Analytic Functionals and Polyakov Blocks

Analytic functionals convert crossing into exact spectral sum rules, while Polyakov blocks reorganize a correlator into crossing-aware exchange terms with controlled Regge behavior. The two constructions are dual descriptions of the same subtraction problem only after their domains, endpoint falloff, and contact-term bases agree.

Required background. CFT dispersion relations fixes the absorptive data and subtraction ambiguity. Crossing and positivity supplies the spectral equation and the conditions under which OPE weights are nonnegative.

Helpful background. One-dimensional crossing and positivity gives the cleanest contour domain. Generalized-free solvable data supplies the spectrum used to normalize a dual functional basis.

Evidence cutoff. Research-sensitive statements about functional completeness, swapping domains, and Polyakov–Regge constructions reflect primary sources available through 2026-08-09. Later bases or completeness results require a renewed source check.

Functionals act on the crossing equation

For identical one-dimensional scalars of dimension $\Delta_\phi$, let

$$G_\Delta(z)=z^\Delta\,{}_2F_1(\Delta,\Delta;2\Delta;z),$$

and define the crossing vector

$$F_\Delta(z)=z^{-2\Delta_\phi}G_\Delta(z) -(1-z)^{-2\Delta_\phi}G_\Delta(1-z).$$

Reflection positivity and a Hermitian identical-scalar OPE give

$$\sum_{\Delta}a_\Delta F_\Delta(z)=0, \qquad a_\Delta\ge0,$$

including the identity at $\Delta=0$. A linear functional $\omega$ produces the sum rule

$$\sum_\Delta a_\Delta\,\omega(\Delta)=0, \qquad \omega(\Delta)\equiv\omega[F_\Delta].$$

The useful analytic functionals are contour or cut integrals, not merely finite derivatives at $z=1/2$. A representative form is

$$\omega[F]=\int_{1/2}^{1/2+i\infty}dz\,f(z)F(z) +\int_{1/2}^{1}dz\,g(z)F(z),$$

with orientation factors understood in $f$ and with $g$ related to discontinuities of $f$ when the contour is deformed. The kernels must make the functional finite on every allowed block and permit swapping with the infinite OPE sum. Sufficient endpoint falloff is checked separately near $z=1$, $z=\infty$, and any subtraction point; crossing symmetry alone does not imply swapping Mazáč and Paulos 2019a, §§2–3 and app. C.

A basis dual to generalized-free data

For a generalized-free bosonic solution, the nonidentity dimensions are

$$\Delta_n=2\Delta_\phi+2n, \qquad n=0,1,2,\ldots .$$

One can construct functionals $\alpha_n$ and $\beta_n$ that are dual to the block and dimension-derivative directions,

$$\alpha_n(\Delta_m)=\delta_{nm}, \qquad \partial_\Delta\alpha_n(\Delta_m)=0,$$ $$\beta_n(\Delta_m)=0, \qquad \partial_\Delta\beta_n(\Delta_m)=\delta_{nm},$$

up to the known lowest-mode modification required by the chosen bosonic or fermionic basis. These relations fix normalization far more strongly than requiring only zeros at $\Delta_m$. Applying the basis to crossing yields one sum rule per $n$ and separates shifts of OPE coefficients from shifts of dimensions. The construction and its completeness domain are developed in Mazáč and Paulos 2019b, §§3–5.

The phrase “basis” is conditional here. Completeness means that every crossing vector in the declared Regge-bounded function space admits the associated expansion. It does not prove that every formal solution has a positive, discrete, local-CFT realization.

Polyakov and Polyakov–Regge blocks

A Polyakov block associated with an exchange $\mathcal O$ is a crossing-symmetric or crossing-covariant function whose physical-channel decomposition contains $G_{\Delta,J}$ plus compensating double-twist terms. A Polyakov–Regge block additionally obeys a specified Regge bound in a chosen channel. Schematically,

$$\mathcal P_{\Delta,J} =G_{\Delta,J} -\sum_n\left[ \alpha_n(\Delta,J)G_{\Delta_n,J_n} +\beta_n(\Delta,J)\partial_\Delta G_{\Delta_n,J_n} \right],$$

where the precise families, spins, and crossed-channel completion depend on dimension and external representations. The derivative blocks cancel spurious double poles or logarithms. Contact terms may be added without changing the exchange discontinuity; Regge falloff and a chosen contact basis are therefore part of the definition, not consequences of the name “Polyakov block.”

A dispersive construction fixes such blocks by prescribing their double discontinuity and subtractions. Expanding the correlator in Polyakov–Regge blocks then yields sum rules from cancellation of unphysical double-twist contributions Caron-Huot et al. 2021, §§3–5.

Distinct conformal objects and their domains

This table prevents common category errors. Every row states an object, not merely a different notation for the same function.

Object	Defining property	Natural domain	What it contains	Ambiguity or limitation
Conformal block $G_{\Delta,J}$	One primary and all descendants in one OPE channel	OPE convergence region, then analytic continuation	Local direct-channel exchange	Multivalued; no shadow completion; not crossing symmetric
Shadow block $G_{d-\Delta,J}$	Block for the shadow representation	Same channel after continuation	The shadow solution of the Casimir equation	Usually not a physical OPE contribution
Conformal partial wave $\Psi_{\Delta,J}$	Single-valued block–shadow combination	Euclidean principal-series harmonic analysis	Both $\Delta$ and $d-\Delta$ solutions with fixed coefficients	Requires Plancherel and shadow normalization
Inversion kernel	Dual projector onto $c(\Delta,J)$	Euclidean domain or a declared Lorentzian cut domain	Poles and residues of one channel	Lorentzian version has Regge and low-spin thresholds
Crossing kernel or conformal $6j$ symbol	Overlap of partial waves from two OPE trees	Principal series, then meromorphic continuation	Representation-theoretic change of channel	Measure, tensor basis, shadows, and discrete residues are convention dependent
Polyakov block	Crossing-completed exchange with canceled spurious terms	Function space fixed by crossing and subtractions	One physical exchange plus compensating double-twist data	Contact-term basis must be fixed
Polyakov–Regge block	Polyakov block with prescribed Regge behavior in a selected channel	Declared Lorentzian sheet and growth class	Exchange absorptive part with a dispersive completion	Different channel choices give different completions
Analytic functional $\omega$	Linear map on crossing vectors with a proved swapping property	Endpoint-controlled block function space	Spectral sum rule or dual-basis coefficient	Positivity and completeness require separate proofs

Generalized-free saturation check

In the reduced-correlator convention

$$\mathcal G_{\mathrm{GFB}}(z) =1+z^{2\Delta_\phi} +\left(\frac{z}{1-z}\right)^{2\Delta_\phi},$$

the OPE contains the generalized-free bosonic dimensions $\Delta_n$. Acting with a properly normalized dual basis on its crossing equation gives zero after the identity and all $a_n>0$ terms are included. There are three independent checks:

1. $\alpha_n$ selects the coefficient direction at $\Delta_n$.
2. $\beta_n$ selects a first-order dimension shift.
3. The corresponding Polyakov expansion has no uncanceled spurious logarithm at any $\Delta_n$.

Passing only the zero conditions is insufficient; the derivative normalization and the swapped sum must also converge.

Failure modes

Swapping a conditionally convergent sum. If the endpoint kernel is too singular, $\omega[\sum a_\Delta F_\Delta]$ need not equal $\sum a_\Delta\omega[F_\Delta]$.

Assuming a sign. A functional useful for an identity may not be nonnegative on the continuum of allowed dimensions. Positivity must be proved over the full claimed range.

Suppressing contact terms. Two Polyakov completions with the same exchange discontinuity can differ by a crossing-compatible contact structure. A sum rule depends on which one was fixed.

Equating a formal expansion with a CFT. Crossing and positive coefficients are necessary data, but locality, convergence, and the rest of the operator algebra remain independent consistency requirements.

Exercises

Let $\mathcal G=\mathcal G_{\mathrm{GFB}}+\varepsilon\,\delta\mathcal G$ perturb the dimensions and coefficients of the generalized-free family by $\gamma_n$ and $\delta a_n$. Apply the duality relations to explain which functional isolates each perturbation to first order.

Solution

Expanding $a_nG_{\Delta_n+\varepsilon\gamma_n}$ gives $a_nG_{\Delta_n}+\varepsilon(\delta a_nG_{\Delta_n}+a_n\gamma_n\partial_\Delta G_{\Delta_n})$. The functional $\alpha_n$ selects the block coefficient direction and hence $\delta a_n$ after known crossed and identity terms are moved to the other side. The functional $\beta_n$ selects the derivative direction and hence $a_n\gamma_n$. The lowest-mode modification of the chosen basis must be included.

References

• Caron-Huot, Simon, Dalimil Mazáč, Leonardo Rastelli, and David Simmons-Duffin. “Dispersive CFT Sum Rules.” Journal of High Energy Physics 2021, no. 5 (2021): 243. doi:10.1007/JHEP05(2021)243.
• Mazáč, Dalimil, and Miguel F. Paulos. “The Analytic Functional Bootstrap. Part I: 1D CFTs and 2D S-Matrices.” Journal of High Energy Physics 2019, no. 2 (2019): 162. doi:10.1007/JHEP02(2019)162.
• Mazáč, Dalimil, and Miguel F. Paulos. “The Analytic Functional Bootstrap. Part II: Natural Bases for the Crossing Equation.” Journal of High Energy Physics 2019, no. 2 (2019): 163. doi:10.1007/JHEP02(2019)163.