[Paper Review] Deformations, Symmetries and Topological Degrees of Freedom of the String

Mark Evans, Ioannis Giannakis|arXiv (Cornell University)|Sep 27, 1991

Computational Physics and Python Applications3 citations

TL;DR

This paper investigates deformations of conformal field theories in bosonic string theory, linking them to symmetries and topological degrees of freedom. It shows that each solution to the linearized equations of motion for massless modes corresponds to a distinct stress tensor deformation, revealing a finite-dimensional space of deformations that may represent free parameters or topological states—potentially analogous to isolated states in 2D systems.

ABSTRACT

We discuss three closely related questions; i)~Given a conformal field theory, how may we deform it? ii)~What are the symmetries of string theory? and iii)~Does string theory have free parameters? We show that there is a distinct deformation of the stress tensor for every solution to the linearised covariant equations of motion for the massless modes of the Bosonic string, and use this result to discuss the symmetries of the string. We also find an additional finite dimensional space of deformations which may correspond to free parameters of string theory, or alternatively may be interpreted as topological degrees of freedom, perhaps analogous to the isolated states found in two dimensions.

Motivation & Objective

• To understand how conformal field theories can be deformed in the context of bosonic string theory.
• To identify the symmetries of string theory through the structure of these deformations.
• To investigate whether string theory contains free parameters or topological degrees of freedom.
• To explore the physical interpretation of a finite-dimensional space of deformations beyond standard massless modes.

Proposed method

• Analyzes deformations of the stress-energy tensor in conformal field theory using linearized equations of motion.
• Relates each deformation to solutions of the linearized covariant equations for massless modes of the bosonic string.
• Identifies a finite-dimensional space of additional deformations beyond those from massless modes.
• Applies techniques from differential geometry and quantum field theory in curved spacetime to analyze symmetries.
• Uses the structure of the stress tensor and its variation to probe the space of possible string theory configurations.
• Draws analogies to topological states in 2D systems to interpret the nature of the additional deformations.

Experimental results

Research questions

• RQ1 How are deformations of the stress tensor in string theory related to the linearized equations of motion for massless modes?
• RQ2 What symmetries of string theory are encoded in the space of such deformations?
• RQ3 Does the existence of an additional finite-dimensional space of deformations imply free parameters in string theory?
• RQ4 Could these extra deformations represent topological degrees of freedom, similar to isolated states in 2D systems?
• RQ5 What is the physical interpretation of the finite-dimensional deformation space beyond standard massless mode contributions?

Key findings

• Each solution to the linearized covariant equations of motion for massless modes of the bosonic string corresponds to a distinct deformation of the stress tensor.
• The space of such deformations is isomorphic to the space of solutions of the linearized field equations for massless modes.
• An additional finite-dimensional space of deformations exists beyond those from massless modes.
• This extra space may represent free parameters of string theory or topological degrees of freedom.
• The authors suggest a possible analogy to isolated states in two-dimensional systems, hinting at topological character.
• The results imply a deeper structure in string theory's moduli space, linking deformations, symmetries, and potential topological invariants.