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On modular transformations of non-degenerate toric conformal blocks

Nikita Nemkov

TL;DR

This work analyzes the modular transformations of non-degenerate toric Virasoro conformal blocks by constructing a non-perturbative series expansion of the toric modular kernel $M_{αα'}(μ)$. Building on the Ponsot–Teschner integral representation, the authors derive and solve a set of consistency-derived difference equations, yielding an explicit series in $e^{4π i b α}$ that matches the integral form and extends its domain of validity. The series provides a non-perturbative completion of the Fourier-type behavior at large internal momenta and clarifies how border cases (notably $c=1$ and $c=25$) fit into the framework via separate analysis. Two independent derivations are presented: a direct solution of the difference equations and a residue-based expansion from the integral representation, highlighting open questions about the precise non-perturbative origin in conformal blocks.

Abstract

We derive and solve the difference equations on the toric modular kernel following from the consistency relations in the fusion algebra. The result is explicit and simple series expansion for the toric modular kernel of non-degenerate Virasoro conformal blocks. We show that this expansion is equivalent to the celebrated integral representation due to B. Ponsot and J. Teschner. We also interpret obtained series representation as a non-perturbative expansion and note that this raises further questions.

On modular transformations of non-degenerate toric conformal blocks

TL;DR

This work analyzes the modular transformations of non-degenerate toric Virasoro conformal blocks by constructing a non-perturbative series expansion of the toric modular kernel . Building on the Ponsot–Teschner integral representation, the authors derive and solve a set of consistency-derived difference equations, yielding an explicit series in that matches the integral form and extends its domain of validity. The series provides a non-perturbative completion of the Fourier-type behavior at large internal momenta and clarifies how border cases (notably and ) fit into the framework via separate analysis. Two independent derivations are presented: a direct solution of the difference equations and a residue-based expansion from the integral representation, highlighting open questions about the precise non-perturbative origin in conformal blocks.

Abstract

We derive and solve the difference equations on the toric modular kernel following from the consistency relations in the fusion algebra. The result is explicit and simple series expansion for the toric modular kernel of non-degenerate Virasoro conformal blocks. We show that this expansion is equivalent to the celebrated integral representation due to B. Ponsot and J. Teschner. We also interpret obtained series representation as a non-perturbative expansion and note that this raises further questions.

Paper Structure

This paper contains 19 sections, 93 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Conformal blocks
  3. General discussion
  4. Exactly solvable examples
  5. Fusion and modular transformations
  6. General discussion
  7. Exactly solvable examples
  8. Toric modular kernel from representation theory of $\mathcal{U}_q(sl_2)$
  9. Main result: series expansion of the toric modular kernel
  10. Series representation as non-perturbative completion
  11. Modular kernel asymptotic expansion from consistency requirements
  12. Pentagon identity and difference equations on the fusion kernel
  13. Consistency constraints on the modular kernel
  14. Solutions to consistency constraints
  15. Special values of $b$
  16. ...and 4 more sections

Figures (7)

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  • ...and 2 more figures