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Fermionic Expressions for Minimal Model Virasoro Characters
Trevor A. Welsh, University of Melbourne, Parkville, Victoria, Australia
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Memoirs of the American Mathematical Society
2005; 160 pp; softcover
Volume: 175
ISBN-10: 0-8218-3656-0
ISBN-13: 978-0-8218-3656-9
List Price: US$67
Individual Members: US$40.20
Institutional Members: US$53.60
Order Code: MEMO/175/827
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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

  • Prologue
  • Path combinatorics
  • The \(\mathcal{B}\)-transform
  • The \(\mathcal{D}\)-transform
  • Mazy runs
  • Extending and truncating paths
  • Generating the fermionic expressions
  • Collating the runs
  • Fermionic character expressions
  • Discussion
  • Appendix A. Examples
  • Appendix B. Obtaining the bosonic generating function
  • Appendix C. Bands and the floor function
  • Appendix D. Bands on the move
  • Appendix E. Combinatorics of the Takahashi lengths
  • Bibliography
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