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Two approaches to the holomorphic modular bootstrap

Suresh Govindarajan, Jagannath Santara

TL;DR

This work presents a complementary framework to the holomorphic modular bootstrap for RCFTs by exploiting vector-valued modular forms to generate new admissible characters from a known RCFT. Starting from a known solution $\\mathbb{X}$ with fixed multiplier, the authors construct additional solutions $\\mathbb{Y}_i$ sharing the same $S$ and $T$ data, and then form strategic linear combinations, including twists by the Hauptmodul $J(\\tau)$, to produce new theories with shifted central charge $c+24m$ and adjusted Wronskian index $\\ell+6m$. Through detailed, increasingly complex examples (up to six characters), they reproduce known two-character results and uncover new admissible families, while also highlighting limitations and the need for higher-order invariants to extend beyond four characters. The approach offers a flexible route to catalog admissible RCFTs beyond the traditional MLDE method and connects to broader structures in modular form theory and modular tensor categories. Overall, the paper expands the toolkit for holomorphic modular bootstrap by leveraging VVMFs to navigate larger character spaces and generate new admissible RCFT candidates.

Abstract

The holomorphic bootstrap attempts to classify rational conformal field theories. The straight ahead approach is hard to implement when the number of characters become large. We combine all characters of an RCFT to form a vector valued modular form with multiplier. Using known results from the theory of vector valued modular forms, given a known RCFT, we obtain new vector valued modular forms that share the same multiplier as the original RCFT. By taking particular linear combinations of the new solutions, we look for and find new admissible solutions. In the well-studied two character case, we reproduce all known admissible solutions with Wronskian indices $6$ and $8$. The method is illustrated with examples with up to six characters. The method using vector valued modular forms thus provides a new approach to the holomorphic modular bootstrap.

Two approaches to the holomorphic modular bootstrap

TL;DR

This work presents a complementary framework to the holomorphic modular bootstrap for RCFTs by exploiting vector-valued modular forms to generate new admissible characters from a known RCFT. Starting from a known solution with fixed multiplier, the authors construct additional solutions sharing the same and data, and then form strategic linear combinations, including twists by the Hauptmodul , to produce new theories with shifted central charge and adjusted Wronskian index . Through detailed, increasingly complex examples (up to six characters), they reproduce known two-character results and uncover new admissible families, while also highlighting limitations and the need for higher-order invariants to extend beyond four characters. The approach offers a flexible route to catalog admissible RCFTs beyond the traditional MLDE method and connects to broader structures in modular form theory and modular tensor categories. Overall, the paper expands the toolkit for holomorphic modular bootstrap by leveraging VVMFs to navigate larger character spaces and generate new admissible RCFT candidates.

Abstract

The holomorphic bootstrap attempts to classify rational conformal field theories. The straight ahead approach is hard to implement when the number of characters become large. We combine all characters of an RCFT to form a vector valued modular form with multiplier. Using known results from the theory of vector valued modular forms, given a known RCFT, we obtain new vector valued modular forms that share the same multiplier as the original RCFT. By taking particular linear combinations of the new solutions, we look for and find new admissible solutions. In the well-studied two character case, we reproduce all known admissible solutions with Wronskian indices and . The method is illustrated with examples with up to six characters. The method using vector valued modular forms thus provides a new approach to the holomorphic modular bootstrap.

Paper Structure

This paper contains 15 sections, 68 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Two approaches
  3. The MMS approach
  4. The VVMF Approach
  5. Generating new solutions from a known solution
  6. Going from quasi-characters to admissible ones
  7. Examples
  8. Two character examples
  9. A $(3,3)$ example
  10. A four-character example
  11. Going beyond four characters
  12. A $(5,0)$ example
  13. A rank six example
  14. Conclusion
  15. Modular Forms