[Paper Review] Finite in All Directions

Gregory W. Moore|ArXiv.org|May 25, 1993

Black Holes and Theoretical PhysicsPhysics and Astronomy41 references69 citations

TL;DR

This paper investigates toroidal compactifications of string theory that include a compactified timelike dimension, revealing that duality acts ergodically on the moduli space, preventing a conventional Narain moduli space manifold. It introduces a flat connection on the moduli space to define parallel transport, enabling a formulation of broken Ward identities for unbroken infinite-dimensional symmetries—such as area-preserving diffeomorphisms—found in special compactifications. The key contribution is a universal symmetry algebra unifying these enhanced symmetries and a contact term prescription for conformal perturbation theory via a preferred coordinate system on moduli space.

ABSTRACT

We study toroidal compactifications of string theories which include compactification of a timelike coordinate. Some new features in the theory of toroidal compactifications arise. Most notably, Narain moduli space does not exist as a manifold since the action of duality on background data is ergodic. For special compactifications certain infinite dimensional symmetries, analogous to the infinite dimensional symmetries of the $2D$ string are unbroken. We investigate the consequences of these symmetries and search for a universal symmetry which contains all unbroken gauge groups. We define a flat connection on the moduli space of toroidally compactified theories. Parallel transport by this connection leads to a formulation of broken symmetry Ward identities. In an appendix this parallel transport is related to a definition of conformal perturbation theory.

Motivation & Objective

• To understand the implications of compactifying a timelike coordinate in string theory, particularly how it alters duality and moduli space structure.
• To investigate the emergence of infinite-dimensional unbroken symmetries—such as area-preserving diffeomorphisms—in special toroidal compactifications.
• To construct a universal symmetry algebra that unifies all enhanced gauge symmetries in toroidally compactified string theories.
• To formulate broken Ward identities for these symmetries using a flat connection on moduli space.
• To clarify conformal perturbation theory in this context via a contact term prescription tied to a preferred coordinate system on Narain moduli space.

Proposed method

• The paper uses BRST cohomology to identify the Lie algebra of inner automorphisms in CFT backgrounds, particularly focusing on ghost number one states.
• It introduces a flat connection on the moduli space of toroidal compactifications, enabling parallel transport of states and defining a covariant formulation of symmetry breaking.
• A key technique is the use of a preferred coordinate system on Narain moduli space, identified via a parametrization of the lattice generator matrices and a specific choice of the matrix $\Delta{\cal E}$.
• The authors derive a contact term prescription in conformal perturbation theory by relating different integral definitions of correlation functions through a reparametrization condition.
• They analyze the behavior of vertex operators under Lorentz transformations and use rotated operators to compute correlation functions via contraction schemes.
• The paper constructs a candidate universal symmetry algebra by identifying a distinguished compactification point with maximal symmetry, based on the factorization of closed strings into open strings.

Experimental results

Research questions

• RQ1 How does compactification of a timelike coordinate alter the structure of duality and moduli space in toroidally compactified string theories?
• RQ2 What conditions lead to the emergence of infinite-dimensional unbroken symmetries in toroidal compactifications, and how can they be unified?
• RQ3 Can a flat connection be defined on the moduli space of toroidal compactifications to consistently describe broken Ward identities?
• RQ4 How is conformal perturbation theory modified in this context, and what role does the choice of coordinates on moduli space play?
• RQ5 What is the significance of the distinguished compactification point where closed strings exactly factorize into open strings?

Key findings

• Duality acts ergodically on the space of toroidally compactified string theories, so Narain moduli space does not exist as a smooth manifold.
• There exists a unique compactification—distinguished by the number of gauged worldsheet supersymmetries—where closed strings exactly factorize into open strings, indicating maximal symmetry.
• A flat connection is defined on the moduli space of lattices, enabling parallel transport of states and a covariant formulation of broken Ward identities.
• The paper constructs a candidate universal symmetry algebra that unifies all enhanced symmetries in toroidal compactifications, particularly those related to area-preserving and volume-preserving diffeomorphisms.
• A contact term prescription for conformal perturbation theory is derived, with a preferred coordinate system on Narain moduli space identified via a specific parametrization of the lattice generator matrices.
• The analysis shows that compactifications with Monster symmetry are not dense in moduli space, and an approximate fundamental domain for duality action is constructed in Euclidean compactifications.