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Fermionic Expressions for Minimal Model Virasoro Characters

Trevor A. Welsh University of Melbourne, Parkville, Victoria, Australia
Available Formats:
Electronic ISBN: 978-1-4704-0428-4
Product Code: MEMO/175/827.E
List Price: $71.00 MAA Member Price:$63.90
AMS Member Price: $42.60 Click above image for expanded view Fermionic Expressions for Minimal Model Virasoro Characters Trevor A. Welsh University of Melbourne, Parkville, Victoria, Australia Available Formats:  Electronic ISBN: 978-1-4704-0428-4 Product Code: MEMO/175/827.E  List Price:$71.00 MAA Member Price: $63.90 AMS Member Price:$42.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1752005; 160 pp
MSC: Primary 82; Secondary 05; 17; 81;

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.

• Chapters
• 1. Prologue
• 2. Path combinatorics
• 3. The $\mathcal {B}$-transform
• 4. The $\mathcal {D}$-transform
• 5. Mazy runs
• 6. Extending and truncating paths
• 7. Generating the fermionic expressions
• 8. Collating the runs
• 9. Fermionic character expressions
• 10. Discussion
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Accessibility – to request an alternate format of an AMS title
Volume: 1752005; 160 pp
MSC: Primary 82; Secondary 05; 17; 81;

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.

• Chapters
• 1. Prologue
• 2. Path combinatorics
• 3. The $\mathcal {B}$-transform
• 4. The $\mathcal {D}$-transform
• 5. Mazy runs
• 6. Extending and truncating paths
• 7. Generating the fermionic expressions
• 8. Collating the runs
• 9. Fermionic character expressions
• 10. Discussion
Review Copy – for reviewers who would like to review an AMS book
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