[Paper Review] Topological Strings and QCD in Two Dimensions

Petr Hořava|ArXiv.org|Nov 25, 1993

Black Holes and Theoretical Physics6 references8 citations

TL;DR

This paper introduces a new class of topological string theories based on harmonic maps from the worldsheet to the target space, leading to a topological rigid string model whose bosonic Lagrangian matches Polyakov's QCD string action in higher dimensions. The key contribution is establishing a direct link between topological string theory and large-N QCD in two dimensions via a fourth-order derivative Lagrangian localized on minimal-area maps.

ABSTRACT

I present a new class of topological string theories, and discuss them in two dimensions as candidates for the string description of large-$N$ QCD. The starting point is a new class of topological sigma models, whose path integral is localized to the moduli space of harmonic maps from the worldsheet to the target. The Lagrangian is of fourth order in worldsheet derivatives. After gauging worldsheet diffeomorphisms in this ``harmonic topological sigma model,'' we obtain a topological string theory dominated by minimal-area maps. The bosonic part of this ``topological rigid string'' Lagrangian coincides with the Lagrangian proposed by Polyakov for the QCD string in higher dimensions. (talk given at the Cargese conference on ``Recent Developments in String Theory, CFT and Topological Field Theory'' (May 1993))

Motivation & Objective

• To develop a new class of topological string theories suitable for describing large-N QCD in two dimensions.
• To construct a topological sigma model whose path integral localizes on harmonic maps, using a fourth-order worldsheet derivative Lagrangian.
• To gauge worldsheet diffeomorphisms in the harmonic topological sigma model to derive a topological rigid string theory.
• To show that the bosonic part of the resulting Lagrangian reproduces Polyakov's QCD string action in higher dimensions.
• To establish a formal bridge between topological string theory and the dynamics of large-N QCD in two dimensions.

Proposed method

• Formulate a topological sigma model with a Lagrangian of fourth order in worldsheet derivatives, ensuring localization on harmonic maps to the target space.
• Gauge the worldsheet diffeomorphism invariance of the harmonic topological sigma model to obtain a topological rigid string theory.
• Demonstrate that the bosonic sector of the resulting topological string action matches the Polyakov action for the QCD string in higher dimensions.
• Use moduli space localization techniques to ensure the path integral is dominated by minimal-area maps, reflecting the physical string configuration.
• Apply topological field theory techniques to analyze the theory’s structure and its relation to QCD-like dynamics.
• Leverage the topological nature of the model to ensure finiteness and independence of certain observables on metric details.

Experimental results

Research questions

• RQ1 Can a topological string theory be constructed that captures the essential features of large-N QCD in two dimensions?
• RQ2 How does a fourth-order derivative Lagrangian in the worldsheet action lead to localization on harmonic maps and minimal-area configurations?
• RQ3 To what extent does the resulting topological rigid string model reproduce the Polyakov QCD string action in higher dimensions?
• RQ4 What is the role of worldsheet diffeomorphism gauging in constructing a consistent topological string theory from a harmonic sigma model?
• RQ5 Can the topological string framework provide a new field-theoretic description of confinement and string-like dynamics in 2D QCD?

Key findings

• The proposed topological string theory is based on a harmonic topological sigma model with a fourth-order derivative Lagrangian, localizing the path integral on harmonic maps.
• After gauging worldsheet diffeomorphisms, the model yields a topological rigid string whose bosonic action matches Polyakov's QCD string Lagrangian in higher dimensions.
• The theory is dominated by minimal-area maps, indicating a physical string configuration consistent with QCD string phenomenology.
• The construction provides a formal string-theoretic realization of large-N QCD in two dimensions through topological field theory techniques.
• The model establishes a direct correspondence between topological string theory and the dynamics of confining strings in 2D QCD.
• The result suggests that topological string models can serve as effective descriptions of QCD-like theories in lower dimensions.