Kac-Moody Algebras and String Theory (THESIS)

TL;DR

This thesis explores the role of Kac-Moody algebras in string theory, develops techniques for higher level algebra models, and investigates fractional superstrings, providing new insights into modular invariants and spacetime supersymmetry.

Contribution

It introduces systematic methods for constructing string models based on higher level Kac-Moody algebras and analyzes fractional superstring theories with novel results on modular invariants and supersymmetry.

Findings

Classified all consistent theories for $SU(2)_{K_A}xSU(2)_{K_B}$ with odd levels.

Constructed asymmetric modular invariants with explicit examples.

Rederived partition functions and analyzed supersymmetry in fractional superstring models.

Abstract

The focus of this thesis is on (1) the role of Ka\v c-Moody (KM) algebras in string theory and the development of techniques for systematically building string theory models based on higher level ($K\geq 2$) KM algebras and (2) fractional superstrings. In chapter two we review KM algebras and their role in string theory. In the next chapter, we present two results concerning the construction of modular invariant partition functions for conformal field theories built by tensoring together other conformal field theories. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type $SU(2)_{K_A}xSU(2)_{K_B}$, finding all consistent theories for $K_A$ and $K_B$ odd. Second we show how known diagonal modular invariants can be used to construct inherently asymmetric invariants where the holomorphic and anti- holomorphic theories do not share the same chiral algebra. Explicit examples are given. Next, in chapter four we investigate some issues relating to proposed fractional superstring theories with $D_{\rm critical}<10$. Using the factorization approach of Gepner and Qiu, we systematically rederive the partition functions of the $K=4,\, 8,$ and $16$ theories and examine their spacetime supersymmetry. Generalized GSO projection operators for the $K=4$ model are found. Uniqueness of the twist field, $\phi^{K/4}_{K/4}$, as source of spacetime fermions, is demonstrated.