$9 and reproduces the entanglement entropy of highly excited states through the replica trick (Hao et al., 29 Oct 2025). The interval is mapped to the cylinder by

$u$0

and the excited-state entropy becomes

$u$1

When the heavy boost charge makes $u$2 imaginary, the entropy assumes a thermal form, which is presented as a realization of ETH (Hao et al., 29 Oct 2025).

Within the present topic, the key point is that the Carrollian limit of the OPE block is not a loss of structure but a semiclassical re-expression of the block in a heavy-state geometry. The same paper also states that OPE blocks constructed in standard CFT$u$3 can be taken to Carrollian or Galilean limits, yielding the C/G blocks obtained in this framework (Hao et al., 29 Oct 2025).

4. Distributional branches, light transforms, and the emergence of 2D CFT-like blocks

A recurring feature of Carrollian OPEs is non-uniqueness. In the operator-algebra analysis on null infinity, the leading OPE structure functions have several branches: a regular power-law branch, chiral and anti-chiral branches localized by one delta function, and an ultra-local branch fully localized by $u$4. The same work emphasizes three distinct features absent from standard relativistic CFT: distributional spacetime support, infinite towers of “ancestor” primaries, and the necessity of composite or massive operators due to Casimir constraints (Nguyen et al., 19 Mar 2025). This is why the Carrollian limit cannot be identified with a naive equal-time reduction of ordinary local OPEs.

Light transformation provides a controlled way to move between these singular and power-law regimes. The chiral light-ray transforms are

$u$5

$u$6

and they map operators to shifted conformal weights: $u$7 produces a primary of weights $u$8, while $u$9 produces $_3$0 (Banerjee et al., 2024). The same paper states that in the special cases considered in $_3$1, light transformation maps the ultra-local “Delta function branch” to the “2D CFT branch” when all insertions are at coincident null time.

This is explicit in the equal-$_3$2 limit of the light-transformed two-point function,

$_3$3

which has standard 2D CFT power-law dependence together with a delta constraint on dimensions (Banerjee et al., 2024). In the same framework, the light-transformed gluon OPE is fixed by symmetry to have channel dimension

$_3$4

and an explicit Beta-function coefficient,

$_3$5

up to the ordering sign convention stated for the reverse ordering (Banerjee et al., 2024).

A refinement of this picture is that translation symmetry does not close on the leading OPE term by itself. In the analysis of light- and shadow-transformed operators, the leading term satisfies translation symmetry only with assistance from the subleading term, so successive orders are entangled (Banerjee et al., 20 Jun 2025). This materially affects the interpretation of the Carrollian limit of OPE blocks: one does not obtain a purely leading singularity controlled by global symmetry alone, but a short-distance structure in which descendant corrections are part of the symmetry data.

5. Smearing along null infinity, mode decompositions, and infinite symmetry towers

In the amplitude-based Carrollian construction, the OPE is obtained directly from the position-space collinear limit of scattering amplitudes at null infinity. The resulting block is an integral over the interpolation parameter $_3$6 and smears the exchanged operator along the null generator, while the fully expanded version is a sum over all $_3$7- and $_3$8-descendants with Beta-function coefficients (Mason et al., 2023). The same paper identifies this OPE block as the ultra-relativistic counterpart of celestial OPE blocks and shows that soft operators defined by $_3$9-moments of Carrollian primaries generate celestial symmetry algebras such as the $\mathfrak{so}(3,2)$0 algebra for Yang–Mills and $\mathfrak{so}(3,2)$1 for gravity (Mason et al., 2023).

A separate line of work formulates Carrollian OPEs through temporal step functions and a contour prescription. In $\mathfrak{so}(3,2)$2 dimensions, the Ward identities contain $\mathfrak{so}(3,2)$3, and the step-function structure is used to relate commutators to OPEs via the so-called $\mathfrak{so}(3,2)$4-prescription. In that setup only certain subsets of fields can be mutually local, and the local holomorphic sector generated by $\mathfrak{so}(3,2)$5, $\mathfrak{so}(3,2)$6, and $\mathfrak{so}(3,2)$7 realizes a Virasoro algebra together with an $\mathfrak{so}(3,2)$8 Kac–Moody sector and an abelian ideal (Saha, 2023). Extending this analysis, consistency of the OPEs for the soft-graviton tower forces an infinite sequence of local fields $\mathfrak{so}(3,2)$9, whose algebra is identified with the Kac–Moody algebra of the wedge subalgebra of $\mathfrak{ccarr}(3)$0 (Saha, 2023). The same work notes, however, that only the first three soft graviton theorems yield independent global symmetries; higher ones follow algebraically (Saha, 2023).

A more representation-theoretic formulation appears in the Carrollian limit of CFT$\mathfrak{ccarr}(3)$1 OPE blocks. Scalar primaries are scaled as

$\mathfrak{ccarr}(3)$2

and admit a Carrollian expansion

$\mathfrak{ccarr}(3)$3

An additional Fourier-type transform defines modes

$\mathfrak{ccarr}(3)$4

which transform as $\mathfrak{ccarr}(3)$5 primaries of dimension $\mathfrak{ccarr}(3)$6 (Gioia et al., 27 Aug 2025). The OPE blocks then decompose into towers of two-dimensional $\mathfrak{ccarr}(3)$7 blocks labeled by negative integer dimensions. For conserved currents these towers reproduce conformally soft photons or gluons and the $\mathfrak{ccarr}(3)$8 algebra; for the stress tensor they reproduce conformally soft gravitons and the $\mathfrak{ccarr}(3)$9 algebra (Gioia et al., 27 Aug 2025).

Taken together, these results show that the Carrollian limit of OPE blocks is a natural source of infinite soft towers. This suggests that the block decomposition is not merely kinematic bookkeeping but a direct mechanism for organizing asymptotic symmetries.

6. Holographic interpretation, bulk correspondence, and conceptual cautions

The holographic role of Carrollian OPE blocks is particularly explicit in two complementary directions. First, in two-dimensional C/G CFT, the heavy-light vacuum block reproduces holographic entanglement entropy computed via the swing surface proposal in three-dimensional Einstein gravity, with a stated dictionary between boundary $_4$0 and bulk $_4$1 (Hao et al., 29 Oct 2025). Second, in the flat limit of AdS correlators written in Bondi coordinates, Witten diagrams reduce directly to flat-space Feynman diagrams and boundary correlators become distributional Carrollian amplitudes supported on the kinematic loci appropriate to null infinity (Alday et al., 2024). Exchange diagrams then acquire the interpretation of Carrollian block-like structures built from null-boundary data.

Several conceptual cautions follow directly from the cited literature. One is that Carrollian OPE blocks are not generically ordinary local power-law blocks with a small deformation. Distributional support, multiple branches, temporal step functions, and null-generator smearing are structural rather than exceptional features (Nguyen et al., 19 Mar 2025, Mason et al., 2023, Saha, 2023). Another is that light- or shadow-transformed blocks should not be treated as mere changes of basis: they can alter the apparent branch structure and, in the equal-null-time limit, produce 2D CFT-like blocks out of otherwise ultra-local Carrollian correlators (Banerjee et al., 2024, Banerjee et al., 20 Jun 2025).

A further caution concerns the meaning of “infinite symmetry.” The appearance of an infinite tower of soft or negative-dimension modes is robust in both the OPE-consistency analysis and the Carrollian limit of CFT$_4$2 blocks, but this does not imply infinitely many independent global constraints. In the soft-graviton tower, only the leading, subleading, and subsubleading theorems are stated to define independent global symmetries (Saha, 2023). The remaining modes are nonetheless algebraically indispensable because they close the OPE structure and realize the full current algebra.

The overall picture is therefore highly structured. In one direction, intrinsic C/G blocks reduce semiclassically to global blocks in heavy-state geometries. In another, Carrollian and celestial OPE blocks arise from null-infinity collinear limits, light transforms, and mode projections. Across these realizations, the Carrollian limit reorganizes conformal families into objects adapted to BMS symmetry, null-time dynamics, and the soft sector of flat-space holography (Hao et al., 29 Oct 2025, Gioia et al., 27 Aug 2025).