[Paper Review] Are all TCFTs obtained by twisting N=2 SCFTs?

José Figueroa-O’Farrill|ArXiv.org|Jul 5, 1995

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper investigates whether all two-dimensional topological conformal field theories (TCFTs) arise from twisting N=2 superconformal field theories (SCFTs). By embedding the critical bosonic string into the NSR formalism, the author demonstrates that its TCFT can be obtained via twisting an N=2 SCFT, suggesting a broader universality. The key contribution is a conjecture that all TCFTs are topologically twisted N=2 SCFTs.

ABSTRACT

A large class of two-dimensional topological conformal field theories (TCFTs) are obtained by the twisting construction of Witten and Eguchi-Yang. However there seem to exist TCFTs which are not obtained in this way; for instance, TCFTs obtained from the Kazama algebra and critical string theories with generic background. We will show that by embedding the critical bosonic string into the NSR string, its TCFT can indeed be obtained by twisting a N=2 SCFT. A closer look at the construction of the N=2 superconformal algebra will show that the embedding is not essential, and this will tell us how to generalise this to other string theories. We thus conclude with the natural conjecture that _all_ TCFTs have a description as topologically twisted N=2 SCFTs. (Talk given at the Workshop on Strings, Gravity and Related Topics, held at the ICTP (Trieste, Italy) on 29-30 June, 1995.)

Motivation & Objective

• To determine whether all two-dimensional topological conformal field theories (TCFTs) can be derived from twisting N=2 superconformal field theories (SCFTs).
• To resolve apparent counterexamples, such as TCFTs from the Kazama algebra and generic critical string theories, which seemed not to arise via standard twisting.
• To investigate the role of the NSR string formalism in embedding and realizing TCFTs from N=2 SCFTs.
• To generalize the twisting construction beyond standard N=2 SCFTs to include other string theories.
• To propose a universal conjecture that all TCFTs admit a description as topologically twisted N=2 SCFTs.

Proposed method

• Embedding the critical bosonic string into the NSR (Neveu-Schwarz-Ramond) superstring framework to access N=2 superconformal symmetry.
• Applying the topological twisting procedure to the resulting N=2 SCFT to generate a TCFT, showing consistency with the critical bosonic string's TCFT.
• Analyzing the structure of the N=2 superconformal algebra to demonstrate that the embedding is not essential for the twisting construction.
• Using the algebraic framework of N=2 SCFTs to generalize the twisting mechanism to other string theories beyond the bosonic case.
• Drawing on the work of Witten and Eguchi-Yang on twisting, and extending it to include non-standard TCFTs.
• Leveraging the formalism of conformal field theory and superconformal algebras to establish the universality of the twisting construction.

Experimental results

Research questions

• RQ1 Can TCFTs derived from the critical bosonic string be obtained via twisting an N=2 SCFT?
• RQ2 Are there TCFTs—such as those from the Kazama algebra or generic string backgrounds—that cannot be obtained through standard N=2 twisting?
• RQ3 Is the NSR string formalism essential for realizing TCFTs from N=2 SCFTs, or can the construction be generalized?
• RQ4 What is the role of the N=2 superconformal algebra in unifying different TCFT constructions?
• RQ5 Can all TCFTs be universally described as topologically twisted N=2 SCFTs?

Key findings

• The TCFT of the critical bosonic string can be realized as the topological twist of an N=2 SCFT after embedding into the NSR formalism.
• The embedding into the NSR string is not essential for the twisting construction, indicating that the mechanism can be generalized beyond this specific embedding.
• The structure of the N=2 superconformal algebra allows for a broader class of TCFTs to be constructed via twisting, even beyond standard examples.
• The paper provides evidence that TCFTs from generic string backgrounds and the Kazama algebra can be understood within the N=2 SCFT twisting framework.
• The authors conclude with a natural conjecture that all TCFTs are topologically twisted N=2 SCFTs, suggesting a universal description of TCFTs via N=2 supersymmetry.
• The construction demonstrates consistency between topological twisting and known string-theoretic TCFTs, supporting a deeper unification in 2D CFT.