 Main
On Conformal Field Theory and Number Theory
 Author(s): Huang, An
 Advisor(s): Borcherds, Richard
 et al.
Abstract
This thesis is a combination of three pieces of work: 1, We explore some axioms of divergent series and their relations with conformal field theory. As a consequence we obtain another way of calculating $L(0,\chi)$ and $L(1,\chi)$ for $\chi$ being a Dirichlet character. We hope this discussion is also of interest to physicists doing renormalization theory for a reason indicated in the Introduction section. We consider a twist of the oscillator representation of the Virasoro algebra by a group of Dirichlet characters first introduced by Bloch in \cite{Bloch}, and use this to give a 'physical interpretation' of why the values of certain divergent series should be given by special L values. Furthermore, we use this to show that some fractional powers which are crucial for some infinite products to have peculiar modular transformation properties are expressed explicitly by certain linear combinations of $L(1, \chi)$'s for appropriately chosen $\chi$'s, and can be understood physically as a kind of 'vacuum Casimir energy' in our settings. 2, We give an attempt to reinterpret Tate's thesis by a sort of conformal field theory on a number field. Based on this and the existence of a hypothetical 3dimensional gauge theory, we give a physical interpretation of the Gauss quadratic reciprocity law by a sort of Sduality. 3, We find a complete list of positive definite symmetric matrices with integer entries $\begin{bmatrix}a&b\b&d\end{bmatrix}$ such that all complex solutions to the system of equations
\begin{align}
1x_1=x_1^ax_2^b\nonumber \
1x_2=x_1^bx_2^d\nonumber
\end{align}
are real. This result is related to Nahm's conjecture in rank 2 case. (In each chapter, except for equations, chapter number is omitted from quotation. For example, theorem 1.1.1 will be quoted as theorem 1.1.)
Main Content
Enter the password to open this PDF file:













