Chiral Algebra Bootstrap (CAB)

• Chiral Algebra Bootstrap (CAB) is a strategy that leverages 2D chiral-algebra consistency conditions, such as OPE singularities and associativity, to derive higher-dimensional correlators, form factors, and spectra.
• It employs techniques like Ward identities, null state constraints, and modular covariance to simplify complex computations in gauge theories, SCFTs, and vertex operator algebras.
• CAB enables the classification of chiral algebras and the reconstruction of observables via protected, twistorial, and factorization mechanisms, providing precise insights into both holographic and algebraic settings.

Chiral Algebra Bootstrap (CAB) denotes, in the literature represented here, a family of bootstrap procedures in which two-dimensional chiral-algebra data are used to determine higher-dimensional observables or to classify the chiral algebra itself. The common inputs are OPE singularities, associativity or Jacobi constraints, protected-sector reductions, Ward identities, null states, modular or quasi-modular covariance, and factorization. The outputs range from protected correlators in supersymmetric CFTs and supergravity, to form factors and amplitudes in twistorial gauge theories, to exact spectra and classifications of vertex operator algebras (VOAs) and related chiral structures (Fernández et al., 2024, Rigatos, 10 Jun 2025, Pan et al., 2024).

1. Scope and principal variants

In the literature represented here, CAB appears in at least three closely related ways. One strand uses a celestial or twistorial chiral algebra to bootstrap form factors and amplitudes in self-dual or twistorial gauge theories. A second uses the protected chiral algebra of a higher-dimensional SCFT to constrain defect correlators, higher-point correlators, or supergravity observables. A third uses VOA-intrinsic data—null states, modular differential equations, fusion rules, associativity, and representation theory—to reconstruct spectra or classify allowed chiral algebras.

CAB regime	Characteristic data	Representative output
Twistorial/self-dual gauge theory	holomorphic collinear OPEs, associativity, supersymmetry	all-orders OPEs, self-dual form factors, two-loop QCD amplitudes (Fernández et al., 2024, Charanya et al., 22 Apr 2026, Morales, 23 Oct 2025)
Protected SCFT correlators	chiral algebra twist, Ward identities, lightcone OPEs	$6$d $\mathcal N=(2,0)$ defect correlators, AdS$_5\times S^5$ six-point bootstrap (Rigatos, 10 Jun 2025, Goncalves et al., 14 Feb 2025)
VOA and algebraic classification	null states, MLDEs, Verlinde data, OPE associativity	flavored spectra, cyclic-orbifold operator algebra, ${\cal W}$-algebra classifications (Pan et al., 2024, Estienne et al., 2022, Gupta et al., 2023)

This breadth explains why CAB is not a single algorithm. The term labels a strategy: translate a problem into chiral-algebraic consistency conditions, solve those conditions as far as possible, and then pull the result back to the original higher-dimensional or algebraic setting.

2. Core mechanisms

A central CAB mechanism is the identification of a kinematic singular limit with a chiral-algebra OPE limit. In twistorial theories, the $2$d OPE limit $z_i\to z_j$ coincides with the $4$d holomorphic collinear limit $\langle ij\rangle\to 0$, after choosing

$\lambda_\alpha=(1,z).$

The basic form-factor dictionary is

$$\langle 0|\mathcal O|p_1\cdots p_n\rangle=\langle \mathcal O|\mathcal V_1(z_1)\cdots \mathcal V_n(z_n)\rangle_{2d},$$

so the singular OPE data of the $\mathcal N=(2,0)$0d algebra become holomorphic collinear splitting data of the $\mathcal N=(2,0)$1d theory. Associativity is then imposed in contour form, for example

$\mathcal N=(2,0)$2

and this recursively fixes higher-loop OPE coefficients (Fernández et al., 2024).

A second mechanism is the protected-sector twist. In the defect correlator problem of the $\mathcal N=(2,0)$3d $\mathcal N=(2,0)$4 theory, the correlator $\mathcal N=(2,0)$5 obeys the Ward identity

$\mathcal N=(2,0)$6

and after the twist $\mathcal N=(2,0)$7 one obtains a meromorphic protected correlator,

$\mathcal N=(2,0)$8

The bootstrap is then performed directly on $\mathcal N=(2,0)$9, rather than on the full unprotected correlator (Rigatos, 10 Jun 2025).

A third mechanism is reconstruction by factorization and Koszul duality. In higher-dimensional chiral algebra theory, the basic equivalence is

$_5\times S^5$0

with diagonal-supported chiral Lie algebras corresponding to factorization coalgebras. This supplies an abstract reconstruction theorem: local chiral/OPE data and global factorization data are two presentations of the same object (Francis et al., 2011).

3. Protected-sector CAB in SCFT and holography

In the $_5\times S^5$1d $_5\times S^5$2 theory with a $_5\times S^5$3-BPS surface defect, the protected sector is described by a $_5\times S^5$4d $_5\times S^5$5-algebra with generators $_5\times S^5$6, $_5\times S^5$7, and central charge

$_5\times S^5$8

The chiral-algebra map sends bulk half-BPS operators to $_5\times S^5$9-generators,

${\cal W}$0

while the defect becomes a pair of vertex operators at ${\cal W}$1 and ${\cal W}$2. The resulting ${\cal W}$3d correlator

${\cal W}$4

is bootstrapped with a meromorphic ansatz and a hidden crossing ${\cal W}$5, or equivalently ${\cal W}$6. A distinctive result is that the coefficients are fixed entirely from bulk-channel data. The final protected answer is expressed through Catalan numbers,

${\cal W}$7

and the defect-channel expansion yields new exact predictions for the protected defect spectrum and OPE coefficients (Rigatos, 10 Jun 2025).

A different protected-sector CAB appears in the tree-level six-supergraviton correlator in AdS${\cal W}$8. There the six-point Mellin amplitude is dissected into triple-pole, double-pole, single-pole, comb, and regular sectors. Multiple lightcone OPE limits reduce the problem to lower-point spinning correlators, and the Beem et al. chiral algebra twist,

${\cal W}$9

imposes holomorphic-independence constraints on the reduced subsectors. In the snowflake channel this sharply constrains the triple-pole sector; in the double-pole sector the twist is implemented directly in Mellin space by converting the condition into difference equations. The correlator is then fixed sector by sector rather than through one monolithic ansatz (Goncalves et al., 14 Feb 2025).

Exact VOA identifications also function as CAB input. For Argyres–Douglas theories engineered from M5-branes, the isolated theory $2$0 is assigned

$2$1

while the matter theory $2$2 is assigned

$2$3

Because these are minimal $2$4-models or affine Kac–Moody algebras at negative level, they provide rigid exact protected data for bootstrap purposes (Xie et al., 2016).

An explicit construction-based version appears in the genus-two $2$5 class $2$6 theory. There the chiral algebra is obtained by BRST reduction, a finite strong generating set is identified, OPE closure is checked, and completeness is supported by matching the vacuum character with the Schur index. An unexpected $2$7 automorphism then imposes additional constraints on the non-scalar sector (Kiyoshige et al., 2020).

4. Twistorial CAB for form factors and amplitudes

The twistorial version of CAB studies massless $2$8d gauge theories admitting a local holomorphic uplift to twistor space. In these theories, generalized towers of soft modes form a $2$9d chiral algebra whose OPE reproduces the holomorphic collinear limit of the $z_i\to z_j$0d theory. The bootstrap target is not usually an ordinary CFT correlator, but a form factor with local operator insertions; the chiral algebra determines the singularities that recursively generate it (Fernández et al., 2024).

For self-dual Yang–Mills coupled to anomaly-canceling matter, the all-orders CAB is organized by symmetry, $z_i\to z_j$1-weight, and $z_i\to z_j$2 representation theory. The allowed loop corrections are first constrained kinematically, and then associativity reduces all coefficients at loop order $z_i\to z_j$3 to a master coefficient $z_i\to z_j$4 obeying a recursion relation. The one-loop seed fixes the entire higher-loop tower. The resulting all-orders OPEs can be viewed as an all-loop result for a subset of collinear splitting functions in non-supersymmetric, massless QCD coupled to special matter content (Fernández et al., 2024).

In self-dual $z_i\to z_j$5 super Yang–Mills, CAB is sharpened by supersymmetry and Koszul duality. The supercurrent on $z_i\to z_j$6 packages the entire $z_i\to z_j$7 multiplet, and the loop-corrected OPEs remain single-pole. Associativity is used to prove the absence of double poles at all loop orders. The same framework reproduces known form factors of $z_i\to z_j$8, proves vanishing statements required by supersymmetry Ward identities, and produces novel form factors up to two loops for operators such as

$z_i\to z_j$9

which compute a supersymmetric version of Higgs amplitudes in the self-dual sector (Charanya et al., 22 Apr 2026).

CAB also enters ordinary QCD indirectly. In the all-plus sector, twistorial two-loop form factors computed by CAB can be combined with supersymmetry Ward identities to determine a previously unknown partial amplitude of the two-loop $4$0-gluon QCD amplitude. More broadly, the full-color two-loop $4$1-gluon amplitude with $4$2 quark flavors can be reconstructed from three twistorial two-loop amplitudes, one-loop axion amplitudes, and a tree-level double-axion amplitude. In this sense the chiral-algebra step lowers the effective loop complexity of the QCD computation by one loop level (Morales, 23 Oct 2025).

5. VOA, modular, and classification forms of CAB

CAB also denotes purely algebraic bootstrap procedures internal to the chiral algebra. A prominent example is the holomorphic quasi-modular bootstrap for flavored characters. Here one postulates a special null state

$4$3

subject to

$4$4

and derives flavored modular linear differential equations from Zhu recursion. The generic unflavored form is

$4$5

The modular $4$6-transform generates a hierarchy of lower-weight equations, interpreted as the modular orbit of the null state. This framework reconstructs the Deligne–Cvitanović series at $4$7 and $4$8, as well as integrable and admissible $4$9 spectra, including twisted sectors produced by spectral flow (Pan et al., 2024).

A related refinement concerns the Macdonald index. The vacuum module is equipped with a filtration

$\langle ij\rangle\to 0$0

and the associated graded space defines a refined character

$\langle ij\rangle\to 0$1

The conjecture is that, after $\langle ij\rangle\to 0$2, this reproduces the $\langle ij\rangle\to 0$3d Macdonald index. This supplies a more stringent grading than the ordinary vacuum character, but the assignment of $\langle ij\rangle\to 0$4-weights is not purely intrinsic when the VOA has multiple generator families; extra $\langle ij\rangle\to 0$5d input is then required (Song, 2016).

In cyclic orbifolds $\langle ij\rangle\to 0$6, the relevant local symmetry is not the full orbifold Virasoro algebra but the maximal local chiral algebra $\langle ij\rangle\to 0$7, the neutral algebra generated by monomials of total Fourier charge zero. Under the assumption that the mother theory is rational and diagonal, the orbifold becomes rational and diagonal with respect to $\langle ij\rangle\to 0$8. Its fusion rules follow from the Verlinde formula, and the fusion multiplicities determine which extended conformal blocks appear in a given four-point function of twisted or untwisted operators. This makes Rényi-twist correlators into a controlled RCFT bootstrap problem (Estienne et al., 2022).

A classification version appears in the determination of all chiral $\langle ij\rangle\to 0$9-algebra extensions of $\lambda_\alpha=(1,z).$0. One chooses an embedding $\lambda_\alpha=(1,z).$1, promotes it to a Virasoro algebra, assigns fields according to the $\lambda_\alpha=(1,z).$2 decomposition, writes the most general OPEs compatible with conformal covariance and the commuting subalgebra $\lambda_\alpha=(1,z).$3, and fixes the structure constants by associativity. The outcome is exactly four inequivalent chiral extensions, including two new algebras, one of which is identified as the finite-central-charge conformal $\lambda_\alpha=(1,z).$4 algebra (Gupta et al., 2023).

6. Mathematical infrastructure, limitations, and significance

The mathematical infrastructure of CAB is unusually rich. On the algebro-geometric side, chiral Koszul duality identifies chiral Lie algebras and chiral commutative coalgebras on $\lambda_\alpha=(1,z).$5, while the factorization condition singles out the diagonal-supported subcategory corresponding to genuine local chiral algebras (Francis et al., 2011). On the holographic and twistorial side, Kaluza–Klein reduction of $\lambda_\alpha=(1,z).$6d holomorphic theories to $\lambda_\alpha=(1,z).$7d holomorphic-topological theories, together with homotopy transfer and Koszul duality, produces boundary chiral algebras whose generators, higher products, and OPE coefficients can be computed explicitly. This provides a common framework for twisted holography and celestial holography (Zeng, 2023).

Several limitations are explicit in the literature. The chiral algebra can be highly constraining without being complete by itself. In the six-point AdS$\lambda_\alpha=(1,z).$8 problem, some tensor structures degenerate in $\lambda_\alpha=(1,z).$9d kinematics, so the chiral algebra twist does not distinguish all $$\langle 0|\mathcal O|p_1\cdots p_n\rangle=\langle \mathcal O|\mathcal V_1(z_1)\cdots \mathcal V_n(z_n)\rangle_{2d},$$0-structures and must be supplemented by current conservation, stress-tensor conservation, and flat-space input (Goncalves et al., 14 Feb 2025). In the $$\langle 0|\mathcal O|p_1\cdots p_n\rangle=\langle \mathcal O|\mathcal V_1(z_1)\cdots \mathcal V_n(z_n)\rangle_{2d},$$1d defect problem, the fact that bulk-channel data alone fix the correlator is emphasized as special to that defect system, because the remaining information is supplied by hidden crossing symmetry (Rigatos, 10 Jun 2025). In refined-character problems, the grading needed for the Macdonald index is ambiguous without additional $$\langle 0|\mathcal O|p_1\cdots p_n\rangle=\langle \mathcal O|\mathcal V_1(z_1)\cdots \mathcal V_n(z_n)\rangle_{2d},$$2d information when several generator families coexist (Song, 2016). In the twistorial setting, the all-orders chiral algebra captures a subset of QCD-like observables, specifically the holomorphic collinear singular terms and the associated normal-ordered-product structure required by associativity (Fernández et al., 2024).

These caveats do not weaken the central role of CAB; they clarify its domain. CAB is most effective when a protected, twistorial, or factorizing subsector isolates meromorphic or quasi-meromorphic data, and when associativity, modularity, or factorization reduce the allowed structures to a finite set. Under those conditions, CAB functions as a reconstruction principle: exact $$\langle 0|\mathcal O|p_1\cdots p_n\rangle=\langle \mathcal O|\mathcal V_1(z_1)\cdots \mathcal V_n(z_n)\rangle_{2d},$$3d chiral-algebra consistency determines higher-dimensional correlators, form factors, amplitudes, or VOA spectra with minimal direct diagrammatic input.