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Recursive Representations of Arbitrary Virasoro Conformal Blocks

Minjae Cho, Scott Collier, Xi Yin

TL;DR

This work develops a comprehensive set of recursive representations for Virasoro conformal blocks on Riemann surfaces of arbitrary genus. By exploiting poles at degenerate representations and the large-$c$ or large-weight limits within a plumbing frame, it extends Zamolodchikov-style recursions to sphere and torus blocks with any number of insertions and to higher genus via pants decompositions. The central results include (i) a generalized $c$-recursion valid in all channels, (ii) an $h$-recursion for torus necklace and sphere linear-channel blocks, and (iii) explicit examples such as sphere 6-point trifundamental and torus 1- and 2-point blocks, together with their global $SL(2)$ and vacuum-block factorization structures. The framework provides efficient, consistent machinery for high-precision bootstrap and string-amplitude computations on complex geometries, tapping into the large-$c$ holographic interpretation via 1-loop gravity on handlebodies and the exact combinatorics of fusion polynomials.

Abstract

We derive recursive representations in the internal weights of N-point Virasoro conformal blocks in the sphere linear channel and the torus necklace channel, and recursive representations in the central charge of arbitrary Virasoro conformal blocks on the sphere, the torus, and higher genus Riemann surfaces in the plumbing frame.

Recursive Representations of Arbitrary Virasoro Conformal Blocks

TL;DR

This work develops a comprehensive set of recursive representations for Virasoro conformal blocks on Riemann surfaces of arbitrary genus. By exploiting poles at degenerate representations and the large- or large-weight limits within a plumbing frame, it extends Zamolodchikov-style recursions to sphere and torus blocks with any number of insertions and to higher genus via pants decompositions. The central results include (i) a generalized -recursion valid in all channels, (ii) an -recursion for torus necklace and sphere linear-channel blocks, and (iii) explicit examples such as sphere 6-point trifundamental and torus 1- and 2-point blocks, together with their global and vacuum-block factorization structures. The framework provides efficient, consistent machinery for high-precision bootstrap and string-amplitude computations on complex geometries, tapping into the large- holographic interpretation via 1-loop gravity on handlebodies and the exact combinatorics of fusion polynomials.

Abstract

We derive recursive representations in the internal weights of N-point Virasoro conformal blocks in the sphere linear channel and the torus necklace channel, and recursive representations in the central charge of arbitrary Virasoro conformal blocks on the sphere, the torus, and higher genus Riemann surfaces in the plumbing frame.

Paper Structure

This paper contains 23 sections, 93 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Review of the sphere 4-point Virasoro block
  3. Definition of Virasoro conformal block
  4. Simple pole structure and its residue
  5. Determining the regular part
  6. $h$-recursion for torus $N$-point Virasoro conformal blocks in the necklace channel (and sphere $N$-point blocks in the linear channel)
  7. Definition of the Virasoro block in the necklace channel
  8. Polar part
  9. Regular part
  10. $h$-recursion representation
  11. Sphere $N$-point block in the linear channel
  12. $c$-recursion for all sphere and torus Virasoro conformal blocks
  13. Factorization of 3-point functions with degenerate representations and the poles of conformal blocks
  14. Large $c$, fixed $h_i$ limit of Virasoro conformal blocks
  15. Global $SL(2)$ blocks
  16. ...and 8 more sections

Figures (8)

  • Figure 1: The sphere six-point block in the trifundamental channel (left) and the torus two-point block in the OPE channel (right). Our $c$-recursion representation for arbitrary sphere and torus $N$-point blocks enables recursive evaluation of these blocks; we work these cases out explicitly in Section \ref{['sec:cRecursionExamples']}.
  • Figure 2: The torus $N$-point block in the necklace channel (left) and the sphere $N$-point block in the linear channel (right).
  • Figure 3: The large-$c$ factorization of a genus-2 two-point block in the plumbing frame, in which the punctured Riemann surface is built by plumbing together two-holed (or punctured) discs using $SL(2)$ maps.
  • Figure 4: $N$-cylinder decomposition of necklace channel for $N=3$ case.
  • Figure 5: Plumbing construction for sphere 6-point conformal block in the trifundamental channel.
  • ...and 3 more figures