Emergent Spacetime from Modular Motives (original) (raw)

String modular motives of mirrors of rigid Calabi–Yau varieties

Modular Forms and String Duality, 2008

The modular properties of some higher dimensional varieties of special Fano type are analyzed by computing the L-function of their Ω−motives. It is shown that the emerging modular forms are string theoretic in origin, derived from the characters of the underlying rational conformal field theory. The definition of the class of Fano varieties of special type is motivated by the goal to find candidates for a geometric realization of the mirrors of rigid Calabi-Yau varieties. We consider explicitly the cubic sevenfold and the quartic fivefold, and show that their motivic L-functions agree with the L-functions of their rigid mirror Calabi-Yau varieties. We also show that the cubic fourfold is string theoretic, with a modular form that is determined by that of an exactly solvable K3 surface. Contents 10 7. A special Fano variety of critical dimension two 16 References 18

Arithmetic of Calabi–Yau varieties and rational conformal field theory

Journal of Geometry and Physics, 2003

It is proposed that certain techniques from arithmetic algebraic geometry provide a framework which is useful to formulate a direct and intrinsic link between the geometry of Calabi-Yau manifolds and the underlying conformal field theory. Specifically it is pointed out how the algebraic number field determined by the fusion rules of the conformal field theory can be derived from the number theoretic structure of the cohomological Hasse-Weil L-function determined by Artin's congruent zeta function of the algebraic variety. In this context a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.

Aspects of conformal field theory from Calabi-Yau arithmetic

Calabi-Yau Varieties and Mirror Symmetry, 2003

This paper describes a framework in which techniques from arithmetic algebraic geometry are used to formulate a direct and intrinsic link between the geometry of Calabi-Yau manifolds and aspects of the underlying conformal field theory. As an application the algebraic number field determined by the fusion rules of the conformal field theory is derived from the number theoretic structure of the cohomological Hasse-Weil L-function determined by Artin's congruent zeta function of the algebraic variety. In this context a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.

Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory

Nuclear Physics B, 1994

We analyze the moduli spaces of Calabi-Yau threefolds and their associated conformally invariant nonlinear σ-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kähler sector of the moduli space of such Calabi-Yau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kähler cones of topologically distinct manifolds.

The Langlands program and string modular K3 surfaces

Nuclear Physics B, 2007

A number theoretic approach to string compactification is developed for Calabi-Yau hypersurfaces in arbitrary dimensions. The motivic strategy involved is illustrated by showing that the Hecke eigenforms derived from Galois group orbits of the holomorphic two-form of a particular type of K3 surfaces can be expressed in terms of modular forms constructed from the worldsheet theory. The process of deriving string physics from spacetime geometry can be reversed, allowing the construction of K3 surface geometry from the string characters of the partition function. A general argument for K3 modularity follows from mirror symmetry, in combination with the proof of the Shimura-Taniyama conjecture.

String Automorphic Motives of nondiagonal Varieties

2013

In this paper automorphic motives are constructed and analyzed with a view toward the understanding of the geometry of compactification manifolds in string theory in terms of the modular structure of the worldsheet theory. The results described generalize a framework considered previously in two ways, first by relaxing the restriction to modular forms, and second by extending the construction of motives from diagonal varieties to nondiagonal spaces. The framework of automorphic forms and representations is described with a view toward applications, emphasizing the explicit structure of these objects.

Arithmetic Space–Time Geometry from String Theory

International Journal of Modern Physics A, 2006

An arithmetic framework for string compactification is described. The approach is exemplified by formulating a strategy that allows to construct geometric compactifications from exactly solvable theories at c = 3. It is shown that the conformal field theoretic characters can be derived from the geometry of space–time, and that the geometry is uniquely determined by the two-dimensional field theory on the worldsheet. The modular forms that appear in these constructions admit complex multiplication, and allow an interpretation as generalized McKay–Thompson series associated to the Mathieu and Conway groups. This leads to a string motivated notion of arithmetic moonshine.

Liouville field, modular forms and elliptic genera

Journal of High Energy Physics, 2007

When we describe non-compact or singular Calabi-Yau manifolds by CFT, continuous as well as discrete representations appear in the theory. These representations mix in an intricate way under the modular transformations. In this article, we propose a method of combining discrete and continuous representations so that the resulting combinations have a simpler modular behavior and can be used as conformal blocks of the theory.