Arithmetic structures in rational conformal field theories
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Abstract
This thesis investigates the arithmetic structures underlying those (supersymmetric) two-
dimensional conformal field theories that are exactly solvable by algebraic methods — so-
called rational conformal field theories. A particular emphasis is placed on their connection
to Hodge theory and number theory. Building on the established correspondence between
N = (2,2) supersymmetric toroidal rational conformal field theories and Hodge structures
of complex multiplication (CM) type, the thesis extends this correspondence to a broader
class of exactly solvable and strongly interacting N = (2,2) rational superconformal field
theories, exemplified by so-called Gepner models.
It is shown in detail how the presence of a Galois symmetry in N = (2,2) rational super-
conformal field theories equips the associated Hodge structures, under certain assumptions,
with sufficient symmetry — precisely in the sense required for them to be of CM-type. As
many N = (2,2) rational superconformal field theories describe infrared fixed points of supersymmetric non-linear sigma models, which play a central role in worldsheet descriptions
of superstring theory, these results provide a controlled setting to study the emergence of
arithmetic structures in strongly coupled regimes in superstring theory moduli spaces.
A substantial portion of the thesis is devoted to a systematic analysis of arithmetic struc-
tures in N = (2,2) toroidal rational conformal field theories. In particular, this thesis
derives an explicit expansion of their partition functions in terms of ray class theta functions, revealing a novel number-theoretic interpretation of rational toroidal partition functions, and elucidating their relation to class field theory. This construction suggests a deep
interplay between modular invariance in conformal field theory and class field theory in
algebraic number theory.
The structure of the thesis reflects these two results. After a detailed review of foundational aspects of two-dimensional conformal field theories — including the role of extended
chiral symmetry algebras as defining properties of exactly solvable rational conformal field
theories, as well as discussions of boundary conditions and Galois theory — a particular
emphasis is placed on N = (2,2) supersymmetric conformal field theories and their relation to Hodge structures. Subsequently, the thesis focuses on arithmetic structures in
N = (2,2) toroidal non-linear sigma models, analysing rational points in their moduli
spaces and deriving their partition functions explicitly as products of generalised theta
functions, which are then related to ray class theta functions. Later chapters explore the
fundamental number-theoretic concept of complex multiplication in the context of elliptic
curves and higher-dimensional abelian varieties, showing how N = (2,2) toroidal rational
conformal field theories naturally endow the associated Hodge structures with the CM-
type property. Finally, the correspondence between N = (2,2) toroidal rational conformal
field theories and CM-type Hodge structures is extended to more general N = (2,2) rational superconformal field theories, exemplified by Gepner models and N= (2,2) minimal
models, establishing that both geometric and non-geometric models can realise arithmetic
structures of CM-type.