New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education

Fermionic Expressions for Minimal Model Virasoro Characters
Trevor A. Welsh, University of Melbourne, Parkville, Victoria, Australia
 SEARCH THIS BOOK:
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

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.

• 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