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Recursion relation for general 3d blocks

Rajeev S. Erramilli, Luca V. Iliesiu, Petr Kravchuk

TL;DR

This work delivers a complete, closed-form setup for the Zamolodchikov-like residue recursion of general spinning conformal blocks in $d=3$ CFTs, deriving explicit expressions for all ingredients needed to recursively build conformal-block tables. By working in momentum space and exploiting null states together with a carefully chosen tensor-structure basis (notably the $ ext{SO}(3)$-based basis), the authors obtain the pole classification, residue matrices, and the leading large-$oldsymbol riangle$ term $h_{ ext{∞}}^{a,b}(z,ar z)$ in a uniform fashion. The results include detailed, explicit formulas for the four families of differential operators $oldsymbol D_{oldsymbol i}$, the corresponding residue matrices $(oldsymbol L_{oldsymbol i},oldsymbol R_{oldsymbol i})$, and the leading asymptotics, along with consistency checks against scalar blocks and differential-operator methods. These closed-form ingredients enable efficient numerical generation of spinning conformal blocks and pave the way for automated bootstrap studies in $d=3$ and potential generalizations to higher dimensions and superconformal theories.

Abstract

We derive closed-form expressions for all ingredients of the Zamolodchikov-like recursion relation for general spinning conformal blocks in 3-dimensional conformal field theory. This result opens a path to efficient automatic generation of conformal block tables, which has immediate applications in numerical conformal bootstrap program. Our derivation is based on an understanding of null states and conformally-invariant differential operators in momentum space, combined with a careful choice of the relevant tensor structures bases. This derivation generalizes straightforwardly to higher spacetime dimensions d, provided the relevant Clebsch-Gordan coefficients of Spin(d) are known.

Recursion relation for general 3d blocks

TL;DR

This work delivers a complete, closed-form setup for the Zamolodchikov-like residue recursion of general spinning conformal blocks in CFTs, deriving explicit expressions for all ingredients needed to recursively build conformal-block tables. By working in momentum space and exploiting null states together with a carefully chosen tensor-structure basis (notably the -based basis), the authors obtain the pole classification, residue matrices, and the leading large- term in a uniform fashion. The results include detailed, explicit formulas for the four families of differential operators , the corresponding residue matrices , and the leading asymptotics, along with consistency checks against scalar blocks and differential-operator methods. These closed-form ingredients enable efficient numerical generation of spinning conformal blocks and pave the way for automated bootstrap studies in and potential generalizations to higher dimensions and superconformal theories.

Abstract

We derive closed-form expressions for all ingredients of the Zamolodchikov-like recursion relation for general spinning conformal blocks in 3-dimensional conformal field theory. This result opens a path to efficient automatic generation of conformal block tables, which has immediate applications in numerical conformal bootstrap program. Our derivation is based on an understanding of null states and conformally-invariant differential operators in momentum space, combined with a careful choice of the relevant tensor structures bases. This derivation generalizes straightforwardly to higher spacetime dimensions d, provided the relevant Clebsch-Gordan coefficients of Spin(d) are known.

Paper Structure

This paper contains 43 sections, 142 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Definition of the conformal blocks
  3. Null states and differential operators in momentum space
  4. Spinning primary operators
  5. Two-point function
  6. Completeness relation and conformal blocks
  7. Null states
  8. Bosons
  9. Fermions
  10. Notation
  11. Differential operators
  12. Relation between $\mathcal{D}_m$ and $\mathcal{B}_m(\Delta,j)$
  13. Summary of differential operators
  14. Family $\mathrm{I}$
  15. Family $\mathrm{II}$
  16. ...and 28 more sections

Figures (1)

  • Figure 1: Zeros and poles coming from $\Gamma$-functions in \ref{['eq:Blocal']}, in bosonic (a) and fermionic (b) cases. The total number of circles indicates the order of the zero coming from denominator $\Gamma$-functions, while the number of filled circles indicates the minimal (over $m$) order of the pole coming from the numerator $\Gamma$-functions.