# Fermionic Expressions for Minimal Model Virasoro Characters

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*Trevor A. Welsh*

Fermionic expressions for all minimal model Virasoro
characters \(\chi^{p, p'}_{r, s}\) are stated and proved. Each such
expression is a sum of terms of fundamental fermionic form type. In
most cases, all these terms are written down using certain trees which are
constructed for \(s\) and \(r\) from the Takahashi lengths and
truncated Takahashi lengths associated with the continued fraction of
\(p'/p\). In the remaining cases, in addition to such terms, the
fermionic expression for \(\chi^{p, p'}_{r, s}\) contains a different
character \(\chi^{\hat p, \hat p'}_{\hat r,\hat s}\), and is thus
recursive in nature.

Bosonic-fermionic \(q\)-series identities for all
characters \(\chi^{p, p'}_{r, s}\) result from equating these fermionic
expressions with known bosonic expressions. In the cases for which
\(p=2r\), \(p=3r\), \(p'=2s\) or \(p'=3s\),
Rogers-Ramanujan type identities result from equating these fermionic
expressions with known product expressions for \(\chi^{p, p'}_{r,
s}\).

The fermionic expressions are proved by first obtaining
fermionic expressions for the generating functions \(\chi^{p, p'}_{a, b,
c}(L)\) of length \(L\) Forrester-Baxter paths, using various
combinatorial transforms. In the \(L\to\infty\) limit, the fermionic
expressions for \(\chi^{p, p'}_{r, s}\) emerge after mapping between the
trees that are constructed for \(b\) and \(r\) from the Takahashi
and truncated Takahashi lengths respectively.

#### Table of Contents

# Table of Contents

## Fermionic Expressions for Minimal Model Virasoro Characters

- Contents v6 free
- 1. Prologue 110 free
- 1.1. Introduction 110
- 1.2. Structure of this paper 514
- 1.3. Continued fractions 615
- 1.4. Fermionic character expressions 615
- 1.5. The Takahashi and string lengths 817
- 1.6. Takahashi trees 817
- 1.7. Takahashi tree vectors 1019
- 1.8. Truncated Takahashi tree 1120
- 1.9. The linear term 1221
- 1.10. The constant term 1221
- 1.11. The quadratic term 1322
- 1.12. The mn-system and the parity vector 1423
- 1.13. The extra term 1423
- 1.14. Finitized fermionic expressions 1524
- 1.15. Fermionic-like expressions 1726

- 2. Path combinatorics 1928
- 3. The B-transform 2938
- 4. The D-transform 4150
- 5. Mazy runs 4554
- 6. Extending and truncating paths 5766
- 7. Generating the fermionic expressions 6574
- 8. Collating the runs 101110
- 9. Fermionic character expressions 111120
- 10. Discussion 127136
- Appendix A. Examples 131140
- Appendix B. Obtaining the bosonic generating function 141150
- Appendix C. Bands and the floor function 145154
- Appendix D. Bands on the move 147156
- Appendix E. Combinatorics of the Takahashi lengths 149158
- Bibliography 159168