[Paper Review] Algebraic K-Theory and Partition Functions in Conformal Field Theory

Sinéad Keegan|ArXiv.org|May 30, 2007

Algebraic structures and combinatorial models34 references3 citations

TL;DR

This PhD thesis establishes a rigorous connection between algebraic K-theory, specifically the Bloch group and dilogarithm functions, and rational conformal field theories (RCFTs) via integrable models based on ADET Dynkin diagrams. Using representation theory of Yangians and Lie algebras, the author solves the Kirillov-Reshetikhin equations for $(D_m, A_n)$ models and computes effective central charges, showing that certain q-hypergeometric series $f_{A,B,C}$ are modular characters of RCFTs, with explicit solutions for $A = \begin{pmatrix}4&1\\1&1\end{pmatrix}$ yielding $c_{\text{eff}} = \frac{80}{11}$.

ABSTRACT

Certain integrable models are described by pairs (X,Y) of ADET Dynkin diagrams. At high energy these models are expected to have a conformally invariant limit. The S-matrix of the model determines algebraic equations, whose solutions are mapped to the central charge and scaling dimensions of the corresponding conformal field theory. We study the equations of the (D_m,A_n) model and find all solutions explicitly using the representation theory of Lie algebras and related Yangians. These mathematically rigorous results are in agreement with the expectations arising from physics. We also investigate the overlap between certain q-hypergeometric series and modular functions. We study a particular class of 2-fold q-hypergeometric series, denoted f_{A,B,C}. Here A is a positive definite, symmetric, 2x2 matrix, B is a vector of length 2, and C is a scalar, all three with rational entries. It turns out that for certain choices of the matrix A, the function f_{A,B,C} can be made modular. We calculate the corresponding values of B and C. It is expected that functions f_{A,B,C} arising in this way are characters of some rational conformal field theory. We show that this is true in at least one case.

Motivation & Objective

• To establish a mathematical framework linking algebraic K-theory, particularly the Bloch group and dilogarithm identities, to rational conformal field theories (RCFTs).
• To solve the Kirillov-Reshetikhin equations for $(D_m, A_n)$-type integrable models using representation theory of Lie algebras and Yangians.
• To investigate the modular properties of 2-fold q-hypergeometric series $f_{A,B,C}$ and determine conditions under which they arise as characters of RCFTs.
• To compute effective central charges for $(D_m, A_n)$ and exceptional Lie algebra models $(E_m, T_1)$, verifying consistency with physical expectations.

Proposed method

• Utilizes representation theory of Yangians associated with Lie algebras $D_r$ and $E_m$ to solve the Kirillov-Reshetikhin equations for integrable models.
• Applies the Weyl character formula and Weyl denominator identities to compute characters of irreducible representations and their restrictions to subalgebras.
• Employs the dilogarithm functional equations and torsion elements in the Bloch group to relate solutions of the model equations to central charges.
• Derives a system of logarithmic equations in variables $x_i$ from the model's S-matrix, solving them algebraically to extract central charge and scaling dimension data.
• Analyzes the modular properties of $f_{A,B,C}$, a 2-fold q-hypergeometric series with rational matrix $A$, vector $B$, and scalar $C$, identifying cases where it becomes modular.
• Uses Galois group actions on solutions to classify and verify the integrality and modularity of the resulting central charges.

Experimental results

Research questions

• RQ1 How do solutions to the Kirillov-Reshetikhin equations for $(D_m, A_n)$ models relate to the Bloch group and dilogarithm identities in algebraic K-theory?
• RQ2 Under what conditions on the parameters $A$, $B$, and $C$ is the 2-fold q-hypergeometric series $f_{A,B,C}$ modular, and does it correspond to a character of a rational conformal field theory?
• RQ3 What is the effective central charge $c_{\text{eff}}$ for $(D_m, A_n)$ models, and how does it compare to predictions from conformal field theory?
• RQ4 Can the function $f_{A,B,C}$ with $A = \begin{pmatrix}4&1\\1&1\end{pmatrix}$ be shown to be a modular character of a rational CFT?
• RQ5 How do the Galois group actions on the solutions of the model equations reflect the structure of the central charge modulo $24\mathbb{Z}$?

Key findings

• For the $(D_m, A_n)$ model, the effective central charge is computed as $c_{\text{eff}} = \frac{24}{5}$ when $x_1 = \frac{1}{2} + \frac{\sqrt{5}}{10}$, matching known RCFT values.
• In the $(E_7, T_1)$ model, the solution $x_1 = -\frac{3}{2} + \frac{\sqrt{21}}{2}$ yields $c_{\text{eff}} = 6$, consistent with minimal model expectations.
• For the $(E_8, T_1)$ model, one solution gives $c_{\text{eff}} = \frac{80}{11}$, derived from $x_1 = -4\cos(\frac{5\pi}{11}) + 2\cos(\frac{4\pi}{11}) - 2\cos(\frac{3\pi}{11}) + 2\cos(\frac{\pi}{11})$, confirming the central charge prediction.
• The 2-fold q-hypergeometric series $f_{A,B,C}$ with $A = \begin{pmatrix}4&1\\1&1\end{pmatrix}$ is shown to be modular, and its parameters $B$ and $C$ are explicitly computed, supporting its interpretation as a character of a rational CFT.
• The Galois group action on the solutions of the model equations is found to preserve the structure of the central charge modulo $24\mathbb{Z}$, linking number theory to CFT data.
• The method successfully computes all solutions to the model equations for $(D_m, A_n)$ and $(E_m, T_1)$ models, confirming the consistency of algebraic K-theory with conformal field theory predictions.