[Paper Review] Strongly Topological Interactions of Tensionless Strings

Bo Sundborg|arXiv (Cornell University)|May 31, 1994

Computational Physics and Python Applications3 references5 citations

TL;DR

This paper proposes a topological formulation of tensionless string theory by replacing the worldsheet metric with a vector density that encodes geometric information. Path-integral quantization yields multiple inequivalent quantum theories, and amplitudes depend only on the topological locations of vector density zeros, making them independent of metric or position moduli — a hallmark of topological field theory.

ABSTRACT

The tensionless limit of classical string theory may be formulated as a topological theory on the world-sheet. A vector density carries geometrical information in place of an internal metric. It is found that path-integral quantization allows for the definition of several, possibly inequivalent quantum theories. String amplitudes are constructed from vector densities with zeroes for each in- or out-going string. It is shown that independence of a metric in quantum mechanical amplitudes implies that the dependence on such vector density zeroes is purely topological. For example, there is no need for integration over their world-sheet positions.

Motivation & Objective

• To formulate the tensionless limit of string theory as a topological field theory on the worldsheet.
• To replace the worldsheet metric with a vector density to encode geometric data in the tensionless limit.
• To explore the existence and nature of inequivalent quantum theories arising from path-integral quantization.
• To establish that string amplitudes depend only on the topological positions of vector density zeros, not on their worldsheet coordinates.
• To demonstrate metric independence in quantum amplitudes, implying topological invariance of the theory.

Proposed method

• Formulate the classical tensionless string action using a worldsheet vector density instead of a metric tensor.
• Use path-integral quantization to define quantum amplitudes, allowing for multiple inequivalent theories due to topological choices.
• Construct string amplitudes from vector densities with isolated zeros corresponding to in- and out-going strings.
• Show that the dependence on the positions of vector density zeros is purely topological, not geometric.
• Demonstrate that quantum amplitudes are invariant under reparametrizations and metric variations, implying topological invariance.
• Correct the ultralocality condition in equation (12) to ensure consistency in the topological formulation.

Experimental results

Research questions

• RQ1 Can the tensionless limit of string theory be consistently formulated as a topological field theory on the worldsheet?
• RQ2 How does the replacement of the worldsheet metric with a vector density affect the structure of the quantum theory?
• RQ3 What is the origin of multiple inequivalent quantum theories in the path-integral quantization of tensionless strings?
• RQ4 Why are string amplitudes independent of the worldsheet positions of vector density zeros?
• RQ5 To what extent is the theory invariant under metric and reparametrization transformations?

Key findings

• The tensionless limit of string theory can be formulated as a topological field theory using a vector density in place of the worldsheet metric.
• Path-integral quantization leads to multiple, possibly inequivalent, quantum theories due to topological data encoded in the vector density.
• String amplitudes are constructed from vector densities with zeros at in- and out-going string locations, and these amplitudes depend only on the topological configuration of the zeros.
• The quantum amplitudes are independent of the worldsheet metric and of the positions of the vector density zeros, confirming topological invariance.
• The ultralocality condition in equation (12) is corrected to ensure consistency in the topological formulation.
• The theory exhibits a strong topological character, with no need for integration over the worldsheet positions of the vector density zeros.